Bayesian Analysis of Ordinal Survey Data Using the Dirichlet Process to Account for Respondent Personality Traits

This article presents a Bayesian latent variable model used to analyze ordinal response survey data by taking into account the characteristics of respondents. The ordinal response data are viewed as multivariate responses arising from continuous latent variables with known cut-points. Each respondent is characterized by two parameters that have a Dirichlet process as their joint prior distribution. The proposed mechanism adjusts for classes of personalities. The model is applied to student survey data in course evaluations. Goodness-of-fit (GoF) procedures are developed for assessing the validity of the model. The proposed GoF procedures are simple, intuitive, and do not seem to be a part of current Bayesian practice.     

Covariate-Adjusted Regression for Time Series

The main purpose of this article is to consider the covariate-adjusted regression (CAR) model for time series. The CAR model was initially proposed by Sentürk and Müller (2005 Sentürk , D. , Müller , H. G. ( 2005 ). Covariate-adjusted regression . Biometrika 92 : 75 – 89 .[Crossref], [Web of Science ®] , [Google Scholar]) for such situations where predictor and response variables are not directly observed, but are distorted by some common observable covariate. Despite CAR being originally designed for independent cross-sectional data, multiple works have extended this method to dependent data setting. In this article, the authors extend CAR to the distorted time series setting. This extension is meaningful in many fields such as econometrics, mathematical finance, and signal processing. The estimates of regression parameters are proposed by establishing connection with functional-coefficient time series model. The consistency and asymptotic normality of the proposed estimates are investigated under the α-mixing conditions. Real data and simulated examples are provided for illustration.     

Goodness-of-Fit Tests for the Gamma Distribution Based on the Empirical Laplace Transform

We propose a class of goodness-of-fit tests for the gamma distribution that utilizes the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity, the test statistics approach limit values related to the first non zero component of Neyman's smooth test for the gamma law. The new tests are compared with other omnibus tests for the gamma distribution.     

Goodness-of-fit tests for the Gompertz distribution

Abstract

While the Gompertz distribution is often fitted to lifespan data, testing whether the fit satisfies theoretical criteria is being neglected. Here four goodness-of-fit measures – the Anderson–Darling statistic, the correlation coefficient test, a statistic using moments, and a nested test against the generalized extreme value distributions – are discussed. Along with an application to laboratory rat data, critical values calculated by the empirical distribution of the test statistics are also presented.     

A general algorithm fob,univariate and multivariate goodness of pit tests based on graphical representation

This paper presents a general algorithm tor assessing the distributional assumptions. Empirical distributions of the corresponding test statistics are obtained and examples are given to illustrate various applications of the proposed test. By using the squared radii and angles, it is shown that the problem of assessing multivariate normality can be reduced to that of testing for a univariate distribution. A limited comparison is made to investigate the power of the proposed test. This work was supported in part by the National Science Foundation under Grant NO.G88135. Support from the Computer Applications ami Software Engineering (CASE) Center of Syracuse University is also gratefully acknowledged     

Asymptotically Distribution-Free Goodness-of-Fit Testing: A Unifying View

We outline a general paradigm for constructing asymptotically distribution-free (ADF) goodness-of-fit tests, which can be regarded as a generalization of Khmaladze (1993 Khmaladze , E. V. ( 1993 ). Goodness of fit problem and scanning innovation martingales . Ann. Statist. 21 ( 2 ): 798 – 829 .[Crossref], [Web of Science ®] , [Google Scholar]). This is achieved by a nonorthogonal projection of a class of functions onto the ortho-complement of the extended tangent space (ETS) associated with the null hypothesis. In parallel with the work of Bickel et al. (2006 Bickel , P. J. , Ritov , Y. , Stoker , T. M. ( 2006 ). Tailor-made tests for goodness of fit to semiparametric hypotheses . Ann. Statist. 34 ( 2 ): 721 – 741 . [Google Scholar]), we obtain transformed empirical processes (TEP) which are the building blocks for constructing omnibus tests such as the usual Kolmogorov–Smirnov type tests and Crámer–von Mise type tests, as well as Portmanteau tests and directional tests. The critical values can be tabulated due to the ADF property. All the tests are capable of detecting local (Pitman) alternative at the root-n scale. We shall illustrate the framework in several examples, mostly in regression model specification testing.     

Graphical comparison of normality tests for unimodal distribution data

A methodology is proposed to compare the power of normality tests with a wide variety of alternative unimodal distributions. It is based on the representation of a distribution mosaic in which kurtosis varies vertically and skewness horizontally. The mosaic includes distributions such as exponential, Laplace or uniform, with normal occupying the centre. Simulation is used to determine the probability of a sample from each distribution in the mosaic being accepted as normal. We demonstrate our proposal by applying it to the analysis and comparison of some of the most well-known tests.     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

Cumulative ratio information based on general cumulative entropy

In this paper, we suggest an extension of the cumulative residual entropy (CRE) and call it generalized cumulative entropy. The proposed entropy not only retains attributes of the existing uncertainty measures but also possesses the absolute homogeneous property with unbounded support, which the CRE does not have. We demonstrate its mathematical properties including the entropy of order statistics and the principle of maximum general cumulative entropy. We also introduce the cumulative ratio information as a measure of discrepancy between two distributions and examine its application to a goodness-of-fit test of the logistic distribution. Simulation study shows that the test statistics based on the cumulative ratio information have comparable statistical power with competing test statistics.     

An omnibus test of goodness-of-fit for conditional distributions with applications to regression models

We introduce an omnibus goodness-of-fit test for statistical models for the conditional distribution of a random variable. In particular, this test is useful for assessing whether a regression model fits a data set on all its assumptions. The test is based on a generalization of the Cramér–von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives. The extension to the case where unknown parameters appear in the model is developed. A simulation study shows that the test has good power against some common departures encountered in regression models. Moreover, its power is comparable to that of other nonparametric tests designed to examine only specific departures.