8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x...
Ouestion 7 (10 points)Suppose Y..... y denote a random sample of
size n from an exponential distribu-| tion with mean 9.a) (5
points)Find the bias and MSE of the estimator B1 = nY().b) (3
points)Consider another estimator B, =Y. Find the efficiency of 6,
relative to 62.e) (7 points)Prove that 2 is a pivotal quantity and
find a 95% confidence interval for 8.
Question 7 (10 points) Suppose Y1, ..., Yn denote a random sample of size n from an...
Suppose Y1, ..., Yn denote a random sample of size n from an exponential distribu- tion with mean 0. a) (5 points) Find the bias and MSE of the estimator ôz = nY1). b) (3 points) Consider another estimator ôz = Y. Find the efficiency of ôı relative to 62. c) (7 points) Prove that 297 Yi is a pivotal quantity and find a 95% confidence interval for 0.
1. Let X1, ..., Xn be a random sample of size n from a normal distribution, X; ~ N(M, 02), and define U = 21-1 X; and W = 2-1 X?. (a) Find a statistic that is a function of U and W and unbiased for the parameter 0 = 2u – 502. (b) Find a statistic that is unbiased for o? + up. (c) Let c be a constant, and define Yi = 1 if Xi < c and...
Let X1, X2, ..., Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) max(X1,X2, ...,Xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for e.
1. Suppose that X1, X2,..., X, is a random sample from an Exponential distribution with the following pdf f(x) = 6, x>0. Let X (1) = min{X1, X2, ... , Xn}. Consider the following two estimators for 0: 0 =nX) and 6, =Ỹ. (a) Show that ő, is an unbiased estimator of 0. (b) Find the relative efficiency of ô, to ô2.
Problem 2: Let (X1,... Xn) denote a random variable from X having density fx(x) = 1/ β,0 < x < β where β > 0 is an unknown param eter. Explain why the Cramer Rao Theorem cannot be applied to show that an unbiased estimator of β is MVU. (Hint: see slides. Condition (A) of Cramer Rao Theorem)
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1) = min(X1, X2, ..., Xn) has the cumulative distribution function given by: αη 1 - ( r> B X F(x) = . x <B 0 Show that X(1) is a consistent estimator of ß.
Let X1, X2, ...,Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) = max(X1, X2, ...,xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for 0.
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with mean p and and variance o2. Let sa be the sample variance. We showed in lectures that S2 is an unbiased estimator of o2. (a) Show that S is not an unbiased estimator of o. (b) Find the constant k such that kS is an unbiased estimator of o. Hint. Use a result from Student's Theorem that (n − 1)52 ~ x?(n − 1)...