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I understand that most statistical methods focus on predicting common occurrences, but are there any tests that would predict outliers? Essentially I would want to know what outside variables stand out for outliers as opposed to "normal" values. An improvised example being predicting a stellar performance on a game (10+ points) rather than 0-9.9 points. I would then want to figure out which physical traits (out of lets say, 200) are abnormal for individuals with a stellar score rather than anything less.

I've gotten some significant results that lower AIC using logistic regression (1 = outlier, 0 = not), but I'm not completely sure if it will hold up in the long run or if it is even appropriate to recode the continuous variables into binary ones.

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Quantile regression can predict various quantiles. If you choose a high (or low) quantile, that might get at what you want. It depends on how "out" the outlier is.

Tree methods could also work, with appropriate options.

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    $\begingroup$ I have no idea how I haven't heard of quantile regression. That seems to be the closest I can get to what I need. Much appreciated! $\endgroup$ – atamalu May 28 '17 at 23:27
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Parametric statistical tests make assumptions about the distributions of the variables involved (nonparametric tests make assumptions about the distributions of ranks of the data). These distributions describe the probability not only of "common occurrences" but of rare ones. If by "outlier" you mean "uncommon occurrence," then the answer to your question is "yes."

If by "outlier" you specifically mean unlikely values in a predictor variable such as the $x$ in $y = \beta_{0} + \beta_{x}x + \varepsilon$; where $\varepsilon \sim \mathcal{N}\left(0,\sigma_{y|x}\right)$, then not really: the aim of such a model is to explain or predict $y$, not $x$. However, there are techniques (e.g., Cook's D, etc.) for examining the sensitivity of such models to the influence of outliers.

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  • $\begingroup$ In the equation for $y$ did you really mean that second equal sign? $\endgroup$ – Lucas Farias May 27 '17 at 5:20
  • $\begingroup$ I am sure the second equal sign should be a plus sign. I think the answer describes outliers well and provides a means for detecting outliers in regression. There are many ways to detect outliers depending on the type of model and the assumptions. But if the OP is asking about predicting that the next observation will be an outlier is not possible if observations occur at random. If the successive observations are highly correlated possibly the occurrence of an outlier might increase the probability that the next observation will be one too. $\endgroup$ – Michael R. Chernick May 27 '17 at 8:24
  • $\begingroup$ @MichaelChernick I have been working on boot-strapping re-infected residuals where outliers effectively contribute to the probability distribution of forecasts via monte-carlo . This also includes probability densities for any significant predictor variables in the model. This approach works for both cross-sectional and time series data. $\endgroup$ – IrishStat May 27 '17 at 11:49
  • $\begingroup$ With my comment I was thinking that the only way to predict an outlier would be through some sort time series model. $\endgroup$ – Michael R. Chernick May 27 '17 at 13:45
  • $\begingroup$ @MichaelChernick So sorry: that was a typo. Thank you. $\endgroup$ – Alexis May 28 '17 at 1:23

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