Root <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>n</mi> </math> </EquationSource> </InlineEquation> estimates of vectors of integrated density partial derivative functionals
Based on a random sample of size <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>n</mi> </math> </EquationSource> </InlineEquation> from an unknown <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$d$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>d</mi> </math> </EquationSource> </InlineEquation>-dimensional density <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$f$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>f</mi> </math> </EquationSource> </InlineEquation>, the nonparametric estimations of a single integrated density partial derivative functional as well as a vector of such functionals are considered. These single and vector functionals are important in a number of contexts. The purpose of this paper is to derive the information bounds for such estimations and propose estimates that are asymptotically optimal. The proposed estimates are constructed in the frequency domain using the sample characteristic function. For every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$d$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>d</mi> </math> </EquationSource> </InlineEquation> and sufficiently smooth <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$f$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>f</mi> </math> </EquationSource> </InlineEquation>, it is shown that the proposed estimates are asymptotically normal, attain the optimal <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$O_p(n^{-1/2})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>O</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> convergence rate and achieve the (conjectured) information bounds. In simulation studies the superior performances of the proposed estimates are clearly demonstrated. Copyright The Institute of Statistical Mathematics, Tokyo 2014
