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Journal articles on the topic 'Conditionally independent random variables'

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Published: 1 June 2024

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1

Foss, Serguei, and Andrew Richards. "On Sums of Conditionally Independent Subexponential Random Variables." Mathematics of Operations Research 35, no. 1 (February 2010): 102–19. http://dx.doi.org/10.1287/moor.1090.0430.

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Zhang, Jiaqi, Yanyan Tang, and Jie Xiong. "Conditional Strong Law of Large Numbers under G-Expectations." Symmetry 16, no. 3 (February 25, 2024): 272. http://dx.doi.org/10.3390/sym16030272.

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In this paper, we investigate two types of the conditional strong law of large numbers with a new notion of conditionally independent random variables under G-expectation which are related to the symmetry G-function. Our limit theorem demonstrates that the cluster points of empirical averages fall within the bounds of the lower and upper conditional expectations with lower probability one. Moreover, for conditionally independent random variables with identical conditional distributions, we show the existence of two cluster points of empirical averages that correspond to the essential minimum and essential maximum expectations, respectively, with G-capacity one.

3

Bayramoglu (Bairamov), Ismihan. "On Conditionally Independent Random Variables, Copula and Order Statistics." Communications in Statistics - Theory and Methods 43, no. 10-12 (April 23, 2014): 2105–17. http://dx.doi.org/10.1080/03610926.2013.818695.

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4

Yuan, De-Mei. "CONVERGENCE RATES FOR SEQUENCES OF CONDITIONALLY INDEPENDENT AND CONDITIONALLY IDENTICALLY DISTRIBUTED RANDOM VARIABLES." Journal of the Korean Mathematical Society 53, no. 6 (November 1, 2016): 1275–92. http://dx.doi.org/10.4134/jkms.j150490.

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5

Gudynas, P. P. "Approximation by distributions of sums of conditionally independent random variables." Lithuanian Mathematical Journal 24, no. 4 (1985): 320–25. http://dx.doi.org/10.1007/bf00969125.

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6

Astashkin, Sergey, Fedor Sukochev, and Chin Pin Wong. "Disjointification of martingale differences and conditionally independent random variables with some applications." Studia Mathematica 205, no. 2 (2011): 171–200. http://dx.doi.org/10.4064/sm205-2-3.

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7

Savinov, Evgeniy Anatolievich. "A Variant of the Necessary Condition for the Absolute Continuity of Symmetric Multivariate Mixture." Mathematics 9, no. 13 (June 27, 2021): 1505. http://dx.doi.org/10.3390/math9131505.

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Sufficient conditions are given under which the absolute continuity of the joint distribution of conditionally independent random variables can be violated. It is shown that in the case of a dimension n>1 this occurs for a sufficiently large number of discontinuity points of one-dimensional conditional distributions.

8

Fristedt, Bert, and Donald A. Berry. "Optimality of myopic stopping times for geometric discounting." Journal of Applied Probability 25, no. 2 (June 1988): 437–43. http://dx.doi.org/10.2307/3214454.

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Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

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Fristedt, Bert, and Donald A. Berry. "Optimality of myopic stopping times for geometric discounting." Journal of Applied Probability 25, no. 02 (June 1988): 437–43. http://dx.doi.org/10.1017/s0021900200041103.

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Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

10

Shervashidze, T. "Local Limit Theorems for Conditionally Independent Random Variables Controlled by a Finite Markov Chain." Theory of Probability & Its Applications 44, no. 1 (January 2000): 131–35. http://dx.doi.org/10.1137/s0040585x97977446.

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11

Hitczenko, Pawel. "On a Domination of Sums of Random Variables by Sums of Conditionally Independent Ones." Annals of Probability 22, no. 1 (January 1994): 453–68. http://dx.doi.org/10.1214/aop/1176988868.

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12

Li, Shuai, Ziqi Chen, Hongtu Zhu, Christina Dan Wang, and Wang Wen. "Nearest-Neighbor Sampling Based Conditional Independence Testing." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 7 (June 26, 2023): 8631–39. http://dx.doi.org/10.1609/aaai.v37i7.26039.

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The conditional randomization test (CRT) was recently proposed to test whether two random variables X and Y are conditionally independent given random variables Z. The CRT assumes that the conditional distribution of X given Z is known under the null hypothesis and then it is compared to the distribution of the observed samples of the original data. The aim of this paper is to develop a novel alternative of CRT by using nearest-neighbor sampling without assuming the exact form of the distribution of X given Z. Specifically, we utilize the computationally efficient 1-nearest-neighbor to approximate the conditional distribution that encodes the null hypothesis. Then, theoretically, we show that the distribution of the generated samples is very close to the true conditional distribution in terms of total variation distance. Furthermore, we take the classifier-based conditional mutual information estimator as our test statistic. The test statistic as an empirical fundamental information theoretic quantity is able to well capture the conditional-dependence feature. We show that our proposed test is computationally very fast, while controlling type I and II errors quite well. Finally, we demonstrate the efficiency of our proposed test in both synthetic and real data analyses.

13

Hadjikyriakou, Milto. "Comparison of conditional expectations of functions of strong N-demimartingales and functions of sums of conditionally independent random variables." Statistics & Probability Letters 83, no. 4 (April 2013): 1282–86. http://dx.doi.org/10.1016/j.spl.2012.12.031.

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14

Hitczenko, Paweł, and Stephen J. Montgomery-Smith. "Tangent sequences in Orlicz and rearrangement invariant spaces." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 1 (January 1996): 91–101. http://dx.doi.org/10.1017/s0305004100074004.

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AbstractLet (fn) and (gn) be two sequences of random variables adapted to an increasing sequence of σ-algebras (ℱn) such that the conditional distributions of fn and gn given ℱn coincide. Suppose further that the sequence (gn) is conditionally independent. Then it is known that where the number C is a universal constant. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities.

15

Branford, A. J. "A self-excited migration process." Journal of Applied Probability 22, no. 1 (March 1985): 58–67. http://dx.doi.org/10.2307/3213748.

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Processes whose entities would be independent but whose behaviour is influenced by environmental variables, which they in turn affect, are processes with mediated interactions. The independent open migration process with which is associated a random variable, the excitation, is such a process. This self-excited (conditionally) independent open migration process, with finite excitation, is related to a derived process, its piecewise-deterministic analogue. The correspondence gives insight into the process, as well as an equilibrium independence result. A simple example illustrates the presence of paroxysmal phenomena.

16

Branford, A. J. "A self-excited migration process." Journal of Applied Probability 22, no. 01 (March 1985): 58–67. http://dx.doi.org/10.1017/s0021900200029016.

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Processes whose entities would be independent but whose behaviour is influenced by environmental variables, which they in turn affect, are processes with mediated interactions. The independent open migration process with which is associated a random variable, the excitation, is such a process. This self-excited (conditionally) independent open migration process, with finite excitation, is related to a derived process, its piecewise-deterministic analogue. The correspondence gives insight into the process, as well as an equilibrium independence result. A simple example illustrates the presence of paroxysmal phenomena.

17

Yuan, De-Mei, Li-Ran Wei, and Lan Lei. "CONDITIONAL CENTRAL LIMIT THEOREMS FOR A SEQUENCE OF CONDITIONAL INDEPENDENT RANDOM VARIABLES." Journal of the Korean Mathematical Society 51, no. 1 (January 1, 2014): 1–15. http://dx.doi.org/10.4134/jkms.2014.51.1.001.

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18

Klein, Thierry, Agnés Lagnoux, and Pierre Petit. "A conditional Berry–Esseen inequality." Journal of Applied Probability 56, no. 01 (March 2019): 76–90. http://dx.doi.org/10.1017/jpr.2019.7.

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AbstractAs an extension of a central limit theorem established by Svante Janson, we prove a Berry–Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.

19

Lenczewski, Romuald. "Conditionally monotone independence and the associated products of graphs." Infinite Dimensional Analysis, Quantum Probability and Related Topics 22, no. 04 (December 2019): 1950023. http://dx.doi.org/10.1142/s0219025719500231.

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We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence on suitably constructed larger algebras. For that purpose, we use the approach developed for a reduction of similar type for boolean, free and monotone independences. We apply the tensor product realization of c-monotone random variables to introduce the c-comb (loop) product of birooted graphs, a generalization of the comb (loop) product of rooted graphs, and we show that it is related to the c-monotone additive (multiplicative) convolution of distributions.

20

Gerber, Hans U. "A Teacher's Remark on Exact Credibility." ASTIN Bulletin 25, no. 2 (November 1995): 189–92. http://dx.doi.org/10.2143/ast.25.2.563247.

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In the classical Bayesian approach to credibility the claims are conditionally independent and identically distributed random variables, with common density f(x, ϑ). The unknown parameter ϑ is a realization of a random variable Θ having initial (prior) density u(ϑ). LetThe initial pure premium isThe premium for X t + 1, given X1 +, …, Xt, is the conditional expectationA central question is for which pairs f(x, ϑ) and u (ϑ) this expression is linear, i.e. of the formwhere is the observed average. This is indeed the case for about half a dozen famous examples. Jewell (1974) has found an elegant and general approach to unify these examples, see also Goovaerts and Hoogstad (1987, chapter 2). The classical examples can be retrieved as special cases; however a preliminary reparameterization has to be performed on a case by case basis. The purpose of this note is to propose an alternative (but of course strongly related) formulation of the general model, from which the classical examples can be retrieved in a straightforward way.

21

Bartlett, Jonathan W., James R. Carpenter, Kate Tilling, and Stijn Vansteelandt. "Improving upon the efficiency of complete case analysis when covariates are MNAR." Biostatistics 15, no. 4 (June 6, 2014): 719–30. http://dx.doi.org/10.1093/biostatistics/kxu023.

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Abstract Missing values in covariates of regression models are a pervasive problem in empirical research. Popular approaches for analyzing partially observed datasets include complete case analysis (CCA), multiple imputation (MI), and inverse probability weighting (IPW). In the case of missing covariate values, these methods (as typically implemented) are valid under different missingness assumptions. In particular, CCA is valid under missing not at random (MNAR) mechanisms in which missingness in a covariate depends on the value of that covariate, but is conditionally independent of outcome. In this paper, we argue that in some settings such an assumption is more plausible than the missing at random assumption underpinning most implementations of MI and IPW. When the former assumption holds, although CCA gives consistent estimates, it does not make use of all observed information. We therefore propose an augmented CCA approach which makes the same conditional independence assumption for missingness as CCA, but which improves efficiency through specification of an additional model for the probability of missingness, given the fully observed variables. The new method is evaluated using simulations and illustrated through application to data on reported alcohol consumption and blood pressure from the US National Health and Nutrition Examination Survey, in which data are likely MNAR independent of outcome.

22

Prakasa Rao, B. L. S. "Characterizations of Probability Distributions Through Linear Forms of Q-Conditional Independent Random Variables." Sankhya A 78, no. 2 (July 22, 2016): 221–30. http://dx.doi.org/10.1007/s13171-016-0086-y.

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23

van Duijn, Marijtje A. J., and Margo G. H. Jansen. "Modeling Repeated Count Data: Some Extensions of the Rasch Poisson Counts Model." Journal of Educational and Behavioral Statistics 20, no. 3 (September 1995): 241–58. http://dx.doi.org/10.3102/10769986020003241.

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We consider data that can be summarized as an N × K table of counts—for example, test data obtained by administering K tests to N subjects. The cell entries yij are assumed to be conditionally independent Poisson-distributed random variables, given the NK Poisson intensity parameters μij. The Rasch Poisson Counts Model (RPCM) postulates that the intensity parameters are products of test difficulty and subject ability parameters. We expand the RPCM by assuming that the subject parameters are random variables having a common gamma distribution with fixed unknown parameters and that the vectors of test difficulty parameters per subject follow a common Dirichlet distribution with fixed unknown parameters. Further, we show how additional structures can be imposed on the test parameters, modeling a within-subjects design. Methods for testing the fit and estimating the parameters of these models are presented and illustrated with the analysis of two empirical data sets.

24

Lin, Peng, Martin Neil, and Norman Fenton. "Risk aggregation in the presence of discrete causally connected random variables." Annals of Actuarial Science 8, no. 2 (August 26, 2014): 298–319. http://dx.doi.org/10.1017/s1748499514000098.

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AbstractRisk aggregation is a popular method used to estimate the sum of a collection of financial assets or events, where each asset or event is modelled as a random variable. Applications include insurance, operational risk, stress testing and sensitivity analysis. In practice, the sum of a set of random variables involves the use of two well-known mathematical operations: n-fold convolution (for a fixed number n) and N-fold convolution, defined as the compound sum of a frequency distribution N and a severity distribution, where the number of constant n-fold convolutions is determined by N, where the severity and frequency variables are independent, and continuous, currently numerical solutions such as, Panjer’s recursion, fast Fourier transforms and Monte Carlo simulation produce acceptable results. However, they have not been designed to cope with new modelling challenges that require hybrid models containing discrete explanatory (regime switching) variables or where discrete and continuous variables are inter-dependent and may influence the severity and frequency in complex, non-linear, ways. This paper describes a Bayesian Factorisation and Elimination (BFE) algorithm that performs convolution on the hybrid models required to aggregate risk in the presence of causal dependencies. This algorithm exploits a number of advances from the field of Bayesian Networks, covering methods to approximate statistical and conditionally deterministic functions to factorise multivariate distributions for efficient computation. Experiments show that BFE is as accurate on conventional problems as competing methods. For more difficult hybrid problems BFE can provide a more general solution that the others cannot offer. In addition, the BFE approach can be easily extended to perform deconvolution for the purposes of stress testing and sensitivity analysis in a way that competing methods do not.

25

Sheikhi, Ayyub, Freshteh Arad, and Radko Mesiar. "Multivariate Asymmetric Distributions of Copula Related Random Variables." Austrian Journal of Statistics 52, no. 4 (July 19, 2023): 1–14. http://dx.doi.org/10.17713/ajs.v52i4.1446.

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It is known that normal distribution plays an important role in analysing symmetric data. However, this symmetric assumption may not hold in many real word and in such cases, asymmetric distribution, including skew normal distribution, are known as the best alternative. Constructing asymmetric distributions is carried out using the conditional/selection approach of several independent variable conditioning on other set of variables and this approach does not work well when the independence between variablesviolated. In this work we construct an asymmetric distribution when variables are dependent using a copula. Specifically, we consider the random vectors X and Y are connected using a copula function CX,Y and we study the selection distribution Z = (X|Y ∈ T ).We present some special cases of our proposed distribution, among them, multivariate skew-normal distribution. Some properties such as moments and moment generating function are investigated. Also, numerical analysis including simulation study as well asa real data set analysis are presented for illustration.

26

Feldman, G. M. "On a characterization theorem on a-adic solenoids." Доклады Академии наук 489, no. 3 (November 29, 2019): 227–31. http://dx.doi.org/10.31857/s0869-56524893227-231.

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According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We prove an analogue of this theorem for linear forms of two independent random variables taking values in an -adic solenoid containing no elements of order 2. Coefficients of the linear forms are topological automorphisms of the -adic solenoid.

27

Speed, T. P. "Cumulants and partition lattices IV: a.s. convergence of generalised k-statistics." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 1 (August 1986): 79–94. http://dx.doi.org/10.1017/s1446788700028093.

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AbstractGeneralised k-statistics associated with multi-indexed arrays of random variables satisfying a generalised form of exchangeability are studied. By showing that they form multi-indexed reversed martingales and that the associated family of σ-fields possesses certain conditional independence properties, conditions for the a.s. convergence of generalised k-statistics are obtained. When the arrays of random variables are sums of independent arrays of independent effects, as is the case with the standard random effects anova models, the limits are identified as the associated generalixed cumulants.

28

Yuan, De-Mei, and Shun-Jing Li. "EXTENSIONS OF SEVERAL CLASSICAL RESULTS FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES TO CONDITIONAL CASES." Journal of the Korean Mathematical Society 52, no. 2 (March 1, 2015): 431–45. http://dx.doi.org/10.4134/jkms.2015.52.2.431.

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29

Broström, Göran, and Leif Nilsson. "Acceptance–Rejection Sampling from the Conditional Distribution of Independent Discrete Random Variables, given their Sum." Statistics 34, no. 3 (January 2000): 247–57. http://dx.doi.org/10.1080/02331880008802716.

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30

Rolski, Tomasz, and Agata Tomanek. "Asymptotics of conditional moments of the summand in Poisson compounds." Journal of Applied Probability 48, A (August 2011): 65–76. http://dx.doi.org/10.1239/jap/1318940456.

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Suppose that N is a ℤ+-valued random variable and that X,X1,X2,… is a sequence of independent and identically distributed ℤ+ random variables independent of N. In this paper we are interested in properties of the conditional variable In particular, we want to know the asymptotic behavior of the conditional mean ENk or the conditional variance varNk as k→∞. We consider the cases when X is Poisson and when X is mixed Poisson. The problem is motivated by modeling loss reserves in nonlife insurance.

31

Rolski, Tomasz, and Agata Tomanek. "Asymptotics of conditional moments of the summand in Poisson compounds." Journal of Applied Probability 48, A (August 2011): 65–76. http://dx.doi.org/10.1017/s0021900200099137.

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Suppose that N is a ℤ+-valued random variable and that X,X 1,X 2,… is a sequence of independent and identically distributed ℤ+ random variables independent of N. In this paper we are interested in properties of the conditional variable In particular, we want to know the asymptotic behavior of the conditional mean EN k or the conditional variance varN k as k→∞. We consider the cases when X is Poisson and when X is mixed Poisson. The problem is motivated by modeling loss reserves in nonlife insurance.

32

Hu, Taizhong, Jing Chen, and Chaode Xie. "REGRESSION DEPENDENCE IN LATENT VARIABLE MODELS." Probability in the Engineering and Informational Sciences 20, no. 2 (March 6, 2006): 363–79. http://dx.doi.org/10.1017/s0269964806060220.

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Three new notions of positive dependence (positive regression dependence, positive left-tail regression dependence, and positive right-tail regression dependence) are studied in this article. Consider a latent variable model where the manifest random variables T1,T2,…,Tn given latent random variable/vector (Θ1,…,Θm) are conditional independent. Conditions are identified under which T1,…,Tn possesses the new dependence notions for different types of latent variable model. Applications of the results are also provided.

33

Sharma, Vidya Sagar. "A Fixed-Parameter Tractable Algorithm for Counting Markov Equivalence Classes with the Same Skeleton." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 18 (March 24, 2024): 20532–39. http://dx.doi.org/10.1609/aaai.v38i18.30038.

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Causal DAGs (also known as Bayesian networks) are a popular tool for encoding conditional dependencies between random variables. In a causal DAG, the random variables are modeled as vertices in the DAG, and it is stipulated that every random variable is independent of its non-descendants conditioned on its parents. It is possible, however, for two different causal DAGs on the same set of random variables to encode exactly the same set of conditional dependencies. Such causal DAGs are said to be Markov equivalent, and equivalence classes of Markov equivalent DAGs are known as Markov Equivalent Classes (MECs). Beautiful combinatorial characterizations of MECs have been developed in the past few decades, and it is known, in particular, that all DAGs in the same MEC must have the same skeleton (underlying undirected graph) and v-structures (induced subgraph of the form a->b

34

Charalambous, Charalambos D., and Jan H. van Schuppen. "A Realization Approach to Lossy Network Compression of a Tuple of Correlated Multivariate Gaussian RVs." Entropy 24, no. 9 (September 1, 2022): 1227. http://dx.doi.org/10.3390/e24091227.

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Examined in this paper is the Gray and Wyner source coding for a simple network of correlated multivariate Gaussian random variables, Y1:Ω→Rp1 and Y2:Ω→Rp2. The network consists of an encoder that produces two private rates R1 and R2, and a common rate R0, and two decoders, where decoder 1 receives rates (R1,R0) and reproduces Y1 by Y^1, and decoder 2 receives rates (R2,R0) and reproduces Y2 by Y^2, with mean-square error distortions E||Yi−Y^i||Rpi2≤Δi∈[0,∞],i=1,2. Use is made of the weak stochastic realization and the geometric approach of such random variables to derive test channel distributions, which characterize the rates that lie on the Gray and Wyner rate region. Specific new results include: (1) A proof that, among all continuous or finite-valued random variables, W:Ω→W, Wyner’s common information, C(Y1,Y2)=infPY1,Y2,W:PY1,Y2|W=PY1|WPY2|WI(Y1,Y2;W), is achieved by a Gaussian random variable, W:Ω→Rn of minimum dimension n, which makes the two components of the tuple (Y1,Y2) conditionally independent according to the weak stochastic realization of (Y1,Y2), and a the formula C(Y1,Y2)=12∑j=1nln1+dj1−dj, where di∈(0,1),i=1,…,n are the canonical correlation coefficients of the correlated parts of Y1 and Y2, and a realization of (Y1,Y2,W) which achieves this. (2) The parameterization of rates that lie on the Gray and Wyner rate region, and several of its subsets. The discussion is largely self-contained and proceeds from first principles, while connections to prior literature is discussed.

35

Bar-Lev, Shaul K., Ernst Schulte-Geers, and Wolfgang Stadje. "Conditional Limit Theorems for the Terms of a Random Walk Revisited." Journal of Applied Probability 50, no. 3 (September 2013): 871–82. http://dx.doi.org/10.1239/jap/1378401242.

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In this paper we derive limit theorems for the conditional distribution ofX1givenSn=snasn→ ∞, where theXiare independent and identically distributed (i.i.d.) random variables,Sn=X1+··· +Xn, andsn/nconverges orsn≡sis constant. We obtain convergence in total variation of PX1∣Sn/n=sto a distribution associated to that ofX1and of PnX1∣Sn=sto a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

36

Behrends, Ehrhard. "On the conditional expectation E(X | X + W) in the case of independent random variables X, W." Statistics & Probability Letters 41, no. 4 (February 1999): 397–400. http://dx.doi.org/10.1016/s0167-7152(98)00192-8.

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Wang, Wensheng, and Anwei Zhu. "Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction." Discrete Dynamics in Nature and Society 2018 (August 1, 2018): 1–9. http://dx.doi.org/10.1155/2018/8968947.

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Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋) (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

38

Holst, Lars. "A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes." Journal of Applied Probability 45, no. 4 (December 2008): 1181–85. http://dx.doi.org/10.1239/jap/1231340241.

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A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

39

Holst, Lars. "A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes." Journal of Applied Probability 45, no. 04 (December 2008): 1181–85. http://dx.doi.org/10.1017/s0021900200005052.

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A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

40

Dimopoulos, Thomas, Hristos Tyralis, Nikolaos P. Bakas, and Diofantos Hadjimitsis. "Accuracy measurement of Random Forests and Linear Regression for mass appraisal models that estimate the prices of residential apartments in Nicosia, Cyprus." Advances in Geosciences 45 (November 29, 2018): 377–82. http://dx.doi.org/10.5194/adgeo-45-377-2018.

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Abstract. The purpose of this article is to examine the prediction accuracy of the Random Forests, a machine learning method, when it is applied for residential mass appraisals in the city of Nicosia, Cyprus. The analysis is performed using transaction sales data from the Cyprus Department of Lands and Surveys, the Consumer Price Index of Cyprus from the Cyprus Statistical Service and the Central Bank of Cyprus' Residential Index (Price index for apartments). The Consumer Price Index and the price index for apartments record quarterly price changes, while the dependent variables for the computational models were the Declared and the Accepted Prices that were conditional on observed values of a variety of independent variables. The Random Forests method exhibited enhanced prediction accuracy, especially for the models that comprised of a sufficient number of independent variables, indicating the method as prominent, although it has not yet been utilized adequately for mass appraisals.

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Hasebe, Takahiro, and Hayato Saigo. "On operator-valued monotone independence." Nagoya Mathematical Journal 215 (September 2014): 151–67. http://dx.doi.org/10.1215/00277630-2741151.

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AbstractWe investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of the central limit theorem for the operator-valued case. Moreover, we prove a generalization of Muraki’s formula for the sum of independent random variables and a relation between generating functions of moments and cumulants.

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Hasebe, Takahiro, and Hayato Saigo. "On operator-valued monotone independence." Nagoya Mathematical Journal 215 (September 2014): 151–67. http://dx.doi.org/10.1017/s002776300001093x.

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AbstractWe investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of the central limit theorem for the operator-valued case. Moreover, we prove a generalization of Muraki’s formula for the sum of independent random variables and a relation between generating functions of moments and cumulants.

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Bar-Lev, Shaul K., Ernst Schulte-Geers, and Wolfgang Stadje. "Conditional Limit Theorems for the Terms of a Random Walk Revisited." Journal of Applied Probability 50, no. 03 (September 2013): 871–82. http://dx.doi.org/10.1017/s0021900200009906.

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In this paper we derive limit theorems for the conditional distribution of X 1 given S n =s n as n→ ∞, where the X i are independent and identically distributed (i.i.d.) random variables, S n =X 1+··· +X n , and s n /n converges or s n ≡ s is constant. We obtain convergence in total variation of P X 1∣ S n /n=s to a distribution associated to that of X 1 and of P nX 1∣ S n =s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

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Dubhashi, Devdatt, and Olle Häggström. "A Note on Conditioning and Stochastic Domination for Order Statistics." Journal of Applied Probability 45, no. 2 (June 2008): 575–79. http://dx.doi.org/10.1239/jap/1214950369.

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For an order statistic (X1:n,…,Xn:n) of a collection of independent but not necessarily identically distributed random variables, and any i ∈ {1,…,n}, the conditional distribution of (Xi+1:n,…,Xn:n) given Xi:n > s is shown to be stochastically increasing in s. This answers a question by Hu and Xie (2006).

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Dubhashi, Devdatt, and Olle Häggström. "A Note on Conditioning and Stochastic Domination for Order Statistics." Journal of Applied Probability 45, no. 02 (June 2008): 575–79. http://dx.doi.org/10.1017/s0021900200004447.

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For an order statistic (X 1:n ,…,X n:n ) of a collection of independent but not necessarily identically distributed random variables, and any i ∈ {1,…,n}, the conditional distribution of (X i+1:n ,…,X n:n ) given X i:n > s is shown to be stochastically increasing in s. This answers a question by Hu and Xie (2006).

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Thomas, A. "The conditional independences between variables derived from two independent identically distributed Markov random fields when pairwise order is ignored." Mathematical Medicine and Biology 27, no. 3 (November 25, 2009): 283–88. http://dx.doi.org/10.1093/imammb/dqp022.

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Falk, Michael, and Diana Tichy. "Asymptotic Conditional Distribution of Exceedance Counts." Advances in Applied Probability 44, no. 1 (March 2012): 270–91. http://dx.doi.org/10.1239/aap/1331216653.

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We investigate the asymptotic distribution of the number of exceedances amongdidentically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of thefragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the firstdRVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence asdincreases.

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Habibi, Reza. "Conditional Beta Approximation: Two Applications." Indonesian Journal of Mathematics and Applications 2, no. 1 (March 31, 2024): 9–23. http://dx.doi.org/10.21776/ub.ijma.2024.002.01.2.

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Suppose that X,Y are two independent positive continuous random variables. Let P=\frac{X}{X+Y} and Z=X+Y. If X, Y have gamma distributions with the same scale parameter, then P distribution will be beta and P,\ Z are independent. In the case that the distributions of these two variables are not gamma, the P distribution is well approximated by the beta distribution. However, P,\ Z are dependent. According to matching moment method, it is necessary to compute the moments of conditional distribution for beta fitting. In this paper, some new methods for computing moments of conditional distribution of P given Z are proposed. First of all, it is suggested to consider the regression method. Then Monte Carlo simulation is advised. The Bayesian posterior distribution of P is suggested. Applications of differential equations are also reviewed. These results are applied in two applications namely variance change point detection and winning percentage of gambling game are proposed. The probability of change in variance in a sequence of variables, as a leading indicator of possible change, is proposed. Similarly, the probability of winning in a sequential gambling framework is proposed. The optimal time to exit of gambling game is proposed. A game theoretic approach to problem of optimal exit time is proposed. In all cases, beta approximations are proposed. Finally, a conclusion section is also given.

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Ross, Sheldon M. "A Note on Approximating Mean Occupation Times of Continuous-Time Markov Chains." Probability in the Engineering and Informational Sciences 2, no. 2 (April 1988): 267–68. http://dx.doi.org/10.1017/s0269964800000796.

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In [1] an approach to approximate the transition probabilities and mean occupation times of a continuous-time Markov chain is presented. For the chain under consideration, let Pij(t) and Tij(t) denote respectively the probability that it is in state j at time t, and the total time spent in j by time t, in both cases conditional on the chain starting in state i. Also, let Y1,…, Yn be independent exponential random variables each with rate λ = n/t, which are also independent of the Markov chain.

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Falk, Michael, and Diana Tichy. "Asymptotic Conditional Distribution of Exceedance Counts." Advances in Applied Probability 44, no. 01 (March 2012): 270–91. http://dx.doi.org/10.1017/s000186780000553x.

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Abstract:

We investigate the asymptotic distribution of the number of exceedances among d identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first d RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as d increases.