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This is a very simple sample problem from ISI MStat PSB 2006 Problem 9. It's based on point estimation and finding consistent estimator and a minimum variance unbiased estimator and recognizing the subtle relation between the two types. Go for it!

Let \(X_1,X_2,......\) be i.i.d. random variables with density \(f_{\theta}(x), \ x \in \mathbb{R}, \ \theta \in (0,1) \), being the unknown parameter. Suppose that there exists an unbiased estimator \(T\) of \(\theta\) based on sample size 1, i.e. \(E_{\theta}(T(X_1))=\theta \). Assume that \(Var(T(X_1))< \infty \).

(a) Find an estimator \(V_n\) for \(\theta\) based on \(X_1,X_2,......,X_n\) such that \(V_n\) is consistent for \(\theta \) .

(b) Let \(S_n\) be the MVUE( minimum variance unbiased estimator ) of \(\theta \) based on \(X_1,X_2,....,X_n\). Show that \(\lim_{n\to\infty}Var(S_n)=0\).

Consistent estimators

Minimum Variance Unbiased Estimators

Rao-Blackwell Theorem

Often, problems on estimation seems a bit of complicated and we feel directionless, but most cases its always beneficiary do go with the flow.

Here, it is given that \(T\) is an unbiased estimator of \(\theta \) based on one observation, and we are to find a consistent estimator for \(\theta \) based on a sample of size \(n\). Now first, we should consider what are the requisition of an estimator to be consistent?

- The required estimator \(V_n\) have to be unbiased for \(\theta \) as \( n \uparrow \infty \) . i.e. \(\lim_{n \uparrow \infty} E_{\theta}(V_n)=\theta \).
- The variance of the would be consistent estimator must converge to 0, as n grows large .i.e. \(\lim_{n \uparrow \infty}Var_{\theta}(V_n)=0 \).

First thing first, let us fulfill the unbiased criteria of \(V_n\), so, from each of the observation from the sample , \(X_1,X_2,.....,X_n\) , of size n, we can get as set of n unbiased estimator of \(\theta \) \( T(X_1), T(X_2), ....., T(X_n)\). So, can we write \(V_n=\frac{1}{n} \sum_{i=1}^n(T(X_i)+a)\) ? where \(a\) is a constant, ( kept for generality). Can you verify that \(V_n\) satisfies the first requirement of being a consistent estimator?

Now, proceeding towards fulfilling the final requirement, that is the variance of \(V_n\) converges to 0 as \(n \uparrow \infty\) . Since we have defined \(V_n\) based on \(T\), and it is given that \(Var(T(X_i)) \) exists for \( i \in \mathbb{N}\), and \(X_1,X_2,...X_n\) are i.i.d. (which is a very important realization here), leads us to

\(Var(V_n)= \frac{Var(T(X_1))}{n}\) , (why ??) . So, clearly, \(Var(V_n) \downarrow 0\) a \( n \uparrow \infty\), fulfilling both required conditions for being a consistent estimator. So, \(V_n= \sum_{i=1}^n(T(X_i)+a)\) is a consistent estimator for \(\theta \).

(b) For this part one may also use Rao-Blackwell theorem, but I always prefer using as less formulas and theorem as possible, and in this case we can do the required problem from the previous part. Since given \(S_n\) is MVUE for \(\theta \) and we found that \(V_n\) is consistent for \(\theta \), so, by the nature of MVUE,

\(Var(S_n) \le Var(V_n) \), so as n gets bigger, \( \lim_{ n \to \infty} Var(S_n) \le \lim{n \to infty} Var(V_n) \Rightarrow \lim_{n \to \infty}Var(S_n) \le 0\)

again, \(Var(S_n) \ge 0\), so, \(\lim_{n \to \infty }Var(S_n)= 0\). Hence, we conclude.

Lets extend this problem a liitle bit just to increase the fun!!

Let, \(X_1,....,X_n\) are independent but not identical, but still \(T(X_1),T(X_2),.....,T(X_n)\), remains unbiased of \(\theta\) , and \(Var(T(X_i)= {\sigma_i}^2 \), and

\( Cov(T(X_i),T(X_j))=0\) if \( i \neq j \).

Can you show that of all the estimators of form \( \sum a_iT(X_i)\), where \(a_i\)'s are constants, and \(E_{\theta}(\sum a_i T(X_i))=\theta\), the estimator,

\(T*= \frac{\sum \frac{T(X_i)}{{\sigma_i}^2}}{\sum\frac{1}{{\sigma_i}^2}} \) has minimum variance.

Can you find the variance ? Think it over !!

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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