Maximum Likelihood Characterization of Distributions
Maximum Likelihood Characterization of Distributions
Henry Teicher
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-. X ) denotes a maximum likelihood estimator ofCT", n X. ^ n X. J h{~) - n log Cr > } h(-^) - n log 0". 1=1 ^ ^ i=l "^ ^ Let y. = x. /^, A =c5/5". Then, ifCT'is a homogeneous function of degree one in x, ,Xp, ... X, the preceding becomes (7) ^Ii: [h(y. ) - h(Vy. )] > log^ \ I for all 'h > and y, ... Y satisfying (8) 0, f, if continuous, is identically zero for y > 0. In the following theorems the indispensable absolute con- tinuity assumption is augmented by a possibly dispensable con- t...inuity assumption of the density. The seemingly ad hoc condition (11) on the other hand appears to be crucial. Theorem 2: Let iF{x/^, ^> 0, P(0) = oj be an absolutely continuous scale parameter family of c.
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