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Q16Section A (Section A)16 of 60

Consistent estimator for the scale of a quartic-type density

Let X1,…,XnX_1,…,X_n be a random sample from a population with probability density function

fθ(x)={4θx3e−x4/θ,x>0, 0,otherwise,f_\theta(x)=\begin{cases}\frac4\theta x^3e^{-x^4/\theta}, & x>0,\ 0, & \text{otherwise,}\end{cases}

where θ>0\theta>0 is unknown. Then, a consistent estimator for θ\theta is

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