In statistics, the method of estimating equations is a way of specifying how the parameters of a

statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, ...

should be estimated. This can be thought of as a generalisation of many classical methods—the method of moments,

least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...

, and

maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statis ...

—as well as some recent methods like

M-estimator In statistics, M-estimators are a broad class of extremum estimators for which the objective function is a sample average. Both non-linear least squares and maximum likelihood estimation are special cases of M-estimators. The definition of M-esti ...

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. 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

likelihood equations The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...

Examples

Consider the problem of estimating the rate parameter, λ of the

exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

which has the

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

: :

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

Suppose that a sample of data is available from which either the

sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger pop ...

\bar{x}

, or the sample

median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fea ...

, ''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 a consistent estimator of the population value, and this provides an intuitive justification for this type of approach to estimation.

References

* * * * {{Statistics, inference, collapsed Estimation methods

Examples

See also

References