. Let Yi.... Yn be a random sample from a distribution with the density function 393 fe(y) =- Is there a UMP test at level α for testing Ho : θ test? vs. Hi : θ > 6? If so, what is the . Let...
Let X be a sample of size 1 from a Lebesgue p.d.f. fe. Find a UMP test of size α (0, ) for Ho : θ-Bo versus Hi : θ-A in the following cases: (a) foo)+( and fo, (x) ) Let X be a sample of size 1 from a Lebesgue p.d.f. fe. Find a UMP test of size α (0, ) for Ho : θ-Bo versus Hi : θ-A in the following cases: (a) foo)+( and fo, (x) )
2. Let Yı, ..., Yn be a random sample from an Exponential distribution with density function e-, y > 0. Let Y(1) minimum(Yi, , Yn). (a) Find the CDE of Y) b) Find the PDF of Y (c) Is θ-Yu) is an unbiased estimator of θ? Show your work. (d) what modification can be made to θ so it's unbiased? Explain.
3. Let Xi, ,X, be i.id. from a normal distribution N(1,0), for θ > 0, Find a UMP test of size α for testing Ho : θ < θο versus H1 : θ > θο. 3. Let Xi, ,X, be i.id. from a normal distribution N(1,0), for θ > 0, Find a UMP test of size α for testing Ho : θ θο.
1. (a) Let Yi,... , Yn be a random sample from a distribution with mean θ and finite variance σ2. Find the BLUE of θ and justify that it is, in fact, the Best Linear Unbiased Estimate. sample variance. 1. (a) Let Yi,... , Yn be a random sample from a distribution with mean θ and finite variance σ2. Find the BLUE of θ and justify that it is, in fact, the Best Linear Unbiased Estimate. sample variance.
3. Let X,,X,,..., X, be a random sample from a Gamma 40distribution, where 6>0. we wish to test H0 : θ-1 vs. Hi : θ #1. Show that the likelihood ratio test statistic, A , can be written as A(V) where a. What is the distribution of V? what is the null distribution of what will be the rejection region for an α level test? b. 20 d. 3. Let X,,X,,..., X, be a random sample from a Gamma 40distribution,...
xn be i.id. from the Pareto distribution Pa(θ, c), θ Let Xi 0 (a) Derive a ÚMP test of size α for testing Ho : θ θ when c is known. versus Hi : xn be i.id. from the Pareto distribution Pa(θ, c), θ Let Xi 0 (a) Derive a ÚMP test of size α for testing Ho : θ θ when c is known. versus Hi :
1. )To test Ho: X ~ N(θ, l), against Hi : X ~ C(1,0), a sample of size 2 is available on X. Find a UMP invariant test of Ho against Hi 2. Let Xi, X2, , Xn be a sample from PA) Find a UMP unbiased size test 1. )To test Ho: X ~ N(θ, l), against Hi : X ~ C(1,0), a sample of size 2 is available on X. Find a UMP invariant test of Ho against...
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
Yi, Y2...., Yn is a random sample from the Uniform distribution ([a, b]). Let u to be the population mean, one wants to test Ho : μ = 1 against Ha : μ 1. Suppose n is large, and both the one-sample t-test and the binomial test can be applied here. Derive the approximate analytic formula for computing the power for each of the test. Besides the sample size n and significance level α, what quantity is essential in the...
Let X be a sample of size 1 from a Lebesgue p.d.f. fo. Find a UMP test of size α (0, ) for Ho : θ--θ : θ-0, in the o versus H1 tollowing case: Let X be a sample of size 1 from a Lebesgue p.d.f. fo. Find a UMP test of size α (0, ) for Ho : θ--θ : θ-0, in the o versus H1 tollowing case: