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 \({\hat \beta}\).

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 |