|WikiProject Statistics||(Rated Start-class, Low-importance)|
What's the point of specifying the value of the density at 0? Densities are not defined pointwise. Changing the value at an isolated point or leaving it undefined at an isolated point does not alter any of the probabilities, since those are given by integrals and a set containing only one point has measure zero. Michael Hardy (talk) 02:54, 4 July 2009 (UTC)
- Good question. In my defence, I was only following my sources: both the NIST page i've cited at present, this paper (introduction) and this one (appendix p423). If you have access to other sources that don't specify the density at 0, i'd be very happy to remove it. The pdf isn't the definition of the distribution though, so assuming the density at 0 can be derived from the definition (haven't tried myself), there would seem to be an argument for including it. By the way, thanks for the formatting and style fixes: in particular, I'd noticed the two cases in the pdf formula were too close together but I didn't know "\\" took an optional spacing argument that would fix it. Qwfp (talk) 14:38, 4 July 2009 (UTC)
- Also in qwfp's defence, the discontinuity is not essential. If L'Hopital's rule is applied to remove the discontinuity, you get the value NIST cites. If the discontinuity isn't essential, why leave it there? By the way, this is consistent with Tukey and Rogers' definition.
- In what particular respects is this less pathological than the Cauchy distribution? (I note in particular that none of the moments of the slash distribution are defined; in that respect it does resemble the Cauchy distribution.)
- The Slash is similar to the Cauchy in that no moments exist. However, the Cauchy does not have exponentially bounded tails. This gives rise to highly pathological behavior (e.g., the distribution of the mean of a SRS of Cauchy variates has the parent distribution) which is not believed by most applied statisticians to commonly exhibited by real data. Since the Cauchy does arise in some real data situations (spectroscopy being the most commonly encountered) the belief is at least arguable.
- Why are the fat tails useful for simulation studies?
- Specifically, why is this distribution useful for simulation studies?
- Lots of experience with real data (see, for example, the NIST series of weighing standard masses cited by Freedman, Pisani and Purves) suggests that the Gaussian distribution isn't a particularly great model for real data. Asymptotic results suggest that under fairly broad conditions commonly used statistics (means broadly defined, which include sample proportions, regression coefficients and variances) have distributions which can be approximated by Gaussian distributions. One important question in applications is, "How large a sample is necessary?" Another is, "What happens if the assumptions are violated?" Since one of the assumptions is exponentially bounded tails (for the Lindeberg CLT) it's nice to know what happens if the parent population has "fat" tails. The answer with Cauchy tails is usually that things go to hell in handbasket. The Slash provides a case intermediate between the Cauchy and exponentially bounded tails.
Phi is fairly standard notation for the cdf (Φ) and pdf (φ) of the standard normal distribution. See, for example, Hogg, Craig and McKean, Introduction to Mathematical Statistics, Prentice-Hall. 188.8.131.52 (talk) 04:17, 28 December 2010 (UTC) Dennis Clason