4. (6 marks) Consider a random sample of size n from a distribution with pdf f(x:0)...
Consider a random sample of size n from the distribution with pdf (In )* f(x; 0) = { 0.c! -, 10, =0,1,... otherwise where 0 > 0. (a) (10 pts) Find a complete sufficient statistic for 0. (b) (10 pts) Using Lehmann-Scheffe theorem, find the UMVUE of Ine. You may need the identity c=
Consider a random sample of size n from a distribution with pdf (In O* S(x; 6) = Ox! x = 0, 1, ...;0 > 1 10 otherwise (a) Find a complete sufficient statistic for 8. (b) Find the MLE of O. (c) Find the CRLB for 6 (d) Find the UMVUE of In e. (e) Find the UMVUE of (In )? (1) Find the CRLB for (In 02
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE. Let Xi , X2,. … X, denote a random sample of size n...
8. Consider a random sample of size n from a distribution with pdf f(x) = 0 else (a) Find the pdf of the smallest order statistic, X(i) b) Find E() and Var(X)) c) Find the pdf of the largest order statistic, X(n)
1. Consider a random sample of size n from a population with pdf: f (x) = (1 -p-p, 0 <p<1, x= 1, 2, ... (a) Show that converges in probability to p. (b) Show that converges in probability to p (1 – p). (c) Find the limiting distribution of
Consider a random sample of size n from a two-parameter exponential dist EXP(e, n). Recall from Exercise 12 that X 1 ., and X are jointly sufficient for O Because Xi:n is complete and sufficient for η for each fixed value of θ, argue from 104.7 that X, and T X1:n X are stochastically independent. ibution, X, 30. Theor (a) Find the MLE θ of θ. (b) Find the UMVUE of η. (c) Show that the conditional pdf of Xi:n...
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
3.4 Let X,, X be a random sample of size n from the U(Q,62) distribution, 6, and let Y, and Yn be the smallest and the largest order statistics of the Xs (i) Use formulas (28) and (29) in Chapter 6 to obtain the p.d.f.'s of Y and Y and then, by calculating depending only on Yi and 1,- Part i. (Note: it is not saying to find the joint pdf of Yi and Yn Find their marginal Theorem 13...
3x2 for 0 < x < θ and zero otherwise. With the parameter θ > 0. We wish to Consider the pdf,f(x) estimate θ using the sample maximum from a random sample (iid) of size n. 0n-maxi Xi. (hint: first find the CDF and PDF of the estimator) Show this estimator is consistent a. b. Show this estimator is biased C. Suggest a better estimator and show that it is UC d. Show that n(9-an) converges (using the original estimator,...
4. A sample of size n-81 is taken from an exponential distribution with the pdf f(x)-Be-6x, θ > 0, x > 0. The sample mean is i-35. Find a 95% large- sample confidence interval for θ using the Central Limit Theorem.