Nonparametric estimation in α-series processes

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Abstract

A counting process {N(t),t0} with the interoccurrence times X1,X2, is an α-series process if there exists a real number α such that (kαXk)k=1,2, forms a renewal process. The nonparametric inference problem in an α-series process is taken into consideration. The Mann test is applied for trend analysis and a graphical technique is presented in order to test whether the data come from an α-series process. Some nonparametric estimators for three important parameters of the α-series process are obtained by using a linear regression method. The consistency and asymptotic normality properties are investigated. The performances of the estimators are evaluated by a simulation study. Some suggestions on the choice of the estimators are made based on the theoretical and simulation results. Further, the method is illustrated through a real-life example.

Highlights

► We propose some nonparametric estimators for the parameters of an α-series process. ► The consistency and asymptotic normality properties are investigated. ► Some suggestions on the choice of the estimators are made. ► The coal-mining disasters data set is used for application purposes.

Introduction

In the statistical analysis of a data set with occurrence times of successive events, a common method is to apply a counting process. If the data consist of independent and identically distributed (i.i.d.) successive interarrival times (i.e., there is no trend), a renewal process can be used. However, this is not always the case, because it is more reasonable to assume that the successive operating times will follow a monotone trend due to the ageing effect and the accumulated wear (see Chan et al. (2004)). Using a nonhomogeneous Poisson process with a monotone intensity function is an approach for modeling these trends (see Cox and Lewis (1966) and Ascher and Feingold (1984)). A more direct approach is to apply a monotone counting process model (see Lam, 1988a, Lam, 1988b, Lam and Chan (1998), and Chan et al. (2004)). Braun et al. (2005), among many others, defined such a monotone process in the following way.

Definition 1

Let Xk be the time between the (k1)th and kth events of a counting process {N(t),t0} for k=1,2,. The counting process {N(t),t0} is said to be an α-series process with the parameter α if there exists a real number α such that kαXk, k=1,2,, are i.i.d. random variables with the distribution function F.

Clearly, the Xk form a stochastically increasing sequence when α<0 and a stochastically decreasing sequence when α>0. When α=0, all the random variables Xk are identically distributed, and the α-series process reduces to a renewal process. If F is an exponential distribution function and α=1, then the α-series process {N(t),t0} turns out to be a linear birth process (Braun et al., 2008).

The α-series process was first introduced as a possible alternative to the geometric process (Lam, 1988a) in cases where the geometric process is inappropriate, and it was applied to some reliability and scheduling problems by Braun et al. (2005). Some theoretical properties of the α-series process were investigated by Braun et al., 2005, Braun et al., 2008. An α-series process has some advantages over a geometric process. For example, the mean value function of a decreasing geometric process does not exist, but the mean value function exists for a decreasing α-series process under fairly general conditions (Braun et al., 2005). Further, a central limit theorem does not hold for a geometric process. The central limit theorem, however, can be applied for an α-series process with α<0.5 (Braun et al., 2008).

For the α-series process, assume that the distribution function F has positive mean μ (i.e., F(0)<1) and finite variance σ2. Then E(Xk)=μkαandV ar(Xk)=σ2k2α,k=1,2,. Thus α, μ, and σ2 are the most important parameters for the α-series process because these parameters completely determine the mean and variance of Xk. However, the parameter estimation problem in the α-series processes has not been investigated in detail, to the best of our knowledge.

For estimation purposes, let us assume that a set of data {X1,X2,,Xn} comes from a counting process. If we want to model this data set by the α-series process, three questions will arise. First, how do we test whether the data of successive interarrival times have a trend? Second, if the data show a trend, how can we justify if the data agree with an α-series process? Finally, if the data come from an α-series process, how can we estimate three important parameters: α, μ, and σ2?

In this paper, we try to answer these questions by using some nonparametric approaches. In Section 2, the Mann test is considered for trend analysis, and a graphical technique is presented in order to test whether the data come from an α-series process. In Section 3, some nonparametric estimators for the parameters α, μ, and σ2 are obtained by using the linear regression method. In Section 4, a simulation study is given to evaluate the performances of the estimators according to their biases and mean square errors. Finally, in Section 5, a real data set is used for application purposes.

Section snippets

The procedure for testing the α-series process

Suppose that a set of data of successive interarrival times {X1,X2,,Xn} is given. In practice, the procedure for testing an α-series process can be performed in two stages, which are presented in the following subsections. First, we carry out a trend analysis of the data. Second, following the observation of a trend in the data, it is checked that whether the data come from an α-series process or not.

Parameter estimation

We use a similar procedure as in Lam (1992). Assume that {X1,X2,,Xn} is a set of data from an α-series process. We obtain some nonparametric estimators for the parameters α, μ, and σ2 of the α-series process by implementing a least squares method for the regression model in (2.5).

The plot of lnXk against lnk gives a rough estimate of α. In addition, a better estimate can be found analytically. The least squares estimators of the parameters α, β, and σe2 are easily obtained from (2.5) as αˆ=k=1

Simulation results

A simulation study is now given to evaluate the performance of the estimator αˆ and to compare the relative performance of the estimators μˆi, i=1,2,,7 and σˆi2, i=1,2,3. The bias and mean square error (MSE) are used to compare the performances. Here, α=2(0.1)2 and n=10(10)100 are selected. We generate a realization {X1,X2,,Xn} of the α-series process as the first occurrence time X1 has the following three distributions:

  • (i)

    the exponential distribution Exp(2) with μ=0.5, σ2=0.25, having the

An illustrative example

In this section, we give real data example to illustrate the data analysis and estimation procedures developed in Sections 2 The procedure for testing the, 3 Parameter estimation, 4 Simulation results. This example uses coal-mining disaster data of the intervals in days between successive coal-mining disasters in Great Britain. The data set has 190 observations (see Andrews and Herzberg (1985)). The data contain a value zero because there were two accidents on the same day. This zero is

Acknowledgments

We are grateful to the editor and referees for their valuable comments and helpful suggestions which enabled us to improve the paper. We also thank Dr. İhsan Karabulut for his valuable contributions.

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