On the characterization of maximum likelihood

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On the characterization of maximum likelihood estimators for location-scale families Werner Hürlimann

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To cite this article: Werner Hürlimann (1998): On the characterization of maximum likelihood estimators for location-scale families, Communications in Statistics - Theory and Methods, 27:2, 495-508 To link to this article: http://dx.doi.org/10.1080/03610929808832108

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COMMUN STATIST -TI-!ECl?Y

METH , 27(2). 495-508 (1998)

T)Y T H E CH4_R4CTEmAT10N

OF MAXIMITM LIKELIHOOD ESTIMATORS FOR LOCATION-SCALE FAMILIES

Werner Hiirlimann

Downloaded by [Werner Huerlimann] at 12:37 10 April 2013

Schonholzweg 24, CH-8409 Wmtenhur, Swrtzeridnd

G v e n m a x i i ~ d m!:ke!:hood ecjuations for location and scale paramc-iers, vile ?,eter:r,ines co:?ditions under which there exists a uniqueiy defined parametric statistical model, whose location and scale maximum likelihood estin~atorsare the given ones. The constructive approach is exemplified at several kinds of mean estimators including the mean, mean square, mean mean and stretched power mean. The possible extension of the method to more generai situations is discussed and illustrated at the sample median maximum likelihood estimator.

INTRODUCTION Statisticians often believe that characterizing results are not of great interest. In the present note we demonstrate the usefulness of maximum likelihood characterizations, and thus contribute to a wider acceptance of "algebraic" ideas in Statistics. We consider real random variables X and Y with absolutely continuous distributions ( x ) and G ( y ) such that the probability density functions f ( x ) = F ' ( x ) and g ( x ) = G t ( x ) are defined. Let 1,- = [a,r,h, Zj = [n,. ,h,] be the smallest closed intervals containing the supports of I.' and G such that a,. = ivf{x : ~ ( x >) 0}, b, = sup{x : F ( x ) < I}, a, = rnf {x . G ( x ) i h, = sup{x : G ( x ) < 1). Let cp(x) be a monotone increasing function with inverse i j i ( x ) = v ' ( x j . In a first appl-oach and for sirnplici~j;,we consider only "smooth" invertible functions such that q ( x ) and i j i ( , ~ ) are twice-differentiable

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Working in an aigebl-aic statistical setting. let us start ~vith an abstract definition Let a , /3 be real numbers, where P z0 is assumed The random -,~';iriable.Y is said io be thc k ) i . ; i t r o i l - ~ ~ ~iiaj?.$jjiji.n~ ~/~~ of I.. if .Y = G P\!: (?-) or cquivaiently I' = @{(.Y -- uj 1 /3j The number a is cailed the location parameter i

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the scale parameter in terms of distribution functions, one has Through var-latlon of and G(y) = l:.(a+ ptl, i..j x) = ~ ( ~ [ (au j/ cp(xj and/or G(yj over some appropriate classes of functions, one obtains a .v.a;iabics .ALL,^ ,:+L , d~sLr,uuL,vt. ; L . , + ; A V ~ P t.r) ~ o ~ . L iii i i .ii i i .ii i i j i l j ~ i , 7 i i ~Of y Our aim is to show that location-scale invariant families are uniquely characterized by the maximum likelihood estimators of the location and scale parameters. This property is then exploited to construct uniquely defined statisticai models with prescribed maximum likelihood estimators A brief outline follows . . In the next Sectio!?, we first c!?arcterize the one-parameter scale (a=!:)) 2nd