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Least‐squares Estimation of an Unknown Number of Shifts in a Time Series
Author(s) -
Lavielle Marc,
Moulines Eric
Publication year - 2000
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00172
Subject(s) - mathematics , consistency (knowledge bases) , mixing (physics) , strong consistency , series (stratigraphy) , least squares function approximation , range (aeronautics) , convergence (economics) , statistics , distribution (mathematics) , asymptotic distribution , rate of convergence , invariance principle , mathematical analysis , estimator , discrete mathematics , paleontology , physics , materials science , quantum mechanics , economics , composite material , biology , economic growth , channel (broadcasting) , linguistics , engineering , philosophy , electrical engineering
In this contribution, general results on the off‐line least‐squares estimate of changes in the mean of a random process are presented. First, a generalisation of the Hajek‐Renyi inequality, dealing with the fluctuations of the normalized partial sums, is given. This preliminary result is then used to derive the consistency and the rate of convergence of the change‐points estimate, in the situation where the number of changes is known. Strong consistency is obtained under some mixing conditions. The limiting distribution is also computed under an invariance principle. The case where the number of changes is unknown is then addressed. All these results apply to a large class of dependent processes, including strongly mixing and also long‐range dependent processes.