de Vylder, F.; Goovaerts, M. J. Optimal parameter estimation under zero-excess assumptions in a classical model. (English) Zbl 0752.62076 Insur. Math. Econ. 11, No. 1, 1-6 (1992). 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. Cited in 5 Documents MSC: 62P05 Applications of statistics to actuarial sciences and financial mathematics 62F10 Point estimation Keywords:credibility theory; zero-excess; minimum variance estimators; coefficient of excess; finite fourth-order moment; Hilbert space techniques; orthogonal projections on finite-dimensional subspaces; existence and uniqueness of solutions PDFBibTeX XMLCite \textit{F. de Vylder} and \textit{M. J. Goovaerts}, Insur. Math. Econ. 11, No. 1, 1--6 (1992; Zbl 0752.62076) Full Text: DOI References: [1] Cramér, H., Mathematical Methods of Statistics (1946), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0063.01014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.