6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi...
Recall that ?-n/ ??-1 log Xi is the mle of ? for a beta(8.1) distribution.Also -_ ? ial log Xi has the gamma distribution ?(n.18) (a) Show that 2eW has a x2 (2n) distribution (b) Using part (a), find c1 and c2 so that 2?? for 0 < ? < 1 . Next, obtain a (1-?)100% confidence interval for ? (c) For ? = 0.05 and n = 10, compare the length of this interval with the lengthof the interval...
5. Let X1,X2,. Xn be a random sample from a Beta(0, 1) distribution. Recall that W -Σ-1 logXi has the gamma distribution Γ(n,1/8) a) Show that 2θW has a χ"(2n) distribution b) Using part a), find c1 and c2 so that P (cı < 쯩 < c2)-1-α, for 0 < α obtain a (1-a) 100% CI for 20n 1, and then
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
, Xn is a sample from a uniform distribution (o, e), you already saw that t-X(n) is the me их1, X2, Of θ. obtain the formula for the confidence interval for θ by using the distribution of Y-X(n)/9. That is, find the α/2 th percentile and the (1-α/2) th percentile of the distribution of w-X(n)/9. hie by solving for w-α/2 Hint: Obtain wi-a/2 in the equation: And obtain Wa/2 by solving for Wa/2 in the equation: Note: the distribution of...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
3. Suppose that ai . ,,an are a random sample from a N( ,02) distribution. Recall that the MLE in this case is [a, σ]T = [x, V (n-1)s2/n]T and the information matrix is Consider the data s2-4.84 with n 16 (a) Use the delta-method to obtain an approximate 95% confidence interval for log(o) (b) Obtain an approximate 95% confidence interval for σ2 using the confidence interval from (a). Compare to the exact interval, [2.21,15.77], and approximate interval [0.43, 10.50...
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).
em 3. Let Xi. A.2. . . . A., be i. i.d. random variables from an exponential diatribatnn-nsmesn be i.i.d. random variables from an exponential distribution with mean Ame and let } samples are independent. Recall that an exponetial random variable with mesn 9 hiss deaity 0 (a) Assuming that θ = θ-θ2, find the MLE of θ when X!, . . , Xn and Yi, ,Yn are observed. (b) Find the LRT to test the hypothesis that θ,-, versus...
Problem 2: Let Xi, X2,..., Xn be i.i.d. random variables with common probability density function 3 -6x21 (i) Calculate the MLE of 0 (ii) Find the limit distribution of Vn(0MLE - 0) and use this result to construct an approximate level 1-α C.I. for θ. [Your confidence interval must have an explicit a form as possible for full credit.] (iii) Calculate μι (0)-E0(Xi) and find a level 1-α C.İ. for μι (0) based on the result in (ii) or by...