Question

a) Find the variance of each unbiased estimator.

b) Use the Central Limit Theorem to create an approximate 95% confidence interval for theta.

c) Use the pivotal quantity Beta(alpha=13, beta=13) to create an approximate 95% confidence interval for theta.

d) Use the pivotal quantity Beta(alpha=25, beta=1) to create an approximate 95% confidence interval for theta.

Suppose that Xi, , x25 are i.i.d. Unifom(0,0), where θ is unknown. Consider three unbiased estimators of 6 25 26 25 25 26 max (X..., X25 225 a = 2 . median ( Х, , x25 ) 2 . Y3 Note the following properties: 1) ,es are ii.d. Uniform(0,1) _, と, 3) --. BetaG-13, β = 13) a) Find the variance of each unbiased estimator Var[6] = pa, [6] = V aar Which is most efficient? Use the central limit theorem to create an approximate 95% confidence interval for θ. Use the notation percentile of a M(0,1) distribution. b) to represent the Y Approximate 95% CI for 0: c) Use the pivotal quantity median(13,-13) to create an exact 95% confidence interval for θ. Use the notation Apr to represent the y percentile of a Beta(&,B) distribution. 95% CI for θ: Use the pivotal quantity max(Ха.xs) 'AS) ~ Beta(α-25, β 1) to create ' ty d) an exact 95% confidence interval for θ 95% CI for :

Answer 1

2.5 So 2 2 12 2 t 5 2 2.

2W 2 12 1.9-6 So, 96 2. 12 v Daka (11, 13 303 099 0 31505t Ч13 115 0-68 61423097 45 wd0315057

凸M , P 0 3623143 2 5 2 5 2 0 8628148

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