We give expressions for the distribution and density of a product of gamma or equivalently chi-square random variables. In particular, we give the distribution of the product of two independent gamma variables of mean k in terms of the Bessel functions K <Subscript>1</Subscript>, … , K <Subscript> k </Subscript>. Copyright Springer...</subscript></subscript>

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>

A family of confidence bands (simultaneous confidence regions) is given for EY=<Emphasis Type="Bold">x′<Emphasis Type="BoldItalic">β that are piecewise-linear in <Emphasis Type="Bold">x. Normality is assumed. These confidence bands are advocated over the usual hyperbolic band when the region of prime interest is distant from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\overline{\bf x}}$$</EquationSource> </InlineEquation>. In...</equationsource></inlineequation></emphasis></emphasis></emphasis>

Given a random sample of size <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> with mean <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\overline{X} $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover> <mi>X</mi> <mo>¯</mo> </mover> </math> </EquationSource> </InlineEquation> and standard deviation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$s$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>s</mi> </math> </EquationSource> </InlineEquation> from a symmetric distribution <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$F(x; \mu , \sigma )=F_{0} (( x- \mu ) / \sigma ) $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi mathvariant="italic">μ</mi> <mo>,</mo> <mi mathvariant="italic">σ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi mathvariant="italic">μ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi mathvariant="italic">σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$F_0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>F</mi> <mn>0</mn> </msub> </math> </EquationSource> </InlineEquation>...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

We show that the correlation between the estimates of two parameters is almost unchanged if they are each transformed in an arbitrary way. To be more specific, the correlation of two estimates is invariant (except for a possible sign change) up to a first order approximation, to smooth...

Consider a sample of independent and identical bivariate observations. Simple consistent confidence intervals for the variances, covariance, and correlation of the underlying population are obtained from their influence functions. They contrast with their confidence intervals obtained under the...