Simplest Nonlinear Estimate of the Mathematical Expectation

S.\,Petersburg State Polytechnic University, Russia

Recently the author has proposed an original method of estimation of the mathematical expectation of the continuous random variable. The method assumes the fulfilment of three natural requirements to the estimation: \begin{description} \item{$1^{\circ}$} unbiasedness of estimate; \item{$2^{\circ}$} the estimation independence from the mathematical expectation; \item{$3^{\circ}$} the minimality of variance linear inhomogeneous estimation of the mathematical expectation. \end{description} \noindent The method can be applied for processing either unordered samples or ordered samples. The problem is reduced to the conditional extremum problem solution in two cases. \smallskip\noindent {\it A practical application of the method to ordered samples leads to superefficient estimates in some instaces.} \smallskip\noindent Whereas one applying to unordered samples supplies {\it the arithmetic average for the sample}. It is clear that the method is applicable to nonlinear estimates. The simplest nonlinear estimate in the unordered sample processing is considered.