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Convergence rates of change‐point estimators and tail probabilities of the first‐passage‐time process
Author(s) -
Baron Michael
Publication year - 1999
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315500
Subject(s) - mathematics , estimator , bernoulli's principle , coverage probability , confidence interval , infinity , conditional probability distribution , statistics , point process , conditional probability , rate of convergence , convergence (economics) , mathematical analysis , physics , channel (broadcasting) , economic growth , electrical engineering , economics , thermodynamics , engineering
In the classical setting of the change‐point problem, the maximum‐likelihood estimator and the traditional confidence region for the change‐point parameter are considered. It is shown that the probability of the correct decision, the coverage probability and the expected size of the confidence set converge exponentially fast as the sample size increases to infinity. For this purpose, the tail probabilities of the first passage times are studied. General inequalities are established, and exact asymptotics are obtained for the case of Bernoulli distributions. A closed asymptotic form for the expected size of the confidence set is derived for this case via the conditional distribution of the first passage times.