Nonparametric Estimation for Associated Sequences

Abstract

In classical statistical inference, the observed random variables of interest are generally assumed to be independent and identically distributed. However, as was mentioned in Chapter 1, in some real life situations, the random variables need not be independent. Study of inference problems for dependent sequences of random variables is of importance due to their applications in fields such as reliability theory, finance and in time series with applications in economics. Statistical inference for stochastic processes was developed for Markov processes by Billingsley (1961) and for stochastic processes in general in Basawa and Prakasa Rao (1980) and Prakasa Rao (1983). Inference for special classes of processes, such as branching processes (cf. Guttorp (1991)), point processes (cf. Karr (1991)), diffusion type processes (cf. Prakasa Rao (1999b), Kutoyants (1984, 2004)), spatial Poisson processes (cf. Kutoyants (1998)), counting processes (cf. Jacobsen (1882)), Semimartingales (cf. Prakasa Rao (1999c)), and fractional diffusion processes (cf. Prakasa Rao (2010)) have been studied extensively. In the examples discussed in Chapter 1, the random variables of interest are not independent but are ‘associated’, a concept we discussed extensively in this book. We gave a review of probabilistic properties of associated sequences of random variables in Chapter 1 and in Chapter 6. One of the important problems of statistical inference is stochastic modelling. In order to understand the evolution of the observed data, it is important to estimate the probabilities of various events in the underlying mechanism which in turn leads to the problem of estimation of the distribution function or the probability density estimation whenever it exists.