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.     

Locking the Wiener Process to its Level-Crossing Time

We consider the specific transformation of a Wiener process {X(t), t ≥ 0} in the presence of an absorbing barrier a that results when this process is “time-locked” with respect to its first passage time T _a through a criterion level a, and the evolution of X(t) is considered backwards (retrospectively) from T _a . Formally, we study the random variables defined by Y(t) ≡ X(T _a  ? t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series “response-locked” to their times of crossing a criterion level.     

Determining the effects of intraclass correlation on factorial experiments

This paper considers the effects of the presence of eqmi-correlation between observations in a k-way factorial experiment, A technique to provide unbiased F-tests is also constructed.     

Estimation of the mean of the exponential distribution with an outlying observation

Given a life testing experiment consisting of n items, n-1 of which have the expected life λ while one could have an expected life λ/α with 0 < α < 1 the problem is. to find a mean square error (MSE) minimizing estimation function. The standard estimators for the homogeneous case (α = 1) overestimate the expected life and their MSE tend to infinity when a tends to 0.

Looking at the estimation problem as an insurance (see Anscombe (1960)) two different “testimators” are compared with respect to their MSE, Numerical results show that an estimation function based on the “Epstein-statistic” x_(n)/[xbar] is the best one.     

Unbiased F Tests for Factorial Experiments for Correlated Data

This paper finds a general form of the correlation matrix that may be used to provide unbiased F tests in a.k-way factorial experiment.     

Differential equation models for statistical functions

Differential equations have been used in statistics to define functions such as probability densities. But the idea of using differential equation formulations of stochastic models has a much wider scope. The author gives several examples, including simultaneous estimation of a regression model and residual density, monotone smoothing, specification of a link function, differential equation models of data, and smoothing over complicated multidimensional domains. This paper aims to stimulate interest in this approach to functional estimation problems, rather than provide carefully worked out methods.     

Convergence of Heavy-tailed Monte Carlo Markov Chain Algorithms

Abstract.  In this paper, we use recent results of Jarner & Roberts ( Ann. Appl. Probab., 12, 2002, 224) to show polynomial convergence rates of Monte Carlo Markov Chain algorithms with polynomial target distributions, in particular random-walk Metropolis algorithms, Langevin algorithms and independence samplers. We also use similar methodology to consider polynomial convergence of the Gibbs sampler on a constrained state space. The main result for the random-walk Metropolis algorithm is that heavy-tailed proposal distributions lead to higher rates of convergence and thus to qualitatively better algorithms as measured, for instance, by the existence of central limit theorems for higher moments. Thus, the paper gives for the first time a theoretical justification for the common belief that heavy-tailed proposal distributions improve convergence in the context of random-walk Metropolis algorithms. Similar results are shown to hold for Langevin algorithms and the independence sampler, while results for the mixing of Gibbs samplers on uniform distributions on constrained spaces are rather different in character.     

The Effect of Drop-Out on the Efficiency of Longitudinal Experiments

It is shown that drop-out often reduces the efficiency of longitudinal experiments considerably. In the framework of linear mixed models, a general, computationally simple method is provided, for designing longitudinal studies when drop-out is to be expected, such that there is little risk of large losses of efficiency due to the missing data. All the results are extensively illustrated using data from a randomized experiment with rats.     

Convergence of Slice Sampler Markov Chains

We analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastically monotone, and we deduce analytic bounds on the total variation distance from stationarity of the method by using Foster–Lyapunov drift condition methodology.