Abstract
In this article, we revisit the problem of estimating the unknown zero-symmetric distribution in a twocomponent location mixture model, considered in previous works, now under the assumption that the zerosymmetric distribution has a log-concave density. When consistent estimators for the shift locations and mixing probability are used, we show that the nonparametric log-concave Maximum Likelihood estimator (MLE) of both the mixed density and that of the unknown zero-symmetric component are consistent in the Hellinger distance. In case the estimators for the shift locations and mixing probability are √n-consistent, we establish that these MLE's converge to the truth at the rate n-2/5 in the L1 distance. To estimate the shift locations and mixing probability, we use the estimators proposed by (Ann. Statist. 35 (2007) 224-251). The unknown zero-symmetric density is efficiently computed using the R package logcondens.mode.
Original language | English (US) |
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Pages (from-to) | 1053-1071 |
Number of pages | 19 |
Journal | Bernoulli |
Volume | 24 |
Issue number | 2 |
DOIs | |
State | Published - May 2018 |
Bibliographical note
Publisher Copyright:© 2018 ISI/BS.
Keywords
- Bracketing entropy
- Consistency
- Empirical processes
- Global rate
- Hellinger metric
- Log-concave
- Mixture
- Symmetric