Bhaskar Bhattacharya

Median of the p Value Under the Alternative Hypothesis

Due to absence of an universally acceptable magnitude of the Type I error in various fields, p values are used as a well-recognized tool in decision making in all areas of statistical practice. The distribution of p values under the null hypothesis is uniform. However, under the alternative hypothesis the distribution of the p values is skewed. The expected p value (EPV) has been proposed by authors to be used as a measure of the performance of the test. This article proposes the median of the p values (MPV) which is more appropriate for this purpose. We work out many examples to calculate the MPVs directly and also compare the MPV with the EPV. We consider testing equality of distributions against stochastic ordering in the multinomial case and compare the EPVs and MPVs by simulation. A second simulation study for general continuous data is also considered for two samples with different test statistics for the same hypotheses. In both cases MPV performs better than EPV.

An iterative procedure for general probability measures to obtain I-projections onto intersections of convex sets

The iterative proportional fitting procedure (IPFP) was introduced formally by Deming and Stephan in 1940. For bivariate densities, this procedure has been investigated by Kullback and Ruschendorf. It is well known that the IPFP is a sequence of successive I-projections onto sets of probability measures with fixed marginals. However, when finding the I-projection onto the intersection of arbitrary closed, convex sets (e.g., marginal stochastic orders), a sequence of successive I-projections onto these sets may not lead to the actual solution. Addressing this situation, we present a new iterative I-projection algorithm. Under reasonable assumptions and using tools from Fenchel duality, convergence of this algorithm to the true solution is shown. The cases of infinite dimensional IPFP and marginal stochastic orders are worked out in this context.

A general duality approach to I-projections

Abstract I-projection problems arise in a myriad of situations and settings. In this paper, it is shown that under reasonable assumptions, a Fenchel type dual optimization problem exists which is equivalent to the stated I-projection problem for very general probability measures. This dual problem is often much more tractable than the original I-projection problem. I-projection problems which are equivalent to least square problems are also identified; primarily through the dual formulation. Several examples are examined to illustrate the duality structure and theorems.

Symmetry Properties of Bi-Normal and Bi-Gamma Receiver Operating Characteristic Curves are Described by Kullback-Leibler Divergences

Receiver operating characteristic (ROC) curves have application in analysis of the performance of diagnostic indicators used in the assessment of disease risk in clinical and veterinary medicine and in crop protection. For a binary indicator, an ROC curve summarizes the two distributions of risk scores obtained by retrospectively categorizing subjects as cases or controls using a gold standard. An ROC curve may be symmetric about the negative diagonal of the graphical plot, or skewed towards the left-hand axis or the upper axis of the plot. ROC curves with different symmetry properties may have the same area under the curve. Here, we characterize the symmetry properties of bi-Normal and bi-gamma ROC curves in terms of the Kullback-Leibler divergences (KLDs) between the case and control distributions of risk scores. The KLDs describe the known symmetry properties of bi-Normal ROC curves, and newly characterize the symmetry properties of constant-shape and constant-scale bi-gamma ROC curves. It is also of interest to note an application of KLDs where their asymmetry—often an inconvenience—has a useful interpretation.

Bayesian inference for multinomial populations under stochastic ordering

In many parametric problems the use of order restrictions among the parameters can lead to improved precision. Our interest is in the study of several multinomial populations under the stochastic order restriction (SOR) for univariate situations. We use Bayesian methods to show that the SOR can lead to larger gains in precision than the method without the SOR when the SOR is reasonable. Unlike frequentist order restricted inference, our methodology permits analysis even when there is uncertainty about the SOR. Our method is sampling based, and we use simple and efficient rejection sampling. The Bayes factor in favor of the SOR is computed in a simple manner, and samples from the requisite posterior distributions are easily obtained. We use real data to illustrate the procedure, and we show that there is likely to be larger gains in precision under the SOR.

Testing equality of scale parameters against restricted alternatives for m≥3 gamma distributions with unknown common shape parameter

Shiue and Bain (1983) proposed an approximate F-test for the equality of the scale parameters of two gamma distributions with equal but unknown shape parameters. In this article, we propose a simple procedure to test equality of scale parameters of m≥3 gamma distributions against nonincreasing order. The test is based on Fishers method of combining p-values. The actual size of the resulting test is investigated through Monte Carlo studies. Also asymptotic results are derived for the nominal test size. These can be used to obtain a test which achieves the desired size. The case of more general partial orders is discussed.

Testing multinomial parameters under order restrictions

Let be a given probability vector .We consider the probability vector , defined by pi = λ iqi , for all i and satisfies an arbitrary quasi-order restriction. Given observations from a multinomial distribution with probability vector p, we consider estimation of p based on the directed divergence criteria subject to such restrictions. We consider the likelihood ratio tests when testing for and against the order restriction, and in both cases the asymptotic null distributions of the test statistics are shown to be of the chi-bar-squared type. When the directed divergence is used as a test criteria, the null asymptotic distribution of the test statistic is shown to be the same as the corresponding null asymptotic distribution of the likelihood ratio statistic in both cases. An example is used to illustrate the technique developed. A simulation is conducted to compare the performance of different test statistics.

A Fenchel duality aspect of iterative I-projection procedures

In this paper we interpret Dykstras iterative procedure for finding anI-projection onto the intersection of closed, convex sets in terms of itsFenchel dual. Seen in terms of its dual formulation, Dykstras algorithm isintuitive and can be shown to converge monotonically to the correctsolution. Moreover, we show that it is possible to sharply bound thelocation of the constrained optimal solution.

Restricted tests for and against the increasing failure rate ordering on multinomial parameters

We consider the likelihood ratio tests for (i) testing a constant failure rate (truncated geometric) against the alternative of increasing (nondecreasing) failure rate ordering of a collection of multinomial parameters, and for (ii) testing the null hypothesis that this parameter vector satisfies increasing failure rate ordering against all alternatives (unrestricted). For both tests the asymptotic distribution of the test statistic under the null hypothesis is shown to be of the chi-bar square type. A numerical example is presented to illustrate the procedure.

CSISZAR DIVERGENCE FROM CONSTANT FAILURE RATE MODEL FOR GROUPED DATA

We propose a measure of divergence in failure rates of a system from the constant failure rate model for a grouped data situation. We use this measure to compare the divergences of several systems from the constant failure rate model and find the asymptotic distributions of the test statistics. Several applications are discussed to illustrate the procedure. In the context of testing the goodness-of-fit with the constant failure rate model, we conduct a simulation study which shows that this procedure compares favorably with the Pearson chi-square test and the likelihood ratio test procedures.