Taylor’s law and heavy-tailed distributions

Lionel Roy Taylor’s law of the mean (TLM), also known as Taylor’s law (1–3), can be described succinctly as

$$\text{sample} \text{variance} \sim a {(\text{sample} \text{mean})}^b,\hspace{1em}a>0, b>0,$$ [1]

for a sample of positive elements. Careful consideration of TLM (2, 4, 5) raises two unanswered fundamental questions in natural and statistical science.

• 1)Is TLM a unique phenomenon in nature, or should alternative extensions and refinements of TLM, exploiting higher moments and various measures for dependence (association), be explored?
• 2)What are the proper measures for the location, spread, asymmetry, and dependence (association) for random samples with infinite mean?

Applying non-Gaussian (stable) limit theorems, Cohen et al. (4) show that the limiting distribution of $[\text{sample} \text{variance}]/{[\text{sample} \text{mean}]}^b$ exists even if the sample is drawn from a distribution with support on $R^{+}$ having infinite mean. A similar limiting result was obtained by Drton and Xiao (6) and Pillai and Meng (7) with the Cauchy distribution as the limiting distribution. To describe how truly bewildering these results are, consider the following scenario from finance. Take a long-only portfolio consisting of $n>1$ risky assets, with a one-period, random portfolio return given by

$$r_p={\displaystyle \sum }_{i=1}^n{w}_i r_i, w_i\ge 0,\hspace{1em}{\displaystyle \sum }_{i=1}^n{w}_i=1.$$ [2]

Assume the asset one-period returns $\mathit{r} = (r_1,\dots , r_n)$ have a joint distribution determined by

$$\left(r_1,\dots ,r_n\right)\stackrel{d}{=}\left(\frac{X_1}{Y_1},\dots ,\frac{X_n}{Y_n}\right),$$ [3]

where ${\mathbb{X}}_n = (X_1,\dots , X_n)$ and ${\mathbb{Y}}_n = (Y_1,\dots , Y_n)$ are two independent and identically distributed (iid) Gaussian vectors with zero mean and covariance matrix $\text{Σ}=\left[{\mathit{\sigma }}_{ij}\right]$, with ${\mathit{\sigma }}_{ii}>0$, $i,j = 1,\dots ,n$. The Drton and Xiao (6)–Pillai and Meng (7) theorem states that $r_p$ has a standard Cauchy distribution with probability density function (pdf)

$$f_{r_p}\left(x\right)=\frac{1}{\pi \left(1+x^2\right)},x\in R,$$ [4]

for any choice of $\text{Σ}$. In [4], the portfolio weights $(w_1,\dots ,w_n)$ can be random as long as they are independent of $\left({\mathbb{X}}_n,{\mathbb{Y}}_n\right)$. A portfolio allocation with random weights can occur if the portfolio manager has prior information not directly related to the fundamentals of the underlying assets, a scenario occurring frequently under active portfolio management. The Drton and Xiao (6)–Pillai and Meng (7) result indicates that such active portfolio management is meaningless when the portfolio return $r_p$ has a distribution with infinite mean as in [4]. One concludes then that portfolio diversification is meaningless when the asset returns do not have a (finite) mean value, leading to a tantamount problem for portfolio risk budgeting (8–11).

It is our firm opinion that the Drton and Xiao (6)–Pillai and Meng (7) and Cohen et al. (4) results are fundamental, as they shed further light on fascinating statistical phenomena occurring when dealing with samples from a distribution with infinite mean.

Is TLM Unique

We address question 1 by starting with a reminder of the usual estimators for sample mean and variance (12).

Mean is “a measure for location or central value for a continuous variable. … For a sample of observations $x_1,x_2,\dots ,x_n$ the measure is calculated as $\overline{x}=n^{-1}{\sum }_{i=1}^n{x}_i$. Most useful when the data have a symmetric distribution and do not contain outliers” (ref. 12, p. 274).

Variance is, “in a population, the second moment about the mean. An unbiased estimator of the population value is provided by $s^2$ given by $s^2={\left(n-1\right)}^{-1}{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2$ where $x_1,x_2,\dots ,x_n$ are the $n$ sample observations and $\overline{x}$ is the sample mean” (ref. 12, p. 445).

Note, particularly, the qualification that the mean estimator is most useful for data that are symmetrically distributed and have no outliers. Consider the statistic $W_n$ for the Taylor’s law ratio,

$$W_n=\frac{\text{sample} \text{variance}}{{\left(\text{sample} \text{mean}\right)}^b}=\frac{{\left(n-1\right)}^{-1}{{\displaystyle \sum }}_{i=1}^n{\left(X_i-\overline{X}\right)}^2}{{\left(n^{-1}{{\displaystyle \sum }}_{i=1}^n{X}_i\right)}^b}.$$ [5]

When the sample ${\mathbb{X}}_n = (X_1,\dots ,X_n)$ consists of iid random variables (RVs) with regularly varying tails having tail index $0 < \mathit{\alpha } < 1,$ neither does the sample variance in the numerator of [5] represent a measure for the dispersion of ${\mathbb{X}}_n$ nor does $\overline{X}$ in the denominator represent a measure for location in ${\mathbb{X}}_n$. It is truly remarkable that the ratio $W_n$ has a limit in distribution.

In PNAS, Brown et al. (5) extend the relationship in [5] to a more general form $M_{h_2}/{\left(M_{h_1}\right)}^b$ where $M_h$ is the $h$ th central moment, and the exponent has the dependence $b=b\left(h_1,h_2,\mathit{\alpha }\right)$. We suggest an approach that is framed in terms of an operator limit theorem (13–17) for the first four central moments. Let $\mathbb{X}{\mathrm{ }}_n^{(\mathit{\alpha })}= \left(X_1^{(\mathit{\alpha })},\dots ,X_n^{(\mathit{\alpha })}\right)$ be an iid random sample drawn from a continuous, unimodal distribution function $F_X^{(\mathit{\alpha })}$ having regularly varying tails with tail index $\mathit{\alpha } > 0.$ Denote the first four central moments as (we are interested in $n\uparrow \infty$)

$$\begin{array}{cc}{\overline{X}}_{n,\mathit{\alpha }}=n^{-1}{\displaystyle \sum }_{i=1}^n{X}_i^{(\mathit{\alpha })}, & s_{n,\mathit{\alpha }}^2=n^{-1}{\displaystyle \sum }_{i=1}^n{\left(X_i^{(\mathit{\alpha })}-{\overline{X}}_{n,\mathit{\alpha }}\right)}^2 ,\\ {\gamma }_{n,\mathit{\alpha }}=n^{-1}{\displaystyle \sum }_{i=1}^n{\left(X_i^{(\mathit{\alpha })}-{\overline{X}}_{n,\mathit{\alpha }}\right)}^3 ,& {\kappa }_{n,\mathit{\alpha }}=n^{-1}{\displaystyle \sum }_{i=1}^n{\left(X_i^{(\mathit{\alpha })}-{\overline{X}}_{n,\mathit{\alpha }}\right)}^4 .\end{array}$$

The operator limit theorem would determine normalizing constants $a_n^{(i,\mathit{\alpha })}>0$ and $b_n^{(i,\mathit{\alpha })}\in R$, $i =1, 2, 3, 4$, governing the vector limit

$$\left[\begin{array}{c}\begin{array}{c}{a}_n^{\left(1,\mathit{\alpha }\right)}{\overline{X}}_{n,\mathit{\alpha }}-b_n^{\left(1,\mathit{\alpha }\right)}\\ a_n^{\left(2,\mathit{\alpha }\right)}{s}_{n,\mathit{\alpha }}^2-b_n^{\left(2,\mathit{\alpha }\right)}\end{array}\\ \begin{array}{c}{a}_n^{\left(3,\mathit{\alpha }\right)}{\gamma }_{n,\mathit{\alpha }}-b_n^{\left(3,\mathit{\alpha }\right)}\\ a_n^{\left(4,\mathit{\alpha }\right)}{\kappa }_{n,\mathit{\alpha }}-b_n^{\left(4,\mathit{\alpha }\right)}\end{array}\end{array}\right]\stackrel{d}{\Rightarrow }{\mathbb{G}}^{(\mathit{\alpha })}=\left[\begin{array}{c}\begin{array}{c}{\mathcal{G}}_1^{(\mathit{\alpha })}\\ {\mathcal{G}}_2^{(\mathit{\alpha })}\end{array}\\ \begin{array}{c}{\mathcal{G}}_3^{(\mathit{\alpha })}\\ {\mathcal{G}}_4^{(\mathit{\alpha })}\end{array}\end{array}\right] .$$ [6]

Equally importantly, the theorem should address the rate of convergence to the vector of limiting distributions. A functional limit theorem [a Donsker (18)–Prokorov (19) type invariance principle] should also be proved. The theorem will have to consider separate ranges for the tail index: $0 <\mathit{\alpha } < 1$ (the mean of ${\overline{X}}_{n,\mathit{\alpha }}$ does not exist), $1 < \mathit{\alpha } < 2$ (the mean of $s_{n,\mathit{\alpha }}^2$ does not exist), $2 < \mathit{\alpha } < 3$ (the mean of ${\gamma }_{n,\mathit{\alpha }}$ does not exist), $3 < \mathit{\alpha } < 4$ (the mean of ${\kappa }_{n,\mathit{\alpha }}$ does not exist), and $4 < \mathit{\alpha }$ [where the limiting distribution can be obtained from the functional central limit theorem (18, 19)], as well as at the critical values $\mathit{\alpha } = 1, 2, 3, 4$.

TLM Moment Measures

To address question 2 for heavy-tailed distributions, robust versions of these four central moments must be considered (20–23), and TLM must be reformulated in terms of the robust moments. One possible set of robust estimators might be (for brevity, we drop the superscript $(\mathit{\alpha })$):* the median, $m\left({\mathbb{X}}_n\right);$ the interquartile range $\text{IQR}\left({\mathbb{X}}_n\right)$; the medcouple (24)*

$${\gamma }_n^{(\text{medcouple})}=m\left\{h\left(X_{i;n},X_{j;n}\right)\right\}, X_{i;n}\le m\left({\mathbb{X}}_n\right)\le X_{j;n},$$ [7]

where $X_{1;n}\le \dots \le X_{n;n}$ denotes the ordered sample, and the function $h(\cdot )$ is defined in Brys et al. (24); and (25)

$${\kappa }_n^{(\text{robust})}=\frac{\left(F_X^{(\text{inv})}\left(3/8\right)-F_X^{(\text{inv})}\left(1/8\right)\right)+\left(F_X^{(\text{inv})}\left(7/8\right)-F_X^{(\text{inv})}\left(5/8\right)\right)}{F_X^{(\text{inv})}\left(6/8\right)-F_X^{(\text{inv})}\left(2/8\right)} .$$ [8]

Then the usual form of the TLM would have the robust estimator

$$W_n^{(\text{robust})}=\frac{\text{IQR}\left({\mathbb{X}}_n\right)}{{\left(m\left({\mathbb{X}}_n\right)\right)}^b} ,$$ [9]

where the constant $b>0$ can be estimated from a robust form of the regression model

$$\text{ln}\left(\text{IQR}\left({\mathbb{X}}_n\right)\right)=b_0+b \text{ln}\left(m\left({\mathbb{X}}_n\right)\right)+{\mathit{\epsilon }}_n,$$ [10]

over the given sample. The ratio [9] is robust with respect to strictly increasing, one-to-one, continuous mappings on $R$ to $R$, making the statistic useful when $F_X$ has heavy tails. It would be of interest to study the limiting behavior (26, 27) of $W_n^{(\text{robust})}$ and the corresponding convergence rate.