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Sektion 8
Dienstag, 19.09.2000, 14.00–14.50 Uhr, POT 106

Testen auf der Basis nichtparametrischer Kurvenschätzungen

Hannelore Liero, Universität Potsdam

We consider the following test problems:

a) Testing, whether a density function f of i.i.d. r.v.’s X1, ..., Xn belongs to a given parametric class of functions, that is

H  : f  (-  F =  {fh |h  (-  Q  (_  Rk} vs.  K : f / (-  F .
b) Testing homoscedasticity of a nonparametric regression model
                V~ ------
Yi =  r(Xi) +    v(Xi) ei,
where r is the regression, v the conditional variance and ei are the error terms:
H : v(t) = v for some  v > 0  vs. K  : v(t) / =_  v' for all v' > 0.
c) Testing cell probabilities pn1, ..., pnkn in sparse multinomial data sets, i.e.
H : p   = p   for all j = 1,...,k   vs.  K : p  '/= p  ' for some j',
     nj     nj                  n            nj    nj
where kn -->  oo .

The test statistics for all these problems are based on a quadratic distance Qn of a nonparametric estimator of the function of interest, i.e. a kernel density estimator f^ n, a kernel estimator of the conditional variance v^ n and smoothed estimators of the cell probabilities ^p nj, respectively, from the smoothed hypothetical functions. Using limit theorems for Qn we derive asymptotic a-tests and investigate the power under local alternatives. Differences and common features and problems of these approaches are considered, comparisons to other test procedures are discussed.