Timeline for Is this hypothesis test somehow "optimal"?
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| Feb 13, 2016 at 17:20 | comment | added | J Li | @DilipSarwate: can you provide a reference on why $\sum_i x_i \mu_i$ is the optimal test? I've encountered a similar problem in another context. Thanks! | |
| Jan 4, 2014 at 13:42 | answer | added | randomperson | timeline score: 3 | |
| Dec 23, 2011 at 2:45 | vote | accept | M.B.M. | ||
| Dec 22, 2011 at 8:01 | answer | added | Elvis | timeline score: 3 | |
| Dec 22, 2011 at 6:53 | history | edited | M.B.M. |
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| Dec 22, 2011 at 5:03 | comment | added | M.B.M. | Re: $\mu_i$'s -- they are not random, but they are unknown. My uncertainty about $S_n$ comes only from the underlying independent normal distribution. The variance of $S_n$ in state 0 is its 4th moment minus the second moment squared. For each squared observation it's 3-1=2. To compute the variance in state 1, I first find the variance of an individual squared observation (4th moment of mean-shifted normal with variance 1) then subtract the squared-mean of the squared observation. Since the individual squared observations are independent, variance is additive... | |
| Dec 22, 2011 at 3:28 | comment | added | whuber♦ | How do you compute a variance of $2/n$ in state 0 and $(4M+2)/n$ in state 1? (I obtain different values for the variances.) Also, how can you be sure $\frac{1}{n}\sum_{i=1}^n\mu_i$ is exactly $0$ and not, say, $O(1/n)$ or $o(1/n)$? How do you know the values of the $\mu_i$ (and, thence, the value of $M$)? Unless you know them exactly, independently of the data, Chebyshev's inequality does not apply here. And if you do know the $\mu_i$ independently of the data, why do you need any kind of test at all? | |
| Dec 22, 2011 at 2:22 | history | migrated | from math.stackexchange.com (revisions) | ||
| Dec 21, 2011 at 18:48 | comment | added | M.B.M. | Dilip, interesting point about energy measurement, yes, I remember the $P=V^2/R$ formula from high school physics. I've posted on stats.SE before, though I thought this question is general enough to be posted here. How can I ask the moderators to move this question to stats.SE? | |
| Dec 21, 2011 at 2:30 | comment | added | Dilip Sarwate | I think you have discovered a form of what is known in the engineering literature as a square-law detector or noncoherent detector. If the sequence of $\mu_i$ is known, the optimum detector uses the statistic $\sum_ix_i\mu_i$, heuristically $\mu_i$ and $x_i$ are more likely to have the same sign. When the $\mu_i$ are not known, the detector essentially makes an "energy measurement" $\sum_i x_i^2$ (you might remember $V^2/R$ from high-school or college physics) with large energy meaning signal is present and little energy meaning signal is absent. | |
| Dec 20, 2011 at 20:51 | history | asked | M.B.M. | CC BY-SA 3.0 |