be i.i.d.r.v.’s each with density function
, and let
be
a sequence (a so-called kernel sequence) of Borel measurable functions defined
on
. Let
be the density function estimate defined by
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
Let
be i.i.d.r.v.’s each with density function
, and let
be
a sequence (a so-called kernel sequence) of Borel measurable functions defined
on
. Let
be the density function estimate defined by
We prove that under general conditions on
and
,
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
belonging to classes of functions defined in terms
of the derivatives of
, 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.