statistic estimators

Question

i need help on this exercise, i can't seem to understand the basis of this problem, what are estimators? and what are the steps you need to do in order to evaluate them?

exercise:

Consider the following estimators for 𝜇, the expected value of X: 𝜇̂ = 𝑋1, 𝜇̃ = 𝑋1 + 𝑋𝑛 and 𝜇̇ = 1 ∑𝑛 𝑋 , where n is the sample size. Evaluate the three estimators and state the one you choose.

Answer 1

I think there's a typo on the second estimator. I think that should be (X1+Xn)/2. If estimator 2 is just X1+Xn, then it's not an estimator since it does not have the same support as the parameter space. :-)

 

Anyway, we can evaluate estimators of the mean using the following criteria: unbiasedness, and minimum variance.

[ In statistics, there's a lot of other criteria such as consistency, efficiency,robustness, resitance, etc.]

 

unbiasedness: E(estimator) = target parameter

 

1. E(X1) = μ

2. E[(X1+Xn)/2]= (1/2)*[E(X1)+E(Xn)] = (1/2) (μ + μ) = μ

3. E[(X1+X2+...+Xn)/n] =(1/n)*[μ + μ + ... + μ] = (1/n)* n*μ = μ 

 

Variance:

1. Var(X1) =σ²

2. Var[(X1+Xn)/2) =(1/4)* [σ²+σ²) = σ²/2

1. Var[(X1+X2+...+Xn)/n) = (1/n²)*(σ² + σ² + ... + σ²) = (1/n²)*(nσ²) =  σ²/n

 

Note that all of the 3 estimators are unbiased but the 3rd estimator, sample mean, gives the smallest variance, which makes it the "best" among the 3.

 

 

 

Answer 2

Mu (μ) is the MEAN.  Which of these three 'estimations' is best??  

 

𝜇̂ = 𝑋1, 𝜇̃ = 𝑋1 + 𝑋𝑛 and 𝜇̇ = 1 ∑𝑛 𝑋 

𝜇̂ is just a guess that the mean happens to be equal to the first value of the set of Xs

𝜇̃ is just a guess that the mean happens to be the sum of the first and last X values

𝜇̇ is intended to mean, I believe, the summation of all of the n values of X (X1 + X2 +...+Xn).

 

Lacking any numerical data for the Xs, it's anybody's guess!  Not the most wonderful problem.