Let X1,..., Xn an iid random sample from Gamma(a, X). Show that [I-1 X; and E X; are jointly sufficient for (a, X).
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- Let X1, X2, ..., Xn ~ exp(A) be a random sample. Show that X is sufficient for 1/A.B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.Let x = (x1, x2 ..., xi, ..., xn) be a data set with a sample mean x̄ and a sample variance Sx2. Also, let y = (y1, y2 ..., yi, ..., yn) be a data set with a sample mean ȳ and sample variance Sy2. Finally, let z = x - y such that z = (x1 - y1, x2 - y2 ..., xi - yi, ... xn - yn). Show that z̄ = x̄ - ȳ.
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- Theorem 3: Let x - (x1,X2 -- . Xp . .., x,) be a data set with a sample mean ĩ and a sample variance S. Also, let y = (y. Y2 ..yp m, Ya) be a data set with a sample mean ỹ and a sample variance S. Finally, let z = x - y such that z- (X1- yı, X2 – y2 -- .X - Ya... . , Xn – Yn). Show that i -i - ỹ.Let a and b be any real numbers and let x = (x1,Xz -- -- Xp m-, X) be a data set with a sample mean f and a sample variance s. Show that if y, = (ax, + b), then S3 = a*s;if x1, x2, ..., Xn be a random sample from Bin(4,0). is T = x UMVUE ? Justify your answer.
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