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The consistency of estimators in finite mixture models

The consistency of estimators in finite mixture models
The consistency of estimators in finite mixture models
The parameters of a finite mixture model cannot be consistently estimated when the data come from an embedded distribution with fewer components than that being fitted, because the distribution is represented by a subset in the parameter space, and not by a single point. Feng & McCulloch (1996) give conditions, not easily verified, under which the maximum likelihood (ML) estimator will converge to an arbitrary point in this subset. We show that the conditions can be considerably weakened. Even though embedded distributions may not be uniquely represented in the parameter space, estimators of quantities of interest, like the mean or variance of the distribution, may nevertheless actually be consistent in the conventional sense. We give an example of some practical interest where the ML estimators are root of n-consistent.

Similarly consistent statistics can usually be found to test for a simpler model vs a full model. We suggest a test statistic suitable for a general class of model and propose a parameter-based bootstrap test, based on this statistic, for when the simpler model is correct.
0303-6898
603-616
Cheng, R.C.H.
a4296b4e-7693-4e5f-b3d5-27b617bb9d67
Liu, W.B.
d3765562-2020-485e-9a1c-61ff5b1f3a90
Cheng, R.C.H.
a4296b4e-7693-4e5f-b3d5-27b617bb9d67
Liu, W.B.
d3765562-2020-485e-9a1c-61ff5b1f3a90

Cheng, R.C.H. and Liu, W.B. (2001) The consistency of estimators in finite mixture models. Scandinavian Journal of Statistics, 28 (4), 603-616. (doi:10.1111/1467-9469.00257).

Record type: Article

Abstract

The parameters of a finite mixture model cannot be consistently estimated when the data come from an embedded distribution with fewer components than that being fitted, because the distribution is represented by a subset in the parameter space, and not by a single point. Feng & McCulloch (1996) give conditions, not easily verified, under which the maximum likelihood (ML) estimator will converge to an arbitrary point in this subset. We show that the conditions can be considerably weakened. Even though embedded distributions may not be uniquely represented in the parameter space, estimators of quantities of interest, like the mean or variance of the distribution, may nevertheless actually be consistent in the conventional sense. We give an example of some practical interest where the ML estimators are root of n-consistent.

Similarly consistent statistics can usually be found to test for a simpler model vs a full model. We suggest a test statistic suitable for a general class of model and propose a parameter-based bootstrap test, based on this statistic, for when the simpler model is correct.

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More information

Published date: December 2001
Organisations: Operational Research

Identifiers

Local EPrints ID: 29719
URI: http://eprints.soton.ac.uk/id/eprint/29719
ISSN: 0303-6898
PURE UUID: 65902004-23ec-49b2-8843-c7e670cb91e2

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Date deposited: 12 May 2006
Last modified: 15 Mar 2024 07:34

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Contributors

Author: R.C.H. Cheng
Author: W.B. Liu

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