The Bayesian and frequentist approaches to testing a one-sided hypothesis about a multivariate mean

This paper compares the Bayesian and frequentist approaches to testing a one-sided hypothesis about a multivariate mean. First, this paper proposes a simple way to assign a Bayesian posterior probability to one-sided hypotheses about a multivariate mean. The approach is to use (almost) the exact posterior probability under the assumption that the data has multivariate normal distribution, under either a conjugate prior in large samples or under a vague Jeffreys prior. This is also approximately the Bayesian posterior probability of the hypothesis based on a suitably flat Dirichlet process prior over an unknown distribution generating the data. Then, the Bayesian approach and a frequentist approach to testing the one-sided hypothesis are compared, with results that show a major difference between Bayesian reasoning and frequentist reasoning. The Bayesian posterior probability can be substantially smaller than the frequentist p-value. A class of example is given where the Bayesian posterior probability is basically 0, while the frequentist p-value is basically 1. The Bayesian posterior probability in these examples seems to be more reasonable. Other drawbacks of the frequentist p-value as a measure of whether the one-sided hypothesis is true are also discussed.     

Growth of tuber crops and almost sure band for quantiles

Some tuber crops are governed by memoryless property of exponential distribution leading to a mixture distribution with heavy tail. Quantile-based estimators may then be appropriate than mean as a measure of central tendency. We prove almost sure representation theorems for sample quantiles in a general setup of U statistics, under slightly stronger assumption than assuming the existence of a continuously differentiable distribution function F for the kernel h. We obtain almost sure (a.s.) upper and lower estimate for F^{? 1}(p), p ∈ (0, 1) as a band for p varying. As an application, dataset arising from two varieties of potato cultivation are analyzed.     

Testing for Persistence in the Error Component Model: A One-Sided Approach

This article proposes new simple testing procedures for the joint null hypothesis of absence of persistent effects, in the form of random effects and first-order serial correlation in the error component model. The fact that the presence of random effects is clearly of a one-sided nature, together with the fact that in many empirical applications researchers worry about positive serial correlation leaves room for a power gain that arises from restricting the parameter space under the alternative hypothesis, compared to existing procedures that allow for two-sided alternatives. A Monte Carlo experiment shows that the proposed statistics have good size and power performance in very small samples like those typically used in applied work in panel data. An empirical example illustrates the usefulness of the proposed statistics.     

Data-Driven Tests for Trend

This article proposes a new nonparametric test for the ordered alternatives problem in the k-sample setting for null hypothesis of lack of trend. This article further elaborates upon and extends the results of Ledwina and Wy?upek (2012a Ledwina , T. , Wy?upek , G. ( 2012a ). Two-sample test against one-sided alternatives . Scand. J. Statist. 39 : 358 – 381 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained for k = 2. Simulations show that the new test has high and stable power and is able to control the Type I error to satisfactory extent, thus solving the problem posed in Terpstra and Magel (2003 Terpstra , J. T. , Magel , R. C. ( 2003 ). A new nonparametric test for the ordered alternative problem . J. Nonparametr. Statist. 15 : 289 – 301 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Our theoretical results say that asymptotic errors of both kinds do not exceed significance level, thus implying that the test is asymptotically unbiased.     

The effect of parameter estimation on the performance of one-sided Shewhart control charts for zero-inflated processes

ABSTRACT

Zero-inflated probability models are used to model count data that have an excessive number of zeros. Shewhart-type control charts have been proposed for the monitoring of zero-inflated processes. Usually their performance is evaluated under the assumption of known process parameters. However, in practice, their values are rarely known and they have to be estimated from an in-control historical Phase I sample. In the present paper, we investigate the performance of Shewhart-type control charts for zero-inflated processes with estimated parameters and propose practical guidelines for the statistical design of the examined charts, when the size of the preliminary sample is predetermined.     

Matching pseudocounts for interval estimation of binomial and Poisson parameters

ABSTRACT

For interval estimation of a binomial proportion and a Poisson mean, matching pseudocounts are derived, which give the one-sided Wald confidence intervals with second-order accuracy. The confidence intervals remove the bias of coverage probabilities given by the score confidence intervals. Partial poor behavior of the confidence intervals by the matching pseudocounts is corrected by hybrid methods using the score confidence interval depending on sample values.     

The ruin probabilities of a discrete time risk model with one-sided linear claim sizes and dependent risks

This article investigates the ruin probabilities of a discrete time risk model with dependent claim sizes and dependent relation between insurance risks and financial risks. The risk-free and risky investments of an insurer lead to stochastic discount factors {θ_n}_{n ? 1}. The claim sizes are assumed to follow a one-sided linear process with independent and identically distributed (i.i.d.) innovations {?_n}_{n ? 1}. The i.i.d. random pairs {(?_n, θ_n)}_{n ? 1} follow a common bivariate Sarmanov-dependent distribution. When the common distribution of the innovations is heavy tailed, we establish some asymptotic estimates for the ruin probabilities of this discrete time risk model.     

ASYMPTOTIC STABILITY OF THE OSCV SMOOTHING PARAMETER SELECTION

The smoothing parameter selection by the one-sided cross-validation (OSCV) method is completely automatic in that it does not require extra parameters estimation. Also it reduces the variability comparable to that of plug-in rules. In this paper we derive analytically the asymptotic variance of the smoothing parameter selected by OSCV. It shows the dependency of the stability on the one-sided kerenl and tells the possibility of the optimal one-sided kernel which minimizes the asymptotic variability.     

Confidence limits for stress–strength reliability involving Weibull models

The problem of interval estimation of the stress–strength reliability involving two independent Weibull distributions is considered. An interval estimation procedure based on the generalized variable (GV) approach is given when the shape parameters are unknown and arbitrary. The coverage probabilities of the GV approach are evaluated by Monte Carlo simulation. Simulation studies show that the proposed generalized variable approach is very satisfactory even for small samples. For the case of equal shape parameter, it is shown that the generalized confidence limits are exact. Some available asymptotic methods for the case of equal shape parameter are described and their coverage probabilities are evaluated using Monte Carlo simulation. Simulation studies indicate that no asymptotic approach based on the likelihood method is satisfactory even for large samples. Applicability of the GV approach for censored samples is also discussed. The results are illustrated using an example.     

Confidence limits and prediction limits for a Weibull distribution based on the generalized variable approach

In this article, we consider the problems of constructing confidence interval for a Weibull mean and setting prediction limits for future samples. Specifically, we construct upper prediction limits that include at least l$l$ of m$m$ samples from a Weibull distribution at each of r$r$ locations. The methods are based on the concept of generalized variable approach. The procedures can be easily extended to the type II censored samples, and they can be used to find approximate inferential procedures for type I censored samples. The proposed methods are conceptually simple and easy to use. The results are illustrated using some practical examples.