An Example on Maximum Likelihood Estimates

LEONARD W. DEATON Naval Postgraduate School Monterey, CalifornioCalifornia

In most introductoryintrodcuctory courses in nlathematicalmatlhematical sta­sta- n/2 n tistics, students see examples and work\vork problems illin :E I f) - lJiI+1-t' I :E If) - 'lit' I, ,,,hichwhich the lnaximummaximum likelihood estimate (IVILE)(MLE) of a ~·=1i=l i=(n/2)+1i= (n1/2) +1 parameter turns out to be either the s[1,mplesample mean,meani, the (3) becomesbeconles sample variance, or the largest, or the smallest sample )/ n item. The purpose of this note is to provide ananiexam examplepIe :E I Yyii --? '777 I ~< :E I Jj -ii -_ f) I' in \\Thichwlhich the IVILEAILE is the sanlplesample nledianmedian and a simple £=1 proof of this fact. But, Suppose a randonlrandom sample of size nit is taken fronlfrom a n n J~; - populatioinpopulation \vithwith the Laplace distribution f(x;f( f))0) = :E 1 'J/iI- - 117 !, = :E I xJ'i - 117III I (!)(2) exp (- I x - f)6 I ). The mean, mode, and nledianmediaii i=l of this distribution is f).0. 1"'1heThe sample median is the lVILEMLE and of f)0 ([lJ,([1] , page 247). n n t1.~ ~ Il~ix-1.(4 Proof: The likelihood functionfuniction is :El,yiI Yi --af) I = :Elxi - f)l \. (4) i=1i=l i=l

L(0) - ( ) exp j x- I The desired results follo\ys.follows. Case II: Let n be odd. Then y(n+l)/2Y(,2+1)/2 is the sanlplesample Now,No\,', I.JL is maxinlummaximum \vhenwhen median. As in (2) above, \\Tewe havelave

n ZKr- I YiI 2- - Y(,+1)/2Y(n+l)/2 I + I Y(n+l)/2Y(-+1)12 - Yn-j-l-i I :E I fi - f) I = i=li=l y-/ Y22+1-2 YII1-2

n n Also, :E I J'i - f) I ~ :E I J'i - 117 I, 0 === I Y(n+l)/2Yn+1)/2 -- Y(n+l)/2Y(22+U/2 I <~ I Y(n+l)/2Y(n+1/2 -- f) I (6) i=l i=l o By summing in (5), \yewe see that where\\There 117mi2is the sample median, holds for every valuevaltue of summing f).0. Consider t\VOtwo cases. (n+l)/2-1 - z (I '/JiY~i- !J(n+1)/2/1(22+1)/2 I + I Y(n+1)/2Y1(n+1)/2 -Y- Y,I+1-i1h2i I) Case I: Let n be even,even. Let Yl,yi, '11'2,-y,y,3, Y3, •••. .. , Un be the :E i=l observed values of the order statistics,statistics. Let '711iimbe anyaiiy real number betweenbet\veen the t\yOtwNomiddle values so that \vewe ((+12)2-1i=l~~~= (n+l/2)-1 have < :E (I Yi - f) I + I f) - Vn+l-i I ), (7) i=l VlYi?Y::;< V2?2<~ •••.. * Un/2/2?nh2~< 177 ~<(2.)+lY(n/2)+1 ~< •••... ~(< Vn' (1)1 ) But, Z /1+i - /122+1-2 0l+l0-,+- 7 (22+1)/2-2i=1~~~= In particular, 171m11may be the salnplesample median. By applica­applica- (n+l)/2-1 tion of (1) and the triangle inequality \vewe see that But,~~~~~~~~~~~~~~~~~~I:E I Y(n+l)/2 - Yn+l-i I 'i=l

- l - Y,1+i-i I Vi - '7111'?1 I + I| III - Yn+1-i I ·n

iJ- - 0-X*+- :E 1 Y(n+l)/2 - Yi I (8) == I Yi'Y - J*Z+I-iYn+1-1~ I <~ I 'ViYi - 0f) I + I f)0 - Yn+l-i I (2)(2.) E I V(,I+)/2-,IJ,+1-iIii=(n+1)/2+1i=(2+1) /2+1 for i = 1, 2, ...... , , n/2 andanid f)0 any real nlunber,number. andanid lIpon summing in (2) \ve get i==(n +1)/2 + Upon summiing in (2) we get (n+l)/2-1( +2-1)/2-1 n i= 1i= n/2a /2 :E I f) - 'Vn+l-i I :E I Yi -'f) I· i=l i=(11+1)/2+1 :EZX((JiI Yi - 111 I + I|1-117 - Vn+1-in--i-| I)) i=l 21~~~~~~~~~~1' I-Ience,IHence, usingusinig (6) andaiid (8) in (7), \vew-e havellave, n/2n2/2 n n < :EZ (Iy( Yi - f) 1++ 18 1-+y,+i-- !/1I+1-i I),). (3) :E I Y·iViY - YCn+l)/2Y(,,+1,,2 I1 ~ :E I Yii - f)0l I. i=1i=i i=l Now,N ~.\V, sincesiinee NNowv,O\Y, after putting Y(n+1)/2Y(,z+1)/2 = 177III andancd using (4), the proof is complete. n/2tl/2 n2

:E 1117 - Yn+l-i I :E 1177 - Yi 1 REFER,ENCEREFERENCE i=l i=(11/2)+1i=(n /2)+1 1. BoggHogg andanid Ctaig,Cfaig, Introduction t·oto MathematicalNlathemat.ical St.atistics,Statistics, and MAlacmillan,Nlacmillan, (~965).(1965).

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