  We apply the results of Andresen A. and Spokoiny V. on profile M-estimators
and the alternating maximization procedure to analyse a sieve profile quasi
maximum likelihood estimator in the single index model with linear index
function. The link function is approximated with \(C^3\)-Daubechies-wavelets
with compact support. We derive results like Wilks phenomenon and Fisher
Theorem in a finite sample setup. Further we show that an alternation
maximization procedure converges to the global maximizer and assess the
performance of a projection pursuit procedure in that context. The approach is
based on showing that the conditions of Andresen A. and Spokoiny V. on profile
M-estimators and the alternating maximization procedure can be satisfied under
a set of mild regularity and moment conditions on the index function, the
regressors and the additive noise. This allows to construct nonasymptotic
confidence sets and to derive asymptotic bounds for the estimator as
corollaries.
