Kolmogorov-type tests of goodness-of-fit against stochastic ordering

This article addresses the problem of testing the null hypothesis H₀ that a random sample of size n is from a distribution with the completely specified continuous cumulative distribution function F_n(x). Kolmogorov-type tests for H₀ are based on the statistics C⁺ _n = Sup[F_n(x)?F₀(x)] and C^? _n=Sup[F₀(x)?F_n(x)], where F_n(x) is an empirical distribution function. Let F(x) be the true cumulative distribution function, and consider the ordered alternative H₁: F(x)≥F₀(x) for all x and with strict inequality for some x. Although it is natural to reject H₀ and accept H₁ if C ⁺ _n is large, this article shows that a test that is superior in some ways rejects F₀ and accepts H₁ if C^mdash _n is small. Properties of the two tests are compared based on theoretical results and simulated results.     

A test for abrupt change in hazard regression models with Weibull baselines

We develop a likelihood ratio test for an abrupt change point in Weibull hazard functions with covariates, including the two-piece constant hazard as a special case. We first define the log-likelihood ratio test statistic as the supremum of the profile log-likelihood ratio process over the interval which may contain an unknown change point. Using local asymptotic normality (LAN) and empirical measure, we show that the profile log-likelihood ratio process converges weakly to a quadratic form of Gaussian processes. We determine the critical values of the test and discuss how the test can be used for model selection. We also illustrate the method using the Chronic Granulomatous Disease (CGD) data.     

Modeling the Distribution of Price Sensitivity and Implications for Optimal Retail Pricing

This article focuses on the distribution of price sensitivity across consumers. We employ a random-coefficient logit model in which brand-specific intercepts and price-slope coefficients are allowed to vary across households. The model is estimated with panel data for two product categories. The implications of the estimated model are deduced through an optimal retail pricing analysis that combines the panel data with chain-level cost figures. We test parametric distributional assumptions using semiparametric density estimates based on series expansions.     

Nonparametric estimation of quantile functions for randomly right censored data

In this paper we compare four nonparametric quantile function estimators for randomly right censored data: the Kaplan–Meier estimator, the linearly interpolated Kaplan–Meier estimator, the kernel-type survival function estimator, and the Bézier curve smoothing estimator. Also, we compare several kinds of confidence intervals of quantiles for four nonparametric quantile function estimators.     

A mixture of beta–Dirichlet processes prior for Bayesian analysis of event history data

In this paper, we propose a mixture of beta–Dirichlet processes as a nonparametric prior for the cumulative intensity functions of a Markov process. This family of priors is a natural extension of a mixture of Dirichlet processes or a mixture of beta processes which are devised to compromise advantages of parametric and nonparametric approaches. They give most of their prior mass to the small neighborhood of a specific parametric model. We show that a mixture of beta–Dirichlet processes prior is conjugate with Markov processes. Formulas for computing the posterior distribution are derived. Finally, results of analyzing credit history data are given.     

Exchange option in a two-state Poisson CAPM

In this paper we derive the pricing formula for the exchange option value in a two-state Poisson CAPM. A two-state Poisson CAPM models the stochastic market environment. We also provide examples and graphs to illustrate our result.     

Hypothesis Testing Problems in an Unbalanced Longitudinal Ophthalmology Study

ABSTRACT

The analysis of clustered data in a longitudinal ophthalmology study is complicated by correlations between repeatedly measured visual outcomes of paired eyes in a participant and missing observations due to the loss of follow-up. In the present article we consider hypothesis testing problems in an ophthalmology study, where eligible eyes are randomized to two treatments (when two eyes of a participant are eligible, the paired eyes are assigned to different treatments), and vision function outcomes are repeatedly measured over time. A large sample-based nonparametric test statistic and a nonparametric Bootstrap test analog are proposed for testing an interaction effect of two factors and testing an effect of a eye-specific factor within a level of the other person-specific factor on visual function outcomes. Both test statistics allow for missing observations, correlations between repeatedly measured outcomes on individual eyes, and correlations between repeatedly measured outcomes on both eyes of each participant. A simulation study shows that these proposed test statistics maintain nominal significance levels approximately and comparable powers to each other, as well as higher powers than the naive test statistic ignoring correlations between repeated bilateral measurements of both eyes in the same person. For illustration, we apply the proposed test statistics to the changes of visual field defect score in the Advanced Glaucoma Intervention Study.     

Approximations in a group sequential test with unequal group sizes

Our concern in this paper is a group sequential test design for which the sample sizes between interim analyses are not identical. First, we consider a repeated significance test for comparing two treatments in a clinical trial, and study asymptotic properties of the test statistic. Using the arguments developed by Siegmund (1985, Chapters 8 and 9), we then obtain approximations for the overall significance level of the test and for the error level at each interim analysis. Simulation studies are performed to assess the accuracy of the approximations and the robustness of the approximations are examined using numerical examples.