In this problem, we will investigate how to find the MLE. Write out the mathemat

Question

In this problem, we will investigate how to find the MLE. Write out the mathematical equation of the log-likelihood function. Why is it difficult to maximize this function directly? It turns out, finding the MLE analytically (via the standard approach of taking derivatives of the log-likelihood, setting it equal to 0, and solving) is impossible (as is finding the mean and variance of the MLE!). Good thing we have R! Write a function logic(alpha, stamp) that takes as input the parameter a and a sample of size n, and returns the log-likelihood of the sample evaluated at a. Write a function called find.mle(samp) that takes as input a sample of size n, and returns the MLE & MLE. If you want to verify that your function is correct, generate a random sample of size n = 20 with the following seed from the above pdf with a = 5 and verify that your find.mle() function returns a MLE = -5.70379:

Explanation / Answer

I write the R-command for a particular distribution. I used inverse weibull distribution to estimate the parameters by using nlm command

Assume that we have a data set or it can be generate because in real life application is a mostly observed things.

> rm(list=ls(all=TRUE))
> x=c(0.84,0.88,0.91,0.93,0.95,1.01,1.1,1.1,1.14,1.15,1.17,1.29,1.29,1.38,1.43,1.86,
+ 1.87,1.89,2.01,2.4,2.55,2.91,3.24,3.73,4.16,4.79,6.67,8.39,8.6,10.77)
> n=length(x)
> ######################### IWD ####################
> LogL1=function(th){
+ bt=th[1]; ld=th[2]
+ z1=n*(log(ld)+log(bt))-(1+bt)*sum(log(x))-ld*sum(x^(-bt))
+ return(-z1)
+ }
> M1=nlm(LogL1, c(3,2), hessian=T)
> Mbt1=M1$estimate[1]; Mld1=M1$estimate[2];
> Mbt1
[1] 1.854932
> Mld1
[1] 1.937413

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