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- 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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- 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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- Classification
***62-09 Graphical methods in statistics**

05C90 Appl. of graph theory

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- 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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- 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: ** Zbl 573.62050;
Zbl495.62005 **

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- 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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- 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 $\theta$, $0<\theta<1$,
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 $X\sim b(n,\theta)$, $Y\sim b(n,\theta) (0<\theta<1)$, 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 $\{b(n,\theta)\}\sb{0<\theta<1}.$
\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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- Classification
***62N10 Statistical quality control**

62F15 Bayesian inference

62L10 Sequential statistical analysis

62D05 Statistical sampling theory

- Keywords
**comparison of sampling procedures**

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