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Recall the lifetime (in months) of a battery is modeled by a random variable X that...

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Basic Math

Recall the lifetime (in months) of a battery is modeled by arandom variable X that has pdf fθ(x)=Kθx1(x>0)where K=ln(1/θ)for an unknown parameter θ∈(0,1) .

Assume instead that we cannot actually observe the lifetime ofthe batteries. Instead, we only observe if the battery is stillworking after Ï„ months for some known Ï„ to be chosen later (this iscalled censored data ).

Let Y1,…,Yn be our observations where Yi=1(Xi>τ) indicatesthat the i th battery is still working after τ months. Our goal isto estimate θ∈(0,1) (the parameter for the pdf of X ) based on thisnew data.

The quantity n−−√(θ~−θ) converges in distribution to N(0,σ2~) .Find the asymptotic variance σ2~ . σ2~=

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