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On asymptotic inference about intensity parameters of a counting process. (English) Zbl 0647.62082
Bull. Int. Stat. Inst. 51, No. 4, Top. No. 23.2, 15 p. (1985).
The Cox regression model may be viewed as a special case of the general model described in this paper via the pair \((\bar A_ t,\psi_ t)\) of predictable characteristics of an r-variate counting process \({\mathbb{N}}_ t=(N^ 1_ t,...,N^ r_ t)\), associated with its compensator \({\mathbb{A}}_ t=(A^ 1_ t,...,A^ r_ t)\) as follows: \(\bar A_ t=A^ 1_ t+...+A^ r_ t\) and \(\psi_ t=d{\mathbb{A}}_ t/d\bar A_ t\). It is supposed that the latter characteristic involves the real valued parameter \(\beta\), i.e. \(\psi_ t=\psi_ t(\beta)\), to be estimated by means of a given sample path of \(\{{\mathbb{N}}_ t\), \(0\leq t\leq 1\}.\)
Treating this problem in its asymptotic setting, we consider our experiment as n-th in a sequence of experiments, and let \(\bar A_ t\) meet a condition of asymptotic stability. Under this and certain additional conditions introduced on demand, we study asymptotic properties of the estimator \({\hat \beta}\) for \(\beta\), which is in fact the Cox estimator extended to our situation.
In particular, we characterize the consistency and asymptotic normality of \({\hat \beta}\) by estimating the probability of large deviations, and then show the convergence in all moments of the distribution of \({\hat \beta}\) to a normal law. Finally, it is shown that \({\hat \beta}\) is the best within a class of (regular) estimators in the sense that neither of them can have an asymptotic distribution that is less spread out than that of \({\hat \beta}\).

MSC:
62M05 Markov processes: estimation; hidden Markov models
62E20 Asymptotic distribution theory in statistics
62M09 Non-Markovian processes: estimation
62B15 Theory of statistical experiments
62F12 Asymptotic properties of parametric estimators