Question

Question 1 (understanding unbiasedness and efficiency) Consider height of adult males. It is a randon variable with mean μ and variance σ2. Our goal is to find the best estimator for μ from a sample of 5 observations {hi,h2,h3,h4,hs). We assume that these observations are randomly selected. The following estimators for μ have been proposed 1.訟リーthe average of the second and fourth observations; 2. İ121 the average of the first, third and fifth observations; 3, μ[3] the average of μ[1] and μ[2]. .the average of the last four observations 5. [Only for groups of five] 5. the average of all five observations. (a) Derive the expected value and the variance of each of the above estimators in terms of μ and σ2 Based on these, determine if each of the above is an unbiased estimator Of μ

Answer 1

h2,h3,h4, hs are randomly selected, so cov(h,,h,)-i.,2,3,4,5; i^ j E(hi)-μ. Var(h.)-σ2; V-1. 2. 3.4.5 h24 1 VarVar alll involves h2 and h4 : μ[21 involves hi and h3 and hs ; thus μ[1] and μΡΙ are independent. var [.pr-var!! 2ml-kvarih-varai, + 2cod" i h-lf+2×リ- 24 σ2 16 25 Thus it can be seen that each of this estimators are unbiased, as all of their observed value equals μ