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 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>

It is pointed out that Hantush’s M(α, β) and M*(α, β) functions, two functions that arise with respect to groundwater pumping modeling, are particular cases of the generalized incomplete exponential functions known in the mathematics/statistics literature. Some relevant computational...

Motivated by hydrological problems, the exact distributions of the sum X + Y, the product X Y and the ratio X/(X + Y) are derived when X and Y are independent Pareto random variables. A detailed application of the results is provided to extreme rainfall data from Florida. Copyright...

In this manuscript we introduce R package <Emphasis FontCategory="NonProportional">Compounding for dealing with continuous distributions obtained by compounding continuous distributions with discrete distributions. We demonstrate its use by computing values of cumulative distribution function, probability density function, quantile...</emphasis>

In this paper we provide a formal yet simple and straightforward proof of the asymptotic χ<Superscript>2</Superscript> distribution for Cochran test statistic. Then, we show that the general form of this type of test statistics is invariant for the choice of weights. This fact is important since in practice many such...</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>