5. Find a method-of-moments estimator (MME) of θ based on a randorn sample Xi, ,Xn from...
5. Find a method-of-moments estimator (MME) of θ based on a random sample XI, , X, from each of the following distributions (a) f(z; θ)-0( 1-0)1-1 , x-1, 2, . . . . 0 (b) f(z; 0) = (0 + 1)2-0-2, x > 1,0 > 0 (c) fr) re, 0, θ 1
,X, from each of the Find a method-of-moments estimator (MME) of θ based on a random sample X1, following distributions (a) f(z; θ) = θ(1-0)x-1, x = 1, 2, . . . . 0 < θ < 1 (c) f(x:0) = θ2xe-ez, x > 0, θ > 0
6. Let Xi,.Xn be a random sample from the pdf Find the method of moments estimator of θ.
5. Let X1,.. ., Xn be a random sample from Uniform(0,0) with an unknown endpoint θ > 0, we want to estimate the parameter θ (a) Find the method of moments estimator (MME) of θ. (b) Find the MLE θ of θ (c) (R) Set the sample size as 25, do a simulation in R to compare these two esti- mators in terms of their bias and variance. Include a side-by-side boxplot that compares their sampling distributions
2. Let X 1, , Xn be iid from the distribution modeled by 8-2 fx (1:0)-(9. θ):r"-"(1-2) dr where 0 < x < 1 and θ > 1 Find the MME (method of moments estimate/estimator) for 0
l. Find the maxinum likelihood estimator (MLE) of θ based on a random sample X1 , xn fronn each of the following distributions (a) f(x:0)-θ(1-0)z-1 , X-1, 2, . . . . 0 θ < 1
Only ques 4 (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , Xn ~ (iid) Uniform(0,0), E(R)-θ . What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively. 4. Let X. Xn be a random sample from a population with pdf xotherwise Let Xa)<..< X(n) be the order statistics. Show that Xa)/X() and X(n) are independent random variables 5....
2. Let Xi,..., Xn be a random sample from the pd f (a) Find the method of moments estimator of θ. (b) Find the maximum likelihood estimator of θ.
2. Let Xi,... ,Xn be a random sample from a distribution with p.d.f for 0 < x < θ f(x; 0) - 0 elsewhere . (a) Find an estimator for θ using the method of moments. (b) Find the variance of your estimator in (a).
2) 6. Let Xi, , xn be i.i.d. Ņ(μ, σ (a) Find the sample analogue estimator of θ (b) Find the ML estimator of θ. 2) 6. Let Xi, , xn be i.i.d. Ņ(μ, σ (a) Find the sample analogue estimator of θ (b) Find the ML estimator of θ.