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Estimation of variance in a classical model when the coefficients of kurtosis of the variables are known. (Estimation de la variance, dans un modèle classique, si les coefficients d’aplatissement des variables sont connus.) (French) Zbl 0972.62507

Summary: The kurtosis of a nondegenerate random variable \(X\) is defined to be \(e(X)=(EY^4/E^2Y^2)-3\), where \(Y=X-EX\). The statistical model considered is defined by the random variables \(X_1,\cdots, X_n\) satisfying \(EX_i=m\), \(\text{Var} X_i=s^2/w_i\), \(e(X_i)=e_i\) \((i=1,\cdots,n)\). When \(m\) is known, the minimum-variance unbiased estimator of \(s^2\) in the family \(\{\sum c_i(X_i-m)^2\: c_1,\cdots, c_n\in R\}\) equals \(\hat s^2=c\sum(w_i/(2+e_i))(X_i-m)^2\), where \(1/c=\sum 1/(2+e_i)\). If \(m\) is not known, an optimal estimator for \(s^2\) can be found, but it is too complicated to be of practical use and we suggest a variant of it. The classical estimator for \(s^2\) is optimal if the kurtosisses vanish, thus certainly when the random variables are normally distributed.

MSC:

62F10 Point estimation
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References:

[1] Bühlmann H. and Straub E. , ( 1970 ), Glaubwürdigkeit für Schadensätze . ( Mitteilungen Schweizerische Vereinigung der Versicherungsmathematiker 70 , n^\circ 1 , 111 - 133 ) Zbl 0197.46502 · Zbl 0197.46502
[2] De Vylder F. and Goovaerts M. , (January 1992 ), Estimation of the heterogeneity parameter in the Bühlmann-Straub credibility theory model . ( Insurance : Mathematics & Economics , Vol. 10 n^\circ 4 , 233 - 238 ) MR 1172685 | Zbl 0745.62097 · Zbl 0745.62097
[3] De Vylder F. and Goovaerts M. , (April 1992 ), Optimal parameter estimation under zero-excess assumptions in a classical model . ( Insurance : Mathematics & Economics , Vol. 11 , n^\circ 1 , 1 - 6 ) MR 1185181 | Zbl 0752.62076 · Zbl 0752.62076
[4] Dubey A. and Gisler A. , ( 1981 ), On parameter estimation in credibility . ( Mitteilungen Schweizerische Vereinigung der Versicherungsmathematiker , Vol. 81 , n^\circ 2 , 187 - 212 ). Zbl 0482.62092 · Zbl 0482.62092
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