Interval estimation using the likelihood function

Abstract

The general properties of two commonly-used methods of interval estimation for population parameters in physics are examined. Both of these methods employ the likelihood function: (i) Obtaining an interval by finding the points where the likelihood decreases from its maximum by some specified ratio; (ii) Obtaining an interval by finding points corresponding to some specified fraction of the total integral of the likelihood function. In particular, the conditions for which these methods give a confidence interval are illuminated, following an elaboration on the definition of a confidence interval. The first method, in its general form, gives a confidence interval when the parameter is a function of a location parameter. The second method gives a confidence interval when the parameter is a location parameter. A potential pitfall of performing a likelihood analysis without understanding the underlying probability distribution is discussed using an example with a normal likelihood function.