Statistical notation in plain English - Cross Validated most recent 30 from stats.stackexchange.com 2019-08-19T01:53:40Z https://stats.stackexchange.com/feeds/question/68877 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/68877 1 Statistical notation in plain English blunders https://stats.stackexchange.com/users/10098 2013-08-31T23:50:31Z 2013-09-01T03:15:53Z <p>I have a very limited understanding of statistics, and am having trouble translating <a href="https://quant.stackexchange.com/a/958/719">an answer</a> to this question from quant.SE, "<a href="https://quant.stackexchange.com/questions/955/separating-the-wheat-from-the-chaff-what-quant-methods-separate-skillful-manage">Separating the wheat from the chaff</a>", the answer being:</p> <blockquote> <p>Larry Harris has a chapter on performance evaluation in <a href="https://quant.stackexchange.com/questions/955/separating-the-wheat-from-the-chaff-what-quant-methods-separate-skillful-manage">Trading and Exchanges</a>. He states that over a long period of time, a skilled asset manager will consistently have excess returns whereas a lucky one will be expected to have random and unpredictable returns. Thus, we start with the portfolio's market-adjusted return standard deviation:</p> <p>\begin{equation} \sigma_{adj} = \sqrt{\sigma^2_{port} + \sigma^2_{mk} - 2\rho\sigma_{port}\sigma_{mk}} \end{equation}</p> <p>where $\rho$ is the correlation between the market and portfolio returns.</p> <p>For a sample size $n$ (generally number of years), the average excess returns, and the adjusted standard deviation from above, we have a <a href="https://quant.stackexchange.com/a/958/719">t-statistic</a>:</p> <p>\begin{equation} t = \frac{\overline{R_{port}} - \overline{R_{mk}}}{\frac{\sigma_{adj}}{\sqrt{n}}} \end{equation}</p> <p>Now we can simply determine the probability that the manager's excess returns were luck by plugging this t-statistic into the <a href="http://en.wikipedia.org/wiki/Student%27s_t-distribution" rel="nofollow noreferrer">t-distribution</a>'s PDF with degrees-of-freedom $n - 1$. The lower the probability, the more we can believe the manager's excess returns were from skill.</p> </blockquote> <p>At this point, I now understand the top formula, though this was based on the finding the <a href="http://books.google.com/books?id=Rd9hDRR1Yx4C&amp;lpg=PP452&amp;pg=PA452#v=onepage&amp;q&amp;f=false" rel="nofollow noreferrer">book's the section on topic</a> referenced in the answer above on Google Books.</p> <p>What I don't get now is this formula:</p> <blockquote> <p>\begin{equation} t = \frac{\overline{R_{port}} - \overline{R_{mk}}}{\frac{\sigma_{adj}}{\sqrt{n}}} \end{equation}</p> </blockquote> <p>I understand it's somehow related to the <a href="http://en.wikipedia.org/wiki/Student%27s_t-distribution" rel="nofollow noreferrer">t-distribution</a>'s "Probability Density Function" but I have no idea what the R or "overline" notation means, or for that matter, where the values are R-port and R-mk come from; "port" being portfolio, and "mk" being market. If it matters, since it took me a bit to figure this out "adj" just flags the sigma (standard deviation) as the "portfolio's market-adjusted return standard deviation."</p> <p>Any suggestions?</p> https://stats.stackexchange.com/questions/68877/-/68878#68878 3 Answer by Glen_b for Statistical notation in plain English Glen_b https://stats.stackexchange.com/users/805 2013-09-01T01:18:28Z 2013-09-01T03:15:53Z <p>To answer the notation questions:</p> <p>The notation "$\bar x$" means the sample mean of $x$'s, $\bar x = \frac{1}{n}(x_1+x_2+\ldots+x_n)$. So just take the (sample) average of whatever is under the bar.</p> <p>$\sigma_{\text{something}}$ <em>should</em> represent a population standard deviation, not a sample standard deviation (for which instead one might use $s_{\text{something}}$).</p> <p>From context, it appears that $R$ is a return, and the subscripts $\text{port}$ and $\text{mk}$ are "portfolio" and "market" respectively.</p> <p>There seem to be a number of problems with the exposition in that post (as it stands, it seems to be wrong in several ways), but as you suggested in comments, that's beyond the scope of the specific question relating to notation.</p>