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Optimal parameter estimation under zero-excess assumptions in a classical model. (English) Zbl 0752.62076

Summary: The coefficient of excess of a random variable \(X\) with finite fourth- order moment equals \((EY^ 4/E^ 2Y^ 2)-3\), where \(Y=X-EX\). It is zero if \(X\) is normally distributed. We consider a sequence \(X_ 1,X_ 2,\dots,X_ n\) of independent random variables with the same expectation, each with coefficient of excess equal to zero and such that \(\text{Var }X_ i=s^ 2/w_ i\) \((i=1,\dots,n)\). Then we prove that among all combinations \[ \sum a_{ijkl}(X_ i-X_ j)(X_ k-X_ l) \] with expectation \(s^ 2\), the classical estimator \(\hat s^ 2\) is the unique one with minimum variance. We use elementary Hilbert space techniques, namely orthogonal projections on finite-dimensional subspaces. They guarantee the existence and uniqueness of solutions to the considered optimization problems.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62F10 Point estimation
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References:

[1] Cramér, H., Mathematical Methods of Statistics (1946), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0063.01014
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