Let X1,.....,Xn be a random sample from N(μ,σ2), and both μ and σ2 are unknown, with -∞<μ<∞ and σ2 > 0.
a. Develop a likelihood ratio test for H0: μ <= μ0 vs. H1: μ > μ0
b. Develop a likelihood ratio test for H0: μ >= μ0 vs. H1: μ < μ0
Let X1,.....,Xn be a random sample from N(μ,σ2), and both μ and σ2 are unknown, with...
Let X1,.....,Xn be a random sample from N(μ,σ2), and both μ and σ2 are unknown, with -∞<μ<∞ and σ2 > 0. a. Develop a likelihood ratio test for H0: μ <= μ0 vs. H1: μ > μ0 b. Develop a likelihood ratio test for H0: μ >= μ0 vs. H1: μ < μ0
Let X1,.....,Xn be a random sample from N(μ,σ2). If μ is unknown but σ2 is known, develop a likelihood ratio test for H0: μ >= μ0 vs. H1: μ < μ0
Let X1,...,Xn be iid N(μ,σ2) with known μ and unknown σ. For α in (0,1), obtain the UMP level α test for H0: σ=σ0 vs. H1: σ>σ0
, Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus , Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus
Let X1, X2, . . . , Xn be IID N(0, σ2 ) variables. Find the rejection region for the likelihood ratio test at level α = 0.1 for testing H0 : σ2 = 1 vs H1 : σ2 = 2.
Suppose that X1,X2, ,Xn are iid N(μ, σ2), where both parameters are unknown. Derive the likelihood ratio test (LRT) of Ho : σ2 < σ1 versus Ho : σ2 > σ.. (a) Argue that a LRT will reject Ho when w(x)S2 2 0 is large and find the critical value to confer a size α test. (b) Derive the power function of the LRT
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
3. Let X1, X2, ,Xn be a random sample from N(μ, σ2), and k be a positive integer. Find E(S2). In particular, find E(S2) and var(s2).
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
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).