| Summary: | 碩士 === 淡江大學 === 數學系 === 82 === When the probability density function of the population is
unknown . We must use the sample datas that we have observed to
estimate . There are many estimate methods about the
probability density function . For example , orthogonal series
, kernel function and the nearest neighbor , etc . Among these
estimate methods , Cencov (1962) first proposed to estimate
probability density function by orthogonal series . After that
, Grelicki and Pawlak (1985) use the method of orthogonal
series to estimate probability density function on the
regression function estimation . Assume that (X1,Y1) , (X2,Y2)
, ... , (Xn,Yn) are independent and identically distributed
random pairs of (X,Y) . Let f(x) be an unknown probability
density function of X and h(x,y) be the joint probability
density function of (X,Y) . Suppose that E.absolute.(Y) < .inf.
, let the regression function of Y on X = x be r(x) = E[Y.
lgvert.X=x] = [g(x)}/[f(x)] , where g(x) = .int. yh(x,y)dy .
Watson and Nadaraya (1964) first proposed to estimate r(x) by
.rnhat.(x) = [.gnhat.(x)]/[.fnhat.(x)] . In this paper , we
want to estimate the regression function by Walsh series ,
where X .in. [0,1) , Y .in. R . Further assume that f(x) , g(x)
.in. L2 . Thus , .rnhat.(x) = [▆YjKN(x,Xj)]/[▆KN(x,Xj)] ,
where KN(.) is the Walsh Kernel function , that is , KN(x,Xj) =
▆.PSI. K(x).PSI.(Xj) = ▆.PSI.K(x.plmin.Xj) , where .PSI.K(.)
is the Walsh series of the Kth term . Under the following
conditions : (i) N(n) .arrr. .inf. as n .arrr. .inf. , (ii) [.
Square.(N(n))]/n .arrr. 0 as n .arrr. .inf. , (iii) .absolute.(
Y) .ltoreq. Cr < .inf. . In chapter 3 , we can show that (1)
.rnhat.(x) conver- gence in probability to r(x) , for all x
.in. [0,1) (2) .rnhat. (x) is Mean Squared Error Consistency ,
that is , .liminj.E .Square.(.rnhat.(x) - r(x)) = 0 . Further
the rate of convergence of mean square error of .rnhat.(x) is
obtained .
|
|---|
