Estimating Equations - Examples

Examples

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

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, or the sample median, m, can be calculated. Then an estimating equation based on the mean is

while the estimating equation based on the median is

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.

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