Computation of the central and noncentral f distributions

Recursion relations suitable for rapid computation are derived for the cumulative distribution of F′ = (X/m)/(Y/n) where X is χ²(λ, m) and Y is independently χ²(n). When n is even no complicated function evaluations are needed. For n odd, a special doubly noncentral t distribution is needed to start the computation. Series representations for this t distribution are given with rigorous bounds on truncation errors. Proper recursion techniques for numerical evaluation of the special functions are given.     

The exact noncentral distribution of the generalized multiple correlation matrix

Given p×n X N(βY, ∑?I), β, ∑ unknown, the noncentral multivariate beta density of the matrix L = [(YY′)^-1/2Y X′ (XX′)^-1XY′ (YY′)^-1/2] is desired. Khatri (1964) finds this density when β is of rank unity. The present paper derives the noncentral density of L and the density of the roots matrix of L for full rank β. The dual case density of L is also obtained. The derivations are based on generalized Sverdrup's lemma, Kabe (1965), and the relationship between primal and dual density of L is explicitly established.     

A stepwise algorithm for selecting category boundaries for the chi-squared goodness-of-fit test

A stepwise algorithm for selecting categories for the chisquared goodness-of-fit test with completely specified continuous null and alternative distributions is described in this paper. The procedure's starting point is an initial partitioning of the sample space into a large number of categories. A second partition with one fewer category is constructed by combining two categories of the original partition. The procedure continues until there are only two categories; the partition in the sequence with the highest estimated power is the one chosen. For illustartive purposes, the performance of the algorithm is evaluated for several hypothesis tests of the from H₀: normal distribution vs. H₁: a specific mixed normal distribution. For each test considered, the partition identified by the algorithm was compared to several equiprobable partitions, including the equiprobable partition with the highest estimated power. In all cases but one, the algorithm identified a parttion with higher estimated power than the best equiprobable partition. Applciations of the procedure are discussed.     

A partitioning of the power in the one sample Student t-test

The power of Student's t-test is partitioned to determine* the proportion of time the sample standard deviation is too small or the mean is misestimated given that rejection of the hypothesis under test has occurred     

The Correlated Bivariate Noncentral F Distribution and Its Application

This article proposes the singly and doubly correlated bivariate noncentral F (BNCF) distributions. The probability density function (pdf) and the cumulative distribution function (cdf) of the distributions are derived for arbitrary values of the parameters. The pdf and cdf of the distributions for different arbitrary values of the parameters are computed, and their graphs are plotted by writing and implementing new R codes. An application of the correlated BNCF distribution is illustrated in the computations of the power function of the pre-test test for the multivariate simple regression model (MSRM).     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

Bayesian tests for matched proportions

A sample of n subjects is observed in each of two states, S₁-and S₂. In each state, a subject is in one of two conditions, X or Y. Thus, a subject may be recorded as showing a change if its condition in the two states is ‘Y,X’ or ‘X,Y’ and, otherwise, the condition is unchanged. We consider a Bayesian test of the null hypothesis that the probability of an ‘X,Y’ change exceeds that of a ‘Y,X’ change by amount k_O. That is, we develop the posterior distribution of k_O, the difference between the two probabilities and reject the null hypothesis if k lies outside the appropriate posterior probability interval. The performance of the method is assessed by Monte Carlo and other numerical studies and brief tables of exact critical values are presented     

Score tests in the two - way layout of counts

For the balanced two-way layout of a count response variable Y classified by fixed or random factors A and B, we address the problems of (i) testing for individual and interactive effects on Y of two fixed factors, and (ii) testing for the effect of a fixed factor in the presence of a random factor and conversely. In case (i), we assume independent Poisson responses with µ_ij= E(Y| A=i,B=j) = α_iβ_jγ_ij corresponding respectively to the multiplicative

interactive and non-interactive cases. For case (ii) with factor A random, we derive a multivariate gamma-Poisson model by mixing on the random variable associated with each level of A. In each case Neyman C(α) score tests are derived. We present simulation results,and apply the interaction test to a data set, to evaluate and compare the size and power of the score test for interaction between two fixed factors, the competing Poisson-based likelihood ratio test, and the F-tests based on the assumptions that √Y+1 or log(Y+1) are approximately normal. Our results provide strong evidence that the normal-theory based F-tests typically are very far from nominal size, and that the likelihood ratio test is somewhat more liberal than the score test.     

A computer program for the design of group testing experiments

This paper describes a computer program GTEST for designing group testing experiments for classifying each member of a population of items as “good” or “defective”. The outcome of a test on a group of items is either “negative” (if all items in the group are good) or “positive” (if at least one of the items is defective, but it is not known which). GTEST is based on a Bayesian approach. At each stage, it attempts to maximize (nearly) the expected reduction in the “entropy”, which is a quantitative measure of the amount of uncertainty about the state of the items. The user controls the procedure through specification of the prior probabilities of being defective, restrictions on the construction of the test group, and priorities that are assigned to the items. The nominal prior probabilities can be modified adaptively, to reduce the sensitivity of the procedure to the proportion of defectives in the population.     

Testing equality of conditionally jfclhwdqw exponential distributions

For the models given V = v (a common random stress), X and Y are independently exponentially distributed with failure rates λ₁and λ₂v, testing H₀λ₁λ₂using a random ‘paired’ sample is considered. It is shown that a uniformly most powerful invariant test does not exist even for one sided alternatives; locally most powerful invariant tests are derived and compared with existing procedures. The method is illustrated with reliability data. Finally, the robustness of the tests when the relationships of the failure rates to V is more complex are established.     

An optimal search procedure

Consider the following problem. There are exactly two defective (unknown) elements in the set X={x₁, x₂,…,x_n}, all possibilities occuring with equal probabilities. We want to identify the unknown (defective) elements by testing some subsets A of X, and for each such set A determining whether A contains any of them. The test on an individual subset A informs us that either all elements of the tested set A are good, or that at least one of them is defective (but we do not know which ones or how many). A set containing at least one defective element is said to be defective. Our aim is to minimize the maximal number of tests. For the optimal strategy, let the maximal test length be denoted by l₂(n). We obtain the value of this function for an infinite sequence of values of n.     

Tables of confidence limits on the tail area of the normal distribution

Tables are given of confidence limits on tail areas, γ, of the normal distribution, where γ = P{Y ≥ L}, and where L is a given number, and Y is normally distributed with unknown mean, μ, and unknown variance, σ².     

Two new estimators for the proportion of true null hypotheses in multiple test

In multiple hypothesis test, an important problem is estimating the proportion of true null hypotheses. Existing methods are mainly based on the p-values of the single tests. In this paper, we propose two new estimations for this proportion. One is a natural extension of the commonly used methods based on p-values and the other is based on a mixed distribution. Simulations show that the first method is comparable with existing methods and performs better under some cases. And the method based on a mixed distribution can get accurate estimators even if the variance of data is large or the difference between the null hypothesis and alternative hypothesis is very small.     

On a new transformation to normality

A Gaussian approximation to the distribution of the nonnegative random variable Y is developed using the Wilson and Hilferty (1931) approach. This approximation uses the symmetrizing transformation ((Y + b)/k₁)^h where k₁ is the first moment of Y and h and b are determined from the first three cumulants of Y. The approximation is illustrated in the case which Y is a non-central chi-square, where numerical evaluations indicate that the new transformation is an improvement over existing ones, especially for small values of k₁.     

Tests for Symmetry Based on the One-Sample Wilcoxon Signed Rank Statistic

ABSTRACT

The one-sample Wilcoxon signed rank test was originally designed to test for a specified median, under the assumption that the distribution is symmetric, but it can also serve as a test for symmetry if the median is known. In this article we derive the Wilcoxon statistic as the first component of Pearson's X ² statistic for independence in a particularly constructed contingency table. The second and third components are new test statistics for symmetry. In the second part of the article, the Wilcoxon test is extended so that symmetry around the median and symmetry in the tails can be examined seperately. A trimming proportion is used to split the observations in the tails from those around the median. We further extend the method so that no arbitrary choice for the trimming proportion has to be made. Finally, the new tests are compared to other tests for symmetry in a simulation study. It is concluded that our tests often have substantially greater powers than most other tests.     

Kolmogorov-type tests of goodness-of-fit against stochastic ordering

This article addresses the problem of testing the null hypothesis H₀ that a random sample of size n is from a distribution with the completely specified continuous cumulative distribution function F_n(x). Kolmogorov-type tests for H₀ are based on the statistics C⁺ _n = Sup[F_n(x)?F₀(x)] and C^? _n=Sup[F₀(x)?F_n(x)], where F_n(x) is an empirical distribution function. Let F(x) be the true cumulative distribution function, and consider the ordered alternative H₁: F(x)≥F₀(x) for all x and with strict inequality for some x. Although it is natural to reject H₀ and accept H₁ if C ⁺ _n is large, this article shows that a test that is superior in some ways rejects F₀ and accepts H₁ if C^mdash _n is small. Properties of the two tests are compared based on theoretical results and simulated results.     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.