## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1053-1071.

### Inference for a two-component mixture of symmetric distributions under log-concavity

Fadoua Balabdaoui and Charles R. Doss

#### Abstract

In this article, we revisit the problem of estimating the unknown zero-symmetric distribution in a two-component location mixture model, considered in previous works, now under the assumption that the zero-symmetric 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 $\sqrt{n}$-consistent, we establish that these MLE’s converge to the truth at the rate $n^{-2/5}$ in the $L_{1}$ 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1053-1071.

**Dates**

Received: November 2014

Revised: May 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980889

**Digital Object Identifier**

doi:10.3150/16-BEJ864

**Mathematical Reviews number (MathSciNet)**

MR3706787

**Zentralblatt MATH identifier**

06778358

**Keywords**

bracketing entropy consistency empirical processes global rate Hellinger metric log-concave mixture symmetric

#### Citation

Balabdaoui, Fadoua; Doss, Charles R. Inference for a two-component mixture of symmetric distributions under log-concavity. Bernoulli 24 (2018), no. 2, 1053--1071. doi:10.3150/16-BEJ864. https://projecteuclid.org/euclid.bj/1505980889

#### Supplemental materials

- Supplement to “Inference for a two-component mixture of symmetric distributions under log-concavity”. In the supplement, we provide the proofs and other technical details that were omitted from the main paper.Digital Object Identifier: doi:10.3150/16-BEJ864SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.