(Pareto MLE and MOME in astronomy) In astronomy, the distribution of asteroid sizes is of great...

 

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(Pareto MLE and MOME in astronomy) In astronomy, the distribution of asteroid sizes is of great interest (ever watched “Armageddon"?). Denoting the size of asteroid i X;, a common model is: a 1 if r>0 fx(x) else Here, 0 is the minimum size of an asteroid; let's call it 1 meter for now. Suppose we observe a sample of n asteroid sizes X1,..., Xn- 1. Holding 0 fixed, what is the likelihood function for a? 2. What is the MLE of a? 3. Now, assume a is fixed. What is the likelihood function for 0? What is its MLE? 4. Now assume that 0 = 3 meters. Find a MOM estimator for a (The mean of a pareto is a, for a > 1). It is ok to assume a is at lcast 1. %3D 5. Draw obscrvations from a Parcto with a = 2 and 0 = 3. Compute the MLE and MOM estimates for a for various sample sizes. Compare the accuracy of the estimates in a plot, as a function of sample size. (Pareto MLE and MOME in astronomy) In astronomy, the distribution of asteroid sizes is of great interest (ever watched “Armageddon"?). Denoting the size of asteroid i X;, a common model is: a 1 if r>0 fx(x) else Here, 0 is the minimum size of an asteroid; let's call it 1 meter for now. Suppose we observe a sample of n asteroid sizes X1,..., Xn- 1. Holding 0 fixed, what is the likelihood function for a? 2. What is the MLE of a? 3. Now, assume a is fixed. What is the likelihood function for 0? What is its MLE? 4. Now assume that 0 = 3 meters. Find a MOM estimator for a (The mean of a pareto is a, for a > 1). It is ok to assume a is at lcast 1. %3D 5. Draw obscrvations from a Parcto with a = 2 and 0 = 3. Compute the MLE and MOM estimates for a for various sample sizes. Compare the accuracy of the estimates in a plot, as a function of sample size.

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1305 Solutim Let x has pdf fxx vor 2 1 Let is fixed Likelihood ... View the full answer