Local linear estimation for stochastic processes driven by <InlineEquation ID="IEq1"> <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>-stable L<InlineEquation ID="IEq2"> <EquationSource ...
The <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>-stable L<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\acute{\mathrm{e}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mi mathvariant="normal">e</mi> <mo>´</mo> </mover> </math> </EquationSource> </InlineEquation>vy motion together with the Poisson process and Brownian motion are the most important examples of L<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\acute{\mathrm{e}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mi mathvariant="normal">e</mi> <mo>´</mo> </mover> </math> </EquationSource> </InlineEquation>vy processes, which form the first class of stochastic processes being studied in the modern spirit. In this paper, the stochastic processes driven by <InlineEquation ID="IEq8"> <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>-stable L<InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\acute{\mathrm{e}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mi mathvariant="normal">e</mi> <mo>´</mo> </mover> </math> </EquationSource> </InlineEquation>vy motion are considered, local linear estimator of the drift function for these processes is discussed. Under mild conditions, we derive consistency of the local linear estimator of the drift function. The performance of the proposed estimator is assessed by simulation study. Copyright Springer Science+Business Media Dordrecht 2013
