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In the one-dimensional case it is easy to obtain random numbers that are not too far away from the mean by checking if they are within one or two standard deviations. Is there some similar way to obtain random numbers that are not to far away from the mean vector of a multivariate normal distribution?

Thank you very much

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    $\begingroup$ Draw vectors from this multivariate normal as you usually would, and then check the Mahalanobis distance. $\endgroup$ – assumednormal May 19 '12 at 20:11
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    $\begingroup$ What software are you using? $\endgroup$ – Peter Flom - Reinstate Monica May 19 '12 at 20:11
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The Mahalanobis distance is the natural distance measure as Max said. But maybe you are really interested in a more general question that doesn't have a unique answer. The similar sounding question is "In one dimension outliers are defined by being much larger or smaller than the bulk of the distribution that the data was drawn from. For multivariate data how are outliers defined." One answer is to define a distance and say that outlier are observations with large distances. But the distance function should depend on the shape of the multivariate distribution. So there are many distance functions that can be defined. In more that one dimension there are other ways to look at outliers. One way is in terms of using the data for inference. Outliers adversely effect parameter estimates but in different ways depending on what parameter we are interested in. Each parameter has a functional defined which is called the influence function. It measures how points in the multidimensional space can chnage the estimate when comapring including the point versus leaving it out. Outliers can be defined in terms of the magnitude of their influence on the parameter. In two dimensions the pearson correlation has contours of constant influence determined by hyperbolae. So as you move out in teo dimensions along these hyperbolae you increase the influence of the pointand do it most rapidly. So an outlier with respect to correlation can be defined as any point beyond a hyperbolic contour that is considered to have high influence. Other parameters would have different contours and hence a different definition for the outlier.

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