Checking for Biased/Consistency

In summary, the conversation discusses 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 $x \geq 0$, $\theta > 0$, where $\theta$ is unknown. The Method of Moments estimator is found to be $\hat{\theta}_1 = \frac{4\bar X^2}{\pi}$ and the Maximum Likelihood Estimator is $\hat{\theta}_2 = 2\bar Y$, where $Y=X_i^2$. To check for bias, the
  • #1
Jmath
1
0
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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  • #2
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
 

Related to Checking for Biased/Consistency

What is bias in scientific research?

Bias in scientific research refers to any systematic error or deviation from the true value of a variable that can affect the results of a study. This can occur in various stages of the research process, such as study design, data collection, and data analysis.

How can I identify bias in my research?

To identify bias in your research, you can start by critically examining the study design and potential sources of bias, such as selection bias, measurement bias, or publication bias. You can also assess the consistency of your findings with previous studies and consider any potential conflicts of interest.

How can I minimize bias in my research?

To minimize bias in your research, you can use rigorous study designs, such as randomized controlled trials, and carefully select your study population to avoid selection bias. You can also use standardized and validated measurement tools, blind data collection and analysis, and disclose any potential conflicts of interest.

Why is consistency important in scientific research?

Consistency is important in scientific research because it ensures that the results obtained are reliable and can be replicated by other researchers. When a study is consistent, it means that the results are not affected by bias or random chance, and they are more likely to reflect the true value of the variable being studied.

What are some strategies for maintaining consistency in research?

To maintain consistency in research, you can use standardized protocols and procedures, carefully document all steps of the research process, and conduct pilot studies to test the reliability of your methods. It is also important to regularly review and update your methods to ensure consistency throughout the research process.

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