K. Krishnamoorthy

Statistical Tolerance Regions: Theory, Applications, and Computation

List of Tables. Preface. 1 Preliminaries. 1.1 Introduction. 1.2 Some Technical Results. 1.3 The Modified Large Sample (MLS) Procedure. 1.4 The Generalized P-value and Generalized Confidence Interval. 1.5 Exercises. 2 Univariate Normal Distribution. 2.1 Introduction. 2.2 One-Sided Tolerance Limits for a Normal Population. 2.3 Two-Sided Tolerance Intervals. 2.4 Tolerance Limits for X 1 - X 2 . 2.5 Simultaneous Tolerance Limits for Normal Populations. 2.6 Exercises. 3 Univariate Linear Regression Model. 3.1 Notations and Preliminaries. 3.2 One-Sided Tolerance Intervals and Simultaneous Tolerance Intervals. 3.3 Two-sided Tolerance Intervals and Simultaneous Tolerance Intervals. 3.4 The Calibration Problem. 3.5 Exercises. 4 The One-Way Random Model with Balanced Data. 4.1 Notations and Preliminaries. 4.2 Two Examples. 4.3 One-Sided Tolerance Limits for N( , sigma^2tau + sigma^2tau e) . 4.4 One-Sided Tolerance Limits for N( , sigma^2tau..). 4.5 Two-Sided Tolerance Intervals for N( , sigma^2tau + sigma^2tau e ). 4.6 Two-Sided Tolerance Intervals for N( , sigma^2tau..). 4.7 Exercises. 5 The One-Way Random Model with Unbalanced Data. 5.1 Notations and Preliminaries. 5.2 Two Examples. 5.3 One-Sided Tolerance Limits for N( , sigma^2tau + sigma^2 e ). 5.4 One-Sided Tolerance Limits for N( , sigma^2tau). 5.5 Two-Sided Tolerance Intervals. 5.6 Exercises. 6 Some General Mixed Models. 6.1 Notations and Preliminaries. 6.2 Some Examples. 6.3 Tolerance Intervals in a General Setting. 6.4 A General Model with Two Variance Components. 6.5 A One-Way Random Model with Covariates and Unequal Variances. 6.5 Testing Individual Bioequivalence. 6.6 Exercises. 7 Some Non-Normal Distributions. 7.1 Introduction. 7.2 Lognormal Distribution. 7.3 Gamma Distribution. 7.4 Two-Parameter Exponential Distribution. 7.5 Weibull Distribution. 7.6 Exercises. 8 Nonparametric Tolerance Intervals. 8.1 Notations and Preliminaries. 8.2 Order Statistics and Their Distributions. 8.3 One-Sided Tolerance Limits and Exceedance Probabilities. 8.4 Tolerance Intervals. 8.5 Confidence Intervals for Population Quantiles. 8.6 Sample Size Calculation. 8.7 Nonparametric Multivariate Tolerance Regions. 8.8 Exercises. 9 The Multivariate Normal Distribution. 9.1 Introduction. 9.2 Notations and Preliminaries. 9.3 Some Approximate Tolerance Factors. 9.4 Methods Based on Monte Carlo Simulation. 9.5 Simultaneous Tolerance Intervals. 9.6 Tolerance Regions for Some Special Cases. 9.7 Exercises. 10 The Multivariate Linear Regression Model. 10.1 Preliminaries. 10.2 Approximations for the Tolerance Factor. 10.3 Accuracy of the Approximate Tolerance Factors. 10.4 Methods Based on Monte Carlo Simulation. 10.5 Application to the Example. 10.6 Multivariate Calibration. 10.7 Exercises. 11 Bayesian Tolerance Intervals. 11.1 Notations and Preliminaries. 11.2 The Univariate Normal Distribution. 11.3 The One-Way Random Model With Balanced Data. 11.4 Two Examples. 11.5 Exercises. 12 Miscellaneous Topics. 12.1 Introduction. 12.2 beta-Expectation Tolerance Regions. 12.3 Tolerance Limits for a Ratio of Normal Random Variables. 12.4 Sample Size Determination. 12.5 Reference Limits and Coverage Intervals. 12.6 Tolerance Intervals for Binomial and Poisson Distributions. 12.7 Tolerance Intervals Based on Censored Samples. 12.8 Exercises. Appendix A: Data Sets. Appendix B: Tables. References. Index.

Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals

Abstract The lognormal distribution is widely used to describe the distribution of positive random variables; in particular, it is used to model data relevant to occupational hygiene and to model biological data. A problem of interest in this context is statistical inference concerning the mean of the lognormal distribution. For obtaining confidence intervals and tests for a single lognormal mean, the available small sample procedures are based on a certain conditional distribution, and are computationally very involved. Occupational hygienists have in fact pointed out the difficulties in applying these procedures. In this article, we have first developed exact confidence intervals and tests for a single lognormal mean using the ideas of generalized p -values and generalized confidence intervals. The resulting procedures are easy to compute and are applicable to small samples. We have also developed similar procedures for obtaining confidence intervals and tests for the ratio (or the difference) of two lognormal means. Our work appears to be the first attempt to obtain small sample inference for the latter problem. We have also compared our test to a large sample test. The conclusion is that the large sample test is too conservative or too liberal, even for large samples, whereas the test based on the generalized p -value controls type I error quite satisfactorily. The large sample test can also be biased, i.e., its power can fall below type I error probability. Examples are given in order to illustrate our results. In particular, using an example, it is pointed out that simply comparing the means of the logged data in two samples can produce a different conclusion, as opposed to comparing the means of the original data.

A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models

This article is about testing the equality of several normal means when the variances are unknown and arbitrary, i.e., the set up of the one-way ANOVA. Even though several tests are available in the literature, none of them perform well in terms of Type I error probability under various sample size and parameter combinations. In fact, Type I errors can be highly inflated for some of the commonly used tests; a serious issue that appears to have been overlooked. We propose a parametric bootstrap (PB) approach and compare it with three existing location-scale invariant tests-the Welch test, the James test and the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test is the best among the four tests with respect to Type I error rates. The PB test performs very satisfactorily even for small samples while the Welch test and the GF test exhibit poor Type I error properties when the sample sizes are small and/or the number of means to be compared is moderate to large. The James test performs better than the Welch test and the GF test. It is also noted that the same tests can be used to test the significance of the random effect variance component in a one-way random model under unequal error variances. Such models are widely used to analyze data from inter-laboratory studies. The methods are illustrated using some examples.

Inferences on the Common Mean of Several Normal Populations Based on the Generalized Variable Method

This article presents procedures for hypothesis testing and interval estimation of the common mean of several normal populations. The methods are based on the concepts of generalized p-value and generalized confidence limit. The merits of the proposed methods are evaluated numerically and compared with those of the existing methods. Numerical studies show that the new procedures are accurate and perform better than the existing methods when the sample sizes are moderate and the number of populations is four or less. If the number of populations is five or more, then the generalized variable method performs much better than the existing methods regardless of the sample sizes. The generalized variable method and other existing methods are illustrated using two examples.

A parametric bootstrap solution to the MANOVA under heteroscedasticity

In this article, we consider the problem of comparing several multivariate normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. We propose a parametric bootstrap (PB) approach and develop an approximation to the distribution of the PB pivotal quantity for comparing two mean vectors. This approximate test is shown to be the same as the invariant test given in [Krishnamoorthy and Yu, Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher problem, Stat. Probab. Lett. 66 (2004), pp. 161–169] for the multivariate Behrens–Fisher problem. Furthermore, we compare the PB test with two existing invariant tests via Monte Carlo simulation. Our simulation studies show that the PB test controls Type I error rates very satisfactorily, whereas other tests are liberal especially when the number of means to be compared is moderate and/or sample sizes are small. The tests are illustrated using an example.

Exact Confidence Intervals for the Common Mean of Several Normal Populations

The problem of interval estimation of the common mean of several normal populations when the variances are unknown and unequal is considered. Some new confidence intervals are proposed. One of them is centered at the well-known Graybill-Deal estimator of the common mean. These new intervals and an existing exact confidence interval are numerically compared with respect to their expected lengths. Based on comparison studies, some recommendations are made regarding the choice of the intervals to be used in applications.

Model-Based Imputation Approach for Data Analysis in the Presence of Non-detects

A model-based multiple imputation approach for analyzing sample data with non-detects is proposed. The imputation approach involves randomly generating observations below the detection limit using the detected sample values and then analyzing the data using complete sample techniques, along with suitable adjustments to account for the imputation. The method is described for the normal case and is illustrated for making inferences for constructing prediction limits, tolerance limits, for setting an upper bound for an exceedance probability and for interval estimation of a log-normal mean. Two imputation approaches are investigated in the paper: one uses approximate maximum likelihood estimates (MLEs) of the parameters and a second approach uses simple ad hoc estimates that were developed for the specific purpose of imputations. The accuracy of the approaches is verified using Monte Carlo simulation. Simulation studies show that both approaches are very satisfactory for small to moderately large sample sizes, but only the MLE-based approach is satisfactory for large sample sizes. The MLE-based approach can be calibrated to perform very well for large samples. Applicability of the method to the log-normal distribution and the gamma distribution (via a cube root transformation) is outlined. Simulation studies also show that the imputation approach works well for constructing tolerance limits and prediction limits for a gamma distribution. The approach is illustrated using a few practical examples.

Computing discrete mixtures of continuous distributions: noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient

In this article, we address the problem of computing the distribution functions that can be expressed as discrete mixtures of continuous distributions. Examples include noncentral chisquare, noncentral beta, noncentral F, noncentral t, and the distribution of squared sample multiple correlation. We illustrate the need for improved algorithms by pointing out situations where existing algorithms fail to compute meaningful values of the cumulative distribution functions (cdf) under study. To address this problem we recommend an approach that can be easily incorporated to improve the existing algorithms. For the distributions of the squared sample multiple correlation coefficient, noncentral t, and noncentral chisquare, we apply the approach and give a detailed explanation of computing the cdf values. We present results of comparison studies carried out to validate the calculated values and computational times of our suggested approach. Finally, we give the algorithms for computing the distributions of the squared sample multiple correlation coefficient, noncentral t, and noncentral chisquare so that they can be coded in any desired computer language.

Generalized P-values and confidence intervals: A novel approach for analyzing lognormally distributed exposure data

The problem of assessing occupational exposure using the mean of a lognormal distribution is addressed. The novel concepts of generalized p-values and generalized confidence intervals are applied for testing hypotheses and computing confidence intervals for a lognormal mean. The proposed methods perform well, they are applicable to small sample sizes, and they are easy to implement. Power studies and sample size calculation are also discussed. Computational details and a source for the computer program are given. The procedures are also extended to compare two lognormal means and to make inference about a lognormal variance. In fact, our approach based on generalized p-values and generalized confidence intervals is easily adapted to deal with any parametric function involving one or two lognormal distributions. Several examples involving industrial exposure data are used to illustrate the methods. An added advantage of the generalized variables approach is the ease of computation and implementation. In fact, the procedures can be easily coded in a programming language for implementation. Furthermore, extensive numerical computations by the authors show that the results based on the generalized p-value approach are essentially equivalent to those based on the Lands method. We want to draw the attention of the industrial hygiene community to this accurate and unified methodology to deal with any parameter associated with the lognormal distribution.

Comparison of approximation methods for computing tolerance factors for a multivariate normal population

In this article, we compare several approximation methods for computing the tolerance factors of a multivariate normal population. These approximate methods are evaluated by comparing the Monte Carlo estimates of the coverage probabilities with those of the specified ones. Numerical studies indicate that, in general, the tolerance factors based on an approximate method given by John, which is commonly used in the literature, are inaccurate when the number of variates is greater than or equal to 2, but another approximation (which is also due to John) with slight modification gives better results. Using Johns idea, we also suggest some new approximations, which give satisfactory results. Two of the new approximations emerge as satisfactory candidates for practical use. Our fairly extensive numerical results provide guidelines regarding the choice of the tolerance factor for practical applications.