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- Classification
***62F15 Bayesian inference**

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- Classification
***62D05 Statistical sampling theory**

62B05 Sufficient statistics

- Keywords
**maximum likelihood predictor; nuisance parameter; pivotal quantity; prediction; minimal sufficient reduction; specific sufficiency; predictive intervals; superpopulation models; examples**

Superpopulation models are transformed
in predictive models in order to permit the use of standard classical statistics
techniques. Confidence intervals based on predictive models replace the predictive
intervals based on superpopulation models. The ideas are illustrated by various
examples and the normal case turns out to produce intervals that are also obtained
by the standard classical survey sampling techniques.

Publ. Year: ** 1993 **

Document Type: ** Journal **

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- Classification
***62A99 Foundations of statistics**

62G10 Nonparametric hypothesis testing

62F03 Parametric hypothesis testing

- Keywords
**Bayes factor; likelihood principle; null and alternative hypotheses; weighted likelihood ratio; $P$-value; prior distributions; significance tests**

Simple examples illustrate how misleading
a $p$-value constructed with no regard to the alternative hypothesis can be.
A $p$-value which regards the alternative hypothesis, called here $P$-value,
is precisely defined. It is shown that the use of the $P$-value avoids the kind
of inconsistencies illustrated by the examples. Although $P$-values could be
considered useless by Bayesians, the use of prior distributions (to obtain weighted
likelihoods) is a way by which classical statisticians could regard alternative
composite hypotheses when performing significance tests.

Publ. Year: ** 1993 **

Document Type: ** Journal **

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- Classification
***60C05 Combinatorial probability**

- Keywords
**lotteries; expected waiting time to observe specific numbers in a sequence of lottery draws**

Questions related to lotteries are
usually of interest to the public since people think there is a magic formula
which will help them to win lottery draws. This note shows how to compute the
expected waiting time to observe specific numbers in a sequence of lottery draws
and show that surprising facts are expected to occur.

Publ. Year: ** 1993 **

Document Type: ** Journal **

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- Classification
***62B05 Sufficient statistics**

62B15 Comparison of statistical experiments

- Keywords
**informative experiments; transition functions for sample spaces; Blackwell sufficiency; influence diagrams; geometrical solution; comparing Bernoulli experiments**

The intuition behind the Blackwell
sufficiency concept is discussed using influence diagrams. A simple geometrical
solution for the problem of comparing Bernoulli experiments is presented.

Publ. Year: ** 1990 **

Document Type: ** Journal **

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- Classification
***62D05 Statistical sampling theory**

62F15 Bayesian inference

60F15 Strong limit theorems

- Keywords
**population size estimation; finite second moments; capture/recapture sequential sampling process; supermartingale; maximum likelihood estimator; sufficient statistic; finite, closed population; without replacement; Bayes estimator; sample properties; almost sure convergence; Bayes risk**

Consider a finite, closed population
of size N. Select a first random sample of size $m\sb 1$ without replacement.
Mark its units and return them to the population. Let $U\sb 1=m\sb 1$. The j
th random sample of size $m\sb j$ is drawn without replacement and the units
already marked are returned to the population. The remaining $U\sb j$ unmarked
sample units are then marked and returned to the population. After k samples
are drawn, the data (random) vector is ${\bbfD}\sb k=(U\sb 1,U\sb 2,...,U\sb
k)$, and the statistic $T\sb k=U\sb 1+U\sb 2+...+U\sb k$ is the number of distinct
units in the whole sampling process. \par Using a prior $\pi$ for N, a Bayes
estimator of N is derived. Large sample properties of the Bayes estimator are
also obtained using standard martingale results. The almost sure convergence
of the Bayes estimator to N and of the Bayes risk to zero are established.

* T.J.Rao (Santa Barbara) *

Publ. Year: ** 1990 **

Document Type: ** Journal **

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- Classification
***62P10 Appl. of statistics to biology**

62F15 Bayesian inference

- Keywords
**Bayes factor; clinical indicant; Dirichlet distribution; divergence; weights of evidence; diagnosability**

The diagnostic probabilities of having
a disease based on possible responses (indicants) to a clinical question (tests,
signs or symptoms) are generally given without reference to their precision.
Here, a Bayesian approach is used to provide a full analysis of the diagnostic
probabilities, the weights of evidence provided by each indicant and the average
weight of evidence (diagnosability) provided by the question. The method is
extended to a sequence of questions in which a particular response may influence
whether a subsequent question is asked. The role of imprecise diagnostic probabilities
in decision making is discussed.

Publ. Year: ** 1990 **

Document Type: ** Journal **

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- Classification
***62P10 Appl. of statistics to biology**

92C50 Medical appl. of mathematical biology

90B50 Multiple-criteria decision making

62F15 Bayesian inference

- Keywords
**Influence diagrams; clinical tests; Bayesian approach**

Influence diagrams are used to illustrate
how the probability of having a disease can be updated given the results from
two or more clinical tests. The problem of calibrating a register using results
from a survey, as discussed by {\it J. Heldal} and {\it E. Spjoetvoll} [Int.
Stat. Rev. 56, 153-164 (1988)] is solved using a Bayesian approach.

Publ. Year: ** 1990 **

Document Type: ** Journal **

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- Classification
***62L12 Sequential estimation**

62D05 Statistical sampling theory

62L99 Sequential statistical methods

- Keywords
**sufficient statistic; general capture-recapture sequential sampling process; maximum likelihood estimate; population size; bounded likelihood functions; maxima**

Using data obtained by the general
capture-recapture sequential sampling process, an analytical expression for
the maximum likelihood estimate of the population size is introduced. It is
shown that the bounded likelihood functions have at most two maxima. For the
simple one-to-one case the estimate is unique.

Publ. Year: ** 1988 **

Document Type: ** Journal **

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- Classification
***62D05 Statistical sampling theory**

62F15 Bayesian inference

- Keywords
**robust linear prediction; weak robustness set; robustness set; balanced sample; shrinkage factor; multiple regression model; finite population; normal priors; posterior distribution; noninformative priors; omission of additional auxiliary regression variables; ratio estimator**

The multiple regression model is
used to describe relationships among quantities associated to finite population
units. Postulating normal priors for the regressor parameters and for the error
vector, after observing a sample, a posterior distribution for the unsampled
part of the population is obtained. The case of noninformative priors is covered
as a limit of the normal priors. We describe the general conditions under which
omission of additional auxiliary regression variables does not affect the posterior
prediction. \par Some standard situations are discussed under this Bayesian
approach. A general class of predictors suggested by such robustness conditions
is considered and some well known predictors (the ratio estimator for example)
are shown to be elements of this class, proving that there are situations where
they are robust predictors.

Publ. Year: ** 1987 **

Document Type: ** Journal **

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- Classification
***62P10 Appl. of statistics to biology**

62D05 Statistical sampling theory

- Keywords
**random allocation; allocation process; sufficient statistic; capture-mark- release-recapture sampling process; multisample CMRR generalized sequential sampling process; single urn model**

The probability distribution associated
with the multisample CMRR generalized sequential sampling process is obtained
by using an analogy with a single urn model. Some statistical features are also
discussed.

Publ. Year: ** 1987 **

Document Type: ** Journal **

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- Classification
***62D05 Statistical sampling theory**

62J05 Linear regression

- Keywords
**balanced sample; overbalanced sample; xi-model; robustness of linear predictors; survey sampling; superpopulation approach; linear models; best linear predictor**

The paper deals with robustness of
linear predictors in survey sampling under the superpopulation approach. With
the help of general results of the theory of linear models, robustness in linear
prediction is characterized. \par The inference is based on an assumed model
(the $\xi$-model), which not necessarily coincides with the true model (the
$\xi\sp*$-model). Necessary and sufficient conditions are given under which
the $\xi$-best linear predictor also is $\xi\sp*$-best. Some known examples
are used to illustrate the results.

* B.Ranneby *

Publ. Year: ** 1983 **

Document Type: ** Journal **

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