Semimartingales and Their Statistical Inference by Rao B. L. S. Prakasa;

Author:Rao, B. L. S. Prakasa;
Language: eng
Format: epub
Publisher: Routledge
Published: 2018-03-05T16:00:00+00:00

5.5 Examples

Example 5.1

Consider the multivariate counting process Xt = (X1t, …, Xpt)T such that each Xit is of the form

with multiplicative intensity Ai(t) = θiJi(t), Ji(t) > 0 almost surely being a predictable process and mit a square integrable martingale (cf. [1, 2 Note that

is a nondecreasing compensator and (mi˙, mj}t = 0 for i≠j[1, Theorem 3.2]. Note that

where

so that

in (5.3.1). Here

and

The quasi-likelihood estimator is given by

which is also the maximum likelihood estimator (cf. [1]).

In the above example and the earlier examples discussed in Section 4.6, one can compute the likelihood function and obtain the maximum likelihood estimators, which are interalia also quasi likelihood estimators. Let us now consider an example where it seems to be difcult to compute the likelihood, and the quasi likelihood approach is possibly applicable.

Example 5.2

Consider the one-dimensional process

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