60. Here is a result that allows for easy identi cation of a minimal suf cient statistic: Suppose there is a function t(x1, . . . , xn) such that for any two sets of observations x1, . . . , xn and y1, . . . , yn, the likelihood ratio f (x1, . . . , xn; u)/f (y1, . . . , yn; u) does not depend on u if and only if t(x1, . . . , xn)  t(y1, . . . , yn). Then T  t(X1, . . . , Xn) is a minimal suf cient statistic.

The result is also valid if u is replaced by u1, . . . , uk, in which case there will typically be several jointly minimal suf cient statistics. For example, if the underlying pdf is exponential with parameter l, then the likelihood ratio is , which will not depend on l if and only if , so T  , is a minimal suf cient statistic for l (and so is the sample mean).

a. Identify a minimal suf cient statistic when the Xi s are a random sample from a Poisson distribution.

b. Identify a minimal suf cient statistic or jointly minimal suf cient statistics when the Xi s are a random sample from a normal distribution with mean u and variance u.

c. Identify a minimal suf cient statistic or jointly minimal suf cient statistics when the Xi s are a random sample from a normal distribution with mean u and standard deviation u.