ESTIMATION OF A CHANGE-POINT (BAYES ESTIMATE, ASYMPTOTICS).

摘要

Let X(,n(,1)), ...,X(,n(,2)) be a sequence of independent random variables such that the p.d.f. of X(,i) (i = n(,1), ...,r) is f(x(,i), (theta)) and the p.d.f. of X(,i) (i = r+1, ...,n(,2)) is g(x(,i), (phi)). The primary goal is to estimate the unknown change-point r.;A recurrence procedure is found to compute the finite sample distribution of r(,b). If n(,1) (--->) -(INFIN) and n(,2) (--->) (INFIN), then under certain conditions, r(,b) - r converges almost surely to (Z(,1) - Z(,2))/(1 + W(,1) + W(,2)) where (W(,1), Z(,1)) and (W(,2), Z(,2)) are independent; and the densities of (W(,1), Z(,1)) and (W(,2), Z(,2)) are unique solutions of integral equations that depend on f and g. The integral equation can be used to compute the asymptotic distribution numerically. When f and g are normal densities, the asymptotic distributions are found to be close to some finite sample empirical distributions.;Since the above asymptotic result depends on the specific distributions of the data, a second type of asymptotics of the Bayes estimate is considered; an additional assumption is that the amount of change goes to zero as the sample sizes before and after the change-point go to infinity. Let X(,1), ..., X(,n) be a sequence of random variables such that X(,i) (i = 1, ...,r(,n)) is f(x(,i), (theta)(,0)) and the p.d.f. of X(,i) (i = r(,n) +1, ...,n) is f(x(,i), (theta)(,0) + (delta)(,n)), where r(,n) = (lamda)n , 0 < (lamda) < 1; (theta)(,0), (lamda) and (delta)(,n) are.;known constants. Let r(,n) be the Bayes estimate of r(,n) with respect to the uniform prior and expected quadratic loss.;When f(x,(theta)) and g (x,(phi)) are known, the finite sample distribu- tion and the asymptotic distribution of the Bayes estimate r(,b) of the change-point r with respect to the uniform prior and expected quadratic loss are derived. This contrasts with David Hinkley's work (1970) on the maximum likelihood estimate of the change-point.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;and f(x,(theta)) satisfies certain regularity conditions, then as n (--->) (INFIN), (delta)(,n)('2)I((theta)(,0))(r(,n) - r(,n)) converges in distribution to.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;where B(,1)(t) and B(,2)(t) are two independent Brownian motions and I((theta)(,0)) is the Fisher information of f(x,(theta)) at (theta)(,0). (Abstract shortened with permission of author.).