peter j bickel, kjell a doksum mathematical statistics basic ideas and selected topics, vol i 2nd edition 2000

[...]... rellected in this hook Kjell Doksum 1976 ' ! • Peter J Bickel Berkeley ! Mathematical Statistics Basic Ideas and Selected Topics Volume I Second Edition ; ' j ' 1 j 1 ' ' I ' ' i I i \ Chapter 1 STATISTICAL MODELS , GOALS , AND PERFORMANCE CRITERIA 1.1 DATA, MODELS, PARA METERS AND STATISTICS 1 1.1 Data and Models Most studies and experiments, scientific or industrial, large scale or small, produce data whose... interest and nuisance parameters into a single grand parameter (), which indexes the family P, that is, make B -+ Po into a parametrization of P Implicit in this description is the assumption that () is a parameter in the sense we have just defined But given a parametrization (} -+ Po, (} is a parameter if and only if the parametrization is identifiable Formally, we can define (} : P -+ 8 as the inverse... first Our one long book has grown to two volumes, each I I i • to be only a little shorter than the first edition Volume I, which we present in 2 000, covers material we now view as important for all beginning graduate students in statistics and science and engin eering graduate students whose research will involve statistics intrinsically rather than as an aid in drawing conclu• SIOnS In this edition. .. subject matter experts arc an essential element in all model building However, insofar as possible we prefer to take the frequentist point of view in validating statistical statements and avoid making final claims in terms of sub jective posterior probabilities (see later) However, by giving () a distribution purely as a theoretical tool to which no subjective significance is attached, we can obtain important... distributions P ranging over models P The notions of parametrization and identifiability are introduced The gen­ eral definition ofparameters and statistics is given and the connection between parameters and pararnetrizations elucidated This is done in the context of a number of classical exam­ ples, the most important of which is the workhorse of statistics, the regression model W e view statistical... patients to whom drugs A and B are to be administered may be haphazard rather than a random sample from the population of sufferers from a disease In this situation (and generally) it is important to randomize That is, we use a random number table or other random mechanism so that the m patients administered drug A are a sample without replacement from the set of m + navailable patients Without this... Fujimura, and our families for support, encouragement, and active participation in an enterprise that at times seemed endless, appeared gratifyingly ended in 1976 but has, with the field, taken on a new life ' Peter J Bickel bickel@ stat.berkeley.edu Kjell Doksum doksum@stat.berkeley.edu I • • I l I i I I PREFACE TO THE FIRST EDITION This book presents our view of what an introduction to mathematical statistics. .. final draft) through ' which this book passed E L Lehmann's wise advice has played a decisive role at many points R Pyke's careful reading of a next-to-final version caught a number of infelicities of style and content Many careless mistakes and typographical errors in an earlier version were caught by D Minassian who sent us an exhaustive and helpful listing W Cannichael, in proofreading the final... assumptions (1)-{4) In fact by varying our assumptions this class of models includes any situation in which we have independent but not necessarily identically distributed obser­ vations By varying the assumptions we obtain parametric models as with (l), (3) and (4) above, semiparametric as with (l) and (2) with F arbitrary, and nonparametric if we drop (I) and simply treat the Zi as a label of the completely... in Volume 2 , the conceptual issues of stationarity, ergodicity, and the associated probability theory models and inference for dependent data are beyond the scope of this book 0 I 12 Statistical Models, Goals, and Performance Criteria Summary Chapter 1 In this section we introduced the first basic notions and formalism of mathe­ matical statistics, vector observations X with unknown probability distributions .                                                   I        II t        F.                    I   I               .               I       I ( 4)        