Ishwar V. Basawa

Asymptotic inference for stochastic processes

This is a survey of some aspects of large-sample inference for stochastic processes. A unified framework is used to study the asymptotic properties of tests and estimators parameters in discrete-time, continuous-time jump-type, and diffusion processes. Two broad families of processes, viz, ergodic and non-ergodic type are introduced and the qualitative differences in the asymptotic results for the two families are discussed and illustrated with several examples. Some results on estimation and testing via Bayesian, nonparametric, and sequential methods are also surveyed briefly.

Asymptotically minimax tests of composite hypotheses for nonergodic type processes

Asymptotically efficient tests satisfying a minimax type criterion are derived for testing composite hypotheses involving several parameters in nonergodic type stochastic processes. It is shown, in particular, that the analogue of the usual Neymans C ([alpha]) type test (i.e., the score test) is not efficient for the nonergodic case. Moreover, the likelihood-ratio statistic is not fully efficient for the model discussed in the paper. The efficient statistic derived here is a modified version of the score-statistic discussed previously by Basawa and Koul (1979).

Simple Linear Models

This chapter discusses problems of prediction, that is, estimation of future random variables; filtering, that is, estimation of random variables in the presence of noise or super-imposed error; and parameter estimation for some simple discrete-time linear stochastic processes. By Wolds Decomposition Theorem, any regular stationary process without a singular component can be written as a linear process. The chapter describes only simple linear models such as autoregressive, moving average, autoregressive with moving average residuals, and autoregressive with superimposed error, that is, noise.

Optimal Asymptotic Tests

In this chapter we discuss optimal asymptotic tests for simple and composite hypotheses involving a scalar or vector parameter. The basic model is as given in §2 of Chapter 1 and we assume the LAMN condition is satisfied. This general model is used in §§3 and 4. In later sections more restrictive conditions are required. It turns out that the usual statistics such as the Rao’s score statistic, the Neyman statistic, and the likelihood-ratio (LR) statistic exhibit non-standard asymptotic behaviour in the non-ergodic case, as regards efficiency and limit distributions.

A General Model and Its Local Approximation

This chapter is concerned with the formulation of a model which generalises the classical Fisher-Rao-Le Cam model as previewed in Chapter 0, and a discussion of an asymptotic model which approximates the proposed general model.

Efficiency of Estimation

In this chapter we examine the notion of efficiency in estimation of θ. The view is taken here that an efficient estimator should be defined as one which attains the maximal possible concentration about the true value of the parameter. It is easy to show that such an estimator also has minimum mean square error, so the theory incorporates the classical notions of estimation efficiency. Of course it is not in general possible to obtain an estimator with maximum concentration for all values of θ, without in some way restricting the class of competing estimators. For example, in the classical theory, the existence of so-called “super-efficient estimators” (which are consistent and have asymptotic variance less than or equal to the Cramer-Rao lower bound at all θ values and strictly less for some θ) confirms this statement. It can however be shown, that without restriction on the class of estimators, there is an upper bound for the asymptotic concentration, such that the set of parameter values on which any particular estimator has higher concentration is of Lebesgue measure zero. The restriction placed on the class of competing estimators in order to assert the validity of the upper bound for all values of the parameter has generally been that the estimator’s asymptotic behaviour be locally uniform.

Mixture Experiments and Conditional Inference

Consider a simple mixture experiment conducted in two stages as follows: V is a positive random variable with density pV(v); in the first stage of the experiment the value v of V is observed; at the second stage, a random sample X(n) = (X1, X2,..., Xn) is drawn from a normal density with an unknown mean α and variance v. Our aim is to draw inference about α. If we consider (X(n),v) as our sample the joint likelihood is given by

Basic Principles and Methods of Statistical Inference

\left\{ {{p_{{X(n)}}}|{V^{{(x(n)|V = v;\alpha )}}}} \right\}{p_{V}}(v)