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I Checking for Biased/Consistency

  1. May 29, 2017 #1
    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.
     
    Last edited: May 29, 2017
  2. jcsd
  3. May 30, 2017 #2

    andrewkirk

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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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