Problem 13.2 Assume that Xi, X2,. Xa form a random sample from a normal distribution for which th...
Exercise 5 Suppose that Xi, X2 X3 denote a random sample from a normal distribution with an unknown mean μ and a variance of 1. That is Xi ~ N(μ, σ-1). Consider two estimators, and μ2 9 For what values of μ doos μ2 obtain a lower MSE than μι, if any?
please answer with full soultion. with explantion. (4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the unknown parameter. Let X denote the sample mean. (Note that the mean and the variance of a normal N(μ, σ2) distribution is μ and σ2, respectively.) Is X2 an unbiased estimator for 112? Explain your answer. (Hint: Recall the fornula E(X2) (E(X)Var(X) and apply this formula for X - be careful on the...
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n). Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (c) Show that there is an appropriate statistic T T(X) that has monotone likelihood ratio. (d) Derive the uniformly most powerful (UMP) level α test for
6. Suppose that X1,X2 , Xn form a random sample from a normal distribution N(μ, σ 2), both unknown. consider the hypotheses Construct a likelihood ratio test and show that this LRT is equivalent to a t-test 6. Suppose that X1,X2 , Xn form a random sample from a normal distribution N(μ, σ 2), both unknown. consider the hypotheses Construct a likelihood ratio test and show that this LRT is equivalent to a t-test
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what is the distribution of 2 Σ x/P Follow the procedure outlined in Exercise 6.28. 6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what...
2. Suppose that Xi, , Xn, n-: 25, form a random sample from a normal distribution with mean θ and variance 4. Consider the following hypotheses at α-0.05 Ho : θ-0 versus H1 : θ > 0. Derive the power function, π( 5), and evaluate it at θ--04,-02, 0,02, 0.4, 0.6, 0.8, 1. 2. Suppose that Xi, , Xn, n-: 25, form a random sample from a normal distribution with mean θ and variance 4. Consider the following hypotheses at...
4) Let Xi , X2, . . . , xn i id N(μ, σ 2) RVs. Consider the problem of testing Ho : μ- 0 against H1: μ > 0. (a) It suffices to restrict attention to sufficient statistic (U, v), where U X and V S2. Show that the problem of testing Ho is invariant under g {{a, 1), a e R} and a maximal invariant is T = U/-/ V. (b) Show.that the distribution of T has MLR,...
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters μ and σ2 a) Devise a significance level α test of the null hypothesis 0 versus the alternative hypothesis for a given positive value σ1. b) Explain how the test would be modified if the population mean μ were known in advance 8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters μ and σ2 a) Devise a...