I Checking for Biased/Consistency

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The discussion focuses on evaluating the unbiasedness and consistency of the Method of Moments and Maximum Likelihood Estimators for the parameter θ from a given population density function. The Method of Moments estimator is derived as \(\hat{\theta}_1 = \frac{4\bar{X}^2}{\pi}\), while the Maximum Likelihood Estimator is found to be \(\hat{\theta}_2 = 2\bar{Y}\), where \(Y\) represents the square of the sample \(X_i\). To check for bias, the expected value \(E[\hat{\theta}_2 - \theta]\) needs to be evaluated, and for consistency, the probability \(Prob(|\hat{\theta}_2 - \theta| < \epsilon)\) should be analyzed as \(\epsilon\) approaches zero. Clarifications on notation and formatting issues were also noted, emphasizing the importance of consistent variable representation. The discussion highlights the need for assistance in verifying the properties of these estimators.
Jmath
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Hello I am trying to check if the Method of Moments and Maximum Likelihood Estimators for parameter $\theta$ from a sample with population density $$f(x;\theta) = \frac 2 \theta x e^{\frac {-x^2}{\theta}} $$
for $$x \geq 0$, $\theta > 0$$ with $\theta$ being unknown.

Taking the first moment of this function I found the Method of Moments estimator to be $$\hat{\theta}_1 = \frac{4\bar X^2}{\pi}$$ and solving for the Maximum Likelihood Estimator the Estimator to be $$\hat{\theta}_2 = 2\bar Y$$ where Y is just square of the Sample X_i, i.e. $$Y = X_i^2$$.

Steps in Solving for Method of Moments:
I took the first moment, i.e.

M_1 = E[x] = $$\int_0^\infty{\frac 2 \theta x^2 e^{\frac {-x^2}{\theta}}dx}$$

Solving this integral with $u$ substitution with $$u = \frac{-x}{2}, du = \frac{-1}{2}, v = e^\frac{x^2}{\theta}, dv = -2xe^\frac{-x^2}{\theta}$$

$$\int_0^\infty{\frac 2 \theta x^2 e^{\frac {-x^2}{\theta}}dx} = [-\frac{xe^\frac{-x^2}{\theta}}{2\theta} - \frac{\sqrt{\pi \theta}}{4}]^\infty_0 = \frac{\sqrt{\pi} {\sqrt{\theta}}}{2}$$

So that $$E[x] = \bar{x} = \frac{\sqrt{\pi} {\sqrt{\theta}}}{2}$ gives the Method of Moments Estimator $\hat{\theta_1} = \frac{4\bar{X}^2}{\pi}$$

Steps in Solving for Maximum Likelihood:

$$lnL(\theta)=(\prod_{i=1}^n\frac 2 \theta x e^{\frac {-x^2}{\theta}}) = -n ln((2\theta)) + \sum_{i=1}^nx_i - \frac {1} {\theta} \sum_{i=1}^nx^2_i$$

$$\frac {dlnL(\theta)}{d\theta} = \frac{-n}{2\theta} + \frac{1}{\theta^2} \sum_{i=1}^nx^2_i$$

Setting $\frac {dL(\theta)}{d\theta} = 0$, I found the Maximum Likelihood Estimator $\hat{\theta_2}$ to be $$\hat{\theta_2} = \frac{2\sum_{i=1}^nx^2_i}{n}$ , so that if $Y = X_i^2$ then $\hat{\theta_2} = 2\bar{Y}$$.

I am trying to check if these estimators for $\theta$ from this density function are unbiased and/or consistent but am lost on how to go about doing so, any help would be much appreciated.
 
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To check for bias, evaluate ##E[\hat\theta_2-\theta]## by integrating. The estimate is unbiased iff this evaluates to zero.

To check for consistency, evaluate ##Prob(|\hat\theta_2-\theta|<\epsilon)##. If the result is a function that goes to zero as ##\epsilon\to 0## the estimator is consistent.

A few points by the by:
  • on physicsforums the $ delimiter for latex in-line maths is not recognised. That's why the formatting is all mucked up in places above. Use a double-# instead.
  • your estimates from the two methods are the same, as ##2\bar Y##. Is that what you intended?
  • the statement ##Y=X_i{}^2## occurs twice. This should be ##Y_i=X_i{}^2## as both sides depend on ##i##
By the way, you seem to conclude
 
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