Simple transformations are given for reducing/stabilizing bias, skewness and kurtosis, including the first such transformations for kurtosis. The transformations are based on cumulant expansions and the effect of transformations on their main coefficients. The proposed transformations are...

We consider the mixed AR(1) time series model <Equation ID="Equa"> <EquationSource Format="TEX">$$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$</EquationSource> </Equation>for −1  β <Superscript> p </Superscript> ≤ 0 ≤ α <Superscript>...</superscript></superscript></equationsource></equation>

We consider the (possibly nonlinear) regression model in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathbb{R }^q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>q</mi> </msup> </math> </EquationSource> </InlineEquation> with shift parameter <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> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\mathbb{R }^q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>q</mi> </msup> </math> </EquationSource> </InlineEquation> and other parameters <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\beta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">β</mi> </math> </EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\mathbb{R }^p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>p</mi> </msup> </math> </EquationSource> </InlineEquation>. Residuals are assumed to be from an unknown...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

Two methods are given for adapting a kernel density estimate to obtain an estimate of a density function with bias O(h <Superscript> p </Superscript>) for any given p, where h=h(n) is the bandwidth and n is the sample size. The first method is standard. The second method is new and involves use of Bell polynomials. The...</superscript>