Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Research and Training School (RTS)


Mitra, S. K. (RTS; ISI)

Abstract (Summary of the Work)

The problem of conbining several estimates of an unknown quentity to obtain an eetimete of Improved precieion arises in meny spheres of application of etatistics, To begin with let us consider the following simple inadelt1 =+4, 1- 1,...,kwhere u ie an unknown quantity and e, e ere errora with a common menn zero end a common veriance o. If we make no further asoumption about the distribution of e,s, the GAuas-Markoff theorem tells us that among sll unbtaved lineer combinationa ofr ya, the least squAre estimator i * Iy /k of u hes the minimum variance. If ,s are jointly hormally dietributed, then the lenst square estimator ts slao the max imum 34kolihood eutihator and kee mindmum veriance in the class of sll unbiened estimators [Rao (1952)]. Under the aasumption of normality the estimator enjoys yet anather property thet it is admiasible in the cleas of all entimators with rapect to any loss function which ie monotonic incrosing function of the ebaolute orror [(Blyth (1951)). All these important resulta schit of Immediete exteneion to the case when e,'s are correlsted and have inequal variances provided we know the relative values of the elemente of the dieperslon matrix of g (e1.e. V(e) = o?H, where H is a known matrix. In thie CBee an well known madi fication of the ordinery Ieest equares procedure provides en estimator with all the properties stated sbove. In meeny canes it is not ureasohable to aseume thet y1s have independent normal distributione but it is unreesoneble to assume thet the relative valuves of the variances of ys are known. For an example, Buppone that two laboratories have made separate determinations ya of the Beme phyeical or chemical quantity. It is ensy to conceive situations where it in unreeeonable to assume that the two labaratories da not differ in precision. In general, the relntive precisione are not known but can be entimated from the current or previous data. Thus in the above exemple each laboretory may provide us with an estimated etandard error s, for the estimatey, of p. The problem of obtaining a good combined estimator in Buch practical situstions is not straightforward. The mathemetical model generally esaumed for the problem is as follows:(1) 1 Nu,o), i = 1,...,k, are independent(11) o- 21 = 21,...,k, are independent The problem of estimating u of the above model, traditionally known es the weighted mean problem has been of conaiderable interest to both theoreticians and practitioners of statistics. The theoretical interest erises because of the difficulty of eliminettintg the unknown variance parameters from inference about see Hinkley(1979)for a recent discussion].


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