Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where...
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a),...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
Conditional on θ, the random variables X1, X2, ,Xn are îid from In turn, the parameter θ is best regarded as random with prior distribution αθ where a 0 is known (a) Find the posterior mean of θ (b) Discuss how you would formulate the Bayesian test of versus Conditional on θ, the random variables X1, X2, ,Xn are îid from In turn, the parameter θ is best regarded as random with prior distribution αθ where a 0 is known...
Suppose that X1, X2,..., Xn are iid from where a 1 is a known constant and θ > 0 is an unknown parameter. (a) Show that the likelihood ratio rejection region for testing Ho : θ θο versus H : θ > θο can be written in terms of X(n), the maximum order statistic. (b) Derive the power function of the test in part (a). (c) Derive the most powerful (MP) level α test of Ho : θ-5 versus H1...
Specifically, suppose that Xi, X2, .., Xn denote n payments, modeled as iid random variables with common Weibull pdf 0, otherwise, where m > 0 is known and θ is unknown. In turn, suppose that θ ~ IG(α, β), that is, θ has an inverted gamma (prior) pdf 0, otherwise (a) Prove that the inverted gamma IG(α, β) prior is a conjugate prior for the Weibull family above. (b) Suppose that m-2, α-05, and β-2. Here are n-10 insurance payments...
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (a) Find the maximum likelihood estimator (MLE) of θ (b) Give the form of the likelihood ratio test for Ho : θ-Bo versus H1: θ > θο. (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 versusS You must give an explicit expression...
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
Suppose X1, X2, .., Xn is an iid sample from where >0. (a) Derive the size α likelihood ratio test (LRT) for Ho : θ-Bo versus H : θ θο. Derive the power function of the LRT (b) Suppose that n 10, Derive the most powerful (MP) level α-0.10 test of Ho : θ 1 versus Hi: 0-2. Calculate the power of your test
Suppose that Xi, X2,..., Xn is an iid sample from 20 for x R and σ 〉 0. (a) Derive a size α likelihood ratio test (LRT) of H0 : σ (b) Derive the power function β(o) of the LRT 1 versus H1 : σ 1.
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise 7.5: Suppose X1, X2, . .. , sufficient for θ. a) Show that the smallest and largest of Xi, ..., Xn are jointliy (b) If p@-constant, θ e (-00, oo), is the prior distribution of θ, find its posterior distribution xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+...