Explicit expressions for moments of gamma order statistics

Explicit closed form expressions are derived for the moments of order statistics from the gamma and generalized gamma distributions. The expressions involve the Lauricella functions of type A and type B. The usefulness of the result is illustrated through two quality control data sets.     

Bivariate gamma distributions, sums and ratios

Exact distributions of R = X +Y and W = X/(X +Y ) and the corresponding moment properties are derived when X and Y follow five flexible bivariate gamma distributions. The expressions turn out to involve several special functions.     

Generalized Cornish-Fisher expansions

Cornish-Fisher expansions about the normal distribution provide accurate approximations for distributions of estimates and also for the level in the nominal error of confidence intervals. However, there is an advantage is expanding about a skew distribution like the chi-square, since the first order approximations become second order if the skewness is matched. Higher order approximations are also simplified. We demonstrate the method by approximating the distribution of standardized and Studentized linear combinations of means.     

Covariance information based on the <Emphasis Type="Italic">t</Emphasis> prior

The exact distribution of the covariance between two populations is derived. It is assumed that one of the population means has a Student's t prior while the other is taken to come from one of normal, Student's t, Laplace, logistic or Bessel families (the five well-known symmetric distributions). The exact distribution is given in terms of the characteristic function. The calculations involve several special functions.     

Expansions about the Gamma for the Distribution and Quantiles of a Standard Estimate

We give expansions for the distribution, density, and quantiles of an estimate, building on results of Cornish, Fisher, Hill, Davis and the authors. The estimate is assumed to be non-lattice with the standard expansions for its cumulants. By expanding about a skew variable with matched skewness, one can drastically reduce the number of terms needed for a given level of accuracy. The building blocks generalize the Hermite polynomials. We demonstrate with expansions about the gamma.     

A double generalized Pareto distribution

Papastathopoulos and Tawn [Papastathopoulos, I., Tawn, J.A., 2013. A generalized Student’s t$t$-distribution. Statistics & Probability Letters 83, 70–77] proposed a generalization of Student’s t$t$ distribution to account for negative degrees of freedom. Here, an alternative distribution that has simpler mathematical properties is discussed. Several advantages are established for using the alternative distribution over Papastathopoulos and Tawn’s generalization.     

A truncated Cauchy distribution

A truncated version of the Cauchy distribution is introduced. Unlike the Cauchy distribution, this possesses finite moments of all orders and could therefore be a better model for certain practical situations. One such situation in finance is discussed. Explicit expressions for the moments of the truncated distribution are also derived.     

A New Robust Class of Skew Elliptical Distributions

Methodology and Computing in Applied Probability - A new robust class of multivariate skew distributions is introduced. Practical aspects such as parameter estimation method of the proposed class...     

Expansions for the distribution and the maximum from distributions with an asymptotically gamma tail when a trend is present

We give expansions about the Gumbel distribution in inverse powers of n and log n for Mn, the maximum of a sample size n or n + 1 when the j-th observation is μ(j/n) + ej, μ is any smooth trend function and the residuals {ej } are independent and identically distributed with P(e r) ≈ exp(-δx)xd0∑∞k=1ckx-kβ as x →∞. We illustrate practical value of the expansions using simulated data sets.