1.5 If Xi, ..., X, are independent r.v.'s distributed as B (k, θ), θ e Ω2(0.1),...
2.4 Consider the independent r.v.'s x,..,.X with the Weibull p.d.f. (i) Show that- is the MLE of θ.
5.7 Let X, X, be independent r.v.'s from the u(e -a, o+ b) distribution, where a and b are (known) positive constants and θ Ω M. Determine the moment estimate θ of θ, and compute its expectation and variance.
P(8), θ eQa(0 4.6 Let X, ..., X, be independent r.v.'s distributed as estimate δ(x , , x )-r and the loss function L ( , δ)-[8-6(5- ,o0 ), and consider the -5)]218. E,[e-5(X, ,X,)],andshow that it isin dependent ofe. R(0:δ)- (i) Calculate the risk (ii) Can you conclude that the estimate is minimax by using Theorem 9?
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
Practice Exam Questions 2 Let X be a r.v. with density function x 2 1 a. Determine the distribution function of X, i.e. F(x). Find E(X) and V(X) b. Find the MLE estimator of θ constructed from a sample Xi,Xn c. Is the estimator find in (b) biased? Practice Exam Questions 2 Let X be a r.v. with density function x 2 1 a. Determine the distribution function of X, i.e. F(x). Find E(X) and V(X) b. Find the MLE...
Exercise 3.16: A sample of n independent observations is taken on a rv. X having a logarithmic series distribution, x=1, 2, EWT-0), , x In . Show that the MLE θ of θ where θ is an unknown parameter in the range (0,1) satisfies the equation e+ ž(1-0) ln(1-9-0, Fuercio ti tample mean. Find the asymptotie distribution oftå. Exercise 3.16: A sample of n independent observations is taken on a rv. X having a logarithmic series distribution, x=1, 2, EWT-0),...
Let Xi, , xn be a randon sannple fron f,(z0)-e-(z-0),0 є (-00,00), z > θ a. Show that X(1) is a complete statistic for θ. Hint: First find the PDF of XI) b. Show that the sample variance S is an ancllar statistic,and use this result to show that Xa) and S2 are independent.
Please show every step, thank you. Let Xi ~ N(μ, σ?), where ơỈ are known and positive for i-1, are independent. Let /- (a) Find the mean and variance of μ. (b) Compare μ to X,-n-Σί.i Xi as an estimator of μ. , n, and Xi, X, , E-1(1/o .m be the MLE of μ. Let Xi ~ N(μ, σ?), where ơỈ are known and positive for i-1, are independent. Let /- (a) Find the mean and variance of μ....
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
We have n independent observations from a geometric distribution with unknown parameter θ. Po(X,-k-θ(1-0)4-1 for k-1, 2, 3, . . . We wish to test the null hypothesis θ-1/2 versus the alternative θ 7|/2. we can show that the MLE θ-1/2. Write out the appropriate LRT statistic as a function of the r, the mean of the observations