Harmonic means, reciprocals, and ratios of random variables

In my last few posts, I have considered “long-tailed” distributions whose probability density decays much more slowly than standard distributions like the Gaussian.  For these slowly-decaying distributions, the harmonic mean often turns out to be a much better (i.e., less variable) characterization than the arithmetic mean, which is generally not even well-defined theoretically for these distributions.  Since the harmonic mean is defined as the reciprocal of the mean of the reciprocal values, it is intimately related to the reciprocal transformation.  The main point of this post is to show how profoundly the reciprocal transformation can alter the character of a distribution, for better or worse.   One way that reciprocal transformations sneak into analysis results is through attempts to characterize ratios of random numbers.  The key issue underlying all of these ideas is the question of when the denominator variable in either a reciprocal transformation or a ratio exhibits non-negligible probability in a finite neighborhood of zero.  I discuss transformations in Chapter 12 of Exploring Data in Engineering, the Sciences and Medicine, with a section (12.7) devoted to reciprocal transformations, showing what happens when we apply them to six different distributions: Gaussian, < place w:st="on">Laplace, Cauchy, beta, Pareto, and lognormal. 

In the general case, if a random variable x has the density p(x), the distribution g(y) of the reciprocal y = 1/x has the density:

            g(y) = p(1/y)/y²

As I discuss in greater detail in Exploring Data, the consequence of this transformation is typically (though not always) to convert a well-behaved distribution into a very poorly behaved one.  As a specific example, the plot below shows the effect of the reciprocal transformation on a Gaussian random variable with mean 1 and standard deviation 2.  The most obvious characteristic of this transformed distribution is its strongly asymmetric, bimodal character, but another non-obvious consequence of the reciprocal transformation is that it takes a distribution that is completely characterized by its first two moments into a new distribution with Cauchy-like tails, for which none of the integer moments exist.

The implications of the reciprocal transformation for many other distributions are equally non-obvious.  For example, both the badly-behaved Cauchy distribution (no moments exist) and the well-behaved lognormal distribution (all moments exist, but interestingly, do not completely characterize the distribution, as I have discussed in a previous post) are invariant under the reciprocal transformation.  Also, applying the reciprocal transformation to the long-tailed Pareto type I distribution (which exhibits few or no finite moments, depending on its tail decay rate) yields a beta distribution, all of whose moments are finite.  Finally, it is worth noting that the invariance of the Cauchy distribution under the reciprocal transformation lies at the heart of the following result, presented in the book Continuous Univariate Distributions by Johnson, Kotz, and Balakrishnan (Volume 1, 2^nd edition, Wiley, 1994, page 319).  They note that if the density of x is positive, continuous, and differentiable at x = 0 – all true for the Gaussian case – the distribution of the harmonic mean of N samples approaches a Cauchy limit as N becomes infinitely large.

As noted above, the key issue responsible for the pathological behavior of the reciprocal transformation is the question of whether the original data distribution exhibits nonzero probability of taking on values within a neighborhood around zero.  In particular, note that if x can only assume values larger than some positive lower limit L, it follows that 1/x necessarily lies between 0 and 1/L, which is enough to guarantee that all moments of the transformed distribution exist.  For the Gaussian distribution, even if the mean is large enough and the standard deviation is small enough that the probability of observing values less than some limit L > 0 is negligible, the fact that this probability is not zero means that the moments of any reciprocally-transformed Gaussian distribution are not finite.  As a practical matter, however, reciprocal transformations and related characterizations – like harmonic means and ratios – do become better-behaved as the probability of observing values near zero become negligibly small.

To see this point, consider two reciprocally-transformed Gaussian examples.  The first is the one considered above: the reciprocal transformation of a Gaussian random variable with mean 1 and standard deviation 2.  In this case, the probability that x assumes values smaller than or equal to zero is non-negligible.  Specifically, this probability is simply the cumulative distribution function for the distribution evaluated at zero, easily computed in R as approximately 31%:

> pnorm(0,mean=1,sd=2)

[1] 0.3085375

In contrast, for a Gaussian random variable with mean 1 and standard deviation 0.1, the corresponding probability is negligibly small:

> pnorm(0,mean=1,sd=0.1)

[1] 7.619853e-24

If we consider the harmonic means of these two examples, we see that the first one is horribly behaved, as all of the results presented here would lead us to expect.  In fact, the qqPlot command in the car package  in R allows us to compute quantile-quantile plots for the Student’s t-distribution with one degree of freedom, corresponding to the Cauchy distribution, yielding the plot shown below.  The Cauchy-like tail behavior expected from the results presented by Johnson, Kotz and Balakrishnan is seen clearly in this Cauchy Q-Q plot, constructed from 1000 harmonic means, each computed from statistically independent samples drawn from a Gaussian distribution with mean 1 and standard deviation 2.  The fact that almost all of the observations fall within the – very wide – 95% confidence interval around the reference line suggest that the Cauchy tail behavior is appropriate here.

To further confirm this point, compare the corresponding normal Q-Q plot for the same sequence of harmonic means, shown below.  There, the extreme non-Gaussian character of these harmonic means is readily apparent from the pronounced outliers evident in both the upper and lower tails.

In marked contrast, for the second example with the mean of 1 as before but the much smaller standard deviation of 0.1, the harmonic mean is much better behaved, as the normal Q-Q plot below illustrates.  Specifically, this plot is identical in construction to the one above, except it was computed from samples drawn from the second data distribution.  Here, most of the computed harmonic mean values fall within the 95% confidence limits around the Gaussian reference line, suggesting that it is not unreasonable in practice to regard these values as approximately normally distributed, in spite of the pathologies of the reciprocal transformation.

One reason the reciprocal transformation is important in practice – particularly in connection with the Gaussian distribution – is that the desire to characterize ratios of uncertain quantities does arise from time to time.  In particular, if we are interested in characterizing the ratio of two averages, the Central Limit Theorem would lead us to expect that, at least approximately, this ratio should behave like the ratio of two Gaussian random variables.  If these component averages are statistically independent, the expected value of the ratio can be re-written as the product of the expected value of the numerator average and the expected value of the reciprocal of the denominator average, leading us directly to the reciprocal Gaussian transformation discussed here.  In fact, if these two averages are both zero mean, it is a standard result that the ratio has a Cauchy distribution (this result is presented in the same discussion from Johnson, Kotz and Balakrishnan noted above).  As in the second harmonic mean example presented above, however, it turns out to be true that if the mean and standard deviation of the denominator variable are such that the probability of a zero or negative denominator are negligible, the distribution of the ratio may be approximated reasonably well as Gaussian.  A very readable and detailed discussion of this fact is given in the paper by George Marsaglia in the May 2006 issue of Journal of Statistical Software.

Finally, it is important to note that the “reciprocally-transformed Gaussian distribution” I have been discussing here is not the same as the inverse Gaussian distribution, to which Johnson, Kotz and Balakrishnan devote a 39-page chapter (Chapter 15).  That distribution takes only positive values and exhibits moments of all orders, both positive and negative, and as a consequence, it has the interesting characteristic that it remains well-behaved under reciprocal transformations, in marked contrast to the Gaussian case.