Hammou El Barmi

Inferences Under a Stochastic Ordering Constraint: The k-Sample Case

If X1 and X2 are random variables with distribution functions F1 and F2, then X1 is said to be stochastically larger than X2 if F1 ≤F2. Statistical inferences under stochastic ordering for the two-sample case has a long and rich history. In this article we consider the k-sample case; that is, we have k populations with distribution functions F1, F2, … , Fk,k ≥ 2, and we assume that F1 ≤ F2 ≤ ˙˙˙ ≤ Fk. For k = 2, the nonparametric maximum likelihood estimators of F1 and F2 under this order restriction have been known for a long time; their asymptotic distributions have been derived only recently. These results have very complicated forms and are hard to deal with when making statistical inferences. We provide simple estimators when k ≥ 2. These are strongly uniformly consistent, and their asymptotic distributions have simple forms. If and are the empirical and our restricted estimators of Fi, then we show that, asymptotically, for all x and all u > 0, with strict inequality in some cases. This clearly shows a uniform improvement of the restricted estimator over the unrestricted one. We consider simultaneous confidence bands and a test of hypothesis of homogeneity against the stochastic ordering of the k distributions. The results have also been extended to the case of censored observations. Examples of application to real life data are provided.

Empirical likelihood ratio test for or against a set of inequality constraints

Abstract We use the empirical likelihood ratio approach introduced by Owen ( Biometrika 75 (1988), 237–249) to test for or against a set of inequality constraints when the parameters are defined by estimating functions. Our objective in this paper is to show that under fairly general conditions, the limiting distributions of the empirical likelihood ratio test statistics are of chi-bar square type (as in the parametric case) and give the expression of the weighting values. The results obtained here are similar to those in El Barmi and Dykstra (1995) where a full distributional model is assumed. This work presents also an extension of the results in Qin and Lawless (1995).

Transformation-Based Density Estimation for Weighted Distributions

In this paper we consider the estimation of a density f on the basis of random sample from a weighted distribution G with density g given by ,where w(u) > 0 for all u and . A special case of this situation is that of length-biased sampling, where w(x) = x. In this paper we examine a simple transformation-based approach to estimating the density f. The approach is motivated by the form of the nonparametric estimator maximum likelihood of f in the same context and under a monotonicity constraint. Since the method does not depend on the specific density estimate used (only the transformation), it can be used to construct both simple density estimates (histograms or frequency polygons) and more complex methods with favorable properties (e.g., local or penalized likelihood estimates). Monte Carlo simulations indicate that transformation-based density estimation can outperform the kernel-based estimator of Jones (1991) depending on the weight function w, and leads to much better estimation of monotone densities than the nonparametric maximum likelihood estimator.

Nonparametric estimation in selection biased models in the presence of estimating equations

Consider two independent samples, one sample of size m from a distribution F and the other of size n from a weighted distribution G where with w(.)≤0 and Assume that there is a parameter θ∊R d associated with F through and consider the nonparametric estimators of F and of G on the basis of these two samples when θ is known and Φ is a real valued function and when θ is unknown and Φ is a rector valued function of dimension r<d. We show that converge weakly to pinned Gaussian processes as m+n goes to +∞ and m/n converges to a constant and provide the expressions of the covariance functions. In the case where θ is unknown and Φ is a vector valued function of dimension r<d, we propose an approximate chi-square test for testing θ = θ0 against all alternatives. This work is an extension of Vardi (1982a,b) and is closely connected to the work of Qin (1993) and Qin and Lawless (1995).

Empirical likelihood-based tests for stochastic ordering

This paper develops an empirical likelihood approach to testing for the presence of stochastic ordering among univariate distributions based on independent random samples from each distribution. The proposed test statistic is formed by integrating a localized empirical likelihood statistic with respect to the empirical distribution of the pooled sample. The asymptotic null distribution of this test statistic is found to have a simple distribution-free representation in terms of standard Brownian bridge processes. The approach is used to compare the lengths of rule of Roman Emperors over various historical periods, including the “decline and fall” phase of the empire. In a simulation study, the power of the proposed test is found to improve substantially upon that of a competing test due to El Barmi and Mukerjee.

Likelihood ratio test against a set of inequality constraints

The first part of this article investigates the joint behaviour of the maximum likelihood estimators and lagrange multipliers corresponding to a null and alternative hypotheses defined by equality constraints. The results are then used to test against a set of inequality constraints using the likelihood ratio approach. This work is an extention of the results in Silvey (1959), El Barmi and Dykstra (1995) and Silvapulle (1994), and is closely connected with the work of Chernoff(1954); Robertson et al. (1988) and Self and Liang (1987). We show that the limiting distribution of the likelihood ratio test statistic under quite general conditions is of chi-bar square type and give an expression for the weighting values.

Likelihood ratio tests for bivariate symmetry against ordered alternatives in a square contingency table

Let (X1, X2) be a bivariate random variable of the discrete type with joint probability density function pij = pr[X1 = i, X2 = j], i, J = 1, ..., k. Based on a random sample from this distribution, we discuss the properties of the likelihood ratio test of the null hypothesis of bivariate symmetry Ho: pij = pji [for all](i, j) vs. the alternative H1: pij [greater-or-equal, slanted] pji, [for all]i> j, in a square contingency table. This is a categorised version of the classical one-sided matched pairs problem. This test is asymptotically distribution-free. We also consider the problem of testing H1 as a null hypothesis against the alternative H2 of no restriction on pijs. The asymptotic null distributions of the test statistics are found to be of the chi-bar square type. Finally, we analyse a data set to demonstrate the use of the proposed tests.

Estimation of Cumulative Incidence Functions in Competing Risks Studies Under an Order Restriction

Abstract In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t for a particular type of failure in the presence of other risks. Its estimation and asymptotic distribution theory have been studied by many. In some cases there are reasons to believe that the CIFs due to two types of failure are order restricted. Several procedures have appeared in the literature for testing for such orders. In this paper we initiate the study of estimation of two CIFs subject to a type of stochastic ordering, both when there are just two causes of failure and when there are more than two causes of failure, treating those other than the two of interest as a censoring mechanism. We do not assume independence of the two types of failure of interest; however, these are assumed to be independent of the other causes in the censored case. Weak convergence results for the estimators have been derived. It is shown that when the order restriction is strict, the asymptotic distributions are the same as those for the empirical estimators without the order restriction. Thus we get the restricted estimators “free of charge”, at least in the asymptotic sense. When the two CIFs are equal, the asymptotic MSE is reduced by using the order restriction. For finite sample sizes simulations seem to indicate that the restricted estimators have uniformly smaller MSEs than the unrestricted ones in all cases.

A note on estimating a non-increasing density in the presence of selection bias

Abstract In this paper we construct the non-parametric maximum likelihood estimator (NPMLE) f n of a non-increasing probability density function f with distribution function F on the basis of a sample from a weighted distribution G with density given by g(x)=w(x)f(x)/μ(f,w), where w(u)>0 for all u and μ(f,w)=∫w(u)f(u) d u is the normalizing constant. We show that the NPMLE of f is proportional to the Grenander (Skand. Akt. 39 (1956) 125) estimator of the density of transformed data using a simple transformation based on w. We explore some of the properties of f n and show that the Prakasa Rao Theorem (Sankhya A 31 (1969) 23) extends to the weighted case. We also give conditions under which the resulting distribution function F n is strongly uniformly consistent and show that a rate of convergence of order n−1/2 can be achieved under conditions on w. We also investigate estimation of f when a second sample directly from f is available and carry out a small-scale simulation study of the performance of two estimators in this case.

Likelihood ratio tests for peakedness in multinomial populations

Likelihood ratio tests concerning the parameters of two multinomial populations are discussed. A peakedness ordering restriction is considered as a one sided alternative to equality. The asymptotic distributions of the likelihood ratio test statistics are obtained and are shown to be of the chi-bar squared type. Exact expressions for the weights in the mixture of chi-squared distributions are also provided. The procedures are illustrated by applying them to a data set on lung cancer mortality in South Australia.