Stefanyuk, A. R. A method for estimating probability density. (English) Zbl 0579.62029 Statistics and control of stochastic processes, Proc. Steklov Semin., Moscow 1984, Transl. Ser. Math. Eng., 451-465 (1985). [For the entire collection see Zbl 0562.00007.] This paper considers the estimate \(p(t,\lambda)=\sum^{N}_{k=1}\lambda_ k\Phi_ k(t)\) of a probability density function, where \(\{\Phi_ k(t)\}\) is some complete orthonormal set of functions. \(\lambda\) and \(N=1,2,..\). are determined by minimizing a standardized version of the Mahalanobis distance \((y- F(\lambda))^ TR_ y^{-1}(y-F(\lambda))\), where the vector y has coefficients \(y_ i=F_ n(\tau_ i)\), \(\tau_ 1<...<\tau_ l\) being fixed, \(R_ y\) is the covariance matrix of y, and F(\(\lambda)\) has components being suitable integrals of p(t,\(\lambda)\). Estimation is suggested in two steps. The proposed estimator is shown to converge (N\(\to \infty)\) in \(L_ 2\) to the true density. Reviewer: R.Schlittgen MSC: 62G05 Nonparametric estimation Keywords:density estimation; two-step estimators; structural; risk-minimization principle; L sub 2 convergence; complete orthonormal set of functions; Mahalanobis distance Citations:Zbl 0562.00007 PDFBibTeX XML