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Inferential Statisticshard
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Consider a random variable XXX with density f(x∣θ)=θxθ−1f(x|\theta) = \theta x^{\theta-1}f(x∣θ)=θxθ−1 for x∈(0,1)x \in (0, 1)x∈(0,1). Using the method of moments, we find θ^MM=Xˉ1−Xˉ\hat{\theta}_{MM} = \frac{\bar{X}}{1-\bar{X}}θ^MM​=1−XˉXˉ​. How does the variance of this estimator behave as n→∞n \to \inftyn→∞ compared to the Cramer-Rao Lower Bound (CRLB)?