Interval Estimation Our Confidence in the Parameter Estimates

Confidence interval estimation is as important to model fitting as finding the most likely parameter estimates. Confidence intervals indicate the range in a parameter value that has a specified chance of including the true parameter value. The notion of chance refers to the interval itself; if we repeated an experiment or set of observations many times, then 100*(1 — a) percent of the time the confidence interval will include the true value of the parameter. Two popular approaches to estimating confidence intervals are the likelihood ratio distribution and parametric bootstrapping.

Likelihood ratio distribution

The likelihood ratio method provides a straightforward way to calculate confidence intervals, but is an asymptotic result that may not hold for all situations. The log ratio of any two values from a likelihood function tends toward a Chi-squared distribution as the number of observations becomes large. To show how this can be used to construct confidence intervals, it is useful to begin with the definition of the log likelihood ratio

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