ISSN:
1129-6569
Keywords:
Bayes and maximum probability estimators
;
bivariate family
;
consistency
;
expected number of observations
;
loss of efficiency
;
ridge
;
spacings
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Economics
Description / Table of Contents:
Riassunto Si propongono alcuni stimatori di θ che caratterizza la dislocazione del punto di discontinuità per la densità di probabilità: $$f(x,\theta ) = \frac{1}{{2\Gamma (\alpha )}}\exp ( - |x - \theta |) \cdot |x - \theta |^{\alpha - 1} $$ con $$0〈 \alpha〈 1, - \infty〈 x〈 + \infty , - \infty〈 \theta〈 + \infty .$$ Gli stimatori proposti sono basati sulla spaziatura delle osservazioni. Si mostra anche che una famiglia di densità-bivariate, dipendenti dal parametro θ possiede caratteristiche simili a quella di densità univariata.
Notes:
Abstract Here we propose a few estimators of θ, in addition to those studied in Goria (1978), the point of discontinuity of the probability density $$f(x,\theta ) = \frac{1}{{2\Gamma (\alpha )}}e^{ - |x - \theta |} |x - \theta |^{\alpha - 1} ,$$ for $$0〈 \alpha〈 1, - \infty〈 x〈 \infty , - \infty〈 \theta〈 \infty .$$ We establish the consistency and the optimality of the Bayes and the maximum probability estimators. Despite their nice properties, these estimators are not easy to compute in this case and their effective computation depends on the knowledge of the exponent α. Hence, we propose another class of estimators, dependent upon the spacings of the observations, computable without actual knowledge of the value of α as long as it is known that α 〈 α0 〈 1: we show that these estimators converge at the best possible rate. We further demonstrate, using a modified version of the maximum probability estimator's technique, that the tails of the density do not substantially effect their efficiency. Finally a bivariate family of densities, having a ridge dependent on the parameter θ, is considered and it is shown that this family exhibits features similar to the univariate case, and thus, the necessary modifications of the arguments of the univariate case are utilized for the estimation of θ in this bivariate example.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02088709
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