Exact Percentage Points Research Articles

Percentage Points for Testing Homogeneity of Several Bivariate Gaussian Populations

SYNOPTIC ABSTRACTIn this article, the exact distribution and exact percentage points for testing equality of q bivariate Gaussian populations are obtained. The distribution has been derived using the inverse Mellin transformation and the residue theorem. The percentage points have been computed for .

Testing for sphericity in phase I control chart applications

AbstractWhen using x̄–R charts it is a crucial assumption that the observations within samples are independent and have common variance. However, this assumption is almost never checked. We propose to use the samples gathered during the phase I study and the test for distributional sphericity, to check this assumption. We supply a graph of the exact percentage points for the distribution of the test statistics. The test statistics can be computed with standard statistical software. Together with the graph of the exact percentage points, the test can easily be performed during a phase I study. We illustrate the method with examples. Copyright © 2009 John Wiley & Sons, Ltd.

Exact inference and prediction for k-sample exponential case under type-ii censoring

The exact inference and prediction intervals for the K-sample exponential scale parameter under doubly Type-II censored samples are derived using an algorithm of Huffer and Lin [Huffer, F.W. and Lin, C.T., 2001, Computing the joint distribution of general linear combinations of spacings or exponen-tial variates. Statistica Sinica, 11, 1141–1157.]. This approach provides a simple way to determine the exact percentage points of the pivotal quantity based on the best linear unbiased estimator in order to develop exact inference for the scale parameter as well as to construct exact prediction intervals for failure times unobserved in the ith sample. Similarly, exact prediction intervals for failure times of units from a future sample can also be easily obtained.

Exact Percentage Points for Testing Multisample Sphericity in the Bivariate Case

This article deals with the distribution and percentage points of the likelihood ratio statistic for testing q-sample sphericity when samples are drawn from bivariate Gaussian populations. The exact distribution is obtained using the inverse Mellin transform and residue theorem. Tables of the percentage points are given for q = 2(1)6.

Exact prediction intervals for exponential distributions based on doubly Type-II censored samples

In this paper, we make use of an algorithm of Huffer & Lin (2001) in order to develop exact prediction intervals for failure times from one-parameter and two- parameter exponential distributions based on doubly Type-II censored samples. We show that this method yields the same results as those of Lawless (1971, 1977) and Like w (1974) in the case when the available sample is Type-II right censored. We present a computational algorithm for the determination of the exact percentage points of the pivotal quantities used in the construction of these prediction intervals. We also present some tables of these percentage points for the prediction of the ’ th order statistic in a sample of size n for both one- and two-parameter exponential distributions, assuming that the available sample is doubly Type-II censored. Finally, we present two examples to illustrate the methods of inference developed here.

Many-to-one Comparisons in Stratified Designs

CHEUNG and HOLLAND (1992) extended Dunnett's procedure for comparing all active treatments with a control simultaneously within each of r groups while maintaining the Type I error rate at some designated level a allowing different sample sizes for each of the group-treatment categories. This paper shows that exact percentage points can be easily calculated with current available statistical software (SAS). This procedure is compared to resampling techniques and a Bonferroni corrected Dunnett-with-in-group procedure by means of a simulation study.

Exact linear inference for scaled exponential distribution based on doubly type-ii censored samples

In this paper, we make use of an algorithm of Huffer and Lin (2000) in order to develop exact interval estimation for the scale parameter to of an exponential distribution based on doubly Type-II censored samples. We also evaluate the accuracy of a chi-square approximation proposed by Balakrishnan and Gupta (1998). We present the MAPLE program for the determination of the exact percentage points of the pivotal quantity based on the best linear unbiased estimator. Finally, we present a couple of examples to illustrate the method of inference developed here.

Exact percentage points for the Kolmogorov test on truncated versions of known continuous distributions with unknown truncation parameters

In this paper we deal with Kolmogorov-Smirnov testson uniformity with completely or partly unknown limits. Tables of exact percentage points are presented or referred using the wellknown determinant formula given by Steck (1971). It is shown that these tables also give the percentage points for the analogous statistics of the test on truncated versions of known continuous distributions with completely or partly unknown truncation limits.

On singular multivariate normal distribution and its applications

The methods of evaluating the singular multivariate normal distribution have been commonly applied even though the complete analytical proofs are not found. Recently, those evaluation methods are shown to have some errors. In this paper we present a new approach with a complete proof for evaluating the exact two-sided percentage points of a standardized m-variate normal distribution with a singular negative product correlation structure for m = 3 and with a singular negative equi-correlated structure for m ⩾ 3. The results are then applied to modify the existing procedures for estimating joint confidence intervals for multinomial proportions and for determining sample sizes. By extending the results from the multivariate normal distribution to the multivariate t-distribution with the corresponding singular correlation structure, we obtain the corrected two-sided exact critical values for the Analysis of Means for m = 4,5.

Evaluation of the one-sided percentage points of the singular multivariate normal distribution

This paper presents a new theorem, as a substitute for existing results which are shown to have some errors, for evaluating the exact one-sided percentage points of the multivariate normal distribution with a singular negative product correlation structure. By extending the result from the multivariate normal distribution to the multivariate t-distribution with corresponding singular correlation structure, we tabulate the one-sided critical points for the Analysis of Means procedure.

Asymptotic percentage points for Siegel's test statistic for compound periodicities

Fisher's test for a periodicity at a single Fourier frequency is often inadequate in practice where it is necessary to test for compound periodicities, i.e. spectral lines at several frequencies. Siegel (1980) derived a test statistic for compound periodicity and provided percentage points for m not exceeding 50, where N=2m+1 is the length of the time series. Here the distribution of the test statistic is rewritten to enable computation of exact percentage points for m up to 2000, and accurate empirical interpolation formulae derived

A problem with the likelihood ratio test for a change-point hazard rate model

A likelihood ratio test for constant hazard against a step change alternative has received considerable attention in recent years. Additional information in the maximum likelihood estimate of the change-point can seriously affect interpretation of test results. Some simple modifications are considered for which exact percentage points can be derived. Monte Carlo power and mean squared error comparisons are encouraging.

Percentage points of a weighted Kolmogorov-Smirnov statistic

Let F and G be the cumulative distribution functions corresponding to two independent random variables. Define the shift function, Δ(x), by F(x)=G(x+Δ(x)). Doksum and Sievers (1976) compared two confidence bands for Δ(x). The confidence band they found to be best requires the percentage points of a weighted form of the Kolmogorov-Smirnov statistic. The goal in this paper is to supply a table of some of the exact percentage points.

Exact Percentage Points of the Likelihood-Ratio Test for a Change-Point Hazard-Rate Model

We wish to test that the hazard rate of survival or failure-time data is constant against the alternative of a change in hazard after an unspecified time. The likelihood ratio is unbounded but the exact null distribution of a restricted likelihood-ratio test statistic is found. This distribution is not affected by Type II censoring but it does depend very strongly on the interval in which the unknown change-point is assumed to lie. Some exact percentage points are given which are much larger than simulated points that have been reported in the literature.

Percentage points of the product of two correlated t varlates

Motivated by two problems occuring in the biological sciences, Wallgren (1980) recently derived the exact distribution of the product of two correlated t variates. The purpose of this paper is to provide a table of exact percentage points so that the solution proposed by Wallgren can be applied. An approximation of the distribution is also proposed and evaluated.

Tests for simultaneously determining numbers of clusters and their shape with multivariate data

Given a set of data, very little is known about tests to determine number of clusters and/or elements of the clusters. Even in the simplest case of detecting between only one or two clusters with multivariate normal data, theoretically the number of tests needed seems to be infinite. Alternatively, suppose N independent estimates of generalized variances (GVs) are computed from a given set of p-dimensional vector observations. Assuming multivariate normality, tests based on GVs are proposed which objectively and uniquely determine, simultaneously, the number of clusters and their corresponding elements. Only a reasonably small nunber of tests are required for this stepwise procedure. The exact percentage points are either available from existing tables or can be computed from a result presented.

The distribution and the exact percentage points for Wilks'Lmvc criterion

Wilks'Lmvc is the likelihood ratio criterion for testing the hypothesis that the mean values are equal, the variances are equal and the covariances are equal, in ap-variate normal population. In this article the exact null distribution as well as the exact percentage points are given for the first time. The distribution is obtained for the most general cases and the inverse tables, namely, the values ofu for given values ofF(u) are computed for the values ofF(u)=0.01, 0.02, 0.05 and for the various values ofn andp whereF(u) is the exact distribution function of the test statistic,n=N−1 andN is the sample size. The exact tables are given forp=2, 3, 4, 5, 6, 7, 8, 9.

The exact distributions and the exact percentage points for testing equality of variances in independent normal populations†

This paper gives the exact percentage points for the likelihood ratio test for testing the hypothesis that the diagonal elements are equal given that the covariance matrix in a multivariate normal population is diagonal. A brief outline of the theory is also given. These tables are computed from the exact distribution without using any approximation. The table is given for all small values of the sample size and the dimensionality parameter p. For large values of the sample size standard approximations are available and hence these points are not tabulated here.

Some Exact Percentage Points of a Statistic Useful in Analysis of Variance and Principal Component Analysis

Some exact percentage points are given for the ratio of the largest eigenvalue to the trace of Z′Z, where Z is a matrix having independent standard normal variates as entries. These percentage points are useful in certain analysis of variance models for two-way cross-classilied data, for which only approximations have been given to date. They also have implications for principal component analysis.

Some Exact Percentage Points of a Statistic Useful in Analysis of Variance and Principal Component Analysis

Some exact percentage points are given for the ratio of the largest eigenvalue to the trace of Z′Z, where Z is a matrix having independent standard normal variates as entries. These percentage points are useful in certain analysis of variance models for two-way cross-classilied data, for which only approximations have been given to date. They also have implications for principal component analysis.