Solved: Suppose that X1, X2,..., Xn1 , Y1, Y2,..., Yn2 , and W1, W2,..., Wn3 are

Chapter 10, Problem 10.126

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Suppose that X1, X2,..., Xn1 , Y1, Y2,..., Yn2 , and W1, W2,..., Wn3 are independent random samples from normal distributions with respective unknown means1, 2, and3 and common variances 2 1 = 2 2 = 2 3 = 2. Suppose that we want to estimate a linear function of the means: = a11 + a22 + a33. Because the maximum-likelihood estimator (MLE) of a function of parameters is the function of the MLEs of the parameters, the MLE of is = a1X + a2Y + a3W. a What is the standard error of the estimator ? b What is the distribution of the estimator ? c If the sample variances are given by S2 1 , S2 2 , and S2 3 , respectively, consider S2 p = (n1 1)S2 1 + (n2 1)S2 2 + (n3 1)S2 3 n1 + n2 + n3 3 . i What is the distribution of (n1 + n2 + n3 3)S2 p/2 ? ii What is the distribution of T = Sp 0 a2 1 n1 + a2 2 n2 + a2 3 n3 ? d Give a confidence interval for with confidence coefficient 1 . e Develop a test for H0 : = 0 versus Ha : = 0

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