Full paper in PDF:
$% A. J. Viollaz, Nonparametric estimation of probability density functions based on orthogonal expansions, Rev. Mat. Univ. Complut. Madrid 2 (1989), no. 1, 41–82.%$

Nonparametric Estimation of Probability Density Functions Based on Orthogonal Expansions

Aldo José VIOLLAZ
Universidad Nacional de Tucumán
Facultad de Ciencias Exactas y Tecnología
Instituto de Matemática
4000 Tucumán Argentina

Received: June 6, 1988
ABSTRACT

Let     n
(Xj)j=1  be i.i.d.r.v.’s each with density function f  , and let (kn(x,t))  be a sequence (a so-called kernel sequence) of Borel measurable functions defined on R × R  . Let fn(x)  be the density function estimate defined by

          n sum 
fn(x)= n-1   kn(x,Xj).
          j=1

We prove that under general conditions on f  and (kn)  , (fn(x))  is consistent in the mean square sense. We find an asymptotic expression for the variance of the estimate and prove that its asymptotic distribution is Gaussian. These results apply to a large class of density estimates which includes the estimates considered by Parzen (1962), Leadbetter (1963) with kernels with compact support and also those estimates derived from orthogonal expansions.

Density estimates derived from trigonometric and Jacobi orthogonal expansions are studied in detail. For f  belonging to classes of functions defined in terms of the derivatives of f  , we find explicit bounds for the mean square error of the estimates, holding uniformly over the classes. We compare the rates of mean square consistency obtained with the best possible rates found by Farrel and Wahba.

1980 Mathematics Subject Classification (1985 revision): 62G05.