Bayesian Proportional Odds Models for Analyzing Current Status Data: Univariate,Clustered, and Multivariate

Current status data commonly arise in many fields such as epidemiological studies and cross-sectional tumorigenicity studies. In this article, we propose a semiparametric Bayesian approach for analyzing current status data with the proportional odds model. The use of monotone splines for the baseline odds function and a novel data augmentation with Poisson latent variables enable simple updating all of the parameters in the posterior computation. The proposed approach shows good performance and is compared with the approach in Wang and Dunson (2010 Wang , L. , Dunson , D. B. ( 2010 ). Semiparametric Bayes proportional odds models for current status data with under-reporting . Biometrics. Online early, DOI: 10.1111/j.1541-0420.2010.01532.x [Web of Science ®] , [Google Scholar]) in a simulation study. We also generalize the proposed approach to analyze clustered and multivariate current status data under the frailty proportional odds models.     

The influence of the sample on the posterior distribution

In this paper, we present conditions on the likelihood function and on the prior distribution which permit us to assess the effect of the sample on the posterior distribution. Our work is inspired by Whitt (1979) J. Amer. Statist. Assoc. 74, and is based on the notions of multivariate totally positive and (strongly) mul-tivariate reverse rule functions introduced and studied by Karlin and Rinott (1980a, b)     

Monotone Log-Odds Rate Distributions in Reliability Analysis

Abstract

Monotone failure rate models [Barlow Richard, E., Marshall, A. W., Proschan, Frank. (1963 Barlow Richard, E., Marshall, A. W. and Proschan, Frank. 1963. Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics, 34: 375–389.  [Google Scholar]). Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics 34:375–389, and Barlow Richard, E., Proschan, Frank. (1965 Barlow Richard, E. and Proschan, Frank. 1965. Mathematical Theory of Reliability New York: John Wiley.  [Google Scholar]). Mathematical Theory of Reliability. New York: John Wiley & Sons, Barlow Richard, E., Proschan, Frank. (1966a Barlow Richard, E. and Proschan, Frank. 1966a. Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics, 37(6): 1593–1601.  [Google Scholar]). Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics 37(6):1593–1601, Barlow Richard, E., Proschan, Frank. (1966b Barlow Richard, E. and Proschan, Frank. 1966b. Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics, 37(6): 1574–1592.  [Google Scholar]). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics 37(6):1574–1592, Barlow Richard, E., Proschan, Frank. (1975 Barlow Richard, E. and Proschan, Frank. 1975. Statistical Theory of Reliability and Life Testing New York: Holt, Rinehart and Winston, Inc..  [Google Scholar]). Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc.] have become one of the most important models of failure time for reliability practitioners to consider and use. The above authors also developed models and bounds for monotone increasing failure rates (IFR) and for monotone decreasing failure rates (DFR). The IFR models and bounds appear to be especially useful for describing and bounding the hazard of aging. This article extends a new model for time to failure based onthe log odds rate [Zimmer William, J., Wang Yao, Pathak, P. K. (1998 Zimmer William, J., Wang, F. K. and Keats, J. Bert. 1998. The Burr XII distribution in reliability analysis. Journal of Quality Technology, 30(4): 386–394.  [Google Scholar]). Log-odds rate and monotone log-odds rate distributions. Journal of Quality Technology 30(4):376–385.] which is comparable to the model based on the failure rate. It is shown that in the case of increasing log odds rate (ILOR) in terms of log time (ln t), the model is less stringent than the IFR model for aging. The characterization of distributions of failure time by log odds rate is also derived. It is shown that the logistic distribution has the property of constant log odds rate over time and that the log logistic distribution has the property of constant log odds rate with respect to ln t. Some other properties of ILOR distributions are presented and bounds based on the relationship to the log logistic distribution are provided for distributions which are ILOR with respect to ln t. Motivational examples are provided. The ILOR bounds are compared to the more stringent bounds based on the IFR model. Bounds on system reliability are also provided for certain systems.     

A General Model for Repeated Audit Controls Using Monotone Subsampling

Abstract

In categorical repeated audit controls, fallible auditors classify sample elements in order to estimate the population fraction of elements in certain categories. To take possible misclassifications into account, subsequent checks are performed with a decreasing number of observations. In this paper a model is presented for a general repeated audit control system, where k subsequent auditors classify elements into r categories. Two different subsampling procedures will be discussed, named “stratified” and “random” sampling. Although these two sampling methods lead to different probability distributions, it is shown that the likelihood inferences are identical. The MLE are derived and the situations with undefined MLE are examined in detail; it is shown that an unbiased MLE can be obtained by stratified sampling. Three different methods for constructing confidence upper limits are discussed; the Bayesian upper limit seems to be the most satisfactory. Our theoretical results are applied to two cases with r = 2 and k = 2 or 3, respectively.     

An essentially complete class of multiple decision procedures

Let π₁,…, π_k represent k(?2) independent populations. The quality of the ith population π_i is characterized by a real-valued parameter θ_i, usually unknown. We define the best population in terms of a measure of separation between θ_i's. A selection of a subset containing the best population is called a correct selection (CS). We restrict attention to rules for which the size of the selected subset is controlled at a given point and the infimum of the probability of correct selection over the parameter space is maximized. The main theorem deals with construction of an essentially complete class of selection rules of the above type. Some classical subset selection rules are shown to belong to this class.     

Modeling dropouts by conditional distribution, a copula-based approach

In this paper the concept of copulas is implemented into the methodology for solving the imputation problem in correlated incomplete data. We use the Gaussian copula as alternative to the joint distribution for modeling the conditional distribution, conditioned by the observed values of measurements. The general formula for imputation and its application for compound symmetry correlation structure are given.     

Shape constrained kernel density estimation

In this paper, a method for estimating monotone, convex and log-concave densities is proposed. The estimation procedure consists of an unconstrained kernel estimator which is modified in a second step with respect to the desired shape constraint by using monotone rearrangements. It is shown that the resulting estimate is a density itself and shares the asymptotic properties of the unconstrained estimate. A short simulation study shows the finite sample behavior.     

Aggregating the Single Crossing Property

The single crossing property plays a crucial role in economic theory, yet there are important instances where the property cannot be directly assumed or easily derived. Difficulties often arise because the property cannot be aggregated: the sum or convex combination of two functions with the single crossing property need not have that property. We introduce a new condition characterizing when the single crossing property is stable under aggregation, and also identify sufficient conditions for the preservation of the single crossing property under multidimensional aggregation. We use our results to establish properties of objective functions (convexity, logsupermodularity), the monotonicity of optimal decisions under uncertainty, and the existence of monotone equilibria in Bayesian games.     

Best Invariant and Minimax Estimation of Quantiles in Finite Populations

The theoretical literature on quantile and distribution function estimation in infinite populations is very rich, and invariance plays an important role in these studies. This is not the case for the commonly occurring problem of estimation of quantiles in finite populations. The latter is more complicated and interesting because an optimal strategy consists not only of an estimator, but also of a sampling design, and the estimator may depend on the design and on the labels of sampled individuals, whereas in iid sampling, design issues and labels do not exist.We study the estimation of finite population quantiles, with emphasis on estimators that are invariant under the group of monotone transformations of the data, and suitable invariant loss functions. Invariance under the finite group of permutation of the sample is also considered. We discuss nonrandomized and randomized estimators, best invariant and minimax estimators, and sampling strategies relative to different classes. Invariant loss functions and estimators in finite population sampling have a nonparametric flavor, and various natural combinatorial questions and tools arise as a result.     

Generalized Likelihood Ratio Tests and Uniformly Most Powerful Tests

For testing a one-sided hypothesis in a one-parameter family of distributions, it is shown that the generalized likelihood ratio (GLR) test coincides with the uniformly most powerful (UMP) test, assuming certain monotonicity properties for the likelihood function. In particular, the equivalence of GLR tests and UMP tests holds for one-parameter exponential families. In addition, the relationship between GLR and UMPU (UMP unbiased) tests is considered when testing two-sided hypotheses.