# Zentralblatt für Mathematik Mathematics Abstracts

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# 813.60082

Irony, Telba Z.; Pereira, Carlos A.B. :
Motivation for the use of discrete distributions in quality assurance. ( English )
Test 3, No.2, 181-193 (1994).
Classification
*60K10 Appl. of renewal theory
Keywords
discrete distributions; exchangeability; finite populations; principle of indifference; majorization; production process

By considering a finite population of $N$ items and $S$ defects, and observing the way defects should be distributed among the items we provide an interesting motivation to the binomial, negative binomial (geometric) and Poisson distributions for the number of defects in a sample from a production line. The idea is to find out, from physical considerations about the production process, which configurations of defects in items are equally likely. A uniform distribution is assessed on the space generated by these configurations. Then, a distribution for a finite, and subsequently for an infinite population of items is derived.

Publ. Year: 1994
Document Type: Journal

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# 788.62091

Barlow, R.E.; Pereira, C.A.B.; Wechsler, S. :
A Bayesian approach to environmental stress screening. ( English )
Nav. Res. Logist. 41, No.2, 215-228 (1994).
Classification
*62N99 Engineering statistics
62N10 Statistical quality control
62P99 Appl. of statistics
62C10 Bayesian problems
Keywords
environmental stress screening; optimal stress screen durations; optimal stress level; posterior density; rate of early failures

The article presents a Bayesian analysis for the environmental stress screening problem. The decision problem of deriving optimal stress screen durations is solved. Given a screen duration, the optimal stress level can also be determined. Indicators of the quality of a screen of any duration are derived. A statistical model is presented which allows a posterior density for the rate of early failures of the production process to be calculated. This enables the user to update his opinion about the quality of the process.

Publ. Year: 1994
Document Type: Journal

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# 825.62012

Barlow, Richard E.; Pereira, Carlos Alberto de Braganca :
Conditional independence and probabilistic influence diagrams. ( English )
Topics in statistical dependence, Proc. Symp. Depend. Stat. Probab., Somerset/PA (USA) 1987, IMS Lect. Notes, Monogr. Ser. 16, 19-33 (1990).
Classification
*62-09 Graphical methods in statistics
05C90 Appl. of graph theory

Publ. Year: 1990
Document Type: Conference article

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# 705.62076

Zacks, S.; Pereira, C.A.de B.; Leite, J.G. :
Bayes sequential estimation of the size of a finite population. ( English )
J. Stat. Plann. Inference 25, No.3, 363-380 (1990).
Classification
*62L12 Sequential estimation
62F15 Bayesian inference
62D05 Statistical sampling theory
Keywords
distribution of stopping times; size of a finite closed population; capture-recapture sequential sampling; posterior mean and variance; sampling distribution

Bayes estimators for estimating the size of a finite closed population are studied for the capture-recapture sequential sampling. In particular we investigate the properties of the posterior mean and variance. A computing formula is developed to overcome some numerical difficulty. Sequential stopping rules are discussed. The sampling distribution of stopping variables and of associated estimators is determined recursively, and the operating characteristics of some procedures are studied.

Publ. Year: 1990
Document Type: Journal

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# 657.62022

Irony, Telba Z.; Pereira, Carlos A. :
Exact tests for equality of two proportions: Fisher v. Bayes. ( English )
J. Stat. Comput. Simulation 25, 93-114 (1986).
Classification
*62F03 Parametric hypothesis testing
62A15 The Bayesian approach
62A20 Classical approach in statistics
Keywords
stated error probabilities; actual error frequencies; estimated error frequencies; Fisher exact test; binomial experiments; Bayes exact test

{\it F. Yates} [J. R. Stat. Soc., Ser. A147, 426-463 (1984; Zbl. 573.62050)] using theoretical and philosophical arguments claims to have proved that the Fisher exact test for comparing the proportions of two binomial experiments is the best exact test. The present article uses objective and practical arguments to confront the Fisher exact test with a Bayes exact test. Using simulated samples we claim to have proved here the inferiority of the Fisher exact test in relation to a Bayes exact test. The comparison is based on the quality concept of {\it A. P. Dawid} [J. Am. Stat. Assoc. 77, 605-613 (1982; Zbl.495.62005)].

Citations: Zbl573.62050; Zbl495.62005
Publ. Year: 1986
Document Type: Journal

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# 573.62008

Basu, D.; Pereira, Carlos A.B. :
Conditional independence in statistics. ( English )
Sankhya, Ser. A 45, 324-337 (1983).
Classification
*62B05 Sufficient statistics
62B20 Measure-theoretic results in statistics
Keywords
conditional independence; ancillarity; sufficiency; specific sufficiency; bounded completeness; splitting sets

The theory of conditional independence is explained and the relations between ancillarity, sufficiency, and statistical independence are discussed in depth. Some related concepts like specific sufficiency, bounded completeness, and splitting sets are also studied in some details by using the language of conditional independence.

Publ. Year: 1983
Document Type: Journal

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# 536.62002

Basu, D.; Pereira, Carlos A.B. :
A note on Blackwell sufficiency and a Skibinsky characterization of distributions. ( English )
Sankhya, Ser. A 45, 99-104 (1983).
Classification
*62E10 Structure theory of statistical distributions
62B10 Statistical information theory
62B15 Comparison of statistical experiments
Keywords
hypergeometric distribution; Blackwell sufficiency; Blackwell transition functions; Fisher sufficient; Skibinsky's characterization; discrete distributions

Let h(N,n,x) denote the hypergeometric distribution with parameters N,n,x ($N\ge n,x)$. In this paper {\it M. Skibinsky's} [J. Am. Stat. Assoc. 65, 926-929 (1970; Zbl. 196, 224)] characterization of the family $\{h(N,n,x)\}\sb{x=0,1,...,N}$ is recalled: A family of $N+1$ probability distributions (indexed, say by $x=0,1,...,N)$, each supported on a subset of $\{$ 0,1,...,$n\}$ is the hypergeometric family having population and sample size parameters N and n, respectively (the remaining parameter of the x-th member being x), if and only if for each number $\theta$, $0&lt;\theta&lt;1$, the mixture of the family with binomial b(N,$\theta)$ mixing distribution is the binomial b(n,$\theta)$ distribution. \par It is remarked that this result can also be expressed in terms of the concepts of Blackwell sufficiency and Blackwell transition functions, according to the following definitions: A statistical model (${\cal X},\{p\sb{\theta}\}\sb{\theta \in \Theta})$ is (Blackwell) sufficient for a statistical model (${\cal Y},\{q\sb{\theta}\}\sb{\theta \in \Theta})$ if there exists a transition function $\tau\sb x(y)$ such that $q\sb{\theta}(y)=\sum\sb x\tau\sb x(y)\cdot p\sb{\theta}(x)$ where ${\cal X}$, ${\cal Y}$ are discrete sample spaces and $\{p\sb{\theta}\}$, $\{q\sb{\theta}\}$ are families of probability distributions on ${\cal X}$ and ${\cal Y}$, respectively; $\tau\sb x(y)$ is called a Blackwell transition function and it is unique if $\{p\sb{\theta}\}$ is complete. \par Skibinsky's result is then proved along the following line: if $X\sim b(n,\theta)$, $Y\sim b(n,\theta) (0&lt;\theta&lt;1)$, we can easily define random variables $X\sp*$, $Y\sp*$ such that a) $X\sp*=\sp{d}X\vert \theta$, $Y\sp*=\sp{d}Y\vert \theta$, b) $X\sp*$ is Fisher sufficient for $Y\sp*$, c) $Y\sp*\vert X\sp*=x\sim h(N,n,x)$; so that we can write $$P\{Y=y\vert \theta \}=P\{Y\sp*=y\vert \theta \}=P\{\cup\sb{x}[(Y\sp*=y)\cap(X\sp*=x)]\}=$$ $$\sum\sb{x}P\{Y\sp*=y\vert X\sp*=x,\theta \}\cdot P\{X\sp*=x\vert \theta \}=\sum\sb{x}P\{Y\sp*=y\vert X\sp*=x\}\cdot P\{X\sp*=x\vert \theta \},$$ and Skibinsky's characterization of $\{$ h(N,n,x)$\}$ is achieved by the completeness of $\{b(n,\theta)\}\sb{0&lt;\theta&lt;1}.$ \par By following the same line, the authors show afterwards analogous characterizations for a number of unidimensional and multidimensional discrete distributions.
F.Spizzichino

Citations: Zbl.196.224
Publ. Year: 1983
Document Type: Journal

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# 484.62108

Lacayo, Herbert; Pereira, Carlos A.B.; Proschan, Frank; Saerndal, Carl Erik :
Optimal sample depends on optimality criterion. ( English )
Scand. J. Stat., Theory Appl. 9, 47-48 (1982).
Classification
*62N10 Statistical quality control
62F15 Bayesian inference
62L10 Sequential statistical analysis
62D05 Statistical sampling theory
Keywords
comparison of sampling procedures

Publ. Year: 1982
Document Type: Journal

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