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Journal articles on the topic 'Likelihood ratio distributions'

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Author: Grafiati

Published: 10 March 2023

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1

Skolimowska, Magdalena, and Jarosław Bartoszewicz. "Weighting, likelihood ratio order and life distributions." Applicationes Mathematicae 33, no. 3-4 (2006): 283–91. http://dx.doi.org/10.4064/am33-3-4.

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2

Jiang, Tiefeng, and Yongcheng Qi. "Likelihood Ratio Tests for High-Dimensional Normal Distributions." Scandinavian Journal of Statistics 42, no. 4 (March 25, 2015): 988–1009. http://dx.doi.org/10.1111/sjos.12147.

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3

Krishnamoorthy, Kalimuthu, Meesook Lee, and Wang Xiao. "Likelihood ratio tests for comparing several gamma distributions." Environmetrics 26, no. 8 (August 31, 2015): 571–83. http://dx.doi.org/10.1002/env.2357.

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4

Huang, Kai, and Jie Mi. "APPLICATIONS OF LIKELIHOOD RATIO ORDER IN BAYESIAN INFERENCES." Probability in the Engineering and Informational Sciences 34, no. 1 (August 6, 2018): 1–13. http://dx.doi.org/10.1017/s026996481800027x.

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The present paper studies the likelihood ratio order of posterior distributions of parameter when the same order exists between the corresponding prior of the parameter, or when the observed values of the sufficient statistic for the parameter differ. The established likelihood order allows one to compare the Bayesian estimators associated with many common and general error loss functions analytically. It can also enable one to compare the Bayes factor in hypothesis testing without using numerical computation. Moreover, using the likelihood ratio (LR) order of the posterior distributions can yield the LR order between marginal predictive distributions, and posterior predictive distributions.

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5

Anderson, T. W., Huang Hsu, and Kai-Tai Fang. "Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions." Canadian Journal of Statistics 14, no. 1 (March 1986): 55–59. http://dx.doi.org/10.2307/3315036.

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6

Coelho, Carlos A., Barry C. Arnold, and Filipe J. Marques. "Near-Exact Distributions for Certain Likelihood Ratio Test Statistics." Journal of Statistical Theory and Practice 4, no. 4 (December 2010): 711–25. http://dx.doi.org/10.1080/15598608.2010.10412014.

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7

Vu, H. T. V., and R. A. Maller. "The Likelihood Ratio Test for Poisson versus Binomial Distributions." Journal of the American Statistical Association 91, no. 434 (June 1996): 818–24. http://dx.doi.org/10.1080/01621459.1996.10476949.

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8

Catana, Luigi-Ionut, and Vasile Preda. "A New Stochastic Order of Multivariate Distributions: Application in the Study of Reliability of Bridges Affected by Earthquakes." Mathematics 11, no. 1 (December 26, 2022): 102. http://dx.doi.org/10.3390/math11010102.

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In this article, we introduce and study a new stochastic order of multivariate distributions, namely, the conditional likelihood ratio order. The proposed order and other stochastic orders are analyzed in the case of a bivariate exponential distributions family. The theoretical results obtained are applied for studying the reliability of bridges affected by earthquakes. The conditional likelihood ratio order involves the multivariate stochastic ordering; it resembles the likelihood ratio order in the univariate case but is much easier to verify than the likelihood ratio order in the multivariate case. Additionally, the likelihood ratio order in the multivariate case implies this ordering. However, the conditional likelihood ratio order does not imply the weak hard rate order, and it is not an order relation on the multivariate distributions set. The new conditional likelihood ratio order, together with the likelihood ratio order and the weak hazard rate order, were studied in the case of the bivariate Marshall–Olkin exponential distributions family, which has a lack of memory type property. At the end of the paper, we also presented an application of the analyzed orderings for this bivariate distributions family to the study of the effects of earthquakes on bridges.

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9

Visscher, Peter M. "A Note on the Asymptotic Distribution of Likelihood Ratio Tests to Test Variance Components." Twin Research and Human Genetics 9, no. 4 (August 1, 2006): 490–95. http://dx.doi.org/10.1375/twin.9.4.490.

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AbstractWhen using maximum likelihood methods to estimate genetic and environmental components of (co)variance, it is common to test hypotheses using likelihood ratio tests, since such tests have desirable asymptotic properties. In particular, the standard likelihood ratio test statistic is assumed asymptotically to follow a χ2 distribution with degrees of freedom equal to the number of parameters tested. Using the relationship between least squares and maximum likelihood estimators for balanced designs, it is shown why the asymptotic distribution of the likelihood ratio test for variance components does not follow a χ2 distribution with degrees of freedom equal to the number of parameters tested when the null hypothesis is true. Instead, the distribution of the likelihood ratio test is a mixture of χ2 distributions with different degrees of freedom. Implications for testing variance components in twin designs and for quantitative trait loci mapping are discussed. The appropriate distribution of the likelihood ratio test statistic should be used in hypothesis testing and model selection.

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10

Slooten, K. "Likelihood ratio distributions and the (ir)relevance of error rates." Forensic Science International: Genetics 44 (January 2020): 102173. http://dx.doi.org/10.1016/j.fsigen.2019.102173.

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11

Obilade, Titi. "A Weighted Likelihood Ratio of Two Related Negative Hypergeomeric Distributions." Acta Mathematicae Applicatae Sinica, English Series 20, no. 4 (November 2004): 647–54. http://dx.doi.org/10.1007/s10255-004-0202-y.

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12

Obilade, Titi. "A weighted likelihood ratio of two related negative hypergeometric distributions." Journal of Statistical Planning and Inference 136, no. 1 (January 2006): 147–58. http://dx.doi.org/10.1016/j.jspi.2004.06.027.

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13

Marques, Filipe J., and Carlos A. Coelho. "Near-exact distributions for the sphericity likelihood ratio test statistic." Journal of Statistical Planning and Inference 138, no. 3 (March 2008): 726–41. http://dx.doi.org/10.1016/j.jspi.2007.01.002.

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14

Han, Gang, Michael J. Schell, and Jongphil Kim. "Comparing Two Exponential Distributions Using the Exact Likelihood Ratio Test." Statistics in Biopharmaceutical Research 4, no. 4 (October 2012): 348–56. http://dx.doi.org/10.1080/19466315.2012.698945.

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15

Bhavsar, C. D. "Asymptotic distributions of likelihood ratio criteria for two testing problems." Kybernetes 29, no. 4 (June 2000): 510–17. http://dx.doi.org/10.1108/03684920010322262.

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16

Nielsen, Frank. "Revisiting Chernoff Information with Likelihood Ratio Exponential Families." Entropy 24, no. 10 (October 1, 2022): 1400. http://dx.doi.org/10.3390/e24101400.

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The Chernoff information between two probability measures is a statistical divergence measuring their deviation defined as their maximally skewed Bhattacharyya distance. Although the Chernoff information was originally introduced for bounding the Bayes error in statistical hypothesis testing, the divergence found many other applications due to its empirical robustness property found in applications ranging from information fusion to quantum information. From the viewpoint of information theory, the Chernoff information can also be interpreted as a minmax symmetrization of the Kullback–Leibler divergence. In this paper, we first revisit the Chernoff information between two densities of a measurable Lebesgue space by considering the exponential families induced by their geometric mixtures: The so-called likelihood ratio exponential families. Second, we show how to (i) solve exactly the Chernoff information between any two univariate Gaussian distributions or get a closed-form formula using symbolic computing, (ii) report a closed-form formula of the Chernoff information of centered Gaussians with scaled covariance matrices and (iii) use a fast numerical scheme to approximate the Chernoff information between any two multivariate Gaussian distributions.

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17

Hürlimann, Werner. "Multivariate Likelihood Ratio Order for Skew-Symmetric Distributions with a Common Kernel." ISRN Probability and Statistics 2013 (November 28, 2013): 1–4. http://dx.doi.org/10.1155/2013/614938.

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The multivariate likelihood ratio order comparison of skew-symmetric distributions with a common kernel is considered. Two multivariate likelihood ratio perturbation invariance properties are derived.

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18

Lee, Lung-Fei. "Asymptotic Distribution of the Maximum Likelihood Estimator for a Stochastic Frontier Function Model with a Singular Information Matrix." Econometric Theory 9, no. 3 (June 1993): 413–30. http://dx.doi.org/10.1017/s026646660000774x.

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This paper investigates the asymptotic distribution of the maximum likelihood estimator in a stochastic frontier function when the firms are all technically efficient. For such a situation the true parameter vector is on the boundary of the parameter space, and the scores are linearly dependent. The asymptotic distribution of the maximum likelihood estimator is shown to be a mixture of certain truncated distributions. The maximum likelihood estimates for different parameters may have different rates of stochastic convergence. The model can be reparameterized into one with a regular likelihood function. The likelihood ratio test statistic has the usual mixture of chi-square distributions as in the regular case.

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19

Basu, A. K., and Debasis Bhattacharya. "On the Asymptotic Non-Null Distribution of Randomly Stopped Log-Likelihood Ratio Statistic." Calcutta Statistical Association Bulletin 42, no. 3-4 (September 1992): 255–60. http://dx.doi.org/10.1177/0008068319920310.

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Non-null asymptotic distribution of randomly stopped loglikelihood ratio statistic for general dependent process has been obtained. We observed that under certain regularity conditions the limiting distribution of randomly stopped log-likelihood ratio statistic under alternative hypothesis is again a mixture of Normal distributions. Some possible applications of the result has also been pointed out in this note.

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20

Wong, Augustine. "Comparing Several Gamma Means: An Improved Log-Likelihood Ratio Test." Entropy 25, no. 1 (January 5, 2023): 111. http://dx.doi.org/10.3390/e25010111.

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The two-parameter gamma distribution is one of the most commonly used distributions in analyzing environmental, meteorological, medical, and survival data. It has a two-dimensional minimal sufficient statistic, and the two parameters can be taken to be the mean and shape parameters. This makes it closely comparable to the normal model, but it differs substantially in that the exact distribution for the minimal sufficient statistic is not available. A Bartlett-type correction of the log-likelihood ratio statistic is proposed for the one-sample gamma mean problem and extended to testing for homogeneity of k≥2 independent gamma means. The exact correction factor, in general, does not exist in closed form. In this paper, a simulation algorithm is proposed to obtain the correction factor numerically. Real-life examples and simulation studies are used to illustrate the application and the accuracy of the proposed method.

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21

Mantelle, Lily Llorens, and Malay Ghosh. "Generalised Likelihood Ratio Tests for Two-Parameter Exponentals Under Type I Censoring." Calcutta Statistical Association Bulletin 36, no. 3-4 (September 1987): 125–40. http://dx.doi.org/10.1177/0008068319870302.

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The paper considers generalized likelihood ratio tests for the equality of the location, parameters and⁄or the failure rates of k independent location and scale parameter exponentials when observations are censored in time. For testing the equality of the failure rates, asymptotic null distributions of the generalized likelihood ratio test (GLRT) criteria are obtained. Also, asymptotic distributions of GLRT criteria are obtained under local alternatives. For testing the equality of the location parameters, asymptotic null distributions of the GLRT criteria are obtained.

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22

Barreto-Souza, Wagner, and Rodrigo B. Silva. "A likelihood ratio test to discriminate exponential–Poisson and gamma distributions." Journal of Statistical Computation and Simulation 85, no. 4 (October 21, 2013): 802–23. http://dx.doi.org/10.1080/00949655.2013.847097.

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23

Grimshaw, Scott D., David G. Whiting, and Thomas H. Morris. "Likelihood Ratio Tests for a Mixture of Two von Mises Distributions." Biometrics 57, no. 1 (March 2001): 260–65. http://dx.doi.org/10.1111/j.0006-341x.2001.00260.x.

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24

Jiang, Dandan, Tiefeng Jiang, and Fan Yang. "Likelihood ratio tests for covariance matrices of high-dimensional normal distributions." Journal of Statistical Planning and Inference 142, no. 8 (August 2012): 2241–56. http://dx.doi.org/10.1016/j.jspi.2012.02.057.

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25

Tikhov, M. S. "Limiting distributions of the likelihood ratio and estimators from censored samples." Journal of Soviet Mathematics 53, no. 6 (March 1991): 607–14. http://dx.doi.org/10.1007/bf01095369.

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26

Jarrahiferiz, Jalil, G. R. Mohtashami Borzadaran, and A. H. Rezaei Roknabadi. "Glaser’s function and stochastic orders for mixture distributions." International Journal of Quality & Reliability Management 33, no. 8 (September 5, 2016): 1230–38. http://dx.doi.org/10.1108/ijqrm-04-2013-0072.

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Purpose The purpose of this paper is to study likelihood ratio order for mixture and its components via their Glaser’s functions for weighted distributions. So, some theoretical examples using exponential family and their mixtures are presented. Design/methodology/approach First, Glaser’s functions of mixture and its components for weighted distributions in different scenarios are computed. Then by them the likelihood ratio order is investigated between mixture and its components. Findings The authors find conditions for weight functions under which the mixture random variable is between of its components in likelihood ratio order. Originality/value Results are obtained for weight function in general. It is well known that the some special weights are order statistics, up and down records, hazard rate, reversed hazard rate, moment generating function, etc. So, the results are valid for all of them.

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27

Lee, Sunbok. "Detecting Differential Item Functioning Using the Logistic Regression Procedure in Small Samples." Applied Psychological Measurement 41, no. 1 (September 24, 2016): 30–43. http://dx.doi.org/10.1177/0146621616668015.

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The logistic regression (LR) procedure for testing differential item functioning (DIF) typically depends on the asymptotic sampling distributions. The likelihood ratio test (LRT) usually relies on the asymptotic chi-square distribution. Also, the Wald test is typically based on the asymptotic normality of the maximum likelihood (ML) estimation, and the Wald statistic is tested using the asymptotic chi-square distribution. However, in small samples, the asymptotic assumptions may not work well. The penalized maximum likelihood (PML) estimation removes the first-order finite sample bias from the ML estimation, and the bootstrap method constructs the empirical sampling distribution. This study compares the performances of the LR procedures based on the LRT, Wald test, penalized likelihood ratio test (PLRT), and bootstrap likelihood ratio test (BLRT) in terms of the statistical power and type I error for testing uniform and non-uniform DIF. The result of the simulation study shows that the LRT with the asymptotic chi-square distribution works well even in small samples.

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28

Lucas, André. "Cointegration Testing Using Pseudolikelihood Ratio Tests." Econometric Theory 13, no. 2 (April 1997): 149–69. http://dx.doi.org/10.1017/s0266466600005703.

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This paper considers pseudomaximum likelihood estimators for vector autoregressive models. These estimators are used to determine the cointegration rank of a multivariate time series process using pseudolikelihood ratio tests. The asymptotic distributions of these tests depend on nuisance parameters if the pseudolikelihood is non-Gaussian. This even holds if the likelihood is correctly specified. The nuisance parameters have a natural interpretation and can be consistently estimated. Some simulation results illustrate the usefulness of the tests: non-Gaussian pseudolikelihood ratio tests generally have a higher power than the Gaussian test of Johansen if the innovations demonstrate leptokurtic behavior.

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29

Kokko, Jan, Ulpu Remes, Owen Thomas, Henri Pesonen, and Jukka Corander. "PYLFIRE: Python implementation of likelihood-free inference by ratio estimation." Wellcome Open Research 4 (December 10, 2019): 197. http://dx.doi.org/10.12688/wellcomeopenres.15583.1.

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Likelihood-free inference for simulator-based models is an emerging methodological branch of statistics which has attracted considerable attention in applications across diverse fields such as population genetics, astronomy and economics. Recently, the power of statistical classifiers has been harnessed in likelihood-free inference to obtain either point estimates or even posterior distributions of model parameters. Here we introduce PYLFIRE, an open-source Python implementation of the inference method LFIRE (likelihood-free inference by ratio estimation) that uses penalised logistic regression. PYLFIRE is made available as part of the general ELFI inference software http://elfi.ai to benefit both the user and developer communities for likelihood-free inference.

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30

Kang, Sang-Gil, Jeong-Hee Lee, and Woo-Dong Lee. "Likelihood based inference for the ratio of parameters in two Maxwell distributions." Journal of the Korean Data and Information Science Society 23, no. 1 (January 31, 2012): 89–98. http://dx.doi.org/10.7465/jkdi.2012.23.1.089.

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31

Coelho, Carlos A., and Filipe J. Marques. "Near-exact distributions for the independence and sphericity likelihood ratio test statistics." Journal of Multivariate Analysis 101, no. 3 (March 2010): 583–93. http://dx.doi.org/10.1016/j.jmva.2009.09.012.

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32

Jiang, Jenny, and Min Tsao. "Mixture Distributions Based Methods of Calibration for the Empirical Log-Likelihood Ratio." Communications in Statistics - Simulation and Computation 36, no. 3 (May 4, 2007): 505–17. http://dx.doi.org/10.1080/03610910701208973.

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33

Nemes, Szilárd. "Likelihood Confidence Intervals When Only Ranges are Available." Stats 2, no. 1 (February 6, 2019): 104–10. http://dx.doi.org/10.3390/stats2010008.

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Research papers represent an important and rich source of comparative data. The change is to extract the information of interest. Herein, we look at the possibilities to construct confidence intervals for sample averages when only ranges are available with maximum likelihood estimation with order statistics (MLEOS). Using Monte Carlo simulation, we looked at the confidence interval coverage characteristics for likelihood ratio and Wald-type approximate 95% confidence intervals. We saw indication that the likelihood ratio interval had better coverage and narrower intervals. For single parameter distributions, MLEOS is directly applicable. For location-scale distribution is recommended that the variance (or combination of it) to be estimated using standard formulas and used as a plug-in.

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34

Clancy, Damian, and Philip K. Pollett. "A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic." Journal of Applied Probability 40, no. 03 (September 2003): 821–25. http://dx.doi.org/10.1017/s002190020001977x.

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For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martínez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, …}. In the case of a birth–death process, the components of Φ(ν) can be written down explicitly for any given distributionν. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.

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35

Al Mohamad, Diaa, Erik W. Van Zwet, Eric Cator, and Jelle J. Goeman. "Adaptive critical value for constrained likelihood ratio testing." Biometrika 107, no. 3 (May 4, 2020): 677–88. http://dx.doi.org/10.1093/biomet/asaa013.

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Summary We present a new general method for constrained likelihood ratio testing which, when few constraints are violated, improves upon the existing approach in the literature that compares the likelihood ratio with the quantile of a mixture of chi-squared distributions; the improvement is in terms of both simplicity and power. The proposed method compares the constrained likelihood ratio statistic against the quantile of only one chi-squared random variable with data-dependent degrees of freedom. The new test is shown to have a valid exact significance level $\alpha$. It also has more power than the classical approach against alternatives for which the number of violations is not large. We provide more details for testing a simple order $\mu_1\leqslant\cdots\leqslant\mu_p$ against all alternatives using the proposed approach and give clear guidelines as to when the new method would be advantageous. A simulation study suggests that for testing a simple order, the new approach is more powerful in many scenarios than the existing method that uses a mixture of chi-squared variables. We illustrate the results of our adaptive procedure using real data on the liquidity preference hypothesis.

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36

Atwood, LD, AF Wilson, JE Bailey-Wilson, JN Carruth, and RC Elston. "On the distribution of the likelihood ratio test statistic for a mixture of two normal distributions." Communications in Statistics - Simulation and Computation 25, no. 3 (January 1996): 733–40. http://dx.doi.org/10.1080/03610919608813339.

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37

Chen, Huijun, and Tiefeng Jiang. "A study of two high-dimensional likelihood ratio tests under alternative hypotheses." Random Matrices: Theory and Applications 07, no. 01 (January 2018): 1750016. http://dx.doi.org/10.1142/s2010326317500162.

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Let [Formula: see text] be a [Formula: see text]-dimensional normal distribution. Testing [Formula: see text] equal to a given matrix or [Formula: see text] equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension [Formula: see text] is fixed, it is known that the LRT statistics go to [Formula: see text]-distributions. When [Formula: see text] is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that [Formula: see text] and [Formula: see text] are proportional to each other. The condition [Formula: see text] suffices in our results.

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38

Bruno, John, Edward G. Coffman, and Peter Downey. "Scheduling Independent Tasks to Minimize the Makespan on Identical Machines." Probability in the Engineering and Informational Sciences 9, no. 3 (July 1995): 447–56. http://dx.doi.org/10.1017/s026996480000396x.

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In this paper we consider scheduling n tasks on m parallel machines where the task processing times are i.i.d. random variables with a common distribution function F. Scheduling is done by an a priori assignment of tasks to machines. We show that if the distribution function F is a Pólya frequency function of order 2 (decreasing reverse hazard rate) then the assignment that attempts to place an equal number of tasks on each machine achieves the stochastically smallest makespan among all assignments. The condition embraces many important distributions, such as the gamma and truncated normal distributions. Assuming that the task processing times have a common density that is a Pólya frequency function of order 2 (increasing likelihood ratio), then we find that flatter schedules have stochastically smaller makespans in the sense of the “joint” likelihood ratio.

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39

Lee, Jaeun, Woo Dong Lee, and Sang Gil Kang. "Likelihood-based inference for the ratio of shape parameters of generalized exponential distributions." Journal of the Korean Data And Information Science Sociaty 29, no. 3 (May 31, 2018): 795–806. http://dx.doi.org/10.7465/jkdi.2018.29.3.795.

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40

Kourouklis, Stavros. "ASYMPTOTIC OPTIMALITY OF LIKELIHOOD RATIO TESTS FOR EXPONENTIAL DISTRIBUTIONS UNDER TYPE II CENSORING." Australian Journal of Statistics 30, no. 1 (March 1988): 111–14. http://dx.doi.org/10.1111/j.1467-842x.1988.tb00617.x.

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41

Wu, Jianrong, Guoyong Jiang, A. C. M. Wong, and Xiang Sun. "Likelihood Analysis for the Ratio of Means of Two Independent Log-Normal Distributions." Biometrics 58, no. 2 (June 2002): 463–69. http://dx.doi.org/10.1111/j.0006-341x.2002.00463.x.

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42

Jiang, Hui, and Shaochen Wang. "Moderate deviation principles for classical likelihood ratio tests of high-dimensional normal distributions." Journal of Multivariate Analysis 156 (April 2017): 57–69. http://dx.doi.org/10.1016/j.jmva.2017.02.004.

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43

Fu, Yuejiao, Jiahua Chen, and Pengfei Li. "Modified likelihood ratio test for homogeneity in a mixture of von Mises distributions." Journal of Statistical Planning and Inference 138, no. 3 (March 2008): 667–81. http://dx.doi.org/10.1016/j.jspi.2007.01.003.

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44

Vexler, Albert, Guogen Shan, Seongeun Kim, Wan-Min Tsai, Lili Tian, and Alan D. Hutson. "An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions." Journal of Statistical Planning and Inference 141, no. 6 (June 2011): 2128–40. http://dx.doi.org/10.1016/j.jspi.2010.12.024.

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45

Jiang, Tiefeng, and Fan Yang. "Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions." Annals of Statistics 41, no. 4 (August 2013): 2029–74. http://dx.doi.org/10.1214/13-aos1134.

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46

Chen, Yunfei, Yue Wu, Ning Chen, Wei Feng, and Jie Zhang. "New Approximate Distributions for the Generalized Likelihood Ratio Test Detection in Passive Radar." IEEE Signal Processing Letters 26, no. 5 (May 2019): 685–89. http://dx.doi.org/10.1109/lsp.2019.2903632.

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47

Haraksim, Rudolf, Daniel Ramos, and Didier Meuwly. "Validation of likelihood ratio methods for forensic evidence evaluation handling multimodal score distributions." IET Biometrics 6, no. 2 (November 9, 2016): 61–69. http://dx.doi.org/10.1049/iet-bmt.2015.0059.

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48

Caro-Lopera, Francisco J., Graciela González-Farías, and N. Balakrishnan. "On Generalized Wishart Distributions - I: Likelihood Ratio Test for Homogeneity of Covariance Matrices." Sankhya A 76, no. 2 (January 4, 2014): 179–94. http://dx.doi.org/10.1007/s13171-013-0047-7.

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49

Andersson, Steen A., and Jesse B. Crawford. "A Likelihood Ratio Test for Equality of Natural Parameters for Generalized Riesz Distributions." Sankhya A 77, no. 1 (May 27, 2014): 186–210. http://dx.doi.org/10.1007/s13171-014-0052-5.

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50

Fang, Biqi, and Kaitai Fang. "Maximum likelihood estimates and likelihood ratio criteria for location and scale parameters of the multivariatel 1-norm symmetric distributions." Acta Mathematicae Applicatae Sinica 4, no. 1 (February 1988): 13–22. http://dx.doi.org/10.1007/bf02018709.

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