2.11 Refer to Exercise 1.11, and determine: (i) A sufficient statistic for a whenỗ is known...
1.11 function p.d.f., , is a On the basis of a random sample of size n from this p.d.f., determine the MLE of: (ii) a when β is known.
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a. with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
a) Show that Σ.1X, and Σηι x? are jointly sufficient statistics for two un known parameters of the normal distribution N(01,02) (based on the data sample X1,..., Xn) in two ways: by factorization theorem and using the property of the exponential family. b) Show that X and s2 are jointly sufficient statistics for the same distribution. c) Give yet another example of a couple of jointly sufficient statistics. Hint: Example 6.7-5 in Hogg et al. Anyway, make sure to include...
2. Let Xi, , Х, be a random sample gamma(a, β). In parts (a-(d) assume a is known. 30 points a. Consider testing H. : β--βο. Derive Wald statistic for testing H, using the MLE of B both in the numerator and denominator of the statistic. b. Derive a test statistic for testing H, using the asymptotic distribution of the MLE of β. What is the relation between the two statistics in parts (a) and (b)? c. Derive the Score...
Problem 7 a) Show that n 1 Xi and n 1 X? are jointly sufficient statistics for two un- known parameters of the normal distribution N(01, 02) (based on the data sample Xi,..., Xn) in two ways: by factorization theorem and using the property of the exponential family. c) Give yet another example of a couple of jointly sufficient statistics. solution into your homework (not just refer to the aforementi statistic, have you checked that this function is invertible? b)...
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1 where I(.) denotes the indicator function. (a) Find a 2-dimensional sufficient statistic for the mode (b) Suppose θ is a known constant. Find the MLE for v. (d) Suppose v-1. Find the MLE for and determine its asymptotic distribution. Carefully justify your answer and state any theorems that you use. (e) Suppose1. Find the asymptotic distribution of the MLE estimator of exp[- Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1...
3.18 Let the r.v. X has the Geometric p.d.f. (i) Show that X is both sufficient and complete. U(X )-1 (ii) Show that the estimate U defined by: estimate of 6 if X-1, and U(X) -0 if X 2 2, is an unbiased (iii) Conclude that U is the UNU estimate of θ and also an entirely unreasonable estimate.
Problem 6. Gamma distribution is defined by its density function T(a) when t 0 and 0 otherwise. Here, θ is unknown parameter, a is a known param- eter Г(a) is a normalization constant (like in chi-square distribution discussed in class, see also p. 99 of the textbook) a) Check that Gamma distribution can be written in the exponential form. b) Use this fact to give a sufficient statistic for θ (you can do it with no additional computations after part...
A complex function is given by the following power seres in ǐy: i. Using cither the root formula test or the ratio test, determine a condition on r that is a sufficient condition for the series to be convergent. ii. Explain your result with reference to the Taylor series of some function, and find a value of y for which this function is defined for all values of x. A complex function is given by the following power seres in...