On asymptotic inference about intensity parameters of a counting process
Kacha Dzhaparidze · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1985
The Cox regression model may be viewed as a special case (see ( 1.3)) of the general model described in this paper via the pair CAt,tt> of predictable characteristics of an r-variate counting process :Nt = (N£, .••,NE), associated with its ~ompensator A.t = (Al, .••,AE) as follows: At = Al+ ••• +AE and '11, = dA,ldA, • It is supposed that the latter characteristic involves the real valued parameter a, i.e.•t = •t<al, to be estimated by means of a given sample path of {Nt, 0St~1}.Treating this problem in its asymptotic setting, we consider our experiment (2.1) as n-th in a sequence of experiments, and let At meet Condition I of asymptotic stability.Under this and certain additional conditions introduced on demand, we study asymptotic properties of the estimator S for a defined by (1.4), which is in fact the Cox estimator extended to our situation.In particular, we characterize the consistency and asymptotic normality of B by estimating the probability of large deviations, and then showing the convergence in all moments of the distribution of S to a normal law.Finally, it is shown that B 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 B. LI 23.2-/ RESUME Le modele de regression de cox peut etre considlire comme un cas special (voir (1.3)) du modele general decrit dans cet article via les caracteristi~es previsibles (At,1fltl d'une processus de comptage r-dimensionnel :Nt = (N•L .••,NE) avec sa CO!!!pensatrice At= (Af, ... ,At' .lcomme suit: At= A£+ ••.+A~ et +", = dA,tdA, • Il est suppose que la derniere caracteristique depend d'un parametre reel 6, c'est a dire ll't = •t