Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data

In this paper, we investigate a nonparametric estimation of the conditional density of a scalar response variable given a random variable taking values in separable Hilbert space. We establish under general conditions the uniform almost complete convergence rates and the asymptotic normality of the conditional density kernel estimator, when the variables satisfy the strong mixing dependency, based on the single-index structure. The asymptotic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(1-\zeta )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi mathvariant="italic">ζ</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> confidence intervals of conditional density function are given, for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$0 > \zeta > 1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>0</mn> <mo>></mo> <mi mathvariant="italic">ζ</mi> <mo>></mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>. We further demonstrate the impact of this functional parameter to the conditional mode estimate. Simulation study is also presented. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed, but not tackled. Copyright Springer-Verlag Berlin Heidelberg 2014