Consistency of a nonparametric conditional mode estimator for random fields

Given a stationary multidimensional spatial process <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\left\{ Z_{\mathbf{i}}=\left( X_{\mathbf{i}},\ Y_{\mathbf{i}}\right) \in \mathbb R ^d\right. \left. \times \mathbb R ,\mathbf{i}\in \mathbb Z ^{N}\right\} $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mfenced close="" open="{" separators=""> <msub> <mi>Z</mi> <mi mathvariant="bold">i</mi> </msub> <mo>=</mo> <mfenced close=")" open="(" separators=""> <msub> <mi>X</mi> <mi mathvariant="bold">i</mi> </msub> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>Y</mi> <mi mathvariant="bold">i</mi> </msub> </mfenced> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> </mfenced> <mfenced close="}" open="" separators=""> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mi mathvariant="bold">i</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">Z</mi> <mi>N</mi> </msup> </mfenced> </mrow> </math> </EquationSource> </InlineEquation>, we investigate a kernel estimate of the spatial conditional mode function of the response variable <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$Y_{\mathbf{i}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>Y</mi> <mi mathvariant="bold">i</mi> </msub> </math> </EquationSource> </InlineEquation> given the explicative variable <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X_{\mathbf{i}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mi mathvariant="bold">i</mi> </msub> </math> </EquationSource> </InlineEquation>. Consistency in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$L^p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi>L</mi> <mi>p</mi> </msup> </math> </EquationSource> </InlineEquation> norm and strong convergence of the kernel estimate are obtained when the sample considered is a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation>-mixing sequence. An application to real data is given in order to illustrate the behavior of our methodology. Copyright Springer-Verlag Berlin Heidelberg 2014