OPTIMAL CRITICAL VALUES FOR A PRELIMINARY TEST IN POOLING MEANS

Given two random samples from normal populations with the same known variance, the experimenter wishes to estimate the mean of the fist population. Whether to pool the two samples or not is made to depend on the result of a preliminary normal test. The bias and mean square error of the pre-test estimator are presented. Based on two different criteria, a minimax regret and a minimum expected regret ones, the optimal critical values for the pre-test are given. The minimax regret value is unique (about 1.37), while the alternative values vary depending on the mean and precision included in the prior distribution.     

Normality Test Based on a Truncated Mean Characterization

This article generalizes a characterization based on a truncated mean to include higher truncated moments, and introduces a new normality goodness-of-fit test based on the truncated mean. The test is a weighted integral of the squared distance between the empirical truncated mean and its expectation. A closed form for the test statistic is derived. Assuming known parameters, the mean and the variance of the test are derived under the normality assumption. Moreover, a limiting distribution for the proposed test as well as an approximation are obtained. Also, based on Monte Carlo simulations, the power of the test is evaluated against stable, symmetric, and skewed classes of distributions. The test proves compatibility with prominent tests and shows higher power for a wide range of alternatives.     

A family of test statistics for trend change in mean residual life with unknown turning point using censored data

In this paper, we develop a family of test statistics for testing exponentiality against increasing then decreasing mean residual life alternatives to accommodate the randomly censored data when neither the change point nor the proportion is known. We establish the asymptotic null distribution of test statistics. Monte Carlo simulations are conducted to investigate the speed of convergence of test statistics to the asymptotic null distribution and study the performance of test statistics by the power of tests.     

Conditional interval estimation of the mean following rejection of a two sided test

The effect of rejecting a two-sided preliminary test of significance for the mean of a normal distribution upon subsequent interval estimation of the mean is examined. For the case where the variance is known, conditional confidence intervals may be shorter than unconditional intervals, in contrast to the one-sided preliminary test case examined by Meeks and D’Agostino (1983, The American Statistician, 7, 134-136) . For the case where the variance is unknown and must be estimated by the sample variance, it is shown that customary intervals do not offer uniformly greater or lesser coverage than the nominal level.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

Detection of a changepoint,a mean-shift accompanied with a trend change,in short time-series with autocorrelation

In this study, a changepoint model, which can detect either a mean shift or a trend change when accounting for autocorrelation in short time-series, was investigated with simulations and a new method is proposed. The changepoint hypotheses were tested using a likelihood ratio test. The test statistic does not follow a known distribution and depends on the length of the time-series and the autocorrelation. The results imply that it is not possible to detect autocorrelation and that the estimate of the autocorrelation parameter is biased. It is therefore recommended to use critical values from the empirical distribution for a fixed autocorrelation.     

Optimal minimax squared error risk estimation of the mean of a multivariate normal distribution

Minimax squared error risk estimators of the mean of a multivariate normal distribution are characterized which have smallest Bayes risk with respect to a spherically symmetric prior distribution for (i) squared error loss, and (ii) zero-one loss depending on whether or not estimates are consistent with the hypothesis that the mean is null. In (i), the optimal estimators are the usual Bayes estimators for prior distributions with special structure. In (ii), preliminary test estimators are optimal. The results are obtained by applying the theory of minimax-Bayes-compromise decision problems.     

Using historical data for Bayesian sample size determination

Summary.  We consider the sample size determination (SSD) problem, which is a basic yet extremely important aspect of experimental design. Specifically, we deal with the Bayesian approach to SSD, which gives researchers the possibility of taking into account pre-experimental information and uncertainty on unknown parameters. At the design stage, this fact offers the advantage of removing or mitigating typical drawbacks of classical methods, which might lead to serious miscalculation of the sample size. In this context, the leading idea is to choose the minimal sample size that guarantees a probabilistic control on the performance of quantities that are derived from the posterior distribution and used for inference on parameters of interest. We are concerned with the use of historical data—i.e. observations from previous similar studies—for SSD. We illustrate how the class of power priors can be fruitfully employed to deal with lack of homogeneity between historical data and observations of the upcoming experiment. This problem, in fact, determines the necessity of discounting prior information and of evaluating the effect of heterogeneity on the optimal sample size. Some of the most popular Bayesian SSD methods are reviewed and their use, in concert with power priors, is illustrated in several medical experimental contexts.     

Gini index based goodness-of-fit test for the logistic distribution

To model growth curves in survival analysis and biological studies the logistic distribution has been widely used. In this article, we propose a goodness-of-fit test for the logistic distribution based on an estimate of the Gini index. The exact distribution of the proposed test statistic and also its asymptotic distribution are presented. In order to compute the proposed test statistic, parameters of the logistic distribution are estimated by approximate maximum likelihood estimators (AMLEs), which are simple explicit estimators. Through Monte Carlo simulations, power comparisons of the proposed test with some known competing tests are carried. Finally, an illustrative example is presented and analyzed.     

Multivariate Saddlepoint test for the wrapped normal model

In this article we show the effectiveness and the accuracy of the test statistic based on the expnnent of the saddlepoint approximation for the density of M-estimators, proposed by Robinson, Ronchetti and Young (1999), for testing simultaneous hypotheses on the mean and on the variance of a wrapped normal distribution. We base this test statistic on the trigonometric method of moments estimator proposed by Gatto and Jammalamadaka (l999b), which admits the M-estimator representation necessary for this test. This test statistic has an approximate chi-squared distribution, asympiotically up to the second order, and the high accuracy of this approximation is shown by numerical simulations.     

A New Procedure to Monitor the Mean of a Quality Characteristic

The Shewhart, Bonferroni-adjustment, and analysis of means (ANOM) control charts are typically applied to monitor the mean of a quality characteristic. The Shewhart and Bonferroni procedure are utilized to recognize special causes in production process, where the control limits are constructed by assuming normal distribution for known parameters (mean and standard deviation), and approximately normal distribution regarding to unknown parameters. The ANOM method is an alternative to the analysis of variance method. It can be used to establish the mean control charts by applying equicorrelated multivariate non central t distribution. In this article, we establish new control charts, in phases I and II monitoring, based on normal and t distributions having as a cause a known (or unknown) parameter (standard deviation). Our proposed methods are at least as effective as the classical Shewhart methods and have some advantages.     

Testing for a Unit Root in a Time Series With a Changing Mean

This study considers testing for a unit root in a time series characterized by a structural change in its mean level. My approach follows the “intervention analysis” of Box and Tiao (1975) in the sense that I consider the change as being exogenous and as occurring at a known date. Standard unit-root tests are shown to be biased toward nonrejection of the hypothesis of a unit root when the full sample is used. Since tests using split sample regressions usually have low power, I design test statistics that allow the presence of a change in the mean of the series under both the null and alternative hypotheses. The limiting distribution of the statistics is derived and tabulated under the null hypothesis of a unit root. My analysis is illustrated by considering the behavior of various univariate time series for which the unit-root hypothesis has been advanced in the literature. This study complements that of Perron (1989), which considered time series with trends.     

ESTIMATION OF A DISTRIBUTION FUNCTION DOMINATING STOCHASTICALLY A KNOWN DISTRIBUTION FUNCTION

This paper considers the problem of estimating a cumulative distribution function (cdf), when it is known a priori to dominate a known cdf. The estimator considered is obtained by adjusting the empirical cdf using the prior information. This adjusted estimator is shown to be consistent, its limiting distribution is found, and its mean squared error (MSE) is shown to be smaller than the MSE of the empirical cdf. Its asymptotic efficiency (compared to the empirical cdf) is also found.     

Critical Values for the Robust Rank-Order Test

ABSTRACT

The nonparametric Wilcoxon–Mann–Whitney test is commonly used by practitioners for detecting differences in location (mean, median) between two samples. Earlier work has shown this test to have a number of disadvantages, most of which are remedied by use of the alternative robust rank-order test. Use of the robust rank-order test has been limited, perhaps partly because exact critical values have up to now been available for only a small number of sample-size values, and not for all of the commonly used levels of significance. This article expands what is known about the distribution of the robust rank-order test statistic; critical values are given for more sample sizes and for more levels of significance.     

On the linear combinations of the means of inverse gaussian distributions

Consider the two parameter Inverse Gaussian distribution with mean μ and scale parameter λ.

Suppose one is interested in testing a problem on a linear combination for the means of Inverse Gaussian distributions. For this problem a test and confidence intervals are proposed when: (1) λ’s are known and; (2) λ’s are unknown.

Finally an application of the procedures is illustrated with a data set of failure times of high-speed turbine bearings.     

Change-point problems for the von Mises distribution

A generalized likelihood ratio procedure and a Bayes procedure are considered for change-point problems for the mean direction of the von Mises distribution, both when the concentration parameter is known and when it is unknown. These tests are based on sample resultant lengths. Tables that list critical values of these test statistics are provided. These tests are shown to be valid even when the data come from other similar unimodal circular distributions. Some empirical studies of powers of these test procedures are also incorporated.     

Testing for superadditivity of the mean value function of a non-homogeneous poisson process

Consider a non-homogeneous Poisson process, N(t), with mean value function Λ(t) and intensity function λ(t). A conditional test of the hypothesis that the process is homogeneous, versus alternatives for which Λ(t) is superadditive, was proposed by Hollander and Proschan (1974). A new test for superadditivity of Λ(t), which is based on a linear combination of the occurrence times of the process N{t) is suggested in this paper. Though this test has the same Pitman efficiency as the Hollander-Proschan test, it is shown by Monte-Carlo simulation that our test has more power for many important alternatives. Tables for the exact null distribution of the test statistic have been given.     

The power of the circular cone test: A noncentral chi-bar-squared distribution

Pincus (1975) derived the null distribution of the likelihood-ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood-ratio test statistic has a chi-bar-squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood-ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi-bar-squared” distribution. To our knowledge, this is the first order-restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear-model setting. That is, the distribution of the likelihood-ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.     

Detecting changes in the mean of functional observations

Summary.  Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as X _i ( t )= μ ( t )+Σ_{1≤ l <∞} η _{i ,  l} +  v _l ( t ), where μ is the common mean, v _l are the eigenfunctions of the covariance operator and the η _{i ,  l} are the scores. Inferential procedures assume that the mean function μ ( t ) is the same for all values of i . If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England.     

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Abstract

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.