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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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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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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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{\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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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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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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