Continuous-time Estimation of a Change-point in a ... - CiteSeerX

Continuous-time Estimation of a Change-point in a Poisson Process By R. Webster West and R. Todd Ogden Department of Statistics University of South Carolina Columbia, SC 29208, U.S.A. Summary

The problem of estimation of change-points in a sequence of Poisson random variables is approached by allowing the change-point  to range over the continuous time interval (0; T ). A maximum-likelihood point estimator is derived, along with a Bayesian-based interval estimator. Simulation studies con rm that the true coverage probability is close to the nominal one for several locations of the change-point within the sequence and various rate changes. Finally, the procedure is applied to the British coal-mining disaster data and the results are shown to perform extremely well in comparison with previous estimates based on more complete information. Some key words : British coal mining disaster data, change-point, continuoustime estimation, Poisson process.

1 Introduction The task of detecting a change-point in the number of daily defects in an industrial process or in the number of annual cases of a particular genetic disease may be considered in the context of a Poisson process. Henderson and Matthews (1993) consider this problem in standard change-point style by restricting the change-point to fall at the endpoint of a time interval. While this is very natural, this restriction only allows for the estimation of the interval in which the change occurred. In order to establish reasons for the 1

industrial or epidemiological shift a more precise estimate of the change-point is desirable. For example, a re ned estimate of the exact time rather than the day in which the number of defects increases in an industrial process allows one to pinpoint and eliminate problems in a more straightforward manner. Methods for obtaining point estimates and con dence intervals for a changepoint which may take on any of a continuum of values are developed in this paper. Suppose data over T unit time periods, X1 ; : : : ; XT , are observed where the observation Xi represents the number of events that occurred in the ith time period. A natural model for Xi is the Poisson distribution. The question of interest is whether there has been an abrupt change in the rate parameter de ning the Poisson distribution over the T periods. Let ; 0 <  < T; represent such a change-point, 0 represent the Poisson rate parameter before the change, and 1 represent the rate parameter after the change. Denote by [x] the greatest integer function of x. If a change occurs at time point  , then the change occurs in the ([ ] + 1)st interval. The observation for this period can be thought of as a sum of two independent Poisson random variables which are not observed directly: X[ ]+1 = Xi0 + Xi1 , where Xi0  Poisson (p ( ) 0 ), Xi1  Poisson ((1 ? p ( )) 1), and p ( ) =  ? [ ]. The sum of independent Poisson random variables is also Poisson so that Xi

8 > Poisson(0) > > <  > Poisson(p ( ) 0 + (1 ? p ( )) 1) > > : Poisson(1)

i

= 1; : : :; [ ]

i

= [ ] + 1

i

= [ ] + 2; : : :; T

where all of the observations are mutually independent.

2 Point Estimation Using the model described in Section 1, it is possible to derive maximum likelihood estimators for the three parameters, ; 0 ; and 1 . The log likelihood is given by log L (0 ; 1 ;  jX1 ; : : :; Xn) =

(1) 2

0 [ ] 1 0 T X X ? 0 + @ XiA log (0) ? (T ? ( + 1)) 1 + @ =1

=[ ]+2

i

i

1 Xi A log (1 )

?p ( ) 0 ? (1 ? p ( )) 1 + X[ ]+1 log (p ( ) 0 + (1 ? p ( )) 1) ? log

T X

=1

i

Xi !:

If  is known, then (1) can be dierentiated with respect to the two parameters 0 and 1 and the maximum likelihood estimators of 0 and 1 can be shown to satisfy

8

9

[ ] = p ( ) ^0 1