. Let Yı, . . . , Ý, be a sample from N(0, σ*) distribution. Show...
. Let Yi, ,Ý, be a sample from N(μ, σ2) distribution, where both μ and σ2 are un known Repeat the argument that was given in class to show that is a pivot (start by representing Yj as a linear function of a N(0, 1) random variable). Use the fact that (n-pe, of freedom") to construct the confidence interval with coverage probability 95% for σ2 (you can state the answer in terms of quantiles of X2-distribution, or find their numerical...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
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).
2) Let Yİ,Ý,, ,y, be independent and identically distributed from the distribution with density where c > 0 is a constant and θ > 0. Find the MLE for 60. 2) Let Yİ,Ý,, ,y, be independent and identically distributed from the distribution with density where c > 0 is a constant and θ > 0. Find the MLE for 60.
1. Let Yı,Y2,..., Yn denote a random sample from a population with mean E (-0,) and variance o2 € (0,0). Let Yn = n- Y. Recall that, by the law of large numbers, Yn is a consistent estimator of . (a) (10 points) Prove that Un="in is a consistent estimator of . (b) (5 points) Prove that Vn = Yn-n is not a consistent estimator of (c) (5 points) Suppose that, for each i, P(Y, - of ? Prove what...
Let Y1,…, Y18 be a random sample from a uniform distribution on the interval (0, θ], where θ is an unknown parameter we want to estimate. Two estimators for θ have been suggested: θ_1 = max {Y1, ..., Y18} and θ _ 2 = 2Y¯ = 2 / 18 ∑i = 1, n =18, Yi a) The expectation value of θ _1 and θ_2 can be expressed respectively E [θ_1] = k_1*θ and E [θ_2] = k_2*θ. What is the...
Let X, , . . ., Xn be a random sample from an N(p, ơ2). (a) Construct a (1-α) 100% confidence interval for μ when the value of σ2 is known. (b) Construct a (1-α) 100% confidence interval for μ when the value of σ2 is unknown.
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...
We have a dataset with n= 10 pairs of observations (2, yi), and n n Σ Ti = 683, Yi = 813, i=1 i=1 72 n n <* = 47,405, 1:4: = 56,089, 4 = 66,731. i=1 i=1 i=1 What is an approximate 99% confidence interval for the mean response at Io = 90?