THE AUSTRALIAN NATIONAL UNIVERSITY RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES AND STATISTICS STAT3040 Statistical Learning Assignment 2: Due at 5pm, Tuesday, 04/05/2021 1. This assignment is worth 25% of your final grade and is compulsory. 2. Maximum points: 50. 3. Assignments can only be submitted via Turnitin on the Wattle. 4. Please type, paste, and combine all of your answers in the answer sheet file on the Wattle, including signing a declaration form in the cover page. Please be aware of the quality of the file when you are preparing the submission, and in particular that the text on each scanned/photographed image is legible. 5. File size limit for Turnitin submission: 40MB. 6. Several trials of submission are strongly recommended before the due date. If there is any problem in your trials, please send an email to report before the due date. Late submission will not be accepted and your assignment will be marked 0. Please prepare to submit your answer sheet at least 30 minutes before the end of due time. 7. While you may use course material, computer software, internet, or other resources, you must complete the assignment individually. Identical submissions even only for one question are treated as cheating. 8. You can use any result, formula or statement from the course material without proof. In fact, doing this will help your assignment. 9. Please round all of your final numeric answers to four decimal places and please do not round in the middle of the computation process. 10. Please include all working out, as marks will not be awarded for those answers which do not include. 11. The significance level is set to be 0.05 throughout the assignment. 1
Question 1 (5 points) If we would like to conduct the standard bootstrap approach in the lecture notes for a dataset with n observations, and generate B bootstrap datasets, please prove the statement "On average, each bootstrap dataset makes use of around two-thirds of the n observations as n is large". Please GIVE the necessary mathematical details about how to show this result in the answer sheet. Hint: (1-1/n)" -> 1/e as n -> 00, and we say (1 -1/n)" is around 1/e as n is large. Question 2 (5 points) In a classification problem, suppose we are interested in a pair (X, Y), and X takes values in Rd while Y takes values in {1, 0}, where Y is the class label of X. The conditional distribution of X, given that the label Y takes the value k is given by X|(Y = k) ~ fk for k = 1,0, where "~" means "is distributed as", and where fk denotes a probability density function. A classifier is a rule that assigns to an observation X = x a guess or estimate of what the unobserved label Y = k actually was. In theoretical terms, a classifier is a function C : Rd > {1,0}, with the interpretation that C classifies the point x to the class C(x).