ABSTRACT
Nok that for every d , S(Old) is ;L continuous function in Q arid it,s graph crmscs the 0-axis orily OIKC a t Q - 0. Further) for any degree of l'rccdom d ( d (X)) tl-ie Suuct,ion S(O/c%) is im~mtlctt and ,S(Oltd) 4 0. as 101 + m. For the PJorriial t1istril)utiorr (d - m): the expected score llmct,ion is S(O X)) 7 U ; mi-ijdi is cont-immw 1)ut ~mlr~ountletl. By numerically il~tegri~.ting (28.3.1 2)- it yicltls thc lollowing Figure 28.1 which illust,~atcs tlie 1)elh;~viors of lhc espectecl score fi~~lctions S(Oic1); for d - 1, 6, 15; X). Irl a,dtlition, it follows from the 1a.n. of large rilm11)ers
as 71, - ) (X?, uritler C2 - For the Ca~ichy tlistribui,ion, the erripirical score S~mctiori
based on oir~scrvetl tlitt,ii (:cl, . . . ; .r,,) is also plottctl against the expected score f~ulctiorr S(O 1 ) ill Figure 28.1. Pcrlman ( 1 W<) arid Rc'tls (1985) proved that. for Cauchy ilistributioii, all the roots of likelihood equation except the nrle (the global rriaxirriu~n) i,end to +X with probal>ilily our:. and (,he rnlc corrvwgcs to Oo ( O o = 0 in Figure 28.1). This corivcrgerlce heliitvior is also olmrved in Figure 28.1 Sor all t-distri1)utiolrs with degrccs OS frcedoru (1 - l , ? , . . ..
