Jul 14, 2015

Student Puzzle Corner 10: deadline August 10, 2015

In this issue, we look at the consequences of having only incomplete data. For example, suppose a random variable $X$ has a normal distribution with mean $\mu $ and variance $\sigma ^2$, and both parameters need to be estimated. With usual data, which we call complete data, namely iid copies $X_1, X_2, \cdots , X_n$ of $X$, we can estimate $\mu $ and $\sigma ^2$ easily; $\bar{X}$ and $s^2$ are consistent and fully efficient. But, if we only have the maximum and the minimum of the $X_i$’s, then although we can still estimate $\mu $ and $\sigma ^2$ consistently, we can no longer estimate them efficiently. This is the price we pay for only having incomplete data. In some cases, data may be so incomplete that even consistent estimation of a parameter may be impossible.
Here is the exact problem of this issue:
For each of the following cases, either prove that consistent estimation of the indicated parameter is not possible, or demonstrate a concrete consistent estimate of the indicated parameter:

(a) $X_i \stackrel {{\it iid }} {\sim } N(\mu , 1), i = 1, 2, \cdots , n, -\infty < \mu < \infty $, and we get to observe only $Y_i, i = 1, 2, \cdots , n$, where $Y_i = |X_i|$; we want to estimate $\mu $; (b) $X_i \stackrel {{\it iid }} {\sim } \mbox{Poisson}(\lambda ), i = 1, 2, \cdots , n, 0 < \lambda < \infty $, and we get to observe only whether each$X_i$ is larger than $k$ or $\leq k$ for some fixed specified positive integer $k$; we want to estimate $\lambda $; (c) $X_i, i = 1, 2, \cdots , n$ are iid exponential with mean $\lambda , 0 < \lambda < \infty $, and we get to observe only the fractional parts of the $X_i$’s; we want to estimate $\lambda $. The solution to the previous Puzzle is here.


1 Comment

Leave a comment




Welcome to the IMS Bulletin website! We are developing the way we communicate news and information more effectively with members. The print Bulletin is still with us (free with IMS membership), and still available as a PDF to download, but in addition, we are placing some of the news, columns and articles on this blog site, which will allow you the opportunity to interact more. We are always keen to hear from IMS members, and encourage you to write articles and reports that other IMS members would find interesting. Contact the IMS Bulletin at bulletin@imstat.org

What is “Open Forum”?

In the Open Forum, any IMS member can propose a topic for discussion. Email your subject and an opening paragraph (to bulletin@imstat.org) and we'll post it to start off the discussion. Other readers can join in the debate by commenting on the post. Search other Open Forum posts by using the Open Forum category link below. Start a discussion today!

About IMS

The Institute of Mathematical Statistics is an international scholarly society devoted to the development and dissemination of the theory and applications of statistics and probability. We have about 4,500 members around the world. Visit IMS at http://imstat.org
Latest Issue