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Let $X_1, \dots, X_n$ be i.i.d random variables with pdf: $f(x|\theta) = \theta x^2$, $0 < \theta \leq x < \infty$.

  1. Find the mle of $\theta$.

  2. Find the mle of $e^{\theta}$.

  3. Find the method of moments estimator of $\theta$.

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2 - yes, I know it doesn't apply exactly to math, but the question sure does. – kahen Sep 30 '12 at 1:13
Let me suggest (1.) to replace $\theta x^2$ by something which integrates to $1$, for example $\theta/x^2$, and (2.) to take @kahen's admonestation seriously. – Did Sep 30 '12 at 5:38

Are you sure you have the correct pdf? The integral of this one diverges to $\infty$. I'm going to assume the pdf is $f(x|\theta)=\theta x^{-2}$ for $x\geq \theta$. Since we know that each $X_{i}$ $\geq \theta$, and since the likelihood function is $\theta^{n}\Pi_{i=1}^{n}x_{i}^{-2}$, this expression is maximised by choosing $\hat{\theta}_{mle}$ = $X_{(1)}$, the smallest observed $X_{i}$ value. Then, since $\exp(\cdot)$ is a continuous function, the MLE of $\exp(\theta)$ is given by $\exp(\hat{\theta}_{mle})$. The Method of Moments estimator for $\theta$ does not exist because $E(X^{k})=\infty$ $\forall k\geq 1$.

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