4. Let Xi,..., Xn be a random sample with density 303 for 0 < θ < x NOTE: We have previously foun...
Let Xi., Xn be a random sample from the distribution with density f(r, θ)-303/2.4 for x > θ and 0 otherwise. Determine the MLE of θ and derive 90% central CI interval for θ. If possible find an exact CI. Otherwise determine an approximate CI. Explain your choice Let Xi., Xn be a random sample from the distribution with density f(r, θ)-303/2.4 for x > θ and 0 otherwise. Determine the MLE of θ and derive 90% central CI interval...
3. Let X1, X2,... , Xn be i.i.dExp(0) NOTE: We have previously found that θMLE X (a) Assuming a large n' derive an approximate 100(1-a)% L for θ that is based on the MLE (b) Now use the pivot Xi ~ χ n to derive an exact 100(1-a)% C.1, for θ. (c) Data was collected on the service time (in minutes) of 100 customers in a bank teller The average service time was 4.382 minutes. Based on the histogram of...
2. Let Xi, X2, . Xn be a random sample from a distribution with the probability density function f(x; θ-829-1, 0 < x < 1,0 < θ < oo. Find the MLE θ
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.
5. Let Xi, . . . , Xn be a random sample from f(x:0) = -| for z > 0. (a) Assume that θ 0.2 Using the Inversion Method of Sampling, write a R function to generate data from f(x; 0). (b) Use your function in (a) to draw a sample of size 100 from f(0 0.2 (c) Find the method of moments estimate of θ using the data in (b). (d) Find the maximum likelihood estimate of θ using...
Let X,...Xn be a random sample from the density fx(x) = 1+θX^θ, 0<x<1 a) Use the Neymar-Pearson lemma to determine the best critical region for testing Ho: θ-θo against H1 θ-θ1 > θo
20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where 0 〈 θく1 is parameter. Show that unbiased estimator of θ for a fixed m. is a uniform minimum variance 20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where...
Let Xi,..., Xn iid from random variables with probability density function, (0+1)x" 1, ?>0 (x)o for 0 < otherwise (a) Find the method of moments estimator for ? (b) Find the mle for (e): Under which condition is the mle valid?
Additional Question i.i.d. ˆ Fix θ > 0 and let X1,...,Xn ∼ Unif[0,θ]. We saw in class that the MLE of θ, θMLE = max(X1, . . . , Xn), is biased. I give two other estimators of θ, which can be made unbiased by appropriate choice of constants C1, C2: ADDITIONAL QUESTION Fix θ 0 and let Xi, . . . , Xn iid. Unifl0.0]. We saw in class that the MLE of θ, θΜ1E- max(Xi,..., Xn), is biased....
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).