Study on Asymptotic Property of Additive-Accelerated Mean Regression Model

Recurrent event data is a kind of important incomplete data existed in survival analysis, biological medicine research, reliability life test and other practical problems. This paper presents an additive-accelerated mean regression model for multiple type recurrent events data, and gives the estimation methods of unknown parameter and non-parameter function. Specially, the asymptotic properties of parameters estimation are proved.


Introduction
Recurrent events data is often observed in applied research fields like biostatistics, clinical experiment, and so on. Recurrent events data refers to the reoccurrence time sequence of interested events observed for individuals [1][2][3][4]. If only one type of resulted data is concerned, it is referred to as single type recurrent events data. Examples are the recurrent time sequence of acute coronary heart disease and machine faults. If the interested results are of several types, and they may occur several times during an observation period, the data is referred to as multiple type recurrent events data [5][6][7][8]. For example, in clinical research, the effect of the infection of pathogens, candida albicans, aspergillosis, and other disease germs on the survival time of kidney-transplantation patients should be studied. In this paper, an additive-accelerated mean regression model for multiple type recurrent events data is presented, and the estimation methods are given. Specially, the asymptotic properties of the estimations are proved.

The Model
Suppose there are n individuals to be observed during an observation period, each individual experiences k different types of recurrent events, and they are mutually independent. Let ikj T represent the occurrence time of the kth-type, jth-time observed event of the ith individual after the experiment begins, and . Let ) ( * t V ik be the number of recurrence of the kth type event of the ith individual at time t, Definition (1) is given as follows: where I(.) is an indicative function. Moreover, suppose the counting process ) with p1, p2 and p3 dimension respectively, and denote as: The following additive-accelerated mean regression model is suggested to be adopted here. ).
where 0 β , 0 γ and ' 0 α are the unknown regression parameter vector of the p1, p2 and p3 dimension respectively, and ) ( 0 ⋅ µ is the unknown benchmark mean continuous function. Denote then the counting process is as follows: and model (2) can be expressed as the following form: In many practical applications, individuals are always observed within a limited period, thus ) ( * t N ik can not be observed completely.

Asymptotic Property and Its Demonstration
To study the asymptotic property of the estimation value of the given model, this chapter assumes the following conditions hold.
(C1) For given k, , n is independently and identically distributed, and ik ik Z X , are linearly independent.
is locally bounded.
(C5) A is a nonsingular matrix, and ) , , ( The above are some commonly used regularity conditions. As to study the asymptotic property of θˆ, generally the asymptotic property of According to consistent strong law of large numbers, the As  normal distribution the mean value of which is zero, and the covariance function is where h is the window width, ) (⋅ K is the kernel function, thus the consistent estimation of the asymptotic covariance matrix of The proof process of Theorem 2 is omitted here.
The estimation of  The proof process of Theorem 3is omitted here.

Conclusions
In this paper, an additive-accelerated mean regression model for multiple type recurrent events is presented, and the estimation methods of unknown parameter and non-parameter function are given. In addition, the asymptotic properties of parameters estimation are proved under the case of large-scale samples.