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Abstract:
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M-estimation, the technique of extracting a parameter estimate by minimizing a loss function is used in almost every statistical problems. We focus on the general theory of such estimators in the presence of dependence in data, a very common feature in time series or econometric applications. Unlike the case of independent and identically distributed observations, there is a lack of an overarching asymptotic theory for M-estimation under dependence. In order to develop a general theory, we have proved a new triangular version of functional central limit theorem for dependent observations, which is useful for broader applications beyond our current paper. We use this general CLT along with standard empirical process techniques to provide the rate and asymptotic distribution of minimizer of a general empirical process. We have used our theory to make inferences for many important problems like change point problems, excess-mass-baseline-inverse problem, different regression settings including maximum score estimator, least absolute deviation regression and censored regression among others.
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