415homework5
.pdf
School
Pennsylvania State University**We aren't endorsed by this school
Course
STAT 415
Subject
Statistics
Date
Jan 11, 2025
Pages
2
Uploaded by whatthehuh
STAT 415 Fall 2024Homework 5Total: 50 pointsDue September 29 at 11:59pmProblem 1.Derive the formula for the MLE of the geometric distribution, i.e. ifX1, . . . , Xn∼Geometric(p), thenP(Xi=xi) = (1−p)xi−1pforxi∈0,1,2, . . ..Problem 2.Derive the formula for the MLE of the Poisson distribution. Recall that ifX1, . . . , XnPoisson(λ), thenP(Xi=xi) =e−λλxixi!forxi∈ {0,1,2, . . .}.Problem 3.Derive the formula for the MLE ofθifX1, . . . , Xn∼F(θ) whereFis adistribution with the following p.d.f.:fXi(xi) =θ32x2ie−θxiover supportxi≥0,Problem 4. (R Problem.)(a) Similar to what we did in class on Monday, visualize the likelihood function ofn= 20samples from a normal distribution with true parameter valuesµtrue= 1 andσ2true= 1. Inother words, plotL(µ|x1, . . . , xn)over a suitable range ofµvalues.Note: When we did this for the binomial distribution, wecould calculate values over a grid of ps between zero and one. Sinceµis not bounded inthis way, you will have to decide on some reasonable range of values that will give anappropriate visualization.(b) Using your plot above, verify that the maximum is equal to the maximum likelihoodestimate.(c) Repeat part (a) but for the parameterσ2; for this you can assume thatµtrue= 1 is known.Bonus Problem1
(a) In your results for Problem 4 part (a), estimate thenumerical derivativeof the likelihoodatµtrueby calculatingL(θi|. . .)−L(θi−1|. . .)θi−θi−1whereiandi−1 are adjacent entries of your grid (in this caseθisµ) andθi≤θtrue≤θi.(b) Repeat the derivative estimation aboveNtimes in a for-loop (or apply statement) forsome decently large value ofNwhere for each iteration you randomly generate a newdatasetx1, . . . , xn. Take the average of the estimated derivatives. What do you noticeabout the average?(c) Repeat part (b) above for different values ofµtrue. What do you observe?2