Let X1, ...,Xn be independentwithmeans E(Xi) = + i and variances V(Xi) = 2 i .

Let X1, ...,Xn be independentwithmeans E(Xi) = μ + βi and variances V(Xi) = σ2 i . Suchasituationcould,forexampleoccurwhen Xi are estimators of μ obtained fromindependentsourcesand βi is the bias of theestimator Xi.

Wenowconsiderpoolingtheestimatorsof μ into acommonestimatorbyusing a linearcombination:

a) Iftheestimatorsareunbiased,i.e.if βi = 0 for all i, showthatalinearcombina-
tion ˜μ as aboveisunbiasedifandonlyif P wi = 1;

b) Inthecasewhen βi = 0 for all i, showthatanunbiasedlinearcombinationhas minimum variancewhentheweights wi are inverselyproportionaltothevari-
ances σ2 i ;

c) Showthatthevarianceof ˜μ for optimalweights wi is V(˜μ) =1/
P i σ−2 i ;

d) Next,considerthecasewheretheestimatorsmaybebiasedsowecouldhave βi ̸= 0. Findthemeansquareerroroftheoptimallinearcombinationobtained above,andcompareitsbehaviouras n → ∞ in thebiasedandunbiasedcase, when σ2 i = σ2, i = 1, ...,n.

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= wX1+w2X2 ++ WnXn. .