Theses

The introduction of the concept of onymptotie efficiency of statistical eatimators in connection with proposing and developing she method of maximu, 1ikelihood by . Fislior (Fisher (1922, 1929)) 1s really the atar ting point of the asymptotie theory of es timation. -istorieally, however, Laplace (1774) and Gausa (1009) had made tuo irferent studieo oarlier then Fisher bo ti connected d th as ymp to tie theory of estination. Fisher considered only consistent a symptotically rewnal untinotoro and meured the asyaptotie performance of an anti- mator by ite aayep totle variaree. Thus, a coreistent aeynp to tieally nomal eetimoator with let possible aayno to tie voriance wes defined to be an erricient estimator. Fisher also clained to have proved that under certain regulari ty condi tiona the naxinum likelhaod estinatar (MLE) 1s offieient in the above sense. In the thirties and forties several authors (Dugue (1936a, 1936b, 1937), Wlks (1938), Neynan (1949) ond ethere) attompted to a btain a Figorous proof of the erricieney of the MLE and there uas a genet al teller that there exinte an afrieient es timator in the general case unich nay be obtained by the method of maximum likelihood. This belief enieted until 3.. Hodges produced in 1951 the " revolutionary exanples of;auper ofricient osti- ma tore (ane ean se, for example, Ghonh (1905) or Le Can (1953) hera It rirot appenred). Hodgen exanple nhowe that in the unual rogular casee there axist neymptotically nomal estinatere uhone aaymptotic variances are alunye lees than or equal to that of tha MLE and are trietly less than that of the MLE at partieuler values of the para- ter and at these particular valuen the asymp to tie variance may even e made equal to zero. Thus, the MLE is net errieient in tha above sense and indeed, within the elass of all symptotically normal esti- natera re entinator ui th minimal symptotic variance exists.However, the ideas of Fiaher and the existence of " super ofricient " otinators greatly influenced the developement of the theory of efficient entimation and a modern approach to the theory of synap totie efrrietent entimation emerged in the fundamantal paper of Le Can (1983). The theory vos further developed in the worke of Le Can (1960, 1964, 1972), Hajek (1970, 1972), unlfouitz (1965), Millar (1903) and othere. Thie approach which renched nore ar lese itarinal form in the papere of Hajek (1972) and Le Can (1972) consi dere all e tina tore in oteed of rontricting to the class of naynptotically nornal en tinn tora only, tut the ofricieney or the performance of these estinn tors is meanur ed in a slightly different uay. Mmiller (1983) prenenta a very clenr expoeitian of thie theory extending sone of the basie results of Le Can (1072). A lucid aneount of this developrent of kajek - La Can theory of afrieient estination is available in Choah (1905).

Theses

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