About This Assignment

This mini-project covers chapter 11 of MS in a more in-depth way than the previous quiz. You will be tasked with answering a challenging word problem. You will be asked to show your work on the next assignment page by submitting HTML.

Format: numeric answer "quiz"
Time limit: none
Number of attempts: 3 total

Academic Integrity

Getting help from your peers is not permitted.
Collaborating, or completing the quiz with others in the class, is not permitted.
Copying or reusing previous work is not permitted.
Open-book research for the quiz is permitted and encouraged.

Maximum Likelihood

The method of maximum likelihood may not work as prescribed sometimes. However the essence and logic of MLE will still help to obtain estimators. This problem appears to fail. You will need to think outside the box and come up with a good estimator.

Introduction

Suppose

y<- c(10.03641, 10.06392, 10.20075, 10.17549, 10.04337, 10.03139, 10.19933, 10.11292, 10.21864, 10.01266)

is a random sample (iid). In addition, suppose that the density takes the following form:

$LaTeX: f_Y(y)=e^{-(y-\theta)},\;y\ge\theta,\;\theta>0$

Calculation

Using the maximum likelihood theory and an RMD document find $LaTeX: \hat{\theta}_{mle}$ by doing the following. As you work, be sure to work things out legibly on paper where indicated for later submission.

Plot the above density and data on the same interface. (Note: Use theta = 10 for the purpose of plotting, remember we don't know this value, use an R chunk - you may use Base R plot(), points(), etc)
Attempt to find the MLE using the 5 steps shown in the resources of Chapter 7. (Do this on paper and then take a picture - use ![](){} in RMD)
What is the value of the derivative $LaTeX: \frac{dl}{d\theta}$ for the given data? (Show your work on paper then embed using a picture)
What order statistic will be the estimator? Y_min or Y_max? (Give the answer with a reason)
Find the density expression of Ymin and Ymax using order statistics theory, using general data $LaTeX: y_1,y_2, \ldots, y_n$ (Use a picture of your working and the final density in Latex)
From the specific data given what is the MLE of $LaTeX: \theta$ ?

Now that you have made your calculations, answer the following problem.