5. Suppose X is a continuous RV modeled by f(x; a) =-e-le-al where-oo < x < 00, If a random sample of size n is drawn with n odd, show the MLE for α is the median of the sample.
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
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e)...
4. Let Xi,..., Xn be a random sample with density 303 for 0 < θ < x NOTE: We have previously found that θMLE-X(1) and that FX(1) (x)-1-(!)3m (a) Using the probability integral transform method, find a pivot for 0 based on the MLE. (b) Use the pivot found in (a) to get an ezact 100(1-a)% C.1. for θ (c) Find an approximate 100(1-a)% C.1. for θ based on our result for the MLE. (d) Suppose that we get n...
R1. Now that you've seen both MME and MLE, we might begin comparing the two worlds. In class, we studied X Uni (0.0) and showed the MLE is aMLE = max Xi One can show the MME for this setup is θΜΜΕ 2 X. As seen in HW2, these estimators are RVs, and each will have its own (sampling) distribution. The sampling distribution gives a good sense of what types of values you'll get from θ when you draw a...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE. Let Xi , X2,. … X, denote a random sample of size n...
R1. Now that you've seen both MME and MLE, we might begin comparing the two worlds. In class, we studied XUnif(0,0) and showed the MLE is OSILE-max Xi. One can show the MME for this setup is OMME -2 X. As seen in HW2, these estimators are RVs, and each will have its own (sampling) distribution. The sampling distribution gives a good sense of what types of values you'll get from θ when you draw a random sample. Use the...
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...
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...