The Annals of Probability

Conditional Boundary Crossing Probabilities, with Applications to Change-Point Problems

Barry James, Kang Ling James, and David Siegmund

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Abstract

For normal random walks $S_1, S_2,\ldots$, formed from independent identically distributed random variables $X_1, X_2,\ldots$, we determine the asymptotic behavior under regularity conditions of $P(S_n > mg(n/m) \text{for some} n < m\mid S_m = m\xi_0, U_m = m\lambda_0), \quad\xi_0 < g(1),$ where $U_m = X^2_1 + \cdots + X^2_m$. The result is applied to a normal change-point problem to approximate null distributions of test statistics and to obtain approximate confidence sets for the change-point.

Article information

Source
Ann. Probab., Volume 16, Number 2 (1988), 825-839.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991789

Digital Object Identifier
doi:10.1214/aop/1176991789

Mathematical Reviews number (MathSciNet)
MR929080

Zentralblatt MATH identifier
0645.62031

JSTOR
links.jstor.org

Subjects
Primary: 60F10: Large deviations
Secondary: 60J15 62F03: Hypothesis testing

Keywords
Boundary crossing probabilities change-point normal random walk

Citation

James, Barry; James, Kang Ling; Siegmund, David. Conditional Boundary Crossing Probabilities, with Applications to Change-Point Problems. Ann. Probab. 16 (1988), no. 2, 825--839. doi:10.1214/aop/1176991789. https://projecteuclid.org/euclid.aop/1176991789


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