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Dissertations / Theses on the topic 'Asymptotic distribution'

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Published: 4 June 2021
Last updated: 20 February 2023

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1

Hofmann, Glenn, Erhard Cramer, N. Balakrishnan, and Gerd Kunert. "An Asymptotic Approach to Progressive Censoring." Universitätsbibliothek Chemnitz, 2002. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200201539.

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Progressive Type-II censoring was introduced by Cohen (1963) and has since been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2x2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.
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2

Baligh, Mohammadhadi. "Analysis of the Asymptotic Performance of Turbo Codes." Thesis, University of Waterloo, 2006. http://hdl.handle.net/10012/883.

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Battail [1989] shows that an appropriate criterion for the design of long block codes is the closeness of the normalized weight distribution to a Gaussian distribution. A subsequent work shows that iterated product of single parity check codes satisfy this criterion [1994]. Motivated by these earlier works, in this thesis, we study the effect of the interleaver on the performance of turbo codes for large block lengths, $N\rightarrow\infty$. A parallel concatenated turbo code that consists of two or more component codes is considered. We demonstrate that for $N\rightarrow\infty$, the normalized weight of the systematic $\widehat{w_1}=\displaystyle\frac{w_1}{\sqrt{N}}$, and the parity check sequences $\widehat{w_2}=\displaystyle\frac{w_2}{\sqrt{N}}$ and $\widehat{w_3}=\displaystyle\frac{w_3}{\sqrt{N}}$ become a set of jointly Gaussian distributions for the typical values of $\widehat{w_i},i=1,2,3$, where the typical values of $\widehat{w_i}$ are defined as $\displaystyle\lim_{N\rightarrow\infty}\frac{\widehat{w_i}}{\sqrt{N}}\neq 0,1$ for $i=1,2,3$. To optimize the turbo code performance in the waterfall region which is dominated by high-weight codewords, it is desirable to reduce $\rho_{ij}$, $i,j=1,2,3$ as much as possible, where $\rho_{ij}$ is the correlation coefficient between $\widehat{w_i}$ and $\widehat{w_j}$. It is shown that: (i)~$\rho_{ij}>0$, $i,j=1,2,3$, (ii)~$\rho_{12},\rho_{13}\rightarrow 0$ as $N\rightarrow\infty$, and (iii)~$\rho_{23}\rightarrow 0$ as $N\rightarrow\infty$ for "almost" any random interleaver. This indicates that for $N\rightarrow\infty$, the optimization of the interleaver has a diminishing effect on the distribution of high-weight error events, and consequently, on the error performance in the waterfall region. We show that for the typical weights, this weight distribution approaches the average spectrum defined by Poltyrev [1994]. We also apply the tangential sphere bound (TSB) on the Gaussian distribution in AWGN channel with BPSK signalling and show that it performs very close to the capacity for code rates of interest. We also study the statistical properties of the low-weight codeword structures. We prove that for large block lengths, the number of low-weight codewords of these structures are some Poisson random variables. These random variables can be used to evaluate the asymptotic probability mass function of the minimum distance of the turbo code among all the possible interleavers. We show that the number of indecomposable low-weight codewords of different types tend to a set of independent Poisson random variables. We find the mean and the variance of the union bound in the error floor region and study the effect of expurgating low-weight codewords on the performance. We show that the weight distribution in the transition region between Poisson and Gaussian follows a negative binomial distribution. We also calculate the interleaver gain for multi-component turbo codes based on these Poisson random variables. We show that the asymptotic error performance for multi-component codes in different weight regions converges to zero either exponentially (in the Gaussian region) or polynomially (in the Poisson and negative binomial regions) with respect to the block length, with the code-rate and energy values close to the channel capacity.
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3

Stewart, Michael. "Asymptotic methods for tests of homogeneity for finite mixture models." Connect to full text, 2002. http://hdl.handle.net/2123/855.

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Thesis (Ph. D.)--University of Sydney, 2002.
Title from title screen (viewed Apr. 28, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliography. Also available in print form.
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4

Unger, William Ramsay. "Asymptotics of increasing trees." Thesis, The University of Sydney, 1993. https://hdl.handle.net/2123/26633.

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This thesis addresses the problem of finding the asymptotic average path length in species of increasing trees. A version of Joyal’s theory of species, L—species, is used to derive power series identities for simple increasing trees and give them bijective proofs. The main identity involved here is an autonomous, nonlinear, ordinary differential equation. Asymptotic results on the number and average path length of these species are derived by analysis of the singularities of these power series when they are treated as analytic functions. The result is a general method for finding the asymptotic number of and asymptotic average path length in increasing trees defined by a single differential equation. In the particular case Where there is a uniform upper bound on the number of descendents a vertex has (and in some other cases) the method can be simply automated using a symbolic algebra package such as MAPLE. It is found that the expected path length of a tree in a species of increasing trees with the above restriction is asymptotically proportional to n log n, where n is the number of vertices in the tree. By contrast, it is shown, using similar combinatorial methods, that labelled, non—increasing trees and unlabelled trees, to which similar analytic methods apply, have path length asymptotically proportional to n*sqrt(n).
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5

Heimbürger, Axel. "Asymptotic Distribution of Two-Protected Nodes in m-ary Search Trees." Thesis, KTH, Matematik (Avd.), 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-151318.

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In this report, the number of two-protected nodes in m-ary search trees is studied i.e., nodes with distance at least two to any leaf in the tree. This is of interest since the protected nodes describe local properties close to the leaves of the m-ary search trees. This is done by using a generalised Pólya urn model and relating this urn model to how the tree evolves after each new key is inserted into the tree. It is proven that the number of two-protected nodes in m-ary search trees is asymptotically normally distributed when m = 4, 5, 6 which is the main result. This is in agreement with previously known results for m = 2, 3, which were obtained by using the same approach. The method and algorithms are presented in such a way that it simpli_es calculations for larger m. Based on the results for m = 2,…, 6 conjectures are made providing a possible way to extend these results for larger m < 26.
I detta examensarbete studeras antalet tvåskyddade noder i m-ära sökträd. En nod kallas tvaskyddad ifall den ar minst två kanter fran ett löv i trädet. Dessa noder är av intresse eftersom de beskriver lokala egenskaper nära löven i de m-ära sökträden. Detta studeras genom att använda en generaliserad Pólya urna och genom att relatera denna urna till hur ett m-ärt sökträd expanderar när nya nycklar placeras in i trädet. Det bevisas att antalet tvåskyddade noder i ett m-ärt sökträd har en asymptotiskt normalfördelad sannolikhetsfördelning för m = 4, 5, 6 när antalet nycklar i trädet går mot oändligheten. Detta stämmer överens med tidigare resultat för m = 2, 3, som har bevisats genom att använda samma metod. Metoden och algoritmerna som används för att beräkna dessa resultat presenteras på ett sådant sätt att de går att applicera på större m utan modifiering. Givet resultaten för m = 2,…, 6 presenteras en möjlig väg för att expandera dessa resultat för större m.
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6

Mwawasi, Grace Makanda. "Approximations and asymptotic expansions for the distribution of quadratic and bilinear forms." Thesis, McGill University, 1992. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=56952.

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In this thesis, approximations and asymptotic expansions to the distribution of quadratic and bilinear forms in normal random variables are discussed.
Chi-square type approximations, normal approximations, the mixture approximation and the laplacian approximation to the exact distribution of positive definite and indefinite quadratic forms and bilinear forms are discussed. Several asymptotic results are also discussed.
Some numerical computations giving probabilities and percentage points and also some simulation for the distribution function of quadratic and bilinear forms are given to give more insight into the approximations.
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7

Breimesser, Sandra Verena. "Asymptotic value distribution for solutions of the Schrödinger equation and Herglotz functions." Thesis, University of Hull, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.272024.

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8

Bulger, Daniel. "The high energy asymptotic distribution of the eigenvalues of the scattering matrix." Thesis, King's College London (University of London), 2013. https://kclpure.kcl.ac.uk/portal/en/theses/the-high-energy-asymptotic-distribution-of-the-eigenvalues-of-the-scattering-matrix(541fc908-ff77-4f0f-b3ba-af1fe53e19dd).html.

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We determine the high energy asymptotic density of the eigenvalues of the scat- tering matrix associated with the operators H0 = −∆ and H = (i∇ + A)2 + V (x), where V : Rd → R is a smooth short-range real-valued electric potential and A = (A1, . . . , Ad) : Rd → Rd is a smooth short-range magnetic vector-potential. Two cases are considered. The first case is where the magnetic vector-potential is non-zero. The spectral density of the associated scattering matrix in this case is expressed as an integral solely in terms of the magnetic vector-potential A. The second case considered is where the magnetic vector-potential is identically zero. Again the spectral density of the scattering matrix is expressed as an integral, this time in terms of the poten- tial V . These results share similar characteristics to results pertaining to semiclassical asymptotics for pseudodifferential operators.
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9

Joyner, James Thomas. "ASYMPTOTIC ANALYSIS OF FRONTAL POLYMERIZATION IN A MEDIUM WITH PERIODIC MONOMER DISTRIBUTION." University of Akron / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=akron1153773428.

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10

Balabdaoui, Fadoua. "Nonparametric estimation of a k-monotone density : a new asymptotic distribution theory /." Thesis, Connect to this title online; UW restricted, 2004. http://hdl.handle.net/1773/8964.

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11

O'Connell, W. Richard Jr. "Estimates for the St. Petersburg game." Diss., Georgia Institute of Technology, 1998. http://hdl.handle.net/1853/28858.

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12

Södergren, Anders. "Asymptotic Problems on Homogeneous Spaces." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-132645.

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This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.
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13

Petersson, Mikael. "Asymptotic Expansions for Perturbed Discrete Time Renewal Equations." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-95490.

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In this thesis we study the asymptotic behaviour of the solution of a discrete time renewal equation depending on a small perturbation parameter. In particular, we construct asymptotic expansions for the solution of the renewal equation and related quantities. The results are applied to studies of quasi-stationary phenomena for regenerative processes and asymptotics of ruin probabilities for a discrete time analogue of the Cramér-Lundberg risk model.
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14

Yaobin, Wen. "Asymptotic Analysis of Interference in Cognitive Radio Networks." Thèse, Université d'Ottawa / University of Ottawa, 2013. http://hdl.handle.net/10393/23997.

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The aggregate interference distribution in cognitive radio networks is studied in a rigorous and analytical way using the popular Poisson point process model. While a number of results are available for this model for non-cognitive radio networks, cognitive radio networks present extra levels of difficulties for the analysis, mainly due to the exclusion region around the primary receiver, which are typically addressed via various ad-hoc approximations (e.g., based on the interference cumulants) or via the large-deviation analysis. Unlike the previous studies, we do not use here ad-hoc approximations but rather obtain the asymptotic interference distribution in a systematic and rigorous way, which also has a guaranteed level of accuracy at the distribution tail. This is in contrast to the large deviation analysis, which provides only the (exponential) order of scaling but not the outage probability itself. Unlike the cumulant-based analysis, our approach provides a guaranteed level of accuracy at the distribution tail. Additionally, our analysis provides a number of novel insights. In particular, we demonstrate that there is a critical transition point below which the outage probability decays only polynomially but above which it decays super-exponentially. This provides a solid analytical foundation to the earlier empirical observations in the literature and also reveals what are the typical ways outage events occur in different regimes. The analysis is further extended to include interference cancelation and fading (from a broad class of distributions). The outage probability is shown to scale down exponentially in the number of canceled nearest interferers in the below-critical region and does not change significantly in the above-critical one. The proposed asymptotic expressions are shown to be accurate in the non-asymptotic regimes as well.
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15

Emberson, E. A. "The asymptotic distribution and robustness of the likelihood ratio and score test statistics." Thesis, University of St Andrews, 1995. http://hdl.handle.net/10023/13738.

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Cordeiro & Ferrari (1991) use the asymptotic expansion of Harris (1985) for the moment generating function of the score statistic to produce a generalization of Bartlett adjustment for application to the score statistic. It is shown here that Harris's expansion is not invariant under reparameterization and an invariant expansion is derived using a method based on the expected likelihood yoke. A necessary and sufficient condition for the existence of a generalized Bartlett adjustment for an arbitrary statistic is given in terms of its moment generating function. Generalized Bartlett adjustments to the likelihood ratio and score test statistics are derived in the case where the interest parameter is one-dimensional under the assumption of a mis-specified model, where the true distribution is not assumed to be that under the null hypothesis.
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16

Zeileis, Achim, and Torsten Hothorn. "Permutation Tests for Structural Change." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2006. http://epub.wu.ac.at/1182/1/document.pdf.

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The supLM test for structural change is embedded into a permutation test framework for a simple location model. The resulting conditional permutation distribution is compared to the usual (unconditional) asymptotic distribution, showing that the power of the test can be clearly improved in small samples. Furthermore, generalizations are discussed for binary and multivariate dependent variables as well as model-based permutation testing for structural change. The procedures suggested are illustrated using both artificial and real-world data (number of youth homicides, employment discrimination data, structural-change publications, and stock returns).
Series: Research Report Series / Department of Statistics and Mathematics
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17

Stewart, Michael Ian. "Asymptotic methods for tests of homogeneity for finite mixture models." Thesis, The University of Sydney, 2002. http://hdl.handle.net/2123/855.

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We present limit theory for tests of homogeneity for finite mixture models. More specifically, we derive the asymptotic distribution of certain random quantities used for testing that a mixture of two distributions is in fact just a single distribution. Our methods apply to cases where the mixture component distributions come from one of a wide class of one-parameter exponential families, both continous and discrete. We consider two random quantities, one related to testing simple hypotheses, the other composite hypotheses. For simple hypotheses we consider the maximum of the standardised score process, which is itself a test statistic. For composite hypotheses we consider the maximum of the efficient score process, which is itself not a statistic (it depends on the unknown true distribution) but is asymptotically equivalent to certain common test statistics in a certain sense. We show that we can approximate both quantities with the maximum of a certain Gaussian process depending on the sample size and the true distribution of the observations, which when suitably normalised has a limiting distribution of the Gumbel extreme value type. Although the limit theory is not practically useful for computing approximate p-values, we use Monte-Carlo simulations to show that another method suggested by the theory, involving using a Studentised version of the maximum-score statistic and simulating a Gaussian process to compute approximate p-values, is remarkably accurate and uses a fraction of the computing resources that a straight Monte-Carlo approximation would.
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18

Stewart, Michael Ian. "Asymptotic methods for tests of homogeneity for finite mixture models." University of Sydney. Mathematics and Statistics, 2002. http://hdl.handle.net/2123/855.

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We present limit theory for tests of homogeneity for finite mixture models. More specifically, we derive the asymptotic distribution of certain random quantities used for testing that a mixture of two distributions is in fact just a single distribution. Our methods apply to cases where the mixture component distributions come from one of a wide class of one-parameter exponential families, both continous and discrete. We consider two random quantities, one related to testing simple hypotheses, the other composite hypotheses. For simple hypotheses we consider the maximum of the standardised score process, which is itself a test statistic. For composite hypotheses we consider the maximum of the efficient score process, which is itself not a statistic (it depends on the unknown true distribution) but is asymptotically equivalent to certain common test statistics in a certain sense. We show that we can approximate both quantities with the maximum of a certain Gaussian process depending on the sample size and the true distribution of the observations, which when suitably normalised has a limiting distribution of the Gumbel extreme value type. Although the limit theory is not practically useful for computing approximate p-values, we use Monte-Carlo simulations to show that another method suggested by the theory, involving using a Studentised version of the maximum-score statistic and simulating a Gaussian process to compute approximate p-values, is remarkably accurate and uses a fraction of the computing resources that a straight Monte-Carlo approximation would.
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19

Etienne, Roland Jean [Verfasser]. "On the asymptotic distribution of the Dirichlet eigenvalues of fractal chains / Roland Jean Etienne." Siegen : Universitätsbibliothek der Universität Siegen, 2015. http://d-nb.info/1068362863/34.

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20

Pielaszkiewicz, Jolanta. "On the asymptotic spectral distribution of random matrices : Closed form solutions using free independence." Licentiate thesis, Linköpings universitet, Matematisk statistik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-92637.

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The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as   where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.
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Pielaszkiewicz, Jolanta Maria. "On the asymptotic spectral distribution of random matrices : closed form solutions using free independence." Licentiate thesis, Linköping University, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-58181.

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The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as   where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.
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22

Osaka, Haruki. "Asymptotics of Mixture Model Selection." Thesis, The University of Sydney, 2021. https://hdl.handle.net/2123/27230.

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In this thesis, we consider the likelihood ratio test (LRT) when testing for homogeneity in a three component normal mixture model. It is well-known that the LRT in this setting exhibits non-standard asymptotic behaviour, due to non-identifiability of the model parameters and possible degeneracy of Fisher Information matrix. In fact, Liu and Shao (2004) showed that for the test of homogeneity in a two component normal mixture model with a single fixed component, the limiting distribution is an extreme value Gumbel distribution under the null hypothesis, rather than the usual chi-squared distribution in regular parametric models for which the classical Wilks' theorem applies. We wish to generalise this result to a three component normal mixture to show that similar non-standard asymptotics also occurs for this model. Our approach follows closely to that of Bickel and Chernoff (1993), where the relevant asymptotics of the LRT statistic were studied indirectly by first considering a certain Gaussian process associated with the testing problem. The equivalence between the process studied by Bickel and Chernoff (1993) and the LRT was later proved by Liu and Shao (2004). Consequently, they verified that the LRT statistic for this problem diverges to infinity at the rate of loglog n; a statement that was first conjectured in Hartigan (1985). In a similar spirit, we consider the limiting distribution of the supremum of a certain quadratic form. More precisely, the quadratic form we consider is the score statistic for the test for homogeneity in the sub-model where the mean parameters are assumed fixed. The supremum of this quadratic form is shown to have a limiting distribution of extreme value type, again with a divergence rate of loglog n. Finally, we show that the LRT statistic for the three component normal mixture model can be uniformly approximated by this quadratic form, thereby proving that that the two statistics share the same limiting distribution.
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23

Kim, Jung-Kyung. "Tail asymptotics of queueing networks with subexponential service times." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29734.

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Thesis (Ph.D)--Industrial and Systems Engineering, Georgia Institute of Technology, 2010.
Committee Chair: Ayhan, Hayriye; Committee Member: Foley, Robert D.; Committee Member: Goldsman, David M.; Committee Member: Reed, Joshua; Committee Member: Zwart, Bert. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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24

Pilon, Paul J. "Gamma type distribution: Maximum likelihood values of the T-year event and their asymptotic variance." Thesis, University of Ottawa (Canada), 1990. http://hdl.handle.net/10393/5837.

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Maximum likelihood and censored sample theory are applied for flood frequency analysis purposes to the Two Parameter Gamma, log Two Parameter Gamma, Pearson Type III, log Pearson Type III (LP3), and Generalized Gamma distributions. The logarithmic likelihood functions are given in terms of the fully specified floods, the historical information, and the parameters to be estimated. Solution of the appropriate transcendental equations yields maximum likelihood estimators of the parameters. T-year floods are expressed as a function of these parameters and the standard normal variate. The asymptotic standard error of estimate of the T-year flood is derived using the general equation for the variance of estimate of a function. The variances and covariances of the parameters are obtained through inversion of Fisher's information matrix. The method is illustrated by application of the LP3 distribution to two sites having historical information. Monte Carlo studies were conducted for the LP3 distribution to analytically verify the accuracy of the derived asymptotic expression for the 10-, 50-, 100-, and 500-year floods. Results indicated that the asymptotic expressions were accurate for both Type I and Type II censored samples, while the bias was less than 2.5%. Subsequently, the Type II censored data were subjected to a random, multiplicative error. Results indicated that historical information contributes greatly to the accuracy of the estimate of the 100-year flood even when the error of its measurement becomes excessive. It is demonstrated that historical information can significantly reduce the standard error of estimate of flood quantiles.
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Yuan, Zhongyi. "Quantitative analysis of extreme risks in insurance and finance." Diss., University of Iowa, 2013. https://ir.uiowa.edu/etd/2422.

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In this thesis, we aim at a quantitative understanding of extreme risks. We use heavy-tailed distribution functions to model extreme risks, and use various tools, such as copulas and MRV, to model dependence structures. We focus on modeling as well as quantitatively estimating certain measurements of extreme risks. We start with a credit risk management problem. More specifically, we consider a credit portfolio of multiple obligors subject to possible default. We propose a new structural model for the loss given default, which takes into account the severity of default. Then we study the tail behavior of the loss given default under the assumption that the losses of the obligors jointly follow an MRV structure. This structure provides an ideal framework for modeling both heavy tails and asymptotic dependence. Using HRV, we also accommodate the asymptotically independent case. Multivariate models involving Archimedean copulas, mixtures and linear transforms are revisited. We then derive asymptotic estimates for the Value at Risk and Conditional Tail Expectation of the loss given default and compare them with the traditional empirical estimates. Next, we consider an investor who invests in multiple lines of business and study a capital allocation problem. A randomly weighted sum structure is proposed, which can capture both the heavy-tailedness of losses and the dependence among them, while at the same time separates the magnitudes from dependence. To pursue as much generality as possible, we do not impose any requirement on the dependence structure of the random weights. We first study the tail behavior of the total loss and obtain asymptotic formulas under various sets of conditions. Then we derive asymptotic formulas for capital allocation and further refine them to be explicit for some cases. Finally, we conduct extreme risk analysis for an insurer who makes investments. We consider a discrete-time risk model in which the insurer is allowed to invest a proportion of its wealth in a risky stock and keep the rest in a risk-free bond. Assume that the claim amounts within individual periods follow an autoregressive process with heavy-tailed innovations and that the log-returns of the stock follow another autoregressive process, independent of the former one. We derive an asymptotic formula for the finite-time ruin probability and propose a hybrid method, combining simulation with asymptotics, to compute this ruin probability more efficiently. As an application, we consider a portfolio optimization problem in which we determine the proportion invested in the risky stock that maximizes the expected terminal wealth subject to a constraint on the ruin probability.
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Reinhold, Küstner. "Asymptotic zero distribution of orthogonal polynomials with respect to complex measures having argument of bounded variation." Nice, 2003. http://www.theses.fr/2003NICE4054.

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On détermine la distribution asymptotique des pôles pour trois types de meilleurs approximants (Padé à l’infini, rationnel en L2 sur le cercle unité, méromorphe dans le disque unité en Lp sur le cercle unité, p>2) de la transformée de Cauchy d’une mesure complexe sous l’hypothèse que le support S de la mesure soit de capacité positive et inclus dans (-1, 1), que la mesure satisfasse une condition de densité et que l’argument de la mesure soit la restriction d’une fonction à variation bornée. Les polynômes dénominateurs des approximants satisfont des relations d’orthogonalité. Au moyen d’un théorème de Kestelman, on obtient des contraintes géométriques pour les zéros qui impliquent que chaque mesure limite faible des mesures de comptage associées à son support inclus dans S. Puis, à l’aide de résultats de la théorie du potentiel dans le plan, on montre que les mesures de comptage convergent faiblement vers la distribution d’équilibre logarithmique respectivement hyperbolique de S
We determine the asymptotic pole distribution for three types of best approximants (Padé at infinity, rational in L2 on the unit circle, meromorphic in the unit disk in Lp on the unit circle, p>2) of the Cauchy transform of a complex measure under the hypothesis that the support S of the measure is of positive capacity and included in (-1 1), that the measure satisfies a density condition and that the argument of the measure is the restriction of a function of bounded variation ? The denominator polynomials of the approximants satisfay orthogonality relations ? By means of a theorem of Kestelman we obtain geometric constraints for the zeros which imply that every weak limit measure of the associated counting measures has support included in S. Then, with the help of results from potential theory in the plane, we show that the counting measures converge weakly to the logarithmic respectively hyperbolic equilibrium distribution of S
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27

Cartailler, Jérôme. "Asymptotic of Poisson-Nernst-Planck equations and application to the voltage distribution in cellular micro-domains." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066297/document.

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Dans cette thèse j’étudie l’impact de la géométrie de micro et nano-domaines biologiques sur les propriétés d'électrodiffusion, ceci à l'aide des équations aux dérivées partielles de Poisson-Nernst-Planck. Je considère des domaines non-triviaux ayant une forme cuspide ou elliptique. Mon objectif est de développer des modèles ainsi que des méthodes mathématiques afin d'étudier les caractéristiques électriques de ces nano/micro-domaines, et ainsi mieux comprendre comment les signaux électriques sont modulés à ces échelles. Dans la première partie j’étudie le voltage à l'équilibre pour un électrolyte dans un domaine borné, et ayant un fort excès de charges positives. Je montre que le premier temps de sortie dans une boule chargée dépend de la surface et non du volume. J’étudie ensuite la géométrie composées d'une boule à laquelle est attachée un domaine cuspide. Je construis ensuite une solution asymptotique pour le voltage dans les cas 2D et 3D et je montre qu’ils sont donnés au premier ordre par la même expression. Enfin, j’obtiens la même conclusion en considérant une géométrie formée d'une ellipse, dont je construis une solution asymptotique du voltage en 2D et 3D. La seconde partie porte sur la modélisation de la compartimentalisation électrique des épines dendritiques. A partir de simulations numériques, je mets en évidence le lien entre la polarisation de concentration dans l'épine et sa géométrie. Je compare ensuite mon modèle à des données de microscopie. Je développe une méthode de déconvolution pour extraire la dynamique rapide du voltage à partir des données de microscopie. Enfin j’estime la résistance du cou et montre que celle-ci ne suit pas la loi d'Ohm
In this PhD I study how electro-diffusion within biological micro and nano-domains is affected by their shapes using the Poisson-Nernst-Planck (PNP) partial differential equations. I consider non-trivial shapes such as domains with cusp and ellipses. Our goal is to develop models, as well as mathematical tools, to study the electrical properties of micro and nano-domains, to understand better how electrical neuronal signaling is regulated at those scales. In the first part I estimate the steady-state voltage inside an electrolyte confined in a bounded domain, within which we assume an excess of positive charge. I show the mean first passage time in a charged ball depends on the surface and not on the volume. I further study a geometry composed of a ball with an attached cusp-shaped domain. I construct an asymptotic solution for the voltage in 2D and 3D and I show that to leading order expressions for the voltage in 2D and 3D are identical. Finally, I obtain similar conclusion considering an elliptical-shaped domain for which I construct an asymptotic solution for the voltage in 2D and 3D. In the second part, I model the electrical compartmentalization in dendritic spines. Based on numerical simulations, I show how spines non-cylindrical geometry leads to concentration polarization effects. I then compare my model to experimental data of microscopy imaging. I develop a deconvolution method to recover the fast voltage dynamic from the data. I estimate the neck resistance, and we found that, contrary to Ohm's law, the spine neck resistance can be inversely proportional to its radius
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28

Hothorn, Torsten, and Achim Zeileis. "Generalized Maximally Selected Statistics." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2007. http://epub.wu.ac.at/1252/1/document.pdf.

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Maximally selected statistics for the estimation of simple cutpoint models are embedded into a generalized conceptual framework based on conditional inference procedures. This powerful framework contains most of the published procedures in this area as special cases, such as maximally selected chi-squared and rank statistics, but also allows for direct construction of new test procedures for less standard test problems. As an application, a novel maximally selected rank statistic is derived from this framework for a censored response partitioned with respect to two ordered categorical covariates and potential interactions. This new test is employed to search for a high-risk group of rectal cancer patients treated with a neo-adjuvant chemoradiotherapy. Moreover, a new efficient algorithm for the evaluation of the asymptotic distribution for a large class of maximally selected statistics is given enabling the fast evaluation of a large number of cutpoints.
Series: Research Report Series / Department of Statistics and Mathematics
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29

Szyszkowicz, B. (Barbara) Carleton University Dissertation Mathematics. "Weak convergence of stochastic processes in weighted metrics and their applications to contiguous changepoint analysis." Ottawa, 1992.

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30

Kimura, Tatsuaki. "Studies on Asymptotic Analysis of GI/G/1-type Markov Chains." 京都大学 (Kyoto University), 2017. http://hdl.handle.net/2433/225742.

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31

Hudson, Richard James Frederick. "Long memory spectral regression : an approach using generalised least squares." Thesis, Queensland University of Technology, 2002.

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32

Minsker, Stanislav. "Non-asymptotic bounds for prediction problems and density estimation." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44808.

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This dissertation investigates the learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the dimension of the problem. The first part answers some open questions for the binary classification problem in the framework of active learning. Given a random couple (X,Y) with unknown distribution P, the goal of binary classification is to predict a label Y based on the observation X. Prediction rule is constructed from a sequence of observations sampled from P. The concept of active learning can be informally characterized as follows: on every iteration, the algorithm is allowed to request a label Y for any instance X which it considers to be the most informative. The contribution of this work consists of two parts: first, we provide the minimax lower bounds for the performance of active learning methods. Second, we propose an active learning algorithm which attains nearly optimal rates over a broad class of underlying distributions and is adaptive with respect to the unknown parameters of the problem. The second part of this thesis is related to sparse recovery in the framework of dictionary learning. Let (X,Y) be a random couple with unknown distribution P. Given a collection of functions H, the goal of dictionary learning is to construct a prediction rule for Y given by a linear combination of the elements of H. The problem is sparse if there exists a good prediction rule that depends on a small number of functions from H. We propose an estimator of the unknown optimal prediction rule based on penalized empirical risk minimization algorithm. We show that the proposed estimator is able to take advantage of the possible sparse structure of the problem by providing probabilistic bounds for its performance.
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33

Hoshaw-Woodard, Stacy. "Large sample methods for analyzing longitudinal data in rehabilitation research /." free to MU campus, to others for purchase, 1999. http://wwwlib.umi.com/cr/mo/fullcit?p9946263.

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34

Litherland, Trevis J. "On the limiting shape of random young tableaux for Markovian words." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/26607.

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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009.
Committee Chair: Houdre, Christian; Committee Member: Bakhtin, Yuri; Committee Member: Foley, Robert; Committee Member: Koltchinskii, Vladimir; Committee Member: Lifshitz, Mikhail; Committee Member: Matzinger, Heinrich; Committee Member: Popescu, Ionel. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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35

Köhnlein, Dieter. "Asymptotisches Verhalten von Lösungen stochastischer linearer Differenzengleichungen im Rd." Bonn : [s.n.], 1988. http://catalog.hathitrust.org/api/volumes/oclc/20267120.html.

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36

Pailden, Junvie Montealto. "Applications of Empirical Likelihood to Zero-Inflated Data and Epidemic Change Point." Bowling Green State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1367579613.

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37

Lee, Sungwook. "Semiparametric regression with random effects /." free to MU campus, to others for purchase, 1997. http://wwwlib.umi.com/cr/mo/fullcit?p9842547.

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38

Calhoun, Grayson Ford. "Limit theory for overfit models." Diss., [La Jolla] : University of California, San Diego, 2009. http://wwwlib.umi.com/cr/ucsd/fullcit?p3359804.

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Thesis (Ph. D.)--University of California, San Diego, 2009.
Title from first page of PDF file (viewed July 23, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 104-109).
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39

Liu, Xi. "Bayesian Designing and Analysis of Simple Step-Stress Accelerated Life Test with Weibull Lifetime Distribution." Ohio University / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1283365980.

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40

Lynch, O'Neil. "Mixture distributions with application to microarray data analysis." Scholar Commons, 2009. http://scholarcommons.usf.edu/etd/2075.

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The main goal in analyzing microarray data is to determine the genes that are differentially expressed across two types of tissue samples or samples obtained under two experimental conditions. In this dissertation we proposed two methods to determine differentially expressed genes. For the penalized normal mixture model (PMMM) to determine genes that are differentially expressed, we penalized both the variance and the mixing proportion parameters simultaneously. The variance parameter was penalized so that the log-likelihood will be bounded, while the mixing proportion parameter was penalized so that its estimates are not on the boundary of its parametric space. The null distribution of the likelihood ratio test statistic (LRTS) was simulated so that we could perform a hypothesis test for the number of components of the penalized normal mixture model. In addition to simulating the null distribution of the LRTS for the penalized normal mixture model, we showed that the maximum likelihood estimates were asymptotically normal, which is a first step that is necessary to prove the asymptotic null distribution of the LRTS. This result is a significant contribution to field of normal mixture model. The modified p-value approach for detecting differentially expressed genes was also discussed in this dissertation. The modified p-value approach was implemented so that a hypothesis test for the number of components can be conducted by using the modified likelihood ratio test. In the modified p-value approach we penalized the mixing proportion so that the estimates of the mixing proportion are not on the boundary of its parametric space. The null distribution of the (LRTS) was simulated so that the number of components of the uniform beta mixture model can be determined. Finally, for both modified methods, the penalized normal mixture model and the modified p-value approach were applied to simulated and real data.
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41

Hitz, Adrien. "Modelling of extremes." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:ad32f298-b140-4aae-b50e-931259714085.

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This work focuses on statistical methods to understand how frequently rare events occur and what the magnitude of extreme values such as large losses is. It lies in a field called extreme value analysis whose scope is to provide support for scientific decision making when extreme observations are of particular importance such as in environmental applications, insurance and finance. In the univariate case, I propose new techniques to model tails of discrete distributions and illustrate them in an application on word frequency and multiple birth data. Suitably rescaled, the limiting tails of some discrete distributions are shown to converge to a discrete generalized Pareto distribution and generalized Zipf distribution respectively. In the multivariate high-dimensional case, I suggest modeling tail dependence between random variables by a graph such that its nodes correspond to the variables and shocks propagate through the edges. Relying on the ideas of graphical models, I prove that if the variables satisfy a new notion called asymptotic conditional independence, then the density of the joint distribution can be simplified and expressed in terms of lower dimensional functions. This generalizes the Hammersley- Clifford theorem and enables us to infer tail distributions from observations in reduced dimension. As an illustration, extreme river flows are modeled by a tree graphical model whose structure appears to recover almost exactly the actual river network. A fundamental concept when studying limiting tail distributions is regular variation. I propose a new notion in the multivariate case called one-component regular variation, of which Karamata's and the representation theorem, two important results in the univariate case, are generalizations. Eventually, I turn my attention to website visit data and fit a censored copula Gaussian graphical model allowing the visualization of users' behavior by a graph.
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42

Chen, Ying-Ju. "Jackknife Empirical Likelihood And Change Point Problems." Bowling Green State University / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1430823961.

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43

Ruppert, Julia [Verfasser], Tobias [Akademischer Betreuer] Kaiser, and Jean-Philippe [Akademischer Betreuer] Rolin. "Asymptotic Expansion for the Time Evolution of the Probability Distribution Given by the Brownian Motion on Semialgebraic Sets / Julia Ruppert ; Tobias Kaiser, Jean-Philippe Rolin." Passau : Universität Passau, 2017. http://d-nb.info/1144611504/34.

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44

Rydén, Patrik. "Statistical analysis and simulation methods related to load-sharing models." Doctoral thesis, Umeå universitet, Matematisk statistik, 2000. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-46772.

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We consider the problem of estimating the reliability of bundles constructed of several fibres, given a particular kind of censored data. The bundles consist of several fibres which have their own independent identically dis-tributed failure stresses (i.e.the forces that destroy the fibres). The force applied to a bundle is distributed between the fibres in the bundle, accord-ing to a load-sharing model. A bundle with these properties is an example of a load-sharing system. Ropes constructed of twisted threads, compos-ite materials constructed of parallel carbon fibres, and suspension cables constructed of steel wires are all examples of load-sharing systems. In par-ticular, we consider bundles where load-sharing is described by either the Equal load-sharing model or the more general Local load-sharing model. In order to estimate the cumulative distribution function of failure stresses of bundles, we need some observed data. This data is obtained either by testing bundles or by testing individual fibres. In this thesis, we develop several theoretical testing methods for both fibres and bundles, and related methods of statistical inference. Non-parametric and parametric estimators of the cumulative distribu-tion functions of failure stresses of fibres and bundles are obtained from different kinds of observed data. It is proved that most of these estimators are consistent, and that some are strongly consistent estimators. We show that resampling, in this case random sampling with replacement from sta-tistically independent portions of data, can be used to assess the accuracy of these estimators. Several numerical examples illustrate the behavior of the obtained estimators. These examples suggest that the obtained estimators usually perform well when the number of observations is moderate.
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45

Hu, Yanling. "SOME CONTRIBUTIONS TO THE CENSORED EMPIRICAL LIKELIHOOD WITH HAZARD-TYPE CONSTRAINTS." UKnowledge, 2011. http://uknowledge.uky.edu/gradschool_diss/150.

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Empirical likelihood (EL) is a recently developed nonparametric method of statistical inference. Owen’s 2001 book contains many important results for EL with uncensored data. However, fewer results are available for EL with right-censored data. In this dissertation, we first investigate a right-censored-data extension of Qin and Lawless (1994). They studied EL with uncensored data when the number of estimating equations is larger than the number of parameters (over-determined case). We obtain results similar to theirs for the maximum EL estimator and the EL ratio test, for the over-determined case, with right-censored data. We employ hazard-type constraints which are better able to handle right-censored data. Then we investigate EL with right-censored data and a k-sample mixed hazard-type constraint. We show that the EL ratio test statistic has a limiting chi-square distribution when k = 2. We also study the relationship between the constrained Kaplan-Meier estimator and the corresponding Nelson-Aalen estimator. We try to prove that they are asymptotically equivalent under certain conditions. Finally we present simulation studies and examples showing how to apply our theory and methodology with real data.
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46

Cassart, Delphine. "Optimal tests for symmetry." Doctoral thesis, Universite Libre de Bruxelles, 2007. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210693.

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Dans ce travail, nous proposons des procédures de test paramétriques et nonparamétrique localement et asymptotiquement optimales au sens de Hajek et Le Cam, pour trois modèles d'asymétrie.

La construction de modèles d'asymétrie est un sujet de recherche qui a connu un grand développement ces dernières années, et l'obtention des tests optimaux (pour trois modèles différents) est une étape essentielle en vue de leur mise en application.

Notre approche est fondée sur la théorie de Le Cam d'une part, pour obtenir les propriétés de normalité asymptotique, bases de la construction des tests paramétriques optimaux, et la théorie de Hajek d'autre part, qui, via un principe d'invariance permet d'obtenir les procédures non-paramétriques.

Nous considérons dans ce travail deux classes de distributions univariées asymétriques, l'une fondée sur un développement d'Edgeworth (décrit dans le Chapitre 1), et l'autre construite en utilisant un paramètre d'échelle différent pour les valeurs positives et négatives (le modèle de Fechner, décrit dans le Chapitre 2).

Le modèle d'asymétrie elliptique étudié dans le dernier chapitre est une généralisation multivariée du modèle du Chapitre 2.

Pour chacun de ces modèles, nous proposons de tester l'hypothèse de symétrie par rapport à un centre fixé, puis par rapport à un centre non spécifié.

Après avoir décrit le modèle pour lequel nous construisons les procédures optimales, nous obtenons la propriété de normalité locale asymptotique. A partir de ce résultat, nous sommes capable de construire les tests paramétriques localement et asymptotiquement optimaux. Ces tests ne sont toutefois valides que si la densité sous-jacente f est correctement spécifiée. Ils ont donc le mérite de déterminer les bornes d'efficacité paramétrique, mais sont difficilement applicables.

Nous adaptons donc ces tests afin de pouvoir tester les hypothèses de symétrie par rapport à un centre fixé ou non, lorsque la densité sous-jacente est considérée comme un paramètre de nuisance.

Les tests que nous obtenons restent localement et asymptotiquement optimaux sous f, mais restent valides sous une large classe de densités.

A partir des propriétés d'invariance du sous-modèle identifié par l'hypothèse nulle, nous obtenons les tests de rangs signés localement et asymptotiquement optimaux sous f, et valide sous une vaste classe de densité. Nous présentons en particulier, les tests fondés sur les scores normaux (ou tests de van der Waerden), qui sont optimaux sous des hypothèses Gaussiennes, tout en étant valides si cette hypothèse n'est pas vérifiée.

Afin de comparer les performances des tests paramétriques et non paramétriques présentés, nous calculons les efficacités asymptotiques relatives des tests non paramétriques par rapport aux tests pseudo-Gaussiens, sous une vaste classe de densités non-Gaussiennes, et nous proposons quelques simulations.
Doctorat en sciences, Orientation statistique
info:eu-repo/semantics/nonPublished

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47

Nold, Mariana Saskia Verfasser], Susanne [Akademischer Betreuer] [Rässler, and Georg [Akademischer Betreuer] Heinze. "Behavior of convergence in logistic regression models - Assessing the drop of the Kolmogorov distance between the sampling distribution and the asymptotic distribution of estimators and test statistics in logistic regression analysis / Mariana Saskia Nold. Betreuer: Susanne Rässler ; Georg Heinze." Bamberg : Otto-Friedrich-Universität Bamberg, 2014. http://d-nb.info/1058554395/34.

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48

Gao, Zhenguo. "Variance Change Point Detection under A Smoothly-changing Mean Trend with Application to Liver Procurement." Diss., Virginia Tech, 2018. http://hdl.handle.net/10919/82351.

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Literature on change point analysis mostly requires a sudden change in the data distribution, either in a few parameters or the distribution as a whole. We are interested in the scenario that the variance of data may make a significant jump while the mean of data changes in a smooth fashion. It is motivated by a liver procurement experiment with organ surface temperature monitoring. Blindly applying the existing change point analysis methods to the example can yield erratic change point estimates since the smoothly-changing mean violates the sudden-change assumption. In my dissertation, we propose a penalized weighted least squares approach with an iterative estimation procedure that naturally integrates variance change point detection and smooth mean function estimation. Given the variance components, the mean function is estimated by smoothing splines as the minimizer of the penalized weighted least squares. Given the mean function, we propose a likelihood ratio test statistic for identifying the variance change point. The null distribution of the test statistic is derived together with the rates of convergence of all the parameter estimates. Simulations show excellent performance of the proposed method. Application analysis offers numerical support to the non-invasive organ viability assessment by surface temperature monitoring. The method above can only yield the variance change point of temperature at a single point on the surface of the organ at a time. In practice, an organ is often transplanted as a whole or in part. Therefore, it is generally of more interest to study the variance change point for a chunk of organ. With this motivation, we extend our method to study variance change point for a chunk of the organ surface. Now the variances become functions on a 2D space of locations (longitude and latitude) and the mean is a function on a 3D space of location and time. We model the variance functions by thin-plate splines and the mean function by the tensor product of thin-plate splines and cubic splines. However, the additional dimensions in these functions incur serious computational problems since the sample size, as a product of the number of locations and the number of sampling time points, becomes too large to run the standard multi-dimensional spline models. To overcome the computational hurdle, we introduce a multi-stages subsampling strategy into our modified iterative algorithm. The strategy involves several down-sampling or subsampling steps educated by preliminary statistical measures. We carry out extensive simulations to show that the new method can efficiently cut down the computational cost and make a practically unsolvable problem solvable with reasonable time and satisfactory parameter estimates. Application of the new method to the liver surface temperature monitoring data shows its effectiveness in providing accurate status change information for a portion of or the whole organ.
Ph. D.
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49

Kong, Fanhui. "Asymptotic distributions of Buckley-James estimator." Online access via UMI:, 2005.

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50

Yi, Yun. "On the structure of asymptotic distributions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/NQ50004.pdf.

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