The empty set and zero likelihood problems in maximum empirical likelihood estimation

W.P. Bergsma, M.A. Croon, L.A. van der Ark

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We describe a previously unnoted problem which, if it occurs, causes the empirical likelihood method to break down. It is related to the empty set problem, recently described in detail by Grendár and Judge (2009), which is the problem that the empirical likelihood model is empty, so that maximum empirical likelihood estimates do not exist. An example is the model that the mean is zero, while all observations are positive. A related problem, which appears to have gone unnoted so far, is what we call the zero likelihood problem. This occurs when the empirical likelihood model is nonempty but all its elements have zero empirical likelihood. Hence, also in this case inference regarding the model under investigation breaks down. An example is the model that the covariance is zero, and the sample consists of monotonically associated observations. In this paper, we define the problem generally and give examples. Although the problem can occur in many situations, we found it to be especially prevalent in marginal modeling of categorical data, when the problem often occurs with probability close to one for large, sparse contingency tables.
Original languageEnglish
Pages (from-to)2356-2361
JournalElectronic Journal of Statistics
Volume6
DOIs
Publication statusPublished - 2012

Fingerprint

Null set or empty set
Empirical Likelihood
Maximum Likelihood
Likelihood
Zero
Breakdown
Additive identity
Model
Empirical likelihood
Likelihood estimation
Nominal or categorical data
Likelihood Methods
Contingency Table

Cite this

Bergsma, W.P. ; Croon, M.A. ; van der Ark, L.A. / The empty set and zero likelihood problems in maximum empirical likelihood estimation. In: Electronic Journal of Statistics. 2012 ; Vol. 6. pp. 2356-2361.
@article{a002de78709b42e0b20479a3c525d136,
title = "The empty set and zero likelihood problems in maximum empirical likelihood estimation",
abstract = "We describe a previously unnoted problem which, if it occurs, causes the empirical likelihood method to break down. It is related to the empty set problem, recently described in detail by Grend{\'a}r and Judge (2009), which is the problem that the empirical likelihood model is empty, so that maximum empirical likelihood estimates do not exist. An example is the model that the mean is zero, while all observations are positive. A related problem, which appears to have gone unnoted so far, is what we call the zero likelihood problem. This occurs when the empirical likelihood model is nonempty but all its elements have zero empirical likelihood. Hence, also in this case inference regarding the model under investigation breaks down. An example is the model that the covariance is zero, and the sample consists of monotonically associated observations. In this paper, we define the problem generally and give examples. Although the problem can occur in many situations, we found it to be especially prevalent in marginal modeling of categorical data, when the problem often occurs with probability close to one for large, sparse contingency tables.",
author = "W.P. Bergsma and M.A. Croon and {van der Ark}, L.A.",
note = "open access",
year = "2012",
doi = "10.1214/12-EJS750",
language = "English",
volume = "6",
pages = "2356--2361",
journal = "Electronic Journal of Statistics",
issn = "1935-7524",
publisher = "Institute of Mathematical Statistics",

}

The empty set and zero likelihood problems in maximum empirical likelihood estimation. / Bergsma, W.P.; Croon, M.A.; van der Ark, L.A.

In: Electronic Journal of Statistics, Vol. 6, 2012, p. 2356-2361.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - The empty set and zero likelihood problems in maximum empirical likelihood estimation

AU - Bergsma, W.P.

AU - Croon, M.A.

AU - van der Ark, L.A.

N1 - open access

PY - 2012

Y1 - 2012

N2 - We describe a previously unnoted problem which, if it occurs, causes the empirical likelihood method to break down. It is related to the empty set problem, recently described in detail by Grendár and Judge (2009), which is the problem that the empirical likelihood model is empty, so that maximum empirical likelihood estimates do not exist. An example is the model that the mean is zero, while all observations are positive. A related problem, which appears to have gone unnoted so far, is what we call the zero likelihood problem. This occurs when the empirical likelihood model is nonempty but all its elements have zero empirical likelihood. Hence, also in this case inference regarding the model under investigation breaks down. An example is the model that the covariance is zero, and the sample consists of monotonically associated observations. In this paper, we define the problem generally and give examples. Although the problem can occur in many situations, we found it to be especially prevalent in marginal modeling of categorical data, when the problem often occurs with probability close to one for large, sparse contingency tables.

AB - We describe a previously unnoted problem which, if it occurs, causes the empirical likelihood method to break down. It is related to the empty set problem, recently described in detail by Grendár and Judge (2009), which is the problem that the empirical likelihood model is empty, so that maximum empirical likelihood estimates do not exist. An example is the model that the mean is zero, while all observations are positive. A related problem, which appears to have gone unnoted so far, is what we call the zero likelihood problem. This occurs when the empirical likelihood model is nonempty but all its elements have zero empirical likelihood. Hence, also in this case inference regarding the model under investigation breaks down. An example is the model that the covariance is zero, and the sample consists of monotonically associated observations. In this paper, we define the problem generally and give examples. Although the problem can occur in many situations, we found it to be especially prevalent in marginal modeling of categorical data, when the problem often occurs with probability close to one for large, sparse contingency tables.

U2 - 10.1214/12-EJS750

DO - 10.1214/12-EJS750

M3 - Article

VL - 6

SP - 2356

EP - 2361

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

SN - 1935-7524

ER -