## Nonconservative estimating functions and

approximate quasi-likelihoods

### Jinfang Wang

*The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu,*

Minato-ku, Tokyo 106-8569, Japan
(Received August 19, 1997; revised September 3, 1998)

**Abstract.** The estimating function approach unifies
two dominant methodologies in statistical inferences:
Gauss's least square and Fisher's maximum likelihood.
However, a parallel likelihood inference is lacking because
estimating functions are in general not integrable, or
nonconservative. In this paper, nonconservative estimating
functions are studied from vector analysis perspective. We
derive a generalized version of the Helmholtz decomposition
theorem for estimating functions of any dimension. Based on
this theorem we propose locally quadratic potentials as
approximate quasi-likelihoods. Quasi-likelihood ratio tests
are studied. The ideas are illustrated by two examples: (a)
logistic regression with measurement error model and (b)
probability estimation conditional on marginal frequencies.

*Key words and phrases*:
Divergence-free vector
fields, generalized Helmholtz decomposition, gradient vector
fields, logistic regression with measurement error,
potentials, quasi-likelihood ratio test, quasi-scores.

**Source**
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