Timeline for Use $\bar{X}^2$ for hypothesis test that $\mu=0$ because faster convergence rate?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2016 at 8:10 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| S Nov 12, 2016 at 17:56 | history | bounty ended | CommunityBot | ||
| S Nov 12, 2016 at 17:56 | history | notice removed | CommunityBot | ||
| Nov 8, 2016 at 14:25 | answer | added | JDL | timeline score: 1 | |
| Nov 7, 2016 at 22:59 | comment | added | jbowman | It doesn't matter whether the approximate variance is larger or smaller, because what counts is the distribution of the statistic. To see this, consider a t-test for $\mu = 0$ with $x \sim N(0,1)$ vs $y \sim N(0,10)$. The statistic $\bar{y}$ always has variance 100x that of $\bar{x}$, but the normalization results in both actual test statistics distributed $t(n-1)$. In your case, remember that squaring a $N(0,1)$ variate gives a $\chi^2$ variate. At the limit, this transformation means that the two tests are identical in terms of their power given a specified level. | |
| Nov 6, 2016 at 3:57 | comment | added | Xu Wang | @whuber thank you for these details. I have been thinking about them but still don't understand. Won't the approximate variance of X-bar^2 eventually be smaller than the approximate variance of X-bar? And isn't that a result of X-bar^2 having a higher rate of convergence than X-bar? I'm sorry for not seeing my fundamental misunderstandings. I know there is something big I am missing and hope to correct such thinking. | |
| Nov 4, 2016 at 23:10 | comment | added | whuber♦ | I think you're focusing on the wrong thing. This rate tells you how quickly the sampling distribution approaches the limiting one--either standard Normal or $\chi^2(1)$. Since $n$ is large, its value makes no practical difference--it's irrelevant. The issue concerns the power of each test, not how well approximated the test statistic is to the limiting distribution. | |
| Nov 4, 2016 at 20:37 | comment | added | Xu Wang | @whuber thanks for questions. I claim "faster rate" because n is larger than square root of n. Is that intuition incorrect? I have in mind test statistic X-bar or X-bar squared. | |
| Nov 4, 2016 at 16:56 | comment | added | whuber♦ | It isn't clear what you are asking. Could you explain in what sense the convergence rate of $\bar X^2$ is "faster" than that of $\bar X$? How are you measuring the rate? What test statistics are you using in the two tests? Clearly these choices can make a difference. | |
| S Nov 4, 2016 at 16:25 | history | bounty started | Xu Wang | ||
| S Nov 4, 2016 at 16:25 | history | notice added | Xu Wang | Draw attention | |
| Oct 29, 2016 at 19:22 | history | tweeted | twitter.com/StackStats/status/792446515730153473 | ||
| Oct 29, 2016 at 15:25 | history | edited | dsaxton | CC BY-SA 3.0 |
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| Oct 29, 2016 at 14:33 | history | asked | Xu Wang | CC BY-SA 3.0 |