## Epi-Convergence in Distribution and Stochastic Equi-Semicontinuity (1997)

Venue: | C o rpus-based wo rk on discourse marke rs such as ‘ a n d ’ ,‘ i f’ , ‘ bu t ’ ,e |

Citations: | 12 - 2 self |

### BibTeX

@INPROCEEDINGS{Knight97epi-convergencein,

author = {Keith Knight},

title = {Epi-Convergence in Distribution and Stochastic Equi-Semicontinuity},

booktitle = {C o rpus-based wo rk on discourse marke rs such as ‘ a n d ’ ,‘ i f’ , ‘ bu t ’ ,e},

year = {1997},

pages = {33--50}

}

### OpenURL

### Abstract

: Epi-convergence in distribution is a useful tool in establishing limiting distributions of "argmin" estimators; however, it is not always easy to find the epi-limit of a given sequence of objective functions. In this paper, we define the notion of stochastic equi-lower-semicontinuity of a sequence of random objective functions. It is shown that epi-convergence in distribution and finite dimensional convergence in distribution (to a given limit) of a sequence of random objective functions are equivalent under this condition. Key words and phrases: argmin estimators, convergence in distribution, epi-convergence, equi-semicontinuity AMS 1991 subject classifications: Primary 62F12, 60F05; Secondary 62E20, 60F17. Running head: Stochastic equi-semicontinuity 1 Introduction Many statistical estimators are defined as the minimizer (or maximizer) of some objective function; common examples include maximum likelihood estimation and M-estimation. Since any maximization problem can be re-exp...