4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwi...
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
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
Question 5. [10 Marks] Suppose . . . ,X, be an SRS from a uniform distribution between θ and 0. a) Į1 Mark] Find the moments estimator (ME) θί of θ. b) [1 Markl Let Y- min(X1,... ,Xn) and its pdf is as follows. -ny"-1 for ye(θ,0); for y E (6,0), -, 0, -, fy(y) otherwise. Show that the maximum likelihood estimator (MLE) θ2 = ntly of θ is unbiased. c) [4 Marks] which one of θ1 and θ2 is...
I need the answer for (ii) 1. A certain continuous distribution has cumulative distribution function (CDF) given by F(a)-0, <0 where θ is an unknown parameter, θ > 0. Let X, be the sample miean and X(n) = max {Xu X2, ,..} 0) Show that n +, is an unbiased stimator of o Find its mean squnare error and check whether θι, is consistent for θ. (ii) Show that 2n- Xn) is a consistent estimator of fe (iii) Assume n...
Let X1, X2, ..., Xn be a random sample with probability density function a) Is ˜θ unbiased for θ? Explain. b) Is ˜θ consistent for θ? Explain. c) Find the limiting distribution of √ n( ˜θ − θ). need only C,D, and E Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
1. (20 points) Let X1....X be a random sample from a uniform distribution over [0,0]. (a) (4 points) Find the maximum likelihood estimator (MLE) 0 MLE for 0. (b) (3 points) Is the MLE ONLE unbiased for 0? If yes, prove it: If not, construct an unbiased estimator 0, based on the MLE. (c) (4 points) Find the method of moment estimator (MME) OM ME for 8. (d) (3 points) Is the MME OMME tnbiased for 6? If yes, prove...
Ql- Let X be a random variable with the following probability distribution: fx) Find the maximum likelihood estimator of θ, based on a random sample size n. (0+1)x -(8 + 1)xe Ql- Let X be a random variable with the following probability distribution: fx) Find the maximum likelihood estimator of θ, based on a random sample size n. (0+1)x -(8 + 1)xe
[20 marks] Let xi, . . . , Xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo 〈 x 〈 oo, θ > 0, 2πθ nx, and r2 - enote T-1 Tn (d) [3 marks] Find the maximum likelihood estimator θ2 of. (You do not need to perform the second derivative test.) (e) 3 marks Find the Fisher information T( (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer....