@MISC{Deaton_anexample, author = {Leonard W. Deaton}, title = {An Example on Maximum Likelihood Estimates}, year = {} }

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Abstract

In most introdcuctory courses in matlhematical sta-tistics, students see examples and work problems in which the maximum likelihood estimate (MLE) of a parameter turns out to be either the sample meani, the sample variance, or the largest, or the smallest sample item. The purpose of this note is to provide ani example in wlhich the AILE is the sample median and a simple proof of this fact. Suppose a random sample of size it is taken from a populatioin with the Laplace distribution f(x; 0) = (2) exp (- x- 6). The mean, mode, and mediaii of this distribution is 0. The sample median is the MLE of 0 ([1] , page 247). Proof: The likelihood funiction is L(0)- ( ) exp j x- I Now, L is maximum when ZKr-i=l is minimum. It will be shown that the inequality, i=l i=l where mi2 is the sample median, holds for every valtue of 0. Consider two cases. Case I: Let n be even. Let yi,-y, y,3,... be the observed values of the order statistics. Let iim be aiiy real number between the twNo middle values so that we have Yi?Y <?2 <.. * /2?nh2 < <(2.)+l <... (< 1) In particular, m11 may be the sample median. By applica-tion of (1) and the triangle inequality we see that- 1'?1 + | l- Y,1+i-i I 'Y iJ- J*Z+I-i I < I Yi- 0 + 0-X*+- (2.) for i = 1, 2,..., n/2 anid 0 any real number. Upon summiing in (2) we get a /2 ZX (Ji- + |1- n--i- |) 21~~~~~~~~~~1' n2/2 < Z y ( + 1-+y,+i-). (3) i=i Now, siinee tl/2 2