Sums of powers of binomial coefficients

An asymptotic series for sums of powers of binomial coefficients is derived, the general term being defined and usable with a computer symbolic language. Sums of squares of coefficients in the symmetric case are shown to have a link with classical moment problems, but this property breaks down for cubes and higher powers. Problems of remainders for the asymptotic series are mentioned. Using the reflection formula for I'(.), a continuous form for a binomial function is set up, and this becomes oscillatory outstde the usual range. A new contmued fraction emerges for the logarithm of an adjusted sum of binomial squares. The note is a contribution to the problem of the interpretation of asymptotic series and processes for their convergence acceleration.     

A stochastic representation and sampling algorithm for nested Archimedean copulas

A general sampling algorithm for nested Archimedean copulas was recently suggested. It is given in two different forms, a recursive or an explicit one. The explicit form allows for a simpler version of the algorithm which is numerically more stable and faster since less function evaluations are required. The algorithm can also be given in general form, not being restricted to a particular nesting such as fully nested Archimedean copulas. Further, several examples are given.     

The asymptotic moment profile and maximum likelihood: Applications to gamma and gamma ratio densities

In previous papers ( Bowman and Shenton, 1998, 1999a ) we have given expressions for the asymptotic skewness and kurtosis for maximum likelihood estimators in the case of several parameters. Skewness is measured by the third standardized central moment, and kurtosis by the fourth standardized central moment. Moments of the basic structure are assumed to exist. The overarching entity is the covariance matrix ( Hessian form ), and elements of its inverse. These entities involve Stieltjes integrals relating to sums of products of multiple derivatives linked to the basic structure. The first paper dealt with skewness and gives a simple expression read¬ily computerized. The second paper is devoted to the forth standardized central moment and although a certain simplification is discovered, the resulting formula is still somewhat complicated, ft is surprising to find that the asymptotic kurtosis in general requires the evaluation of several hundred components. The present paper studies cases involving estimator for two parameter gamma and one, and three pa¬rameter gamma ratio densities, and mentions strategies aimed at avoiding algebraic and numerical errors.     

Matrix‐Exponential Distributions: Calculus and Interpretations via Flows

By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).     

On a general mixed priority queue with server discretion

We consider a single-server queueing system which attends to N priority classes that are classified into two distinct types: (i) urgent: classes which have preemptive resume priority over at least one lower priority class, and (ii) non-urgent: classes which only have non-preemptive priority among lower priority classes. While urgent customers have preemptive priority, the ultimate decision on whether to interrupt a current service is based on certain discretionary rules. An accumulating prioritization is also incorporated. The marginal waiting time distributions are obtained and numerical examples comparing the new model to other similar priority queueing systems are provided.     

The limiting spectral distribution for large sample covariance matrices with unbounded m-dependent entries

ABSTRACT

In this note, the limiting spectral distribution for large sample covariance matrices with unbounded m-dependent structure is obtained under the third moment for the entries. This partially extends the results of Hui and Pan (Comm. Statist. Theory and Methods, 2010, 39: 935–941).     

Estimation of the population spectral distribution from a large dimensional sample covariance matrix

This paper introduces a new method to estimate the spectral distribution of a population covariance matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Mar?enko–Pastur equation, originally defined in the complex plane, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation matrix of S&P 500 daily stock returns is also given.     

Classes of power semicircle laws that are randomly weighted average distributions

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012–1020] that a certain class of power semicircle distributions, parameterized by n, gives the distributions of the average of n independent and identically Arcsine random variables weighted by the cuts of (0,1) by the order statistics of a uniform (0, 1) sample of size n?1, for each n. Then we establish the central limit theorem for this class of distributions. We also use the Demni [On generalized Cauchy–Stieltjes transforms of some beta distributions. Comm Stoch Anal. 2009;3:197–210] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions.     

On the solution of renewal-type integral equations

A simple method (here called the RS-method) established for solving renewal-type integral equations based on direct numerical Riemann-Stieltjes integration is presented and evaluated. In almost all situations it has shown surprisingly good results in terms of simplicity, convergence and applicability compared with the other known methods. The RS-method is particularly useful when the probability density function has singularities.