According to the random sample of froma distribution with probability function ias
for
elsewhere .
(a) Now the likelihood estimation is calculated as
Taking log both side we get
Taking derivative both side with respect to we get
first order condition of optimization.
as
Therefore likelihood of the sample is (maximum likelihood estimator)
(b) According to the random sample of froma distribution with probability function , joint probability density function is
Therefore by replacing we get
Now according to the factorization theorem we can factored the joint proabbility function as one which is depends on the parameter with the statistics and another is independent on the parameter as
Where depends on parameter and is independent of parameter .
Therefore the factorization theorem shows that is a sufficient statistics for and as is one to one relationship with , then also a sufficient statistics for .
QUESTION 3 17) Let Xi. X. X be a random sample from a distribution with probability...
Let XI, X2, , Xn İs a random sample from the probability density function Use factorization theorem to show that X(1) = min(X1 , . . . , Xn) is sufficient for θ Is X(1) minimal sufficient for θ? a. b.
2.a. Let X1, X2, ..., X., be a random sample from a distribution with p.d.f. (39) f( 0) = (1 - 1) if 0 < x <1 elsewhere ( 1 2.) = where 8 > 0. Find a sufficient statistic for 0. Justify your answer! Hint: (2(1-)). b. Let X1, X2,..., X, be a random sample from a distribution with p.d.f. (1:0) = 22/ if 0 < I< elsewhere where 8 >0. Find a sufficient statistic for 8. Justify your...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter and rate parameter λ, Γ(a) Here α > 0 and λ > 0. Let θ sufficient statistic for the data (α, β). Determine the log-likelihood, I(0), and a 2-dimensional b) Suppose that xi,...,In are a random sample from a U(-0,) distribution, 1/(20) if- otherwise x-θ f(x;0)-' 0, Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood...
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
2 Let X1, X2, ...,X, be independent continuous random variables from the following distribution: f(3) = ox-(0-1) where : > 1 and a > 1 You may use the fact: E[X]- .- 2.1 Show that the maximum likelihood estimator of a is ômle = Ei log Xi 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency? 2.4 Show that the fisher information in the whole sample is: 1(a)= 2.5 What Cramer Rao lower bound...
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, . . . , Xn be a random sample from a population with density 8. Let Xi,... ,Xn be a random sample from a population with density 17 J 2.rg2 , if 0<、〈릉 0 , if otherwise ( a) Find the maximum likelihood estimator (MLE) of θ . (b) Find a sufficient statistic for θ (c) Is the above MLE a minimal sufficient statistic? Explain fully.
2. Let Xi, X,.., Xn denote a random sample from the probability density function Show that X(i) = min(X1,X2, . . . , Xn} is sufficient for ?. Hint: use an indictorfunction since the support depends on ?