When is distance covariance less appropriate than linear covariance? I've just been introduced (vaguely) to brownian/distance covariance/correlation. It seems particularly useful in many non-linear situations, when testing for dependence. But it doesn't seem to be used very often, even though covariance/correlation are often used for non-linear/chaotic data.
That has me thinking that there might be some drawbacks to distance covariance. So what are they, and why doesn't everyone just always use distance covariance?
 A: I could well be missing something, but just having a quantification of the nonlinear dependence between two variables doesn't seem to have much of a payoff.  It won't tell you the shape of the relationship.  It won't give you any means to predict one variable from the other.  By analogy, when doing exploratory data analysis one sometimes uses a loess curve (locally weighted scatterplot smoother) as a first step towards seeing whether the data are best modeled with a straight line, a quadratic, a cubic, etc.  But the loess in and of itself is not a very useful predictive tool.  It's just a first approximation on the way to finding a workable equation to describe a bivariate shape.  That equation, unlike the loess (or the distance covariance result), can form the basis of a confirmatory model.
A: I have tried to collect a few remarks on distance covariance based on my impressions from reading the references listed below. However, I do not consider myself an
expert on this topic. Comments, corrections, suggestions, etc. are welcome.
The remarks are (strongly) biased towards
potential drawbacks, as requested in the original question.
As I see it, the potential drawbacks are as follows:


*

*The methodology is new. My guess is that this is the single
 biggest factor regarding lack of popularity at this time. The
 papers outlining distance covariance start in the mid 2000s and
 progress up to present day. The paper cited above is the one that
 received the most attention (hype?) and it is less than three years
 old. In contrast, the theory and results on correlation and
 correlation-like measures have over a century of work already
 behind them.

*The basic concepts are more challenging. Pearson's
 product-moment correlation, at an operational level, can be
 explained to college freshman without a calculus background
 pretty readily. A simple "algorithmic" viewpoint can be laid
 out and the geometric intuition is easy to describe. In contrast, in the case of distance covariance, even the notion of sums of products of pairwise Euclidean
 distances is quite a bit more difficult and the notion of
 covariance with respect to a stochastic process goes far beyond
 what could reasonably be explained to such an audience.

*It is computationally more demanding. The basic algorithm for
 computing the test statistic is $O(n^2)$ in the sample size as
 opposed to $O(n)$ for standard correlation metrics. For small
 sample sizes this is not a big deal, but for larger ones it
 becomes more important.

*The test statistic is not distribution free, even
 asymptotically. One might hope that for a test statistic that is
 consistent against all alternatives, that the
 distribution—at least asymptotically—might be
 independent of the underlying distributions of $X$ and $Y$ under
 the null hypothesis. This is not the case for distance covariance 
 as the distribution under the null depends on the underlying
 distribution of $X$ and $Y$ even as the sample size tends to
 infinity. It is true that the distributions are uniformly
 bounded by a $\chi^2_1$ distribution, which allows for the
 calculation of a conservative critical value.

*The distance correlation is a one-to-one transform of $|\rho|$ in
 the bivariate normal case. This is not really a drawback, and
 might even be viewed as a strength. But, if one accepts a
 bivariate normal approximation to the data, which can be quite
 common in practice, then little, if anything, is gained from
 using distance correlation in place of standard procedures.

*Unknown power properties. Being consistent against all
 alternatives essentially guarantees that distance covariance must
 have very low power against some alternatives. In many cases, one
 is willing to give up generality in order to gain additional
 power against particular alternatives of interest. The original
 papers show some examples in which they claim high power relative
 to standard correlation metrics, but I believe that, going back
 to (1.) above, its behavior against alternatives is not yet well
 understood.


To reiterate, this answer probably comes across quite negative. But,
that is not the intent. There are some very beautiful and interesting
ideas related to distance covariance and the relative novelty of it
also opens up research avenues for understanding it more fully.
References:


*

*G. J. Szekely and M. L. Rizzo (2009), Brownian distance
covariance, Ann. Appl. Statist., vol. 3, no. 4, 1236–1265.

*G. J. Szekely, M. L. Rizzo and N. K. Bakirov (2007), Measuring and
testing independence by correlation of distances, Ann. Statist.,
vol. 35, 2769–2794.

*R. Lyons (2012), Distance covariance in metric spaces,
Ann. Probab. (to appear).

