Statistical inference for uniform stochastic ordering in several populations

tion for k-population problems under uniform stochastic ordering restrictions. We derive closed-form estimates even with right-censored data by a reparameterization which reduces the problem to a well-known isotonic regression problem. We also derive the asymptotic distribution of the likelihood ratio statistic for testing equality of the k populations against the uniform stochastic ordering restriction. This asymptotic distribution is of the chi-bar-square type as discussed by Robertson, Wright and Dykstra. These distributional results are obtained by appealing to elegant results from empirical process theory and showing that the proposed test is asymptotically distribution free. Recurrence formulas are derived for the weights of the chi-bar-square distribution for particular cases. The theory developed in this paper is illustrated by an example involving data for survival times for carcinoma of the oropharynx.

[1]  Tim Robertson,et al.  Likelihood Ratio Tests for and Against a Stochastic Ordering Between Multinomial Populations , 1981 .

[2]  Emad-Eldin A. A. Aly.  Comparing and testing order relations between percentile residual life functions , 1988 .

[3]  Allan R. Sampson,et al.  Estimation of Multivariate Distributions under Stochastic Ordering , 1989 .

[4]  Subhash C. Kochar,et al.  On tail-ordering and comparison of failure rates , 1986 .

[5]  J. Kiefer,et al.  CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS , 1956 .

[6]  R. Madsen,et al.  A nonparametric likelihood ratio test , 1983 .

[7]  Nozer D. Singpurwalla,et al.  A New Approach To Inference From Accelerated Life Tests , 1980, IEEE Transactions on Reliability.

[8]  J. Kalbfleisch.  Statistical Inference Under Order Restrictions , 1975 .

[9]  R. Dykstra,et al.  Nonparametric maximum likelihood estimation of survival functions with a general stochastic ordering and its dual , 1989 .

[10]  Tests for the equality of failure rates , 1985 .

[11]  W. Franck,et al.  A Likelihood Ratio Test for Stochastic Ordering , 1984 .

[12]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[13]  R. Dykstra,et al.  Maximum Likelihood Estimation of the Survival Functions of Stochastically Ordered Random Variables , 1982 .

[14]  R. E. Miles.  THE COMPLETE AMALGAMATION INTO BLOCKS, BY WEIGHTED MEANS, OF A FINITE SET OF REAL NUMBERS , 1959 .

[15]  D. L. Hanson,et al.  Maximum Likelihood Estimation of the Distributions of Two Stochastically Ordered Random Variables , 1966 .

[16]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[17]  E. Kaplan,et al.  Nonparametric Estimation from Incomplete Observations , 1958 .

[18]  S. Kochar.  A new distribution-free test for the equality of two failure rates , 1981 .

[19]  J. Lynch,et al.  Uniform stochastic orderings and total positivity , 1987 .

[20]  F. T. Wright,et al.  On approximation of the level probabilities and associated distributions in order restricted inference , 1983 .

[21]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[22]  F. T. Wright,et al.  On the Maximum Likelihood Estimation of Stochastically Ordered Random Variates , 1974 .

[23]  D. J. Bartholomew,et al.  A TEST OF HOMOGENEITY FOR ORDERED ALTERNATIVES. II , 1959 .

[24]  Tim Robertson,et al.  Order Restricted Statistical Tests on Multinomial and Poisson Parameters: The Starshaped Restriction , 1982 .

[25]  Ward Whitt,et al.  Comparison methods for queues and other stochastic models , 1986 .

[26]  J. Keilson,et al.  Uniform stochastic ordering and related inequalities , 1982 .

[27]  Douglas A. Wolfe,et al.  A Distribution-Free Test for Stochastic Ordering , 1976 .

[28]  Distribution-free comparison of two probability distributions with reference to their hazard rates , 1979 .

[29]  Tail Ordering and Asymptotic Efficiency of Rank Tests , 1988 .

[30]  H. Joe,et al.  Comparison of two life distributions on the basis of their percentile residual life functions , 1984 .