Philip McDunnough

On Some Fourier Methods for Inference

Abstract Common statistical procedures such as maximum likelihood and M-estimation admit generalized representations in the Fourier domain. The Fourier domain provides fertile ground for approaching a number of difficult problems in inference. In particular, the empirical characteristic function and its extension for stationary time series are shown to be fundamental tools which support numerically simple inference procedures having arbitrarily high asymptotic efficiency and certain robustness features as well. A numerical illustration involving the symmetric stable laws is given.

On some sampling schemes for estimating the parameters of a continuous time series

In estimating the mean of certain stationary processes it is shown that it is better to sample at fixed equi-spaced time intervals than to sample randomly according to a renewal process. On the other hand it is shown that the estimation of autocorrelation is sometimes better accomplished by random sampling.

What can or can't be estimated in branching and related processes?

Estimation of the underlying distribution is considered for the incompletely observed random walk and the incompletely observed Galton--Watson branching tree. Based on infrequent observation of a random walk, parameters not completely determined by the first few moments of the underlying distribution cannot be consistently estimated. A similar result is given for the branching tree when observations are sums of family sizes. When the offspring distribution belongs to the power series family MLEs are obtained from an approximate likelihood.

SAMPLING A BRANCHING TREE

A Galton-Watson branching tree is sampled, yielding a derived vector process of family sizes. Exact and asymptotic distributions for this process are derived, rates of convergence given, and the probability of selecting different families is shown to converge rapidly to one. Consistent, asymptotically normal nonparametric estimates of the underlying offspring distribution are obtained and for power series distributions approximate MLEs are shown to be asymptotically normal and efficient.

On generalization of the analysis of variance

The standard analysis of variance procedures were developed and organized primarily in the context of the normal linear model; central to this organization is the orthogonality of components and the use of orthogonal projections. This paper examines two model-type generalizations of the normal linear model: the regression model with nonnormal error and the exponential linear model. Principles of conditioning and measurement are used to develop corresponding analysis-of-variance procedures. In each case a linear fibre or foliation structure replaces orthogonality; however, for the intersection of the two model-types, which is the normal linear model, the two quite-different fibre-foliation structures reduce to a product space structure, which with the appropriate inner product, is the usual orthogonality. For implementation, conditional-marginal densities are involved, the marginalization aspect being the restricting aspect: the marginalization degree is the number of nuisance parameters for the regression model-type and is the complement of the number of free parameters for the exponential model-type. Approximations are available and will be discussed subsequently.

Efficient Estimation for the Stable Laws

This paper is concerned with Fourier procedures in inference which admit arbitrarily high asymptotic efficiency. The problem of estimation for the stable laws is treated by two different approaches. The first involves FFT inversion of the characteristic function. A detailed discussion is given of truncation and discretization effects with reference to the special structure of the stable densities. Some further results are give also concerning a second approach based on the empirical characteristic function (ecf). Finally we sketch an application of this method to testing for independence, and also present a stationary version of the ecf.

Estimating the precision of a bivariate population

Quick efficient estimates are proposed for estimating the standard deviation of a circular bivariate population. Two procedures based on extreme observations are considered. The first of these employs the 100 p percent largest observations, while the second utilizes the extreme observations in k radial sectors.

Estimating an interaction parameter of an infinite particle system

This article deals with the estimation of a parameter in the stochastic motion affecting an infinite number of particles. An estimator, based on a nonstationary time-series is considered and shown to be consistent. A comparison with more well-known estimates, via asymptotic variances, is also carried out.