Instructions
Take Home Midterm (MAT 4374/5182)
February 28, 2022
1) Please submit your solutions to this assignment in one PDF file in Brightspace. Only one file will be accepted.
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2) You can submit a PDF file more than once. However, only the last submission will be saved. If you want to modify your submitted assignment, that is fine as long as it is before the deadline.
3) Late submissions of the assignment are not going to be marked.
4) Please use R markdown to write the solutions for this part.
5) You can submit hand written solutions for the mathematical parts of the assignment, but please combine images of your hand-written solutions with the PDF produced with R markdown as one PDF. (See https://imagetopdf.com/ as a possible solution to combine images as one PDF). Alternatively, you can insert your image in the R markdown file.
6) Deadline: Before 11:59 pm on Friday, March 4
7) Do not work in groups.
1. Let X have a log-normal(μ,σ) distribution. Here is a stochastic representation of the log-normal distribution: let Z have a standard normal distribution, and let X = eσ Z+μ, then we say that X has a log-normal(μ,σ) distribution. Note that ln(X) has a normal distribution with mean μ and standard deviation σ.
Interpretation of the parameters:
• eμ = eE[ln(X)] is called the geometric mean of X.
V [ln(X)] is called the geometric standard deviation of X. • The mean of X is E[X] = eμ+σ2/2.
Suppose that we have a random sample X1, . . . , Xn from a log-normal population. The sample mean X ̄ = (1/n) ni=1 Xi is an estimator of the mean eμ+σ2/2, while the sample geometric mean G = e(1/n) ni=1 ln(Xi)
is an estimator of the geometric mean eμ.
Consider μ = 0, and σ = 1. Perform a simulation study to compare the efficiency of the estimation of the geometric mean and of the mean? Use n = 15, 20, 25, 30, 35, 100. We will say that the estimator with the smallest mean squared error is the most efficient estimator. Also give the estimated MSE and estimated standard error of the MSE for both estimators.
2. Consider the continuous distribution with pdf
f(x) = |sin(x)|, −π