Testing reliability in a stress-strength model when X and Y are normally distributed
TL;DR: In this article, the authors consider situations in which X and Y are independent and have normal distributions or can be transformed to normality and present a test statistics which are exact p values that are represented as one-dimensional integrals.
Abstract: 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.
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TL;DR: In this paper, a generalized version of the confidence interval is defined, and the generalized confidence interval can be applied to the problem of constructing confidence intervals for the difference in two exponential means and for variance components in mixed models.
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, Pratt's paradox is revisited, and a procedure that resolves the paradox is given.
587 citations
TL;DR: In this article, the classical F-test of the one-way ANOVA is extended to the case of unequal error variances, and an exact test for comparing variances of a number of populations is also developed.
Abstract: 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. Scheff6's 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.
180 citations
TL;DR: In this article, the authors present a procedure for obtaining confidence intervals and tests for a single lognormal mean using the ideas of generalized p-values and generalized confidence intervals. But the procedure is computationally very involved.
176 citations
Cites background from "Testing reliability in a stress-str..."
...Several articles have appeared in the literature describing such applications; see Weerahandi and Johnson (1992), Zhou and Mathew (1994), Weerahandi (1995b), Weerahandi and Berger (1999), and Krishnamoorthy and Mathew (2002)....
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...PII: S0378 -3758(02)00153 -2...
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TL;DR: In this paper, the estimation of R = P(Y < X) when Y and X are two independent but not identically distributed Burr-type X random variables is dealt with.
124 citations
TL;DR: The overall conclusion is that the WH normal approximation provides a simple, easy-to-use unified approach for addressing various problems for the gamma distribution.
Abstract: In this article we propose inferential procedures for a gamma distribution using the Wilson–Hilferty (WH) normal approximation. Specifically, using the result that the cube root of a gamma random variable is approximately normally distributed, we propose normal-based approaches for a gamma distribution for (a) constructing prediction limits, one-sided tolerance limits, and tolerance intervals; (b) for obtaining upper prediction limits for at least l of m observations from a gamma distribution at each of r locations; and (c) assessing the reliability of a stress-strength model involving two independent gamma random variables. For each problem, a normal-based approximate procedure is outlined, and its applicability and validity for a gamma distribution are studied using Monte Carlo simulation. Our investigation shows that the approximate procedures are very satisfactory for all of these problems. For each problem considered, the results are illustrated using practical examples. Our overall conclusion is tha...
121 citations
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Book•
27 Nov 2002
TL;DR: Inference procedures for Log-Location-Scale Distributions as discussed by the authors have been used for estimating likelihood and estimating function methods. But they have not yet been applied to the estimation of likelihood.
Abstract: Basic Concepts and Models. Observation Schemes, Censoring and Likelihood. Some Nonparametric and Graphical Procedures. Inference Procedures for Parametric Models. Inference procedures for Log-Location-Scale Distributions. Parametric Regression Models. Semiparametric Multiplicative Hazards Regression Models. Rank-Type and Other Semiparametric Procedures for Log-Location-Scale Models. Multiple Modes of Failure. Goodness of Fit Tests. Beyond Univariate Survival Analysis. Appendix A. Glossary of Notation and Abbreviations. Appendix B. Asymptotic Variance Formulas, Gamma Functions and Order Statistics. Appendix C. Large Sample Theory for Likelihood and Estimating Function Methods. Appendix D. Computational Methods and Simulation. Appendix E. Inference in Location-Scale Parameter Models. Appendix F. Martingales and Counting Processes. Appendix G. Data Sets. References.
4,151 citations
Book•
01 Jan 1975TL;DR: In this paper, the authors define the notion of conditional probability as the probability of a union of events with respect to a given set of variables, and define a set of classes of variables.
Abstract: 1. Introduction to Probability. The History of Probability. Interpretations of Probability. Experiments and Events. Set Theory. The Definition of Probability. Finite Sample Spaces. Counting Methods. Combinatorial Methods. Multinomial Coefficients. The Probability of a Union of Events. Statistical Swindles. Supplementary Exercises. 2. Conditional Probability. The Definition of Conditional Probability. Independent Events. Bayes' Theorem. Markov Chains. The Gambler's Ruin Problem. Supplementary Exercises. 3. Random Variables and Distribution. Random Variables and Discrete Distributions. Continuous Distributions. The Distribution Function. Bivariate Distributions. Marginal Distributions. Conditional Distributions. Multivariate Distributions. Functions of a Random Variable. Functions of Two or More Random Variables. Supplementary Exercises. 4. Expectation. The Expectation of a Random Variable. Properties of Expectations. Variance. Moments. The Mean and The Median. Covariance and Correlation. Conditional Expectation. The Sample Mean. Utility. Supplementary Exercises. 5. Special Distributions. Introduction. The Bernoulli and Binomial Distributions. The Hypergeometric Distribution. The Poisson Distribution. The Negative Binomial Distribution. The Normal Distribution. The Central Limit Theorem. The Correction for Continuity. The Gamma Distribution. The Beta Distribution. The Multinomial Distribution. The Bivariate Normal Distribution. Supplementary Exercises. 6. Estimation. Statistical Inference. Prior and Posterior Distributions. Conjugate Prior Distributions. Bayes Estimators. Maximum Likelihood Estimators. Properties of Maximum Likelihood Estimators. Sufficient Statistics. Jointly Sufficient Statistics. Improving an Estimator. Supplementary Exercises. 7. Sampling Distributions of Estimators. The Sampling Distribution of a Statistic. The Chi-Square Distribution. Joint Distribution of the Sample Mean and Sample Variance. The t Distribution. Confidence Intervals. Bayesian Analysis of Samples from a Normal Distribution. Unbiased Estimators. Fisher Information. Supplementary Exercises. 8. Testing Hypotheses. Problems of Testing Hypotheses. Testing Simple Hypotheses. Uniformly Most Powerful Tests. Two-Sided Alternatives. The t Test. Comparing the Means of Two Normal Distributions. The F Distribution. Bayes Test Procedures. Foundational Issues. Supplementary Exercises. 9. Categorical Data and Nonparametric Methods. Tests of Goodness-of-Fit. Goodness-of-Fit for Composite Hypotheses. Contingency Tables. Tests of Homogeneit. Simpson's Paradox. Kolmogorov-Smirnov Test. Robust Estimation. Sign and Rank Tests. Supplementary Exercises. 10. Linear Statistical Models. The Method of Least Squares. Regression. Statistical Inference in Simple Linear Regression. Bayesian Inference in Simple Linear Regression. The General Linear Model and Multiple Regression. Analysis of Variance. The Two-Way Layout. The Two-Way Layout with Replications. Supplementary Exercises. 11. Simulation. Why is Simulation Useful? Simulating Specific Distributions. Importance Sampling. Markov Chain Monte Carlo. The Bootstrap. Supplementary Exercises.
2,578 citations
TL;DR: In this article, the authors examined some problems of significance testing for one-sided hypotheses of the form H 0 : θ ≤ θ 0 versus H 1: θ > θ 1, where θ is the parameter of interest, and provided a solution to the problem of testing hypotheses about the differences in means of two independent exponential distributions.
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...
428 citations
TL;DR: In this paper, the confidence intervals for P (Y < X) are obtained under the assumption that X and Y are independently normally distributed and t.he distribution of Y is known.
Abstract: Confidence intervals for P (Y < X) are obtained under the assumptions that X and Y are independently normally distributed and t.he distribution of Y is known. The procedures of this paper are compared with a procedure suggested by Z. Govindarajulu.
213 citations