Christopher S. Withers

Sound Field Reproduction With Energy Constraint on Loudspeaker Weights

Audio rendering problems are not always well-posed. An approach is devised for solving ill-posed sound field reproduction problems using regularization, where the Tikhonov parameter is chosen by upper bounding the summed square of the loudspeaker weights. The method ensures that the sound in the room remains at reasonable levels.

A Generalized Suzuki Distribution

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.

Fredholm equations for non-symmetric kernels with applications to iterated integral operators

Abstract We give the Jordan form and the singular value decomposition for an integral operator N with a non-symmetric kernel N ( y , z ) . This is used to give solutions of Fredholm equations for non-symmetric kernels, and to determine the behaviour of N n and ( NN ∗ ) n for large n.

MGFs for Rayleigh Random Variables

Expressions are given for the moment generating functions of the Rayleigh and generalized Rayleigh distributions.

Negentropy as a function of cumulants

Suppose f is a density that is close to that of a standard normal, ϕ. For the first time, exact expressions are given for the negentropy, the differential entropy, and related quantities such as ∫-∞∞fr(x)/ϕr-1(x)dx for r=2,3,… in terms of cumulants. Several advantages of our expressions over known ones are pointed out.

The bias and skewness of M-estimators in regression

Abstract: We consider M estimation of a regression model with a nuisance parameter and a vector of other parameters. The unknown distribution of the residuals is not assumed to be normal or symmetric. Simple and easily estimated formulas are given for the dominant terms of the bias and skewness of the parameter estimates. For the linear model these are proportional to the skewness of the ‘independent’ variables. For a nonlinear model, its linear component plays the role of these independent variables, and a second term must be added proportional to the covariance of its linear and quadratic components. For the least squares estimate with normal errors this term was derived by Box [1]. We also consider the effect of a large number of parameters, and the case of random independent variables.

Moments from cumulants and vice versa

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 Kendalls use of Faa di Brunos chain rule to express cumulants in terms of moments.

A Recurrence Relation for Moments of the Noncentral Chi Square

We give a recurrence relation for the moments of the noncentral chi-square distribution.

Expressions for the distribution and percentiles of the sums and products of chi-squares

Linear combinations of central or non-central chi-squares occur naturally in a variety of contexts. The products of chi-squares occur when a variance has a chi-square prior and in electrical engineering. Here, we give expansions for their distribution and quantiles and also for the products of the powers of chi-squares, including ratios. These provide much more accurate approximations than those based on asymptotic normality. The larger the degrees of freedom or the larger the non-centrality parameters, the better the approximations. We give the first four terms of these expansions. These provide approximations with errors smaller by five magnitudes than those based on asymptotic normality or on Satterthwaites approximation. His method matched the first two moments of the target and a multiple of a chi-square and is only a first-order approximation like that based on the central limit theorem. We show that it can be made second order by matching the first three moments. The appendices show how to obtain analytical expressions for the distribution of weighted sums of chi-squares.

Delta and jackknife estimators with low bias for functions of binomial and multinomial parameters

An estimator is said to be of orders>0 if its bias has magnitude n−s, where n is the sample size. We give delta estimators and jackknife estimators of order four for smooth functions of the parameters of a multinomial distribution. An unbiased estimator is given for its density function. We also give a jackknife estimator of any order for smooth functions of the binomial parameter.