Question

Below are some parameters I'm interest in, and some proposed estimators. Show me whether the estimators are consistent and unbiased. Assume all samples are i.i.d., and cite any theorems that you use Hint: You should only need to use WLLN, CMT, and the i.i.d. assumption.. 1. I want to know E[X], and I estimate it using the sample mean, X 2. I want to know EX], and I estimate it using TL 4 3. I want to know Var[X], and I estimate it using (Xi - X) 4. I want to know E[XY], and I estimate it using 7n 7n -1 5. I want to know E[XY]. However, I collected my sample of Xs and Ys separately, i.e. for any individual i, I only observe Xi or Yi, but not both. So I estimate it using X × Y

Answer 1

1)
Xbar is consistent and unbiased

i-1 1 71

$Var(\overline{X}) = Var(\frac{X1+..Xn}{n})$

$= \frac{1}{n^2} Var(X1+..Xn) = \frac{nVar(X) }{n^2} = \frac{Var(X)}{n}$

when n tend to infinity

Var(Xbar) tend to 0

hence it is consistent
2)
this is biased
E(X tilda) is not equal to E(X)
this is consistent as
when n tend to infinity
4/n^2 goes to 0

$E(\frac{1}{n} \sum (X_i + \frac{4}{n})) = E(\overline{X} + \frac{4}{n})$

$= \mu + \frac{4}{n}$

hence this is biased because E(X tilda) $\neq$$\mu$

Now

$Var(\frac{1}{n} \sum (X_i + \frac{4}{n})) =\frac{1}{n^2}Var(\sum X_i + 4/n)$

$= \frac{1}{n^2} Var( \sum X_i)$ {4/n is constant}

nVar(X) Var(X)

hence this is consistent too

3)
this is biased estimator
as sample variance is unbiased estimator of population variance
if it was n-1 in stead of n, then it would be unbiased

it is consistent though

Proposition 1. The sample mean and variance are consistent and unbiased esti- mators of the mean and variance of the underlying distribution Proof. It is casy to compute that an TL 2 and now expanding the E|X as = EK1EKJ = μ2 we get that the and also using the independence, eg. EK' above equals to We, therefore, obtain that the sample mean and sample variance are unbiased esti mators

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Please post last 2 questions separately