Maximum-Likelihood Estimates
(a) Suppose it is known that a random variable x follows the uniform distribution f(x) = 1/b for 0 = Xmax is the maximumlikelihood estimator of b. (Hint: This result cannot be obtained by differentiation, but from a simple consideration about the likelihood function).
(b) Write down the likelihood equations for the two parameters a and Γ of the Lorentz distribution (see Example 3.5). Show that these do not necessarily have unique solutions. You can, however, easily convince yourself that for |x(j) - a| « Γ the arithmetic mean is an estimator of a.
Example 3.5
Lorentz (Breit-Wigner) distribution With x = (J. = 0 and r = 2 we can write the probability density (3.3.31) of the Cauchy distribution in the form