Suppose that x = (x1,…..,xn) is a sample from the Gamma Distribution. Determine the minimal sufficient statistics for this model.
Suppose that x = (x1,…..,xn) is a sample from the Gamma Distribution. Determine the minimal sufficient...
6.1.12 Suppose that (x1,..., xn) is a sample from a Geometric(θ) distribution, where θ ∈ [0, 1] is unknown. Determine the likelihood function and a minimal sufficient statistic for this model. (Hint: Use the factorization theorem and maximize the logarithm of the likelihood.)
Suppose that X1,..., Xn is a random sample from a gamma distribu- tion, The gamma distribution has parameters r and λ, and also has E(X)-r/λ and Var(X)-r/ P. Calculate the method of moments MOM) estimators of r and λ in terms of the first two sample moments Mi and M2 Suppose that X1,..., Xn is a random sample from a gamma distribu- tion, The gamma distribution has parameters r and λ, and also has E(X)-r/λ and Var(X)-r/ P. Calculate the...
2. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with shape parameter α and rate parameter λ, Here α > 0 and λ > 0. Let θ-(α, β). Determine the log-likelihood, 00), and a 2-dimensional sufficient statistic for the data (b) Suppose that xi, ,Xn are a random sample from a U(-9,0) distribution. f(x; 8) otherwise Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood should...
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
Let X1,..., Xn be a random sample from a distribution. Suppose Ti (X),T2(X) and U(X) respectively are sufficient, minimal sufficient, and unbiased estimators for the parameter θ of the distribution. Let U1(X) = E U(X) T, (X), U2(X) = EU㈤ T2(X)] a. Show that U1(X) and U(X) are unbiased for θ. b. Show that U2(x)-E[Uj(X)ITLX] c. Show that U2 has a smaller variance than U
Let X1, ..., Xn be a random sample from a distribution with pdf 2πσχ (a) If σ and μ are both unknown, find a minimal sufficient statistic T. (b) If σ is known and μ is unknown, is T from last part a sufficient statistic? Is it a minimal sufficient statistic? Prove your answer. (c) Let V (II1 X)/m, what is the distribution of V? Are V andindependently distributed? Let X1, ..., Xn be a random sample from a distribution...
Problem 3 Let X1, X2, ... , Xn be a random sample of size n from a Gamma distribution fr; a,B) 22-12-1/B, 0 < < (a) Find a sufficient statistics for a. (b) Find a sufficient statistics for B.
Q. 4 Let X1, ... , Xn be a random sample from gamma (2,0) distribution. c) Show that it is the regular case of the exponential class of distributions. d) Find a complete, sufficient statistic for 0. e) Find the unique MVUE of 0. Justify each step.
Suppose X1, X2, · · · , Xn form a random sample from a distribution with p.d.f. f(x;?)=(1+?)x?, 0<x<1, ?>0. a. Find the MLE of ?. b. Show that the MLE is sufficient for ?.
Let X1 ,……, Xn be a random sample from a Gamma(α,β) distribution, α> 0; β> 0. Show that T = (∑n i=1 Xi, ∏ n i=1 Xi) is complete and sufficient for (α, β).