that μ,-0 and μ,-3113 where μί denotes the i-th moment of x. that μ,-0 and μ,-3113 where μί denotes the i-th moment of x.
Suppose that X ~ POI(μ), where μ > 0. You will need to use the following fact: when μ is not too close to 0, VR ape x N(VF,1/4). (a) Suppose that we wish to test Ho : μ-710 against Ha : μ μί are given and 10 < μι. m, where 140 and Using 2 (Vx-VHo) as the test statistic, find a critical region (rejection region) with level approximately a (b) Now suppose that we wish to test Ho...
4. Recall the "means ANOVA model Y,-μί +Gj where μί-B( ) is thepopulation mean of the ith group. Now, let μ Σί.th and T-μί-μ to define the "effects" model. Here, Ti is the "effect" of the ith group. Show that Σ-n must be 0.
Problem 2. Consider the one-way layout ANOVA model, where we assume that Yij = μί-cij,に1, . . . , I and J 1, . . . ,J, where μί's are fixed unknown with zero mean treatment means and eiy's are random errors , al such that Σ-lai -0 and E[Yj-μ + ai,1- Show that there exists unique numbers μ, ai, a. b. Show that the null hypothesis Ho : μ,-...- μι is equivalent to Ho : 01 ,-. . .-a1-0...
6. (a) If f(x a) for -al R, show that i-0 fO(a) fori 0,1,2, (where f0 (a) f(a), and fori 1 f(a) denotes the i-th derivative of f at a). (b) If f e* , find f(2014)
6. (a) If f(x a) for -al R, show that i-0 fO(a) fori 0,1,2, (where f0 (a) f(a), and fori 1 f(a) denotes the i-th derivative of f at a). (b) If f e* , find f(2014)
Problem 2. Consider the one-way layout ANOVA model, where we assume that Yij = μί-cij,に1, . . . , I and J 1, . . . ,J, where μί's are fixed unknown with zero mean treatment means and eiy's are random errors , al such that Σ-lai -0 and E[Yj-μ + ai,1- Show that there exists unique numbers μ, ai, a. b. Show that the null hypothesis Ho : μ,-...- μι is equivalent to Ho : 01 ,-. . .-a1-0
Please explain very carefully!
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 > 0 are unknown. (a) (5 marks) Let μ+σ~p denote the p-th quantile of the N(μ, σ*) distribution. What does this mean? (b) (10 marks) Determine a UMVU estimate of,1+ ơZp and justify your answer.
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 >...
Suppose Xi, X2, ,Xn is an iid N(μ, c2μ2 sample, where c2 is known. Let μ and μ denote the method of moments and maximum likelihood estimators of μ, respectively. (a) Show that ~ X and μ where ma = n-1 Σηι X? is the second sample (uncentered) moment. (b) Prove that both estimators μ and μ are consistent estimators. (c) Show that v n(μ-μ)-> N(0, σ ) and yM(^-μ)-+ N(0, σ ). Calculate σ and σ . Which estimator...
Consider the process Y.-μ + et-o, et-1-912 et-12, where {ed denotes a white-noise process with mean 0 and variance σ? > 0. Assume that et ls independent of Yt-1, Yt-2, Find the autocorrelation function for (Yt).
Let y'(x)y(x)g'(x) = g(x)g'(x), y(0) = 0, x e í, where f'(x) denotes ar(X) and g(x) is a given non- 4. dx constant differentiable function on R with g(0) = g(2) = 0. Then find the value of y(2)
Let y'(x)y(x)g'(x) = g(x)g'(x), y(0) = 0, x e í, where f'(x) denotes ar(X) and g(x) is a given non- 4. dx constant differentiable function on R with g(0) = g(2) = 0. Then find the value of y(2)
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,...