Abstract
In this paper, the second order expansions for the first two moments of the minimum point of an unbalanced two-sided normal random walk are obtained when the drift parameters approach zero. The basic technique is the uniform strong renewal theorem in the exponential family. The comparison with numerical values shows that the approximations are very accurate. It is shown, particularly, that the first moment is significantly different from its continuous Brownian motion analog while the second moments are the same in the first order. The results can be used to study properties of the maximum likelihood estimator for the change point.
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REFERENCES
Alsmeyer, G. (1988). Second order approximations for certain stopped sums in extended renewal theory, Advances in Applied Probability, 20, 391-410.
Carlsson, H. (1983). Remainder term estimates of the renewal function, Ann. Probab., 11, 143-157.
Chang, J. T. (1992). On moments of the first ladder height of random walks with small drift, Ann. Appl. Probab., 2, 714-738.
Chow, Y. S., Robbins, H. and Teicher, H. (1965). Moments of randomly stopped sums, Ann. Math. Statist., 36, 789-799.
Gut, A. (1988). Randomly Stopped Random Walks, Springer, Berlin.
Hinkley, D. V. (1971). Inference about the change point from cumulative sum tests, Biometrika, 58, 509-523.
Ibragimov, A. and Khasminskii, R. Z. (1981). Statistical Estimation, Springer, Berlin.
Klass, T. (1983). On the maximum of a random walk with small negative drift, Ann. Probab., 11, 491-505.
Lai, T. L. and Siegmund, D. (1979). A non-linear renewal theory with applications to sequential analysis, II, Ann. Statist., 7, 60-76.
Siegmund, D. (1979). Corrected diffusion approximations in certain random walk problems, Advances in Applied Probability, 11, 701-719.
Siegmund, D. (1985). Sequential Analysis, Springer, New York.
Siegmund, D. (1988). Confidence sets in change point problems, International Statistical Review, 56, 31-48.
Stone, C. (1965). On moment generating functions and renewal theory, Ann. Math. Statist., 36, 1298-1301.
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Wu, Y. Second Order Expansions for the Moments of Minimum Point of an Unbalanced Two-sided Normal Random Walk. Annals of the Institute of Statistical Mathematics 51, 187–200 (1999). https://doi.org/10.1023/A:1003843520980
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DOI: https://doi.org/10.1023/A:1003843520980