D. Morales

Asymptotic divergence of estimates of discrete distributions

Abstract o-divergence D o ( P , Q ) of two estimates P and Q of a discrete distribution Pθ is shown to play an important role in mathematical statistics and information theory. If both P and Q are based on the same sample then special attention is paid to nonparameteric estimates P which are √n-consistent and parametric estimates Q = P gq defined by means of minimum o∗-divergence point estimates gq where o∗ need not be the same as o. Under a standard regularity these point estimates are shown to be efficient and the corresponding distribution estimates Q √n-consistent. But the asymptotics of D o ( P , Q ) is evaluated for arbitrary cn-consistent distribution stimates P and Q with cn → ∞. If the two distribution estimates are based on different samples then the asymptotics of D o ( P , Q ) is evaluated only for the above-mentioned special P and Q .

Asymptotic distribution of (h, φ)-entropies

In this paper, (h,φ)-entropies are presented as a generalization of φ-entropies, Havrda-Charvat entropies and the Renyi entropy among others. For this functional, asymptotic distribution for simple random sampling and stratified .sampling with proportional affixing is obtained.

Uncertainty of discrete stochastic systems: general theory and statistical inference

Uncertainty is defined in a new manner, as a function of discrete probability distributions satisfying a simple and intuitively appealing weak monotonicity condition. It is shown that every uncertainty is Schur-concave and conversely, every Schur-concave function of distributions is an uncertainty. General properties of uncertainties are systematically studied. Many characteristics of distributions introduced previously in statistical physics, mathematical statistics, econometrics and information theory are shown to be particular examples of uncertainties. New examples are introduced, and traditional as well as some new methods for obtaining uncertainties are discussed. The information defined by decrease of uncertainty resulting from an observation is investigated and related to previous concepts of information. Further, statistical inference about uncertainties is investigated, based on independent observations of system states. In particular, asymptotic distributions of maximum likelihood estimates of uncertainties and uncertainty-related functions are derived, and asymptotically /spl alpha/-level Neyman-Pearson tests of hypotheses about these system characteristics are presented.

Asymptotic properties of divergence statistics in a stratified random sampling and its applications to test statistical hypotheses

Abstract In order to study the asymptotic properties of divergence statistics, we propose a unified expression, called ( h , o )-divergence, which includes as particular cases most divergences. Under different assumptions, it is shown that the asymptotic distributions of the ( h , o )-divergence statistics, in a stratified random sampling set up, are either normal or a linear form in chi square variables depending on whether or not suitable conditions are satisfied. The chi square and the likelihood ratio test statistics are particular cases of the ( h , o )-divergence test statistics considered. From the previous results, asymptotic distributions of entropy statistics are also derived. The problem of optimum allocation is studied and the relative precision of stratified and simple random sampling is analyzed. Applications to test statistical hypotheses in multinomial populations are given. To finish, appendices with the asymptotic variances of many well known divergence statistics are presented.

Asymptotic behaviour and statistical applications of divergence measures in multinomial populations: a unified study

Divergence measures play an important role in statistical theory, especially in large sample theories of estimation and testing. The underlying reason is that they are indices of statistical distance between probability distributions P and Q; the smaller these indices are the harder it is to discriminate between P and Q. Many divergence measures have been proposed since the publication of the paper of Kullback and Leibler (1951). Renyi (1961) gave the first generalization of Kullback-Leibler divergence, Jeffreys (1946) defined the J-divergences, Burbea and Rao (1982) introduced the R-divergences, Sharma and Mittal (1977) the (r,s)-divergences, Csiszar (1967) the ϕ-divergences, Taneja (1989) the generalized J-divergences and the generalized R-divergences and so on. In order to do a unified study of their statistical properties, here we propose a generalized divergence, called (h,ϕ)-divergence, which include as particular cases the above mentioned divergence measures. Under different assumptions, it is shown that the asymptotic distributions of the (h,ϕ)-divergence statistics are either normal or chi square. The chi square and the likelihood ratio test statistics are particular cases of the (h,ϕ)-divergence test statistics considered. From the previous results, asymptotic distributions of entropy statistics are derived too. Applications to testing statistical hypothesis in multinomial populations are given. The Pitman and Bahadur efficiencies of tests of goodness of fit and independence based on these statistics are obtained. To finish, apendices with the asymptotic variances of many well known divergence and entropy statistics are presented.

Two approaches to grouping of data and related disparity statistics

Csiszars φ-divergences of discrete distributions are extended to a more general class of disparity measures by restricting the convexity of functions φ(t), t > 0, to the local convexity at t = 1 and monotonicity on intervals (0, 1) and (l,∞). Goodness-of-fit estimation and testing procedures based on the (^-disparity statistics are introduced. Robustness of the estimation procedure is discussed and the asymptotic distributions for the testing procedure are established in statistical models with data grouped according to their values or orders

Bayesian survival estimation for incomplete data when the life distribution is proportionally related to the censoring time

This article presents a nonparametric Bayesian procedure for estimating a survival curve in a proportional hazard model when some of the data are censored on the left and some are censored on the right. The method works under the assumption that there is a Dirichlet process prior knowledge on the variable of interest. An example is given .To finish some simulation is presented to analyze the estimator.

Asymptotic distributions of φ‐divergences of hypothetical and observed frequencies on refined partitions

For a wide class of goodness-of-fit statistics based on φ-divergences between hypothetical cell probabilities and observed relative frequencies, the asymptotic normality is established under the assumption n/mnγ∈(0,∞), where n denotes sample size and mn the number of cells. Related problems of asymptotic distributions of φ-divergence errors, and of φ-divergence deviations of histogram estimators from their expected values, are considered too.

LARGE SAMPLE BEHAVIOR OF ENTROPY MEASURES WHEN PARAMETERS ARE ESTIMATED

In this paper, we study the asymptotic behavior of (h,π)-entropy statistics when the parameters are replaced by some consistent and asymptotically normal (CAN) estimates. In the case of stratified sampling, asymptotic distribution is obtained and the optimum allocation is derived for a fixed cost and for a fixed variance. Some tests of hypotheses are constructed on the basis of these asymptotic distributions and a numerical example is presented.

Theϕ-divergence statistic in bivariate multinomial populations including stratification

Abstractϕ-divergence statistics quantify the divergence between a joint probability measure and the product of its marginal probabilities on the basis of contingency tables. Asymptotic properties of these statistics are investigated either considering random sampling or stratified random sampling with proportional allocation and independence among strata. To finish same tests of hypotheses of independence are presented.