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Chapter 9   More About Estimation

9.1   Estimating two parameters

9.1.1   Method of moments

The method of moments can be generalised to estimate two parameters by setting the distribution's mean and variance equal to those of a random sample.

9.1.2   Maximum likelihood

Maximum likelihood can be used to estimate any number of unknown parameters. The estimates are usually where the partial derivatives of the log-likelihood are zero.

9.1.3   Example

Maximum likelihood estimates of the normal distribution's two parameters are derived.

9.1.4   Grid search for MLEs

When the maximum likelihood estimators cannot be found algebraically, a grid search can be used to maximise the log-likelihood.

9.2   Confidence intervals from pivots

9.2.1   Wald-type confidence intervals

If an estimator's distribution is approximately normal, a confidence interval can be found as the estimate ± a multiple of its standard error.

9.2.2   Pivots

A pivot for a parameter is a function of the data and parameter whose distribution is fully known. A confidence interval can be based on quantiles of this distribution.

9.2.3   Confidence interval for exponential rate

A pivot is defined from the sum of values in an exponential random sample. The pivot has a gamma distribution so a confidence interval for the rate parameter, λ, can be based on quantiles of this distribution.

9.2.4   Confidence interval for binomial probability

An approximate pivot is found for a binomial probability. A confidence interval from it has better properties than the conventional Wald-type confidence interval.