Estimating equations

In statistics, the method ofestimating equationsis a way of specifying how the parameters of astatistical modelshould beestimated.This can be thought of as a generalisation of many classical methods—themethod of moments,least squares,andmaximum likelihood—as well as some recent methods likeM-estimators.

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.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 thelikelihood equations.

Examples

Consider the problem of estimating the rate parameter, λ of theexponential distributionwhich has theprobability density function:

$f(x;\lambda )=\left\{{\begin{matrix}\lambda e^{-\lambda x},&\;x\geq 0,\\0,&\;x<0.\end{matrix}}\right.$

Suppose that a sample of data is available from which either thesample mean,${\bar {x}}$,or the samplemedian,m,can be calculated. Then an estimating equation based on the mean is

${\bar {x}}=\lambda ^{-1},$

while the estimating equation based on the median is

$m=\lambda ^{-1}\ln 2.$

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 aconsistent estimatorof the population value, and this provides an intuitive justification for this type of approach to estimation.

See also

• Generalized estimating equations
• Method of moments (statistics)
• Generalized method of moments
• Maximum likelihood
• Empirical likelihood

References

• Godambe,V. P., ed. (1991).Estimating Functions.New York: Oxford University Press.ISBN0-19-852228-2.
• Heyde, Christopher C. (1997).Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation.New York: Springer-Verlag.ISBN0-387-98225-6.
• McLeish, D. L.; Small, Christopher G. (1988).The Theory and Applications of Statistical Inference Functions.New York: Springer-Verlag.ISBN0-387-96720-6.
• Small, Christopher G.; Wang, Jinfang (2003).Numerical Methods for Nonlinear Estimating Equations.New York: Oxford University Press.ISBN0-19-850688-0.