8.19 Suppose that Y₁, Y2, . . ., Y, denote a random sample of size n from a population with an exponential distribution whose density is given by (1/0)e-y/0, f(y) = 0, y > 0, elsewhere. If Y(1) = min(Y₁, Y2, ..., Yn) denotes the smallest-order statistic, show that Ô = nY (1) is an unbiased estimator for and find MSE(Ô). [Hint: Recall the results of Exercise 6.81.]
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Question 8.19 from Mathematical statistics with applications (Wackerly):
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- a) Let X₁, X₂, X3,... X, be a random sample of size n from population X. Suppose that X-N (0,1) and Y = -√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.17. Let (yı, Y2, Yn) be a random sample from a normal distribution such that *** E(y;) = a + Bxi, Var(y;) =o² where ri, i = 1,2, ..., n are constants, a, likelihood estimators of a , B and o?. B and o? are the parameters of interest. Find the maximum25. Obtain the maximum likelihood estimates for a and ß in terms of the sample observations x1, x2, X taken from the exponential population with density functions : f (x, a, B) = yoe-B(*- ), a0 ", asx0 Y'o being the constant.
- Q3) Let X1,X2,. X, be a random sample from the truncated exponential distribution with pdf: f(x,0) = Please else (a) Derive the method of moment estimator of 0; (b) Derive the MLE of 0. (c) Are the MOME and the MLE unbiased estimators for 0? (d) Compare the MSE of the MOME and the MLE for 0. Which one is a better estimator for 0Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.Let Y1, Y2, Y3, Y4, and Y, be i.i.d. random variables from a population with mean µ and variance o². Further, let Y = (Y, +Y½ +Y3 +Y4+Y;) denote the average of these five random variables. Now consider a different estimator of µ called W w = }Y + }Y½ + }Y, + }Y, + Y, What is the variance of W? That is, calculate Var|W] Hint: Use the properties of expectations, variances, etc. 187 800 800 187
- Let x, x2, ..., Xn be a random sample from a Poisson distribution with parameter 2. Find ML estimator of 2. Also find its variance.c) Let Y₁, Y₂, ..., Yn be a random sample whose probability density function is given by - {7 0, f(y; B) = 0 0 elsewhere 100, Σ?fy and suppose that n = 200, Z=1 yi = : 20, Σ? y 200 200 = ii) Find an approximate 95% Confidence interval for ß. = 250 and 3 = 0.025.Suppose Y1,.,Yn are independent random variables with probability mass function f(y) = p"(1 – p)*-", y = 0,1. Here, p E (0, 1), but unknown. Suppose Y = Lisi is the sample mean. Then, show that the Maximum Likelihood Estimator (MLE) of p is p = Y. What is the V(Y)? Now, give the MLE of V(Y).
- Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.Let Y, represent the th normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁, Y2. Yin, of size n₁, from the ith population with sample mean Y, and sample variance S² = ₁₁-1(Y₁-₁². j=1 (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #₁ and μ2, when o² = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ - Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.Let Y, represent the ith normal population with unknown mean , and unknown variance of for i=1,2. Consider independent random samples, Ya, Y2. the ith population with sample mean Y, and sample variance S² = Yin, of size n,, from (Y-₁². (a) What is the distribution of Y,? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho : 4 = o versus Ha i Pio when of is unknown and n, is small. : (e) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.