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Abstract: Cornish and Fisher (1937) gave a fourth order approximation for the percentile of the F-distribution by approximating the cumulants of . We obtain an alternative formula by approximating the cumulants of F. Hill and Davis (1968) gave Cornish–Fisher type expansions about any asymptotically normal distribution. We specialize their results to give expansions about the F-distribution. [Copyright &y& Elsevier]

Foschini gave a lower bound for the channel capacity of an N-transmit M-receive antenna system in a Raleigh fading environment with independence at both transmitters and receivers. We show that this bound is approximately normal. [ABSTRACT FROM AUTHOR]

Suzuki distribution is a popular model in wireless communications. In this paper, a generalization of it is proposed and two moment estimates derived. The performance of the estimates is assessed by simulation. Finally, applications are discussed for two problems in wireless communications. [ABSTRACT FROM AUTHOR]

Given a normal sample with means $${{\bf x}_{1}^{\prime} {\bf \varphi}, \ldots, {\bf x}_{n}^{\prime} {\bf \varphi}}$$ and variance v, minimum variance unbiased estimates are given for the moments of L, where log L is normal with mean $${{\bf x}^{\prime} {\bf \varphi}}$$ and variance v. These estimates converge to wrong values if the normality assumption is false. In the latter case estimates based on any M-estimate of $${{\bf \varphi}}$$ are available of bias $${O\left(n^{-1}\right)}$$ and $${O\left(n^{-2}\right)}$$. More generally, these are given for any smooth function of $${\left({\bf \varphi}, F\right)}$$, where F is the unknown distribution of the residuals. The regression functions need not be linear. [ABSTRACT FROM AUTHOR]

Motivated by time series analysis, we consider the problem of solving the recurrence relation un+1 = vn+1 + un ⊗ vn for n ≠ 0 and un, given the sequence vn. A solution is given as a Bell polynomial. When vn can be written as a weighted sum of nth powers, then the solution un also takes this form. [ABSTRACT FROM AUTHOR]

Expressions are given for repeated upper tail integrals of the univariate normal density (and so also for the general Hermite function) for both positive and negative arguments. The expressions involve moments of the form E( x + i N) and E1 / ( x + N), where N is a unit normal random variable. Laplace transforms are provided for the Hermite functions and the moments. The practical use of these expressions is illustrated. [ABSTRACT FROM AUTHOR]

Suppose $${\widehat{\theta}}$$ is an estimator of θ in $${\mathbb{R}}$$ that satisfies the central limit theorem. In general, inferences on θ are based on the central limit approximation. These have error O( n), where n is the sample size. Many unsuccessful attempts have been made at finding transformations which reduce this error to O( n). The variance stabilizing transformation fails to achieve this. We give alternative transformations that have bias O( n), and skewness O( n). Examples include the binomial, Poisson, chi-square and hypergeometric distributions. [ABSTRACT FROM AUTHOR]

We give the cumulative distribution function of M, the maximum of a sequence of n observations from an autoregressive process of order 1. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlation is positive, where where { ν} are the eigenvalues of a non-symmetric Fredholm kernel, and ν is the eigenvalue of maximum magnitude. When the correlation is negative The weights β depend on the jth left and right eigenfunctions of the kernel. These are given formally by left and right eigenvectors of an infinite Toeplitz matrix whose eigenvalues are just { ν}. These results are large deviations expansions for extremes, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist. The use of the derived expansion for P( M ≤ x) is illustrated using both simulated and real data sets. [ABSTRACT FROM AUTHOR]

Let M be the rth largest of a random sample of size n from a distribution F of exponential power type on R. That is, 1- F( z) = O( x exp(− x)) as x = ( z/σ) → ∞. For example, the exponential, gamma, chi-square, Laplace and normal distributions are of this type. We obtain an asymptotic expansion in powers of u = −log(1 − u) and u = log u, for the quantile F( u) near u = 1. From this, we obtain a double expansion in inverse powers of (log n, n) for the moments of M/ n/ n, with the coefficient a polynomial in log log n. We also discuss a possible application to an optimal stopping problem. [ABSTRACT FROM AUTHOR]

We show that adding a predictor to a linear model increases the variance of the estimated regression function, and so generally increases the width of a confidence interval. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Suppose that we observe X = σXθi with probability pi, 1 ≤ i ≤ c, where Σi=1c pi = and Xθ ∼ F∼, a one-parameter family. Here, we provide the first parametric test of the hypothesis that F∼ has some particular form; in particular that F∼ is a gamma distribution. [ABSTRACT FROM AUTHOR]

It is well known that Hylleraas''s series for the wave function of He does not satisfy Schrödinger''s equation. has given a class of solutions which do satisfy this equation as well as four constraints. This note extends class of solutions. [Copyright &y& Elsevier]

We extend the well-known identity, log det A = tr log A, for any square non-singular matrix A. [ABSTRACT FROM AUTHOR]

Suppose we have a sample randomly drawn from one of a given number of distributions. We wish to select the distribution based on the optimal maximum likelihood procedure. In this note, various tight estimates are derived under general conditions for the probability of making the wrong selection. The estimates are also extended to the case of many exponential families, where the general conditions fail. Some of the estimates are illustrated by means of simulation. The practical use of the estimates is discussed. [ABSTRACT FROM AUTHOR]

Abstract: Given a random variable X, the moments, , satisfy for . If , then for , there is a random variable such that for . We apply the inequality to to obtain new inequalities for the moments when . An application is illustrated to obtain tighter bounds for skewness and kurtosis. [Copyright &y& Elsevier]

Edgeworth–type expansions are given for the log density and also for the derivatives of the density of an asymptotically normal random variable having the standard expansions for its cumulants. Expansions for the log density are much simpler than for the density. In fact the rth term of the expansion for the log density is a polynomial of degree only r + 2, while that for the density is a polynomial of degree 3 r. [ABSTRACT FROM AUTHOR]

Moments and cumulants are expressed in terms of each other using Bell polynomials. Inbuilt routines for the latter make these expressions amenable to use by algebraic manipulation programs. One of the four formulas given is an explicit version of Kendall's use of Faa di Bruno's chain rule to express cumulants in terms of moments. [ABSTRACT FROM AUTHOR]

Recently, several authors have been interested in obtaining asymptotic expansions for the negative moments of the binomial conditioned to be positive. Here, we show how asymptotic expansions for its moments and cumulants can be easily derived. As an aside, we also give exact results for the mean of the inverse of a binomial plus an integer. [ABSTRACT FROM AUTHOR]

Abstract: Every canonical exponential family generates a regression model that is also a canonical exponential family. For example, for the normal it is the set of say normal observations, with means and variances of the form and for , where is the canonical regression parameter and are known vectors of the same dimension. This is a much richer model than the usual linear regression model with means and variances and for . We give the first few terms of the Edgeworth–Cornish–Fisher expansions for the distribution, density and quantiles of any smooth function of the maximum likelihood estimate of , and the associated expansion for the confidence limit of any smooth function of . [Copyright &y& Elsevier]

A letter to the editor related to an unknown vector and its simultaneous confidence variables is presented.

A letter to the editor is presented in response to the article on factors for central moments of log-normal distribution in the previous issue.

A letter to the editor is presented which offers mathematical information on the engagement of Plackett's identity in the bivariate normal distribution and multivariate normal distribution.

A letter to the editor is presented regarding explicit expressions for moments and cumulants of three distributions, published within the issue.

We give a recurrence relation for the moments of the noncenrtral chi-square distribution. [ABSTRACT FROM AUTHOR]