Additive-Accelerated Mean Regression Model for Multiple Type Recurrent Events

Recurrent events data is often observed in applied research fields like biostatistics, clinical experiment, and so on. In this paper, an additive-accelerated mean regression model is established for multiple type recurrent events data, and the estimation methods of unknown parameter and non-parameter function based on the idea of estimating equation are given.


Introduction
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. Recurrent event data is a large class of important incomplete data existed in survival analysis, biological medicine research, reliability life test and other practical problems, and the statistical analysis for them has been valued all over the world, especially by developed countries [5][6].
Complex statistical analysis of recurrent event data is focal point of research of modern statistics and the important part of the development of various disciplines. Analyzing complex data, establishing the corresponding statistical model, and revealing the internal laws of complex data are the important foundation of their relevant disciplines. Especially in the research on biology, medicine, ecology, demography, environmentology and economics and other disciplines, with the development of experimental techniques, testing methods and means of data analysis, the data obtained are more and more complex and precise in structure, and the information provided is more and more miscellaneous, which put forward higher requirements for the quantitative analysis of data [7][8][9]. How to make statistical modeling and statistical inference has become the frontier topic of biology, medicine, ecology, demography, environmentology and economics and other interdiscipline. In this research field, there still exist some problems to be solved by developing effective statistical methods [10][11][12][13].
This paper discusses the additive-accelerated mean regression model for multiple type recurrent events data, puts forward an estimation method of unknown parameter and non-parameter function based on the idea of estimating equation.

Model construction and estimation method
To describe the observation data of multiple type recurrent events, 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 I(.) is an indicative function. At the same time, we assume 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. ).
In many practical applications, individuals are always observed within a limited period, thus ) ( * t N ik can not be observed completely.
Denote ik C be the censored time of the kth type event of the ith individual, and it is independent from the condition ) ( * t N ik , then according to Anderson and Gill (1982), the number of observed events ) ( * t N ik within the observation period for the kth type event of the ith individual can be defined as follows: . Based on the above hypothesis model, the time scale model of transformation is:

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Information Technology for Manufacturing Systems III and also define the following process: is a stochastic process with zero mean value, therefore, for given ) , , ( ,τ is a given constant that makes 0 (