Samaradasa Weerahandi

Generalized Confidence Intervals

Abstract The definition of a confidence interval is generalized so that problems such as constructing exact confidence regions for the difference in two normal means can be tackled without the assumption of equal variances. Under certain conditions, the extended definition is shown to preserve a repeated sampling property that a practitioner expects from exact confidence intervals. The proposed procedure is also applied to the problem of constructing confidence intervals for the difference in two exponential means and for variance components in mixed models. A repeated sampling property of generalized p values is also given. With this characterization one can carry out fixed level tests of parameters of continuous distributions on the basis of generalized p values. Finally, Pratts paradox is revisited, and a procedure that resolves the paradox is given.

Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters

Abstract This article examines some problems of significance testing for one-sided hypotheses of the form H 0 : θ ≤ θ 0 versus H 1 : θ > θ 0, where θ is the parameter of interest. In the usual setting, let x be the observed data and let T(X) be a test statistic such that the family of distributions of T(X) is stochastically increasing in θ. Define Cx as {X : T(X) — T(x) ≥ 0}. The p value is p(x) = sup θ≤θ0 Pr(X ∈ Cx | θ). In the presence of a nuisance parameter η, there may not exist a nontrivial Cx with a p value independent of η. We consider tests based on generalized extreme regions of the form Cx (θ, η) = {X : T(X; x, θ, η) ≥ T(x; x, θ, η)}, and conditions on T(X; x, θ, η) are given such that the p value p(x) = sup θ≤θ0 Pr(X ∈ Cx (θ, η)) is free of the nuisance parameter η, where T is stochastically increasing in θ. We provide a solution to the problem of testing hypotheses about the differences in means of two independent exponential distributions, a problem for which the fixed-level testing approach...

Exact statistical methods for data analysis

1 Preliminary Notions.- 1.1 Introduction.- 1.2 Sufficiency.- 1.3 Complete Sufficient Statistics.- 1.4 Exponential Families of Distributions.- 1.5 Invariance.- 1.6 Maximum Likelihood Estimation.- 1.7 Unbiased Estimation.- 1.8 Least Squares Estimation.- 1.9 Interval Estimation.- Exercises.- 2 Notions in significance testing of hypotheses.- 2.1 Introduction.- 2.2 Test Statistics and Test Variables.- 2.3 Definition of p-Value.- 2.4 Generalized Likelihood Ratio Method.- 2.5 Invariance in Significance Testing.- 2.6 Unbiasedness and Similarity.- 2.7 Interval Estimation and Fixed-Level Testing.- Exercises.- 3 Review of Special Distributions.- 3.1 Poisson and Binomial Distributions.- 3.2 Point Estimation and Interval Estimation.- 3.3 Significance Testing of Parameters.- 3.4 Bayesian Inference.- 3.5 The Normal Distribution.- 3.6 Inferences About the Mean.- 3.7 Inferences About the Variance.- 3.8 Quantiles of a Normal Distribution.- 3.9 Conjugate Prior and Posterior Distributions.- 3.10 Bayesian Inference About the Mean and the Variance.- Exercises.- 4 Exact Nonparametric Methods.- 4.1 Introduction.- 4.2 The Sign Test.- 4.3 The Signed Rank Test and the Permutation Test.- 4.4 The Rank Sum Test and Allied Tests.- 4.5 Comparing k Populations.- 4.6 Contingency Tables.- 4.7 Testing the Independence of Criteria of Classification.- 4.8 Testing the Homogeneity of Populations.- Exercises.- 5 Generalized p-Values.- 5.1 Introduction.- 5.2 Generalized Test Variables.- 5.3 Definition of Generalized p-Values.- 5.4 Frequency Interpretations and Generalized Fixed-Level Tests.- 5.5 Invariance.- 5.6 Comparing the Means of Two Exponential Distributions.- 5.7 Unbiasedness and Similarity.- 5.7 Comparing the Means of an Exponential Distribution and a Normal Distribution.- Exercises.- 6 Generalized Confidence Intervals.- 6.1 Introduction.- 6.2 Generalized Definitions.- 6.3 Frequency Interpretations and Repeated Sampling Properties.- 6.4 Invariance in Interval Estimation.- 6.5 Interval Estimation of the Difference Between Two Exponential Means.- 6.6 Similarity in Interval Estimation.- 6.7 Generalized Confidence Intervals Based on p-Values.- 6.8 Resolving an Undesirable Feature of Confidence Intervals.- 6.9 Bayesian and Conditional Confidence Intervals.- Exercises.- 7 Comparing Two Normal Populations.- 7.1 Introduction.- 7.2 Comparing the Means when the Variances are Equal.- 7.3 Solving the Behrens-Fisher Problem.- 7.4 Inferences About the Ratio of Two Variances.- 7.5 Inferences About the Difference in Two Variances.- 7.6 Bayesian Inference.- 7.7 Inferences About the Reliability Parameter.- 7.8 The Case of Known Stress Distribution.- Exercises.- 8 Analysis of Variance.- 8.1 Introduction.- 8.2 One-way Layout.- 8.3 Testing the Equality of Means.- 8.4 ANOVA with Unequal Error Variances.- 8.5 Multiple Comparisons.- 8.6 Testing the Equality of Variances.- 8.7 Two-way ANOVA without Replications.- 8.8 ANOVA in a Balanced Two-way Layout with Replications.- 8.9 Two-way ANOVA under Heteroscedasticity.- Exercises.- 9 Mixed Models.- 9.1 Introduction.- 9.2 One-way Layout.- 9.3 Testing Variance Components.- 9.4 Confidence Intervals.- 9.5 Two-way Layout.- 9.6 Comparing Variance Components.- Exercises.- 10 Regression.- 10.1 Introduction.- 10.2 Simple Linear Regression Model.- 10.3. Inferences about Parameters of the Simple Regression Model.- 10.3 Multiple Linear Regression.- 10.4 Distributions of Estimators and Significance Tests.- 10.5 Comparing Two Regressions with Equal Variances.- 10.6 Comparing Regressions without Common Parameters.- 10.7 Comparison of Two General Models.- Exercises.- Appendix A.- Elements of Bayesian Inference.- A.1 Introduction.- A.2 The Prior Distribution.- A.3 The Posterior Distribution.- A.4 Bayes Estimators.- A.5 Bayesian Interval Estimation.- A.6 Bayesian Hypothesis Testing.- Appendix B Technical Arguments.- References.

ANOVA under unequal error variances

By taking a generalized approach to findingp values, the classical F-test of the one-way ANOVA is extended to the case of unequal error variances. The relationship of this result to other solutions in the literature is discussed. An exact test for comparing variances of a number of populations is also developed. Scheff6s procedure of multiple comparison is extended to the case of unequal variances. The possibility and the approach that one can take to extend the results to simple designs involving more than one factor are briefly discussed.

Testing reliability in a stress-strength model when X and Y are normally distributed

We consider the stress-strength problem in which a unit of strength X is subjected to environmental stress Y. An important problem in stress-strength reliability concerns testing hypotheses about the reliability parameter R = P[X > yl. In this article, we consider situations in which X and Y are independent and have normal distributions or can be transformed to normality. We do not require the two population variances to be equal. Our approach leads to test statistics which are exact p values that are represented as one-dimensional integrals. On the basis of the p value, one can also construct approximate confidence intervals for the parameter of interest. We also present an extension of the testing procedure to the case in which both strength and stress depend on covariates. For comparative purposes, the Bayesian solution to the problem is also presented. We use data from a rocket-motor experiment to illustrate the procedure.

TESTING REGRESSION EQUALITY WITH UNEQUAL VARIANCES

On obtient un test exact, similaire a celui de Chow (1960), qui comporte une simple integration unidimensionnelle

Generalized p -values and generalized confidence regions for the multivariate Behrens-Fisher problem and MANOVA

For two multivariate normal populations with unequal covariance matrices, a procedure is developed for testing the equality of the mean vectors based on the concept of generalized p-values. The generalized p-values we have developed are functions of the sufficient statistics. The computation of the generalized p-values is discussed and illustrated with an example. Numerical results show that one of our generalized p-value test has a type I error probability not exceeding the nominal level. A formula involving only a finite number of chi-square random variables is provided for computing this generalized p-value. The formula is useful in a Bayesian solution as well. The problem of constructing a confidence region for the difference between the mean vectors is also addressed using the concept of generalized confidence regions. Finally, using the generalized p-value approach, a solution is developed for the heteroscedastic MANOVA problem.

Testing Variance Components in Mixed Models with Generalized p Values

Abstract A procedure for testing one-sided hypotheses on variance components in a balanced random effects model is developed. The testing is performed on the basis of a generalized p value and involves a one-dimensional numerical integration. In the special case for which the conventional methods provide an exact test, this test is equivalent to the usual F test. The test is also generalized Bayes in the sense that, under a certain diffuse prior, the p value is numerically the same as the posterior probability of the null hypothesis.

Testing the difference of two exponential means using generalized p-values

There are no exact fixed-level tests for testing the null hypothesis that the difference of two exponential means is less than or equal to a prespecified value θ0. For this testing problem, there are several approximate testing procedures available in the literature. Using an extended definition of p-values, Tsui and Weerahandi (1989) gave an exact significance test for this testing problem. In this paper, the performance of that procedure is investigated and is compared with approximate procedures. A size and power comparison is carried out using a simulation study. Its findings show that the test based on the generalized p-value guarantees the intended size and that it is either as good as or outperforms approximate procedures available in the literature, both in power and in size.

A framework for forecasting demand for new services and their cross effects on existing services

Abstract In this article we deal with the problem of forecasting the demand for future new services and existing services simultaneously. When a number of new services are to be introduced into the market, according to our experience, it is very difficult to carry out a market survey and obtain reliable data to facilitate the forecasting. The main difficulty is due to the fact that each new service may have a cross effect on almost every existing service. Moreover, new services are likely to have substitutive and complementary effects on each other. In this article we propose a model which can incorporate these effects. This is accomplished by characterizing each telecommunications service and each telecommunications application by a set of service attributes. The cost (to the customer) of each service is also explicity incorporated. The model parameters are estimated by exploiting the historical customer choice behavior in choosing existing services. The proposed model also allows an investigator to incorporate any foreseen trends of new telecommunications applications (customer needs). This approach has produced reasonable results in our applications.