Intro to Statistical Learning (2nd Ed), Solution to Problem 11.7c

Q11.7c: In this problem, we will derive (11.5) and (11.6), which are needed for the construction of the log-rank test statistic (11.8). Recall the notation in Table 11.1. (a) Assume that there is no difference between the survival functions of the two groups. Then we can think of q1k as the number of failures if we draw r1k observations, without replacement, from a risk set of rk observations that contains a total of qk failures. Argue that q1k follows a hypergeometric distribution. Write the hypergeometric distribution parameters of this distribution in terms of r1k , rk , and qk . (b) Given your previous answer, and the properties of the hypergeometric distribution, what are the mean and variance of q1k ? Compare your answer to (11.5) and (11.6). Download Book: https://www.statlearning.com/ Authors' Lectures (R): https://youtube.com/playlist?list=PLoROMvodv4rOzrYsAxzQyHb8n_RWNuS1e&si=NP0wJ6RjP8XkxU7y Authors' Lectures (Python): https://youtube.com/playlist?list=PLoROMvodv4rPP6braWoRt5UCXYZ71GZIQ&si=0Z8tx4xlPLEyjZ70 https://colab.research.google.com/drive/1mAK5S-EKbq1ECuqb1wv1S9jE7Bk7ZPNA?usp=sharing