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2013-11-28T02:00:17Z

Energy Recovery and Pressure Exchange Systems

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In statistics, the method of '''estimating equations''' is a way of specifying how the parameters of a [[statistical model]] should be [[estimation|estimated]]. This can be thought of as a generalisation of many classical methods --- the [[Method of moments (statistics)|method of moments]], [[least squares]], and [[maximum likelihood]] --- as well as some recent methods like [[M-estimator]]s.

The basis of the method is to have, or to find, a set of simultaneous equations involving both the sample data and the unknown model parameters which are to be solved in order to define the estimates of the parameters.<ref>Dodge, Y. (2003) ''Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9</ref> Various components of the equations are defined in terms of the set of observed data on which the estimates are to be based.

Important examples of estimating equations are the [[Maximum likelihood|likelihood equation]]s.

==Examples==

Consider the problem of estimating the rate parameter, λ of the [[exponential distribution]] which has the [[probability density function]]:

:<math>
f(x;\lambda) = \left\{\begin{matrix}
\lambda e^{-\lambda x}, &\; x \ge 0, \\
0, &\; x < 0.
\end{matrix}\right.</math>
Suppose that a sample of data is available from which either the [[sample mean]], <math>\bar{x}</math>, or the sample [[median]], ''m'', can be calculated. Then an estimating equation based on the mean is

:<math>\bar{x}=\lambda^{-1},</math>

while the estimating equation based on the median is

:<math>m=\lambda^{-1} \ln 2 .</math>

Each of these equations is derived by equating a sample value (sample statistic) to a theoretical (population) value. In each case the sample statistic is a [[consistent estimator]] of the population value, and this provides an intuitive justification for this type of approach to estimation.

== See also ==
*[[Generalized estimating equation]]s
*[[Method of moments (statistics)]]
*[[Generalized method of moments]]
*[[Maximum likelihood]]

==References==
{{Reflist}}

* V. P. Godambe, editor. ''Estimating functions'', volume 7 of Oxford Statistical Science Series. The Clarendon Press Oxford University Press, New York, 1991.

* Christopher C. Heyde. ''Quasi-likelihood and its application: A general approach to optimal parameter estimation''. Springer Series in Statistics. Springer-Verlag, New York, 1997.

* D. L. McLeish and Christopher G. Small. ''The theory and applications of statistical inference functions'', volume 44 of Lecture Notes in Statistics. Springer-Verlag, New York, 1988.

* Parimal Mukhopadhyay. ''An Introduction to Estimating Functions''. Alpha Science International, Ltd, 2004.

* Christopher G. Small and Jinfang Wang. ''Numerical methods for nonlinear estimating equations'', volume 29 of Oxford Statistical Science Series. The Clarendon Press Oxford University Press, New York, 2003.

[[Category:Estimation theory]]

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