Statistics help please?

Suppose that Y1, Y2, . . . , Yn are independent N(渭,sigma square) random variables.


(a) Show that Y bar is an unbiased estimator of 渭.


(b) Is Y bar consistent for 渭?


(c) Show that sum of (from i=1 to n )


1(Yi 鈭?Ybar )^2/n is a biased estimator of sigma^2. Is it asymptotically unbiased?|||(a)





E(Ybar) = E(1/n * 鈭慪i) = 1/n 鈭慐(Y) = n/n * 渭 = 渭





the expectation of Ybar is 渭, thus Ybar is unbiased.





(b)





An estimator is consistent sequence of estimators for a parameter 胃 if for every 蔚 %26gt; 0 and every 胃 in 螛





lim n 鈫?鈭?P( | Xn - 胃 | %26lt; 蔚 ) = 1





for Ybar in this case you have:





P( | Ybar - 渭 | %26lt; 蔚 ) = 鈭?f(ybar) dybar





where f(ybar) is the density function of ybar and the limits of integration are: 渭 - 蔚 to 渭 + 蔚.





Using a couple substitutions you will be able to show the integral reduces to:





P( - 蔚 鈭歯 %26lt; Z %26lt; 蔚 鈭歯 )





where Z is the standard normal.





and the limit of this probability statement as n 鈫?鈭?is 1, thus Ybar is consistent for 渭.





(3)





E( 1/n鈭?Yi 鈭?Ybar )^2 )





= 1/n * E(鈭?Yi 鈭?Ybar )^2 )





= (n-1)/n * E( 1/ (n-1) 鈭?Yi 鈭?Ybar )^2 )





= (n-1) / n * 蟽虏





as n 鈫掆垶 the limit of the expectation of this estimator for the variance becomes unbiased as (n-1) / n 鈫?1. So the estimator is biased but it asympototically unbiased.|||See my answer to Rosie O.