Finite sample distributions of statistics in change point analysis

This paper is concerned with derivation of finite sampling distributions of some statistics which appear frequently in change point analysis. The exact distribution of cusum test statistic is approximated by two methods. Approximations are presented and their accuracies are measured. We first consider the change point in mean problem and we study the exact distribution of change point estimator. Finally, we consider the change point in variance case.     

Statistical Gambler's Ruin Problem

Abstract

The gambler's ruin problem is one of the most important problems in the emergence of probability. The problem has been long considered “solved” from a probabilistic viewpoint. However, we do not find the solution satisfactory. In this paper, the problem is recast as a statistical problem. Bounds of the estimate are derived over wide classes of priors. Interestingly, the probabilistic estimates ω(1/2) are identified as the most conservative solutions while the plug-in estimates are found to be out of range of the bounds. It implies that, although conservative, the probabilistic estimates ω(1/2) are justified by our analysis while the plug-in estimates are too extreme for estimating the ruin probability of gambler.     

Properties of the beta regression model for small area estimation of proportions and application to estimation of poverty rates

Abstract

Linear mixed effects models have been popular in small area estimation problems for modeling survey data when the sample size in one or more areas is too small for reliable inference. However, when the data are restricted to a bounded interval, the linear model may be inappropriate, particularly if the data are near the boundary. Nonlinear sampling models are becoming increasingly popular for small area estimation problems when the normal model is inadequate. This paper studies the use of a beta distribution as an alternative to the normal distribution as a sampling model for survey estimates of proportions which take values in (0, 1). Inference for small area proportions based on the posterior distribution of a beta regression model ensures that point estimates and credible intervals take values in (0, 1). Properties of a hierarchical Bayesian small area model with a beta sampling distribution and logistic link function are presented and compared to those of the linear mixed effect model. Propriety of the posterior distribution using certain noninformative priors is shown, and behavior of the posterior mean as a function of the sampling variance and the model variance is described. An example using 2010 Small Area Income and Poverty Estimates (SAIPE) data is given, and a numerical example studying small sample properties of the model is presented.     

Bayesian local bandwidth selector in multivariate associated kernel estimator for joint probability mass functions

ABSTRACT

This work treats non-parametric estimation of multivariate probability mass functions, using multivariate discrete associated kernels. We propose a Bayesian local approach to select the matrix of bandwidths considering the multivariate Dirac Discrete Uniform and the product of binomial kernels, and treating the bandwidths as a diagonal matrix of parameters with some prior distribution. The performances of this approach and the cross-validation method are compared using simulations and real count data sets. The obtained results show that the Bayes local method performs better than cross-validation in terms of integrated squared error.     

Cryptosporidium Infection Risk: Results of New Dose‐Response Modeling

Cryptosporidium human dose‐response data from seven species/isolates are used to investigate six models of varying complexity that estimate infection probability as a function of dose. Previous models attempt to explicitly account for virulence differences among C. parvum isolates, using three or six species/isolates. Four (two new) models assume species/isolate differences are insignificant and three of these (all but exponential) allow for variable human susceptibility. These three human‐focused models (fractional Poisson, exponential with immunity and beta‐Poisson) are relatively simple yet fit the data significantly better than the more complex isolate‐focused models. Among these three, the one‐parameter fractional Poisson model is the simplest but assumes that all Cryptosporidium oocysts used in the studies were capable of initiating infection. The exponential with immunity model does not require such an assumption and includes the fractional Poisson as a special case. The fractional Poisson model is an upper bound of the exponential with immunity model and applies when all oocysts are capable of initiating infection. The beta Poisson model does not allow an immune human subpopulation; thus infection probability approaches 100% as dose becomes huge. All three of these models predict significantly (>10x) greater risk at the low doses that consumers might receive if exposed through drinking water or other environmental exposure (e.g., 72% vs. 4% infection probability for a one oocyst dose) than previously predicted. This new insight into Cryptosporidium risk suggests additional inactivation and removal via treatment may be needed to meet any specified risk target, such as a suggested 10^?4 annual risk of Cryptosporidium infection.     

Relative Price Changes and Inequality in the Size Distribution of Various Components of Income

This article uses a comprehensive model of economic inequality to examine the impact of relative price changes on inequality in the marginal distributions of various income components in which the marginal distributions are derived from a multidimensional joint distribution. The multidimensional joint distribution function is assumed to be a member of the Pearson Type VI family; that is, it is assumed to be a beta distribution of the second kind. The multidimensional joint distribution is so called because it is a joint distribution of components of income and expenditures on various commodity groups. Gini measures of inequality are devised from the marginal distributions of the various income components. The inequality measures are shown to depend on the parameters of the multidimensional joint distribution. It is then shown that the parameters of the multidimensional joint distribution depend on the relative prices of various commodity groups and several other specified exogenous variables. Thus, knowledge of how changes in relative prices affect the parameters of the multidimensional joint distribution is deductively equivalent to knowledge of how changes in relative prices affect inequality in the marginal distributions of various components of income. It is found that relative price changes have a statistically significant impact on inequality in various components of income.     

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model''s parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.     

A Model for Water Pricing

This note describes a situation in which a simple mathematical model helped solve an important practical problem: how to price water fairly. It is intended as an example, rather than as a mathematical contribution to control theory.     

中国股市收益的横截面实证研究

除β外,公司规模是影响股票平均收益的一个重要的指标。文章选取了1999年1月至2006年12月所有在市A股股票为样本,分析了股票收益与β和公司规模之间的关系。结果表明,无论β作为单独的解释变量,还是和公司规模一起,它在解释股票收益时,都不具有显著性;相反,公司规模都一直具有很强的解释能力。     

A skew extension of the t-distribution, with applications

Summary. A tractable skew t -distribution on the real line is proposed. This includes as a special case the symmetric t -distribution, and otherwise provides skew extensions thereof. The distribution is potentially useful both for modelling data and in robustness studies. Properties of the new distribution are presented. Likelihood inference for the parameters of this skew t -distribution is developed. Application is made to two data modelling examples.