Expansion of scale mixtures of the gamma distribution (Q1116218)

Let \(\sigma\) and X be positive random variables and X follows the gamma distribution G(x;\(\lambda)\) with the density \(g(x;\lambda)\equiv x^{\lambda -1}e^{-x}/\Gamma (\lambda)\), \(x>0\). An expansion of the distribution function F(x) of the variable \(\eta =\sigma X\) around G(x;\(\lambda)\) and its error bounds are obtained. The expansion is given in terms of the Laguerre polynomials and the moments of \(\sigma\) or that of \(\eta\). When \(\lambda =1\), the distribution F has decreasing hazard rate and the present article generalizes inequalities obtained by several authors on the difference \(| F(x)-G(x;1)|\).