A Modified Method For Bayesian Prediction Of Future Order Statistics From Generalized Power Function
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Date
2015-04
Authors
Omar, Almutairi Aned
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Abstract
Bayesian statistics is a statistical method that is widely used in many fields, including
medicine, social and applied sciences. These fields occasionally have little or limited
information about their populations. Therefore, using new techniques that require
fewer samples while providing the same quality as the case of available samples is
necessary. Bayesian prediction is a commonly used tool in Bayesian statistics. This
study modifies three Bayesian prediction methods: one-, two- and multi-sample
predictions. Bayesian prediction modified method does not require the many
samples. Therefore, a future sample is a significant term in this thesis. Our Bayesian
prediction modified method used a prediction for the future order statistics based on
the observed ordered data, and predictive densities provided the Bayesian prediction
intervals for the future order statistics. The standard generalized power function
distribution serves as the basis for the three modified methods by applying Bayes'
theory to achieve close lower and upper limits for the 95% and 99% Bayesian
prediction intervals. The proposed intervals contributed to increasing the precision
for the predictive value. The performance of the three modified methods is evaluated
using the lower and upper limits from the observed sample and for the order statistic
from the future sample. Two types of prior functions are used in Bayesian prediction:
informative and non-informative priors, both of which use Bayes' theory. The
numerical analysis illustrates lower and upper limits for the 95% and 99% Bayesian prediction intervals for the three modified methods, and the data set generated from
the standard generalized power function distribution. Bayesian estimation is used to
determine the shape, scale and location parameters. Bayesian estimators are
suggested as the mean of the posterior distribution based on an informative or noninformative
prior function. Both prior functions use a formula for the posterior
distribution from Bayes theory to combine the likelihood function and prior function.
The proposed Bayesian estimator is the mean of the posterior distribution based on
the standard generalized power function distribution and a squared error loss
function. In addition to this technique, a Bayesian criterion is used. The performance
of the shape, scale and location estimators are evaluated with some types of prior
distributions and used simultaneously with the Bayesian prediction, which, when
compared, confirms the suitability and advantage of some types of prior distributions
for estimation or prediction using the Bayesian method. The numerical analysis
illustrates the proposed estimators derived from the data set generated from the
standard generalized power function distribution.
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Keywords
Bayesian statistical decision theory