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What is the Variance of the estimator $ hatμ_n = frac1n(5X_1+ sum_i=2^n-1X_i-3X_n)$

0

1

$begingroup$

I am given the random sample X1, . . . , Xn, which is identically and independently distributed over f(X), for a population density f(X) with finite mean μ and variance $σ_2$. I am given the estimator
$ hatμ_n = frac1n(5X_1+ sum_i=2^n-1X_i-3X_n)$
for the unknown mean of the population.

I need to calculate the variance of $hatμ_n$ and compare is to $bar X_n$. This is causing me a little trouble.

I haven't manipulated Variance much, so I might have made a simple error here:

$Var(hatμ_n)= Var(frac1n(5X_1+ sum_i=2^n-1X_i-3X_n))\
=frac1n^2 (5^2Var(X_1)+sum_i=2^n-1Var(X_i)+(-3^2Var(X_n)))\
frac1n^2 (25Var(X_1)+sum_i=2^n-1Var(X_i)+9Var(X_n))$

I know I need to get this to compare to $fracσ^2n$ ...I just don't know how to get it into that form.

If anyone could please help to clarify if I am right so far, and how to get it into a comparable form, I'd be very grateful.

estimation variance

share|cite|improve this question

edited Mar 26 at 9:43

MWalters

asked Mar 23 at 21:04

MWaltersMWalters

12

$endgroup$

add a comment | 

0

1

$begingroup$

I am given the random sample X1, . . . , Xn, which is identically and independently distributed over f(X), for a population density f(X) with finite mean μ and variance $σ_2$. I am given the estimator
$ hatμ_n = frac1n(5X_1+ sum_i=2^n-1X_i-3X_n)$
for the unknown mean of the population.

I need to calculate the variance of $hatμ_n$ and compare is to $bar X_n$. This is causing me a little trouble.

I haven't manipulated Variance much, so I might have made a simple error here:

$Var(hatμ_n)= Var(frac1n(5X_1+ sum_i=2^n-1X_i-3X_n))\
=frac1n^2 (5^2Var(X_1)+sum_i=2^n-1Var(X_i)+(-3^2Var(X_n)))\
frac1n^2 (25Var(X_1)+sum_i=2^n-1Var(X_i)+9Var(X_n))$

I know I need to get this to compare to $fracσ^2n$ ...I just don't know how to get it into that form.

If anyone could please help to clarify if I am right so far, and how to get it into a comparable form, I'd be very grateful.

estimation variance

share|cite|improve this question

edited Mar 26 at 9:43

MWalters

asked Mar 23 at 21:04

MWaltersMWalters

12

$endgroup$

add a comment | 

$begingroup$

I am given the random sample X1, . . . , Xn, which is identically and independently distributed over f(X), for a population density f(X) with finite mean μ and variance $σ_2$. I am given the estimator
$ hatμ_n = frac1n(5X_1+ sum_i=2^n-1X_i-3X_n)$
for the unknown mean of the population.

I need to calculate the variance of $hatμ_n$ and compare is to $bar X_n$. This is causing me a little trouble.

I haven't manipulated Variance much, so I might have made a simple error here:

$Var(hatμ_n)= Var(frac1n(5X_1+ sum_i=2^n-1X_i-3X_n))\
=frac1n^2 (5^2Var(X_1)+sum_i=2^n-1Var(X_i)+(-3^2Var(X_n)))\
frac1n^2 (25Var(X_1)+sum_i=2^n-1Var(X_i)+9Var(X_n))$

I know I need to get this to compare to $fracσ^2n$ ...I just don't know how to get it into that form.

If anyone could please help to clarify if I am right so far, and how to get it into a comparable form, I'd be very grateful.

estimation variance

share|cite|improve this question

edited Mar 26 at 9:43

MWalters

asked Mar 23 at 21:04

MWaltersMWalters

12

$endgroup$

share|cite|improve this question

edited Mar 26 at 9:43

MWalters

asked Mar 23 at 21:04

MWaltersMWalters

12

share|cite|improve this question

edited Mar 26 at 9:43

MWalters

asked Mar 23 at 21:04

MWaltersMWalters

12

asked Mar 23 at 21:04

MWaltersMWalters

12

asked Mar 23 at 21:04

MWaltersMWalters

12