Quasi-Likelihood And Its Application by Christopher C. Heyde

Author:Christopher C. Heyde [Heyde, Christopher C.]
Language: eng
Format: epub
Published: 2011-02-19T05:00:00+00:00

116

CHAPTER 7. PROJECTED QUASI-LIKELIHOOD

Then, the estimating function

Gφ − V φψ V −1

ψψ Gψ

= X1 Σ−1 − (X1 Σ−1 X2)(X2 Σ−1 X2)−1X2 Σ−1 (Z − X1 φ) does not involve the nuisance parameter ψ and the estimator for φ is obtained from equating this to zero.

This solution can, of course, be obtained directly from solving the ordinary estimating equation

Gφ

0

=

Gψ 0

and then eliminating ψ from the resultant equations. Indeed, many nuisance parameter problems are amenable to such a direct approach.

The focus in this section has been on projection but there is no need for this geometric interpretation to be emphasized. The first order theory seeks the best combination Gφ − c(φ, ψ)Gψ, which is easy to calculate analytically.

Also, it is not difficult to go beyond the first order theory. Second order theory allows, incorporation, for example, of the result of Godambe and Thompson (1974) who dealt with the case in which the likelihood score is Uθ = (Uφ, Uψ) , φ and ψ being scalars, and showed that an estimating function of the form Uφ − c(φ, ψ)(U 2 − EU 2),

ψ

ψ

if c(φ, ψ) can be chosen to make it free of ψ, is the best choice for estimating φ. An example where this holds is in estimation of φ using a random sample of observations from the N (ψ, φ) distribution.

7.4

Generalizing the E-M Algorithm:

The P-S Method

The E-M algorithm (e.g., Dempster et al. (1977)) is a widely used method for dealing with maximum likelihood calculations where there are missing or otherwise incomplete data. It involves the taking of the expectation of the complete-data likelihood with respect to the available data (the E-step) and then maximizing this over possible distributions (the M-step) and the procedure suggests a simple iterative computing algorithm. It has not been available in contexts where a likelihood is unknown or unavailable.

In this section we extend the E-M algorithm method to deal with estimation via estimating functions, in particular the quasi-score. The transitions to estimating functions is made since there are situations where no quasi-log-likelihood exists. The discussion here follows Heyde and Morton (1995).

In our approach the E-step is replaced by a step which projects the quasi-score rather than taking expectations of a log-likelihood. In many examples the 7.4. GENERALIZING THE E-M ALGORITHM: THE P-S METHOD

117

projection is equivalent to predicting the missing data or terms. The predictor will not in general be a conditional expectation. The M-step is replaced by solving the projected quasi-score set equal to zero. The approach can reasonably be described as the projection-solution (P-S) method.

When the likelihood is available and the score function is included in the class of estimating functions permitted, the standard E-M procedure is recovered from the P-S method.

More broadly, however, we seek to make the point that there is no formal difference between QL estimation for incomplete data and QL estimation for complete data.

7.4.1

From Log-Likelihood to Score Function

As a prelude to generalization of the E-M algorithm formalism from log-likelihoods to estimating functions, we first show how attention can be transferred from operations on the log-likelihood to corresponding ones on its derivative, the likelihood score.

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