1. Let X1,..., Xn 2 iid N(μ, T) where μ is known. (a) Find the UMP size-a test for Ho T≥ To vs H₁T < To. (b) Find the P-value of this test. ANOL
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- Let Xi- NcO, ) andn=2 aud T=X,+ Xz Show that Tis a %3D Sulfficiesry Statistic.b) Let Z₁-N(0,1), and W₁ = Y~N(0,1), for i=1,2,3,...,10, then: dx dy i) State, with parameter(s), the probability distribution of the statistic, T = - 154 ii) Find the mean and variance of the statistic T = ₁² 10 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = ₁2₁² +².We have seen that two different tests for Ho B₁ = 0 versus H₁: B₁0, a t-test and a F-test. : Please show that the F-test statistic is equal to the t-test statistic squared. Hint: F-test statistic = t-test statistic MSreg MSres B₁ s.e.(B₁) " where M.Sreg = ₁(Ŷ₁ – Ý)², MSres = Σ₁₁ (Y₁ – Ŷ₁)²/(n − 2). - =1
- 2.9 Give the joint (bivariate) cdf of X and Y, Fy(x, y), in tabular form. (Copy and complete this joint cdf table on your paper.) Fxy (x, y) y< 5 5sy<6 y26 x< 1 1sy<2 2sy<3 y2 3 ? ? ? ? ? ? 2.10 Use the joint cdf of X and Y, Fx y (x, y), in #2.9 to compute P(X = 2, Y = 5) (show your computation). 2.11 Give the marginal cdf of X, Fx (x).Show that under the assumptions of normal multipleregression analysis(a) E(Bˆ i) = βi for i = 0, 1, ... , k;(b) var(Bˆ i) = ciiσ2 for i = 0, 1, ... , k;(c) cov(Bˆ i, Bˆ j) = cijσ2 for i Z j = 0, 1, ... , k.13. Let Y,, Y2, ..., Yn denote a random sample from a population with pdf f(yl0) = (0 + 1)yº,0 -1 a) Find an estimator for theta by the method of moments. b) Find the maximum likelihood estimator for theta.
- Q. 22 Using Neyman-Pearson lemma, find the best critical region for the test of hypothesis Ho:µ = µ0 vs. Hj : µ = H, for the normal population f (x; µ, o²) when o² is known in the cases, (i) µo Also find the power of the test.Let X1, . . . , Xn be an iid sample from f(x | θ) = θxθ−1 , 0 < x < 1, where the parameter θ is positive. Find the MLE and MOM estimators for θ.Let X1,...,Xn be an iid sample from f(x | θ) = θxθ−1, 0 < x < 1, where the parameter θ is positive. Find the MLE and MOM estimators for θ
- iid iid f(y; Py) where Hy = EY; and X1,..., Xm g(x; µ„) where u = EX; be • Let Y1,...,Yn independent random samples and assume the population variances are not equal. Write the formulae for: .. . a large-sample interval for g – Hy; a small-sample interval for pg y. -6. Let (y1, Y2, ., Yn) be a random sample from the uniform distribution on the interval [A – av5, 1 + av/5], where A is the unknown parameter and a is a known constant. Determine the moment substitution estimator of d.4. Let X - N(u, o²). The characteristic function (ch.f.) of X is given by $(w) = ejHw-o?w?/2, W E R, where j = v-1 is the imaginary unit. Use the ch.f. to find the mean and variance of X.