C. C. Heyde

Quasi-likelihood and Optimal Estimation

Within the framework of estimating function theory, this paper provides a very general definition of quasi-likelihood estimating equations. Applications to stochastic processes are discussed. This work extends the previous results of Godambe (1985) and Hutton & Nelson (1986).

ESTIMATION THEORY FOR GROWTH AND IMMIGRATION RATES IN A MULTIPLICATIVE PROCESS

This paper deals with the simple Galton-Watson process with immigration, {X n } with offspring probability generating function (p.g.f.) F(s) and immigration p.g.f. B(s), under the basic assumption that the process is subcritical (0 < m ≡ F’(l−) < 1), and that 0 < » ≡ B’(l−) < ∞, 0 < B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the rates m and λ on the basis of a single terminated realization of the process {X n } is developed, in that (strongly) consistent estimators for both m and λ are obtained, together with associated central limit theorems in relation to m and μ ≡ λ(1 − m)−1 Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales ; and discusses relation of the above theory to that of a first order autoregressive process.

Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence

In this paper we establish central limit theorems for the smoothed unbiased periodogram [integral operator][pi]-[pi]...[integral operator][pi]-[pi]g([omega],[theta]){I*T,X([omega])-EI*T,X([omega])}d[omega]1...d[omega]r, where {Xt} is a stationary r-dimensional random process or random field, possibly with long-range dependence, which is not necessarily Gaussian. Here I*T,X([omega]) is the unbiased periodogram and g([omega],[theta]) is a smoothing function satisfying modest regularity conditions. This result implies asymptotic normality of the asymptotic quasi-likelihood estimator of a distributional characteristic [theta] of the process {Xt} under very general conditions. In particular, these results show the asymptotic optimality of the Whittle estimation procedure for both short and long-range dependence in the absence of the Gaussian assumption, and extend those of Giraitis and Surgailis (1990) for the case r = 1.

On Defining Long-Range Dependence

Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing the behaviour for various infinite variance models is shown.

A contribution to the theory of large deviations for sums of independent random variables

Let the random variables \(X_i, i = 1, 2, 3, \ldots\) be independent and identically distributed. Write \(S_n = \sum^{n}_{i=1} X_i \, {\rm and}\, Z_n = S_n B^{-1}_n - A_n\) for normed and centered sums. The classical central limit formulation is for \({\rm Pr} (Z_n \leqq x_n)\) where x n = 0(1) as \(n \rightarrow \infty\) and thus gives only trivial information in the case where \(x_n \rightarrow \infty\) as \(n \rightarrow \infty\). However, we often require information on \({\rm Pr} (Z_n > x_n)\) under these circumstances. This type of problem is called a problem on the probability of large deviations and little in the way of comprehensive general results on such problems has so far been obtained. For references to the known results and also to various fields in which large deviation problems arise see, for example, Linnik [3] and Sethuraman [7].

On combining quasi-likelihood estimating functions

The comparison of competing estimating functions for a vector parameter of a stochastic process is discussed and the formation of combined quasi-likelihood estimating functions where this is advantageous. An example is given to illustrate the methodology.

On the Influence of Moments on the Rate of Convergence to the Normal Distribution

Let \(X_i, i = 1, 2, 3, \ldots\) be a sequence of independent and identically distribution random variables with \(X_i=\sigma^2<\infty\) and \(EX_i=0.\) Write

An Invariance Principle and Some Convergence Rate Results for Branching Processes

S_n=\sum^n_{i=1}Xi,\quad n\geqq1.