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THE DISTRIBUTION OF THE LIKELIHOOD RATIO

FOR MIXTURES OF DENSITIES

FROM THE ONE-PARAMETER EXPONENTIAL FAMILY

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DANKMAR BÖHNING^{1}, EKKEHART
DIETZ^{1}, RAINER SCHAUB^{1},

PETER SCHLATTMANN^{1} AND B
RUCE G. LINDSAY^{2}

^{1} *Department of Epidemiology, Free University Berlin,*

Augustastr. 37, 12 203 Berlin, Germany

^{2} *Department of Statistics, The Pennsylvania State University,*

422 Classroom Building, University Park, PA 16802, U.S.A.
(Received November 16, 1992; revised November 29, 1993)

**Abstract.**
We here consider testing the hypothesis
of *homogeneity* against the alternative of a
two-component mixture of densities. The paper focuses on the
asymptotic null distribution of 2 log *lambda*_{n}, where
*lambda*_{n} is the likelihood ratio statistic. The main result,
obtained by simulation, is that its limiting distribution
appears pivotal (in the sense of constant percentiles over the
unknown parameter), but model specific (differs if the model
is changed from Poisson to normal, say), and is not at all well
approximated by the conventional *chi*^{2}_{(2)}-distribution
obtained by counting parameters. In Section 3, the binomial
with sample size parameter 2 is considered. Via a simple
geometric characterization the case for which the likelihood
ratio is 1 can easily be identified and the corresponding
probability is found. Closed form expressions for the
likelihood ratio *lambda*_{n} are possible and the asymptotic
distribution of 2 log *lambda*_{n} is shown to be the mixture
giving equal weights to the one point distribution with all
its mass equal to zero and the *chi*^{2}-distribution with 1
degree of freedom. A similar result is reached in Section 4
for the Poisson with a small parameter value (*theta* __<__
0.1), although the geometric characterization is different.
In Section 5 we consider the Poisson case in full generality.
There is still a positive asymptotic probability that the
likelihood ratio is 1. The upper precentiles of the null
distribution of 2 log *lambda*_{n} are found by simulation for
various populations and shown to be nearly independent of the
population parameter, and approximately equal to the
( 1 - 2 *alpha*)100 percentiles of *chi*^{2}_{(1)}. In Sections 6
and 7, we close with a study of two continuous densities, the
*exponential* and the *normal with known
variance*. In these models the asymptotic distribution of
2 log *lambda*_{n} is pivotal. Selected (1 - *alpha*)100
percentiles are presented and shown to differ between the two
models.

*Key words and phrases*:
Aitken acceleration,
boundary problem, mixtures, asymptotic distribution of
likelihood ratio.

**Source**
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