Maximum likelihood ratio theory contributes tremendous success to parametric inferences, due to the fundamental theory of Wilks (1938). Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to nd and can not be optimal as shown in this paper. In this paper, we introduce the sieve likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks' phenomenon is unveiled. We demonstrate that the sieve likelihood statistics are asymptotically distribution free and follow 2-distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models and generalized varying coefficient models. We further demonstrate that sieve likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by Ingster (1993). They can even be adaptively optimal in the sense of Spokoiny (1996) by using a simple choice of adaptive smoothing parameter. Our work indicates that the sieve likelihood ratio statistics are indeed general and powerful for nonparametric inferences based on function estimation.

The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.

No abstract uploaded!

No abstract uploaded!

Let$L$be a formally selfadjoint differential operator and$p$a real-valued function, both on$a$≤$x$<∞. The deficiency indices are the numbers of solutions of$Lu$=λ$pu$for Im λ>0 and for Im λ<0 which have a certain regularity at$x$=∞. (A) If$p(x)$≥0 this regularity means that the integral of$p(x)$│$u$│^{2}converges at infinity. (B) If$p$changes its sign for arbitrarily large values of$x$but$L$has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl’s classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also when$L$is of odd order. It is then indicated how the method carriers over to (B).