Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

Brown, Mark; Cohen, Joel E.; Tang, Chuan-Fa; Yam, Sheung Chi Phillip

doi:10.1073/pnas.2108031118

Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

Mark Brown [email protected], Joel E. Cohen https://orcid.org/0000-0002-9746-6725 [email protected], Chuan-Fa Tang [email protected], and Sheung Chi Phillip Yam [email protected]Authors Info & Affiliations

Contributed by Joel E. Cohen, September 28, 2021 (sent for review April 28, 2021; reviewed by Svetlozar Rachev and Johannes Ruf)

November 12, 2021

118 (46) e2108031118

https://doi.org/10.1073/pnas.2108031118

This article has been updated

Commentary

December 10, 2021

Taylor’s law and heavy-tailed distributions

W. Brent Lindquist, Svetlozar T. Rachev

PDF/EPUB

Significance

Many quantities are extremely large extremely rarely. Examples include income, wealth, financial returns, insurance losses, firm size, and city population size; earthquake magnitude, hurricane energy, tornado outbreaks, precipitation, and flooding; and pest outbreaks, infectious epidemics, and forest fires. When such a quantity is modeled as a nonnegative random variable with a heavy upper tail, the probability of an observation larger than some threshold falls as a small power (the “tail index”) of the threshold. When the tail index is small enough, the mean and all higher moments of the random quantity are infinite. Surprisingly, the sample mean and the sample higher moments obey orderly scaling laws, which we prove and apply to estimating the tail index.

Abstract

We generalize Taylor’s law for the variance of light-tailed distributions to many sample statistics of heavy-tailed distributions with tail index α in (0, 1), which have infinite mean. We show that, as the sample size increases, the sample upper and lower semivariances, the sample higher moments, the skewness, and the kurtosis of a random sample from such a law increase asymptotically in direct proportion to a power of the sample mean. Specifically, the lower sample semivariance asymptotically scales in proportion to the sample mean raised to the power 2, while the upper sample semivariance asymptotically scales in proportion to the sample mean raised to the power

(2 - α) / (1 - α) > 2

. The local upper sample semivariance (counting only observations that exceed the sample mean) asymptotically scales in proportion to the sample mean raised to the power

(2 - α^{2}) / (1 - α)

. These and additional scaling laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the Omega index, the upside potential ratio, and the Farinelli–Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. Such power-law scaling relationships are known in ecology as Taylor’s law and in physics as fluctuation scaling. We find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of α based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators of α.

Heavy-tailed nonnegative random variables with infinite moments, such as nonnegative stable laws with index α in (0,1), have theoretical and practical importance [e.g., Carmona (1), Feller (2), Resnick (3), and Samorodnitsky and Taqqu (4)]. Heavy-tailed nonnegative random variables with some or all infinite moments have been claimed to arise empirically in finance [operational risks in Nešlehová et al. (5)], economics [income distributions in Campolieti (6) and Schluter (7); returns to technological innovations in Scherer et al. (8) and Silverberg and Verspagen (9)], demography [city sizes in Cen (10)], linguistics [word frequencies in Bérubé et al. (11)], and insurance [economic losses from earthquakes in Embrechts et al. (12) and Ibragimov et al. (13)]. Partial reviews are in Carmona (1) and Ibragimov (14).

Brown et al. (15) (hereafter BCD) showed that when a random sample is drawn from a nonnegative stable law with index

α \in (0, 1)

, the sample variance is asymptotically (as the sample size n goes to

\infty

) proportional to the sample mean raised to a power that is an explicit function of α (Eqs. 11 and 13). This relationship generalizes to stable laws with infinite moments a widely observed power-law relationship between the variance and the mean in families of distributions with finite population mean and finite population variance. This power-law relationship is commonly known as Taylor’s law in ecology [Taylor (16, 17)] and as fluctuation scaling in physics [Eisler et al. (18)].

To the two ingredients combined by BCD (nonnegative stable laws with infinite moments and Taylor’s law), this paper adds two more ingredients. We establish scaling relationships that generalize the usual Taylor’s law, for light-tailed distributions, to many functions of the sample in addition to the variance, including all positive absolute and central moments, upper and lower semivariances, and several measures of risk-adjusted investment performance such as the Sortino, Sharpe, and Farinelli–Tibiletti ratios. In addition, based on these scaling relationships, we propose several estimators of the index α of a nonnegative stable law with infinite first moment.

Section 1 defines most of the sample functions studied here. Section 2 gives background on Taylor’s law, semivariances, and nonnegative stable laws, including key prior results from BCD. Section 3 establishes that the lower sample semivariance, the upper sample semivariance, the local lower sample semivariance, and the local upper sample semivariance are asymptotically each a power of the sample mean with explicitly given exponents. These results are the core of the paper. When investment returns obey a nonnegative heavy-tailed law with index

α \in (0, 1)

, these results reveal the asymptotic behavior of the Sharpe ratio, the Sortino ratio, and the Farinelli–Tibiletti ratio. Section 4 extends these results to higher central and noncentral moments and various indices of volatility. Section 5 analyzes the number of observations from a stable law or an approximately stable (i.e., regularly varying) law that exceed the sample mean. Section 6 proposes and compares estimators of α by simulation. SI Appendix gives all proofs of results stated in the text and additional numerical simulations.

1. Preliminary

Let

\overset{d}{\to}

mean “converges in distribution to.” Let

\overset{p}{\to}

mean “converges in probability to.” Let

\overset{a . s .}{\to}

mean “converges almost surely to.”

Let X be a real-valued nonnegative random variable. Let n be a positive integer and assume that n > 1. For

i = 1, \dots, n

, let X_i be independent and identically distributed as X. For any real

h \geq 0

, the hth (raw) sample moment is defined as

M_{h}' ≔ \frac{1}{n} \sum_{i = 1}^{n} X_{i}^{h} .

[1]

Thus,

M_{1}'

is the sample mean. For any nonnegative integer h, the hth sample central moment is defined as

M_{h} ≔ \frac{1}{n} \sum_{i = 1}^{n} {(X_{i} - M_{1}')}^{h} .

[2]

Clearly,

M_{1} = 0

, and M₂ is the sample variance normalized by n. The sample variance normalized by n – 1 is defined as

v_{n} ≔ \frac{1}{n - 1} \sum_{i = 1}^{n} {(X_{i} - M_{1}')}^{2} .

[3]

Obviously,

v_{n} = M_{2} n / (n - 1)

and

v_{n} / M_{2} \overset{a . s .}{\to} 1

as

n \to \infty

.

The lower sample semivariance and the upper sample semivariance are defined as

\begin{array}{l} v_{n}^{-} ≔ \frac{1}{n - 1} \sum_{i : X_{i} \leq M_{1}'} {(X_{i} - M_{1}')}^{2}, \\ v_{n}^{+} ≔ \frac{1}{n - 1} \sum_{i : X_{i} > M_{1}'} {(X_{i} - M_{1}')}^{2}, \end{array}

[4]

so that

v_{n} = v_{n}^{-} + v_{n}^{+}

. Define

N_{n}^{-}

as the number of values of X_i that do not exceed the sample mean and

N_{n}^{+}

as the number of values of X_i that (strictly) exceed the sample mean:

N_{n}^{-} ≔ # {i : X_{i} \leq M_{1}'}, N_{n}^{+} ≔ # {i : X_{i} > M_{1}'} .

[5]

Then,

N_{n}^{-} N_{n}^{+} > 0

unless

X_{i} = M_{1}'

for all

i = 1, \dots, n

. The local lower sample semivariance and the local upper sample semivariance are defined only when

N_{n}^{-} > 0

and

N_{n}^{+} > 0

, respectively, as

\begin{array}{l} v_{n}^{- *} ≔ \frac{1}{N_{n}^{-}} \sum_{i : X_{i} \leq M_{1}'} {(X_{i} - M_{1}')}^{2}, \\ v_{n}^{+ *} ≔ \frac{1}{N_{n}^{+}} \sum_{i : X_{i} > M_{1}'} {(X_{i} - M_{1}')}^{2} . \end{array}

[6]

The local upper sample semivariance

v_{n}^{+ *}

is the more mathematically challenging sequence to analyze because it depends on the asymptotic behavior of the number of observations that exceed the sample mean. Our result, Theorem 9, may be of independent interest in the study of heavy-tailed distributions.

For the remainder of this article, we assume two restrictions on X without further restatement. First, we assume that X takes only nonnegative values. Second, to assure that

ℙ (N_{n}^{+} = 0) = 0

, we assume that X is not atomic [i.e., for all real a, we assume that

ℙ (X = a) = 0

]. Then,

ℙ (N_{n}^{+} = 0) = 0

and conversely; for otherwise, if

ℙ (X = a) > 0

for some a, then

ℙ (N_{n}^{+} = 0) \geq {ℙ (X = a)}^{n} > 0

. Under the assumption that X is not atomic,

ℙ (N_{n}^{-} N_{n}^{+} > 0) = 1

, and

v_{n}^{- *}

and

v_{n}^{+ *}

are well defined almost surely (a.s.); also,

v_{n}^{-} = N_{n}^{-} v_{n}^{- *} / (n - 1), v_{n}^{+} = N_{n}^{+} v_{n}^{+ *} / (n - 1)

, and

v_{n} = (N_{n}^{-} v_{n}^{- *} + N_{n}^{+} v_{n}^{+ *}) / (n - 1)

a.s. The assumption that X is not atomic also plays an important role in Theorems 5 and 8(3), Remark 2, and Corollaries 6(3) and 8.

Alternatively, we could assume that X is not constant (i.e., not a degenerate random variable with all probability mass concentrated at a single value). If X is atomic but not a constant, then

ℙ (N_{n}^{-} N_{n}^{+} > 0) \to 1

as

n \to \infty

, but

ℙ (N_{n}^{-} N_{n}^{+} > 0) \neq 1

. Nevertheless, similar asymptotic results could still be proved.

The infinite sequences of random variables defined in Eqs. 1 to 6 (one random variable for each

n = 1, 2, \dots

) exist a.s., whether or not X has any finite moments. Our goal here is to show that, if X is a stable distribution (or an approximately stable distribution under Definition 1) with support

(0, \infty)

and index

α \in (0, 1)

, then as

n \to \infty

, the quantities in Eqs. 1 to 6 and other related quantities defined in section 3, when divided by some power b of the sample mean

M_{1}'

, converge in distribution, in probability or almost surely, depending on the case. Here, b may depend on α and on which quantity is being examined.

2. Background and Prior Results

Taylor’s law [Taylor (16)] says that the sample variance v_n scales approximately in direct proportion to a nonzero power b (positive or negative) of the sample mean

M_{1}'

. Taylor’s law is a widely confirmed empirical pattern in ecology and other sciences [Taylor (17)], nearly always with b > 0 and often with

b \in (1, 2)

. Taylor’s law holds also for the mean and variance of some single-parameter probability distributions, in addition to holding for the sample mean and sample variance. For example, for varying values of the population mean μ, the population variance

σ^{2}

varies according to Taylor’s law

σ^{2} = a μ^{b}

with a = 1, b = 1 for the Poisson distribution and a = 1, b = 2 for the exponential distribution.

The semivariances, especially the lower, have important applications in agricultural and financial economics [Berck and Hihn (19), Bond and Satchell (20), Hogan and Warren (21), Jin et al. (22), Liagkouras and Metaxiotis (23), Nantell and Price (24), Porter (25), Turvey and Nayak (26), and van de Beek et al. (27)]. We know no prior proofs that the sample semivariances of a nonnegative stable law satisfy Taylor’s law.

Higher moments include skewness and kurtosis in statistics and the Farinelli–Tibiletti ratio in finance. Power-law scaling relationships for moments other than the sample variance are generalized Taylor’s laws [Giometto et al. (28)]. Generalized Taylor’s laws are less widely studied empirically or theoretically.

Every stable random variable X with support

(0, \infty)

has Laplace transform [Feller (2), pp. 448–449]

L (s) ≔ E (e^{- s X}) = e^{- {(c s)}^{α}},

[7]

for

s \geq 0, 0 < α < 1

, and c > 0. We say that

X \overset{d}{=} F (c, α)

when the distribution of X has Laplace transform Eq. 7, and then we say that X has index α. We have

X \overset{d}{=} F (c, α) \overset{d}{=} c F (1, α)

. Such a heavy-tailed distribution has an infinite mean. Consequently, the sample mean, sample variance, sample semivariances, and sample higher moments are not estimators of population moments, and the normal central limit theorem does not apply.

If

X \overset{d}{=} F (c, α)

for some

0 < α < 1

, c > 0, the survival function of X evaluated at

t \in (0, \infty)

is defined as

\bar{F} (c, α) (t) ≔ 1 - F (c, α) (t)

. By Feller (2, p. 448), if

0 < α < 1

and c > 0, then as

t \to \infty

,

\bar{F} (c, α) (t) / \frac{c^{α} t^{- α}}{Γ (1 - α)} \to 1 .

[8]

Many distributions on

(0, \infty)

satisfy Eq. 8 but are not of the special form

F (c, α)

in Eq. 7.

Definition 1.

X \overset{d}{\approx} F (c, α)

and

F_{X} \overset{d}{\approx} F (c, α)

both mean that a nonnegative random variable X has a distribution function F_X that satisfies Eq. 8: that is, as

t \to \infty

,

{1 - F_{X} (t)} / \frac{c^{α} t^{- α}}{Γ (1 - α)} \to 1 .

[9]

When Eq. 9 holds, we say that X is approximately stable.

For

α \in (0, 1)

and real

g > α, h > α

, define

α (g, h) ≔ \frac{g - α}{h - α}, α^{*} ≔ α (2, 1) = \frac{2 - α}{1 - α} .

[10]

If g > h, then

α (g, h) > g / h

. Consequently,

α^{*} > 2

. If g < h, then

α (g, h) < g / h < 1

. Thus if, as we shall prove below,

α (g, h)

is the exponent b in Taylor’s law for a stable nonnegative law with index

α \in (0, 1)

and if

g \geq 2 h

or g < h, then the exponent b must fall outside the interval

(1, 2)

that is commonly (although not universally) observed in many ecological applications [Cohen et al. (29, 30)].

Among other results, BCD (ref. 15, p. 663, proposition 2) showed that if

X \overset{d}{=} F (1, α)

, then as

n \to \infty

,

W_{n} ≔ \frac{v_{n}}{M_{1}'^{α^{*}}} \overset{d}{\to} W,

[11]

where

E (W_{n}) = 1 - α, Var (W_{n}) = {E (W_{n})}^{2} {1 + 2 α / (n - 1)}

, and the limiting random variable W has

ℙ (0 < W < \infty) = 1

. W has a finite mean and a finite SD, both of which equal

1 - α

. Moreover, for all

h = 1, 2, \dots, E (W_{n}^{h}) \to E (W^{h})

. The second and third moments of W are

E (W^{2}) = 2 {E (W)}^{2}, E (W^{3}) = (6 - \frac{α}{(5 - 2 α)}) {E (W)}^{3},

[12]

while for an exponentially distributed random variable Y,

E (Y^{3}) = 6 {E (Y)}^{3}

(ref. 15, p. 666).

For general c > 0 in Eq. 7, BCD showed that

v_{n} / M_{1}'^{α^{*}} \overset{d}{\to} c^{- \frac{α}{1 - α}} W

, where W is the limiting random variable in Eq. 11. Consequently, for any c > 0, BCD showed that as

n \to \infty

,

\frac{\log v_{n}}{\log M_{1}'} \overset{p}{\to} α^{*} .

[13]

Thus, for large n, with arbitrarily high probability,

(\log v_{n}) / (\log M_{1}')

will be close to

α^{*}

, regardless of c > 0. This scaling relationship is an asymptotic form of Taylor’s law with exponent

b = α^{*} > 2

. BCD further argued without detailed proofs that

X \overset{d}{\approx} F (c, α)

satisfies Eq. 13.

A common sample statistic used to compare the effectiveness of investments is the well-known Sharpe ratio [Sharpe (31)]

(M_{1}' - r_{f}) / v_{n}^{1 / 2}

for the period rates of return of a security, where r_f is a zero-risk reference: for example, the London interbank offered rate. In signal processing, the Sharpe ratio (with

r_{f} = 0

) is a useful but biased estimator of the signal-to-noise ratio [Miller and Gehr (32)]. In statistics, the reciprocal of the Sharpe ratio (with r_f = 0) is called the coefficient of variation. If the period rate of return has a distribution

X \overset{d}{\approx} F (c, α)

, where

0 < c < \infty

and

0 < α < 1

, then the Sharpe ratio converges in probability to zero as

n \to \infty

. Why? Eq. 11 implies that, as

n \to \infty, M_{1}'^{α^{*}} / v_{n} \overset{d}{\to} 1 / W

, so

M_{1}'^{α^{*} / 2} / v_{n}^{1 / 2} \overset{d}{\to} 1 / W^{1 / 2}

. However,

M_{1}'^{α^{*} / 2} = M_{1}' \times M_{1}'^{(α^{*} / 2) - 1}

, and because

α^{*} > 2

(as noted just after Eq. 10), the second factor

M_{1}'^{(α^{*} / 2) - 1}

goes a.s. to

\infty

. Therefore, the Sharpe ratio

(M_{1}' - r_{f}) / v_{n}^{1 / 2}

must converge in probability to zero. Asymptotically, for large n, the Sharpe ratio reveals no information about the distribution.

Inspired by Taylor’s law in Eq. 13, one may consider

\log (M_{1}' - r_{f}) / \log v_{n}

as a modified financial ratio, which converges to

1 / α^{*} = (1 - α) / (2 - α)

in probability. Because

(1 - α) / (2 - α)

is decreasing in α over (0, 1), the smaller α is, the heavier the distribution, so the larger the risk. The original Sharpe ratio is quasiconcave, scale invariant, and distribution based [Eling et al. (33)]. The modified ratio is also distribution based and reveals the tail index α for large-enough n. Because of the logarithmic transformation, the modified ratio is not scale invariant. However, both numerator and denominator diverge to infinity. The effect of finite scaling becomes negligible for large sample sizes, and hence, the ratio is

F_{α}

-asymptotically scale invariant.^* In other words, when

X \overset{d}{\approx} F (c, α)

, the modified ratio is asymptotically invariant with respect to c. The modified Sharpe ratio is

F_{α}

-asymptotically quasiconcave.^† The proof is in SI Appendix. Thus, asymptotically with large sample size n, the modified Sharpe ratio inherits all the properties of the original Sharpe ratio. We discuss this using semivariances and partial moments for the financial ratios in the following sections.

3. Taylor’s Laws for Semivariances

A. Lower Semivariances and Sortino Ratio

The lower semivariance of any nonnegative random variable with infinite expectation is almost surely asymptotic to the square of the sample mean.

Theorem 1

(Taylor’s law for the lower semivariance). Let X be a nonnegative random variable with

E (X) = \infty

. Then, as

n \to \infty

,

\frac{v_{n}^{-}}{M_{1}^{' 2}} \overset{a . s .}{\to} 1 .

[14]

This theorem does not assume X is stable or approximately stable.

The Sortino ratio [Sortino and Price (34)] is another sample statistic used to compare the risks and rewards in some period of a set of investments such as individual equities, mutual funds, trading systems, or investment managers. It is defined as

(M_{1}' - r_{f}) / s_{d}

, where

M_{1}'

is the sample mean of the period rate of return X, r_f is a threshold or reference point or target return, the zero-risk rate of return or minimal acceptable return, which we take to be zero, and

s_{d} ≔ {(v_{n}^{-})}^{1 / 2}

is the downside risk, equal to the square root of the lower sample semivariance

v_{n}^{-}

of the period rate of return [e.g., Sortino and Price (34) and Rollinger and Hoffman (35)]. Under our assumption that

ℙ (0 < X < \infty) = 1

, one might interpret X as the ratio of final price to initial price, so that

0 < X < 1

would represent a loss, while X > 1 would represent a gain. The possible use of n instead of

n - 1

in the denominator of Eq. 4 is immaterial for large samples. Eq. 14 shows that if the period rate of return X is a nonnegative random variable with an infinite mean, then the Sortino ratio converges a.s. to one as

n \to \infty

. When the mean is infinite, asymptotically, for large n, the Sortino ratio reveals no information about the distribution.

Similar to our modified Sharpe ratio for heavy-tailed distributions, for the Sortino ratio, we consider the ratio between the logarithm of the sample mean minus r_f and the logarithm of the sample lower semivariance, namely

\log (M_{1}' - r_{f}) / \log v_{n}^{-}

. Theorem 1 and Slutsky’s theorem imply that a power law with exponent 2 relates the lower semivariance to the sample mean. So Taylor’s law holds between the sample mean and the lower semivariance.

Corollary 1.

Let X be a nonnegative random variable with

E (X) = \infty

. As

n \to \infty

,

\frac{\log v_{n}^{-}}{\log M_{1}'} \overset{a . s .}{\to} 2 .

[15]

The modified Sortino ratio is

F_{α}

-asymptotically quasiconcave and

F_{α}

-asymptotically scale invariant, like the original Sortino ratio; proofs are in SI Appendix. However, from Corollary 1, the limiting value of the modified Sortino ratio is independent of the tail index α.

We now extend Taylor’s law to the local lower semivariance

v_{n}^{- *}

. The local lower semivariance differs from the lower semivariance by a factor equal to the ratio

N_{n}^{-} / n

. We show that

N_{n}^{-} / n \to 1

almost surely if

E (X) = \infty

.

Lemma 1.

Let X be a nonnegative random variable with

E (X) = \infty

. Then, with

N_{n}^{-}

defined in Eq. 5, as

n \to \infty

,

\frac{N_{n}^{-}}{n} \overset{a . s .}{\to} 1 .

[16]

Corollary 1 and Lemma 1 imply that a power law with exponent 2 relates the local lower semivariance to the sample mean.

Corollary 2.

Let X be a nonnegative random variable with

E (X) = \infty

. Then, as

n \to \infty

,

\frac{\log v_{n}^{- *}}{\log M_{1}'} \overset{a . s .}{\to} 2 .

[17]

If X is approximately stable with infinite expectation, then Lemma 1 and Corollaries 1 and 2 imply further results that will be useful later for studying the local upper semivariance and upper semivariance.

Corollary 3.

Let

X \overset{d}{\approx} F (1, α), 0 < α < 1

. Let

α^{*} ≔ (2 - α) / (1 - α)

as defined in Eq. 10. Then, as

n \to \infty

,

\frac{v_{n}^{-}}{M_{1}'^{α^{*}}} \overset{a . s .}{\to} 0 and \frac{v_{n}^{- *}}{M_{1}'^{α^{*}}} \overset{a . s .}{\to} 0 .

[18]

B. Upper Semivariances

Although the asymptotic values of the ratios in Eqs. 15 and 17 are both two, which is independent of α, if one replaces the lower or local lower semivariances by the upper or local upper semivariances, respectively, Taylor’s law continues to hold, and it depends on α.

Theorem 2.

Let

X \overset{d}{\approx} F (1, α), 0 < α < 1

. Then, as

n \to \infty

,

\frac{\log v_{n}^{+}}{\log M_{1}'} \overset{p}{\to} α^{*} and \frac{\log v_{n}^{+ *}}{\log M_{1}'} \overset{p}{\to} α^{*} + α = \frac{2 - α^{2}}{1 - α} .

[19]

Inspired by Taylor’s law in Eq. 19, one may consider ratios between the logarithm of the sample mean minus r_f and the logarithm of either the sample upper or local upper semivariances, namely

\log (M_{1}' - r_{f}) / \log v_{n}^{+}

and

\log (M_{1}' - r_{f}) / \log v_{n}^{+ *}

, respectively, which converge in probability to

1 / α^{*} = (1 - α) / (2 - α)

and

(1 - α) / (2 - α^{2})

, respectively. Because

(1 - α) / (2 - α)

and

(1 - α) / (2 - α^{2})

are both decreasing in α, the smaller α is, the heavier the distribution is, and the larger these ratios are asymptotically. The asymptotic properties and proofs are in SI Appendix, Proposition D.3.

4. Fluctuation Scaling for Higher Moments

In this section, we show that the sample higher moments are proportional to a power of the sample mean. These relations imply power-law relations between sample higher moments used in financial ratios such as the Farinelli–Tibiletti ratio (36).

A. Higher Sample Moments, Skewness, and Kurtosis

Theorem 3.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1

, and

h > α

, then, as

n \to \infty

,

\frac{M_{h}'}{{(M_{1}')}^{α (h, 1)}} \overset{d}{\to} {Γ (1 - α)}^{\frac{h - 1}{1 - α}} \frac{U_{h}}{V^{α (h, 1)}},

where the random vector

(U_{h}, V)

has the joint Laplace transform

E (e^{- s U_{h} - t V}) = \exp {- \int_{0}^{\infty} {r_{h} (y, s, t)}^{- α} e^{- y} d y},

for

s, t, y > 0

, and

r_{h} (y, s, t)

is the unique positive root of the equation

s x^{h} + t x - y = 0 .

The ratio in Theorem 3 may not be a practically useful financial ratio since α is usually unknown. However, the following Theorem 4 and its corollaries heavily depend on it. The following remark uses the joint moment-generating function to give the marginal distributions of U_h and V.

Remark 1.

In the joint Laplace transform defined in Theorem 3, if we set t = 0, then

r_{h} (y, s, 0) = {(y / s)}^{1 / h}

and

E (e^{- s U_{h}}) = \exp {- \int_{0}^{\infty} {{(y / s)}^{1 / h}}^{- α} e^{- y} d y} .

Hence, U_h follows the distribution

F ({Γ (1 - α / h)}^{h / α}, α / h)

. On the other hand, if we set s = 0, then

r_{h} (y, 0, t) = y / t

and

E (e^{- t V}) = \exp {- \int_{0}^{\infty} {(y / t)}^{- α} e^{- y} d y} .

Hence, V follows the distribution

F ({Γ (1 - α)}^{1 / α}, α)

.

These results follow Albrecher et al. (ref. 37, remark 2.1) by the arguments in their proof. The following theorem shows that Taylor’s law holds for raw moments.

Theorem 4.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1, h_{1} > α

, and

h_{2} > α

, then as

n \to \infty

,

\frac{\log M_{h_{2}}'}{\log M_{h_{1}}'} \overset{p}{\to} α (h_{2}, h_{1}) .

In particular, for

h > α

, as

n \to \infty

,

\frac{\log M_{h}'}{\log M_{1}'} \overset{p}{\to} α (h, 1) .

For a positive integer h > 1, the ratio between the central moment M_h and the

α (h, 1)

power of the sample mean

M_{1}'

converges to a distribution given in Corollary 4.

Corollary 4.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1

, and h > 1 is a positive integer, then as

n \to \infty

,

\frac{M_{h}}{{(M_{1}')}^{α (h, 1)}} \overset{d}{\to} {Γ (1 - α)}^{\frac{h - 1}{1 - α}} \frac{U_{h}}{V^{α (h, 1)}},

where the random vector

(U_{h}, V)

is specified in Theorem 3.

Theorem 5.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1

, and h > 1 is a positive integer, then as

n \to \infty

,

\frac{\log | M_{h} |}{\log M_{1}'} \overset{p}{\to} α (h, 1) .

For any positive integers

h_{1} > 1

and

h_{2} > 1

, as

n \to \infty

,

\frac{\log | M_{h_{2}} |}{\log | M_{h_{1}} |} \overset{p}{\to} α (h_{2}, h_{1}) .

For the raw moments, we have generalized Theorem 3 for the ratio of two raw moments with orders both larger than α.

Theorem 6.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1

, and both

h_{1}, h_{2} > α

, then as

n \to \infty

,

\frac{M_{h_{2}}'}{{(M_{h_{1}}')}^{α (h_{2}, h_{1})}} \overset{d}{\to} {Γ (1 - α)}^{\frac{h_{2} - h_{1}}{h_{1} - α}} \frac{U_{h_{2}}}{{(U_{h_{1}})}^{α (h_{2}, h_{1})}},

where

(U_{h_{1}}, U_{h_{2}})

has the joint Laplace transform

E (e^{- s U_{h_{2}} - t U_{h_{1}}}) = \exp {- \int_{0}^{\infty} {r_{h_{2}, h_{1}} (y, s, t)}^{- α} e^{- y} d y},

with y > 0, s > 0, t > 0, and

r_{h_{2}, h_{1}} (y, s, t)

is the unique positive root x of

s x^{h_{2}} + t x^{h_{1}} - y = 0 .

Moreover, as

n \to \infty

,

\frac{\log M_{h_{2}}'}{\log M_{h_{1}}'} \overset{p}{\to} α (h_{2}, h_{1}) .

Corollary 5.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1

, and

h_{2} \geq h_{1} > 1

are positive integers, then as

n \to \infty

,

n^{\frac{h_{1} - h_{2}}{h_{1}}} \frac{M_{h_{2}}}{{(M_{h_{1}})}^{h_{2} / h_{1}}} \overset{d}{\to} \frac{U_{h_{2}}}{{(U_{h_{1}})}^{h_{2} / h_{1}}},

where

(U_{h_{1}}, U_{h_{2}})

is defined in Theorem 6.

Remark 2.

From Corollary 5, it is clear that the skewness

M_{3} / {(v_{n})}^{3 / 2}

and the kurtosis

M_{4} / {(v_{n})}^{2}

diverge to infinity, yet the scaled skewness and the scaled kurtosis have distributions, asymptotically as

n \to \infty

,

\frac{M_{3}}{n^{1 / 2} {(v_{n})}^{3 / 2}} \overset{d}{\to} \frac{U_{3}}{{(U_{2})}^{3 / 2}} and \frac{M_{4}}{n {(v_{n})}^{2}} \overset{d}{\to} \frac{U_{4}}{{(U_{2})}^{2}},

where the joint distributions of (U₂, U₃) and

(U_{2}, U_{4})

are defined in Theorem 6. The limiting distribution of

M_{4} / {n {(v_{n})}^{2}}

matches the result derived in Cohen et al. (ref. 38, equation 3.9). Moreover, by Slutsky’s theorem, as

n \to \infty

,

\frac{\log | M_{3} |}{\log [{(v_{n})}^{3 / 2}]} \overset{p}{\to} \frac{2}{3} α (3, 2) and \frac{\log M_{4}}{\log [{(v_{n})}^{2}]} \overset{p}{\to} \frac{1}{2} α (4, 2) .

B. Central Lower and Local Lower Partial Moments

Definition 2.

Define

c_{+} ≔ \max {0, c}

for

c \in R

. For h > 0, define

M_{h}^{-} ≔ \frac{1}{n} \sum_{i = 1}^{n} {[{(M_{1^{'}} - X_{i})}_{+}]}^{h}, M_{h}^{- *} ≔ \frac{n M_{h}^{-}}{N_{n}^{-}} .

Theorem 7.

Let X be a nonnegative random variable with

E (X) = \infty

, and let h > 0. Then, as

n \to \infty

,

M_{h}^{-} / {(M_{1}')}^{h} \overset{a . s .}{\to} 1 and \log M_{h}^{-} - h \log M_{1}' \overset{a . s .}{\to} 0 .

Corollary 6.

Let X be a nonnegative random variable with

E (X) = \infty

. Then, as

n \to \infty

,

1)

M_{1}^{-} / M_{1}' \overset{a . s .}{\to} 1

;

2)

for h > 1,

M_{h}^{-} / {(M_{1}')}^{α (h, 1)} \overset{a . s .}{\to} 0

;

3)

for h > 0,

\frac{\log M_{h}^{-}}{\log M_{1}'} \overset{a . s .}{\to} h and \frac{\log M_{h}^{- *}}{\log M_{1}'} \overset{a . s .}{\to} h .

C. Central Upper Moments and Local Upper Moments

Definition 3.

For h > 0, define the hth central upper moments and central local upper moments:

M_{h}^{+} ≔ \frac{1}{n} \sum_{i = 1}^{n} {[{(X_{i} - M_{1^{'}})}_{+}]}^{h}, M_{h}^{+ *} ≔ \frac{n M_{h}^{+}}{N_{n}^{+}} .

Theorem 8

(central upper moments). Let

X \overset{d}{\approx} F (1, α)

,

0 < α < 1

. Then, as

n \to \infty

,

1)

for

0 < h < 1, M_{h}^{+} / {(M_{1}')}^{h} \overset{p}{\to} 0

;

2)

for

h \geq 1

,

\frac{M_{h}^{+}}{{(M_{1}')}^{α (h, 1)}} \overset{d}{\to} {Γ (1 - α)}^{\frac{h - 1}{1 - α}} \frac{U_{h}}{V^{α (h, 1)}},

where the random vector

(U_{h}, V)

has the joint Laplace transform defined in Theorem 3;

3)

for

h \geq 1

,

\frac{\log M_{h}^{+}}{\log M_{1}'} \overset{p}{\to} α (h, 1) and \frac{\log M_{h}^{+ *}}{\log M_{1}'} \overset{p}{\to} \frac{h - α^{2}}{1 - α} .

[20]

D. Omega Index, Upside Potential Ratio, and Farinelli–Tibiletti Ratio

Farinelli–Tibiletti (36) extended the Sharpe ratio to an index including asymmetrical information on the volatilities above and below the benchmark

r_{f} \in R

. Their index

Φ_{FT}

is defined by

Φ_{FT} (r_{f}, p, q) ≔ \frac{{[E {[{(X - r_{f})}_{+}]}^{p}]}^{1 / p}}{{[E {[{(r_{f} - X)}_{+}]}^{q}]}^{1 / q}} .

The Omega index, introduced by Cascon et al. (39), is

Φ_{FT} (r_{f}, 1, 1)

with

p = q = 1

. The upside potential index, introduced by Sortino et al. (40), is

Φ_{FT} (r_{f}, 1, 2)

with p = 1 and q = 2. The ratio

Φ_{FT} (r_{f}, p, q)

may not be well defined since the expectations may not exist for the heavy-tailed distributions. However, one can define an empirical version of the Farinelli–Tibiletti ratio by

Φ_{FT}^{n} (r_{f}, p, q) ≔ \frac{{[\frac{1}{n} \sum_{i = 1}^{n} {[{(X_{i} - r_{f})}_{+}]}^{p}]}^{1 / p}}{{[\frac{1}{n} \sum_{i = 1}^{n} {[{(r_{f} - X_{i})}_{+}]}^{q}]}^{1 / q}} .

The following corollary shows that both

Φ_{FT}^{n} (r_{f}, p, q)

and

Φ_{FT}^{n} (M_{1}', p, q)

converge to

\infty

in probability.

Corollary 7.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1, r_{f} > 0

, p > 1, and q > 1, then as

n \to \infty, Φ_{FT}^{n} (r_{f}, p, q) \overset{p}{\to} \infty

and

Φ_{FT}^{n} (M_{1}', p, q) \overset{p}{\to} \infty

.

A modification of the usual Farinelli–Tibiletti ratio might have the ratio of the logarithm of the numerator to the logarithm of the denominator in

Φ_{FT} (r_{f}, p, q)

. However, for a fixed

r_{f} > 0

, the numerator converges to infinity in probability, while the denominator is bounded above with probability one. Therefore, this ratio diverges to infinity.

We propose as an alternative to the Farinelli–Tibiletti ratio:

Φ_{FTlog} (p, q) ≔ p \log M_{q}^{-} / (q \log M_{p}^{+}),

which is the ratio of the logarithm of the numerator to that of the denominator in

Φ_{FT} (M_{1}', p, q)

. The following corollary describes generalized Taylor’s laws for the ratio of the logarithm of the upper central moment to the logarithm of the lower central moment.

Corollary 8.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1, p \geq 1

, and

q \geq 1

, then as

n \to \infty

,

\frac{\log M_{p}^{+}}{\log M_{q}^{-}} \overset{p}{\to} \frac{p - α}{q (1 - α)} .

Corollary 8 implies that

Φ_{FTlog} (p, q) \overset{p}{\to} p (1 - α) / (p - α),

which is decreasing in α for

p \geq 1, q \geq 1

. Therefore, the smaller α is, the heavier the distribution is, and the larger the risk is. Our modified Farinelli–Tibiletti ratio

Φ_{FTlog} (p, q)

is asymptotically scale invariant and distribution based, like the original Farinelli–Tibiletti ratio, and satisfies

F_{α}

-asymptotic quasiconcavity (SI Appendix).

5. Number of Observations Exceeding Sample Mean of Stable Law

A. Asymptotic Distributions and Moments of $N_{n}^{+} / n^{α}$

In a sample of size n from an approximately stable law with index

α \in (0, 1)

, asymptotically the number of observations above the sample mean scales as

n^{α}

and has a distribution given by Theorem 9. To prove this result, we use Einmahl (ref. 41, corollary 2.1) together with SI Appendix, Lemma C.1.

Theorem 9.

If

X \overset{d}{\approx} F (1, α), 0 < α < 1

, and

U \overset{d}{=} F (1, α)

, then as

n \to \infty

,

\frac{N_{n}^{+}}{n^{α}} \overset{d}{\to} V ≔ \frac{U^{- α}}{Γ (1 - α)} .

The asymptotic moments of

N_{n}^{+} / n^{α}

are the moments of V defined in Theorems 9 and 10.

Theorem 10.

Let

U \overset{d}{=} F (1, α), 0 < α < 1, V ≔ U^{- α} / Γ (1 - α)

, and

ε \overset{d}{=} Exp (1)

(an exponential random variable with mean and parameter 1), where ε is independent of U.

1)

U^{- α} ε^{α} \overset{d}{=} Exp (1)

.

2)

For integer K > 0,

\begin{array}{l} E [U^{- K α}] = \frac{K!}{Γ (1 + K α)}, \\ E [V^{K}] = \frac{K!}{Γ (1 + K α) {Γ (1 - α)}^{K}} . \end{array}

Specifically, when K = 1, then

E [U^{- α}] = {Γ (1 + α)}^{- 1}

and

E [V] = {Γ (1 + α) Γ (1 - α)}^{- 1}

; when K = 2, then

E [U^{- 2 α}] = 2 {Γ (1 + 2 α)}^{- 1}, E [V^{2}] = 2 {Γ (1 + 2 α) {Γ (1 - α)}^{2}}^{- 1}

. Hence

\begin{array}{l} Var (U^{- α}) = \frac{2}{Γ (1 + 2 α)} - \frac{1}{{Γ (1 + α)}^{2}}, \\ Var (V) = \frac{1}{{Γ (1 - α)}^{2}} Var (U^{- α}) . \end{array}

3)

SD (V) < E [V]

. For example, when

α = 1 / 2, E [V^{2}] = 2 / π, E [V] = 2 / π, Var (V) = \frac{2}{π} (1 - \frac{2}{π})

. Numerically,

SD (V) \approx 0.48097, E [V] \approx 0.63662

, where here “

\approx

” means the numerical approximation is inexact.

4)

For

K \geq 2, E [V^{K}] < K! {(E [V])}^{K}

.

5)

V \leq_{s t} ε

[i.e., by the definition of the stochastic ordering

\leq_{s t}, ℙ (V > t) \leq ℙ (ε > t)

for all

t \in R

].

Part 1 of Theorem 10 is not well known. The moment results in part 2 of Theorem 10 are derived using fractional calculus by Wolfe (42). Because the logarithm of the moment-generating function of a nonnegative random variable is a convex function of the moment (by Artin’s theorem) [Marshall and Olkin (ref. 43, theorem B.8)], it follows that

\log E (U^{- x α}) = \log Γ (1 + x) - \log E (W^{x})

is concave in

x \in [1, \infty)

.

The distribution of

U^{- α}

approximates the standard exponential distribution

Exp (1)

when

α \to 0

.

Corollary 9.

Let

U \overset{d}{=} F (1, α)

. Then, as

α \to 0

,

U^{- α} \overset{d}{\to} Exp (1) .

6. Numerical Experiments

A. Tail Estimators

The preceding results describe the asymptotic ratio of the logarithm of the sample mean to the logarithm of various forms of the sample variance, such as the ordinary sample variance v_n, the upper semivariance

v_{n}^{+}

, the local upper semivariance

v_{n}^{+ *}

, and the lower semivariance

v_{n}^{-}

when a random sample is from an approximately stable

F (1, α)

satisfying Eq. 9. Most of these ratios (apart from that for the lower semivariance) depend asymptotically only on α. Based on these results, we propose estimators of the index α. We define the ratios

R_{1}, R_{2}, R_{3}

, and R_L where

\begin{array}{l} R_{1} ≔ \frac{\log v_{n}}{\log M_{1}'} \overset{p}{\to} \frac{2 - α}{1 - α}, R_{2} ≔ \frac{\log v_{n}^{+}}{\log M_{1}'} \overset{p}{\to} \frac{2 - α}{1 - α}, \\ R_{3} ≔ \frac{\log v_{n}^{+ *}}{\log M_{1}'} \overset{p}{\to} \frac{2 - α^{2}}{1 - α}, R_{L} ≔ \frac{\log v_{n}^{-}}{\log M_{1}'} \overset{a . s .}{\to} 2 . \end{array}

The results generalize to

F (c, α)

for c > 0 because as noted after Eq. 9,

X / c \overset{d}{\approx} F (1, α)

if and only if

X \overset{d}{\approx} F (c, α)

for c > 0. Applying the continuous mapping theorem to the above results for the variance, the upper semivariance, and the local upper semivariance yields three consistent estimators of α:

\begin{array}{l} B_{1} ≔ \frac{2 - R_{1}}{1 - R_{1}}, B_{2} ≔ \frac{2 - R_{2}}{1 - R_{2}}, \\ B_{3} ≔ \frac{R_{3} - \sqrt{R_{3}^{2} - 4 (R_{3} - 2)}}{2} . \end{array}

The Hill estimator [Hill (44)] is a traditional tail-index estimator, which requires the largest k observations where

k \to \infty

and

k / n \to 0

as

n \to \infty

. However, k depends on the unknown parameters such as α and the series representation of the survival function [Hall (45)]. In practice, the number k is based on the “stable” point in the Hill plot, which may not always be available. Gomes and Guillou (46) give a comprehensive review.

Theorem 9 implies that

N_{n}^{+} / n

converges to zero in probability, which motivates the choice of

k = N_{n}^{+} + 1

in the Hill estimator:

{(\frac{1}{k} \sum_{i = n - k + 1}^{n} \log (X_{(i)}) - \log (X_{(n - k + 1)}))}^{- 1},

where

X_{(i)}

is the ith-order statistic,

1 \leq i \leq n

. We evaluate this choice of

k = N_{n}^{+} + 1

in the Hill estimator, denoted by HI.N, numerically. We also replace the smallest

(n - k)

order statistics in the original Hill estimator by the sample mean

M_{1}'

to obtain a new Hill-type estimator:

HI . M ≔ {(\frac{1}{N_{n}^{+}} \sum_{X_{i} > M_{1}'} \log (X_{i} / M_{1}'))}^{- 1} .

From Bergström (47), the survival function of the stable law for

0 < α < 1

is

\begin{array}{l} \bar{F} (1, α) (x) = \int_{x}^{\infty} - \frac{1}{π} \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k}}{k!} (\sin π α k) \frac{Γ (a k + 1)}{t^{a k + 1}} d t \\ = \frac{1}{π} \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k + 1}}{k!} (\sin π α k) \frac{Γ (a k)}{x^{a k}} \\ = C x^{- α} [1 + D x^{- α} + o (x^{- α})], \end{array}

where C > 0 and

D \neq 0

. From Hall (45), it is optimal to choose k tending to infinity at a rate of order

n^{2 α / (2 α + α)} = n^{2 / 3}

. We also consider this choice

k = n^{2 / 3}

for another Hill-type estimator, denoted by HI.Opt, and we compare the behavior with other estimators.

In our simulations, we generate 10⁴ independent random samples, each with sample size n, from

F (1, α)

by using the rstable function from the R package stabledist with arguments for the tail-index parameter alpha

= α

, the skewness parameter beta

= 1

, the scale parameter gamma

= | 1 - i \tan (π α / 2) |^{- 1 / α}

, the location parameter delta

= 0

, and parameterization pm

= 1

. Setting pm

= 1

specifies that we use the parameterization of stable laws in Samorodnitsky and Taqqu (4). For each random sample, we calculate the six estimators

B_{1}, B_{2}, B_{3}

, HI.N, HI.M, and HI.Opt. Then, we estimate the bias as the average of the 10⁴ differences between each estimator of α and the true α. We estimate the mean squared error (MSE) as the average of 10⁴ squared differences between each estimator of α and the true α.

In Table 1 for bias and Table 2 for MSE, the sample size is

n = 10^{4}

. According to the bias estimates in Table 1, B₁ tends to underestimate α, while B₂ and B₃ reduce the bias from B₁ by introducing the upper semivariance, which focuses more on larger numbers. B₃ has smaller bias than B₂ for most of the α except

α = 0.7

and 0.8. In Table 2, B₃ has smaller MSE than B₁ and B₂. Estimators HI.N and HI.M do not perform as well as B₃.

Table 1

Bias (

\times 10^{3}

; average of [estimate minus true α]) for tail-index estimators

B_{1}, B_{2}, B_{3}

, HI.N, HI.M, HI.Opt, and MHB3 with sample size

n = 10^{4}

from

F (1, α)

α	B1	B2	B3	HI.N	HI.M	HI.Opt	MHB3
0.1	–5.24	–3.87	–3.00	10.25	135.16	–0.92	–5.82
0.2	– 11.96	– 6.88	– 3.79	– 9.31	73.52	– 1.73	– 9.65
0.3	–19.43	–8.55	–2.38	–25.60	30.89	–2.05	–12.03
0.4	– 27.72	– 9.75	0.63	– 32.82	4.87	– 1.54	– 13.44
0.5	–35.03	–8.91	5.96	–29.40	–5.30	1.42	–12.56
0.6	–43.76	–10.44	9.41	–24.21	–8.21	6.67	–10.26
0.7	–50.27	–11.28	12.19	–10.06	0.13	19.37	–3.26
0.8	–53.49	–12.55	11.48	31.58	37.82	51.80	7.30
0.9	–50.31	–13.69	5.46	204.26	208.27	153.44	5.45

Table 2

MSE

(\times 10^{3})

(mean squared [estimate minus true α]) for tail-index estimators

B_{1}, B_{2}, B_{3}

, HI.N, HI.M, HI.Opt, and MHB3 with sample size

n = 10^{4}

from

F (1, α)

α	B1	B2	B3	HI.N	HI.M	HI.Opt	MHB3
0.1	0.14	0.15	0.11	2.61	20.06	0.02	0.10
0.2	0.53	0.58	0.35	4.73	9.13	0.09	0.31
0.3	1.13	1.23	0.71	7.33	6.31	0.19	0.57
0.4	1.86	1.96	1.16	9.15	6.39	0.34	0.85
0.5	2.60	2.66	1.76	9.12	6.76	0.54	1.15
0.6	3.47	3.20	2.32	7.93	6.13	0.84	1.53
0.7	4.15	3.38	2.60	6.46	5.30	1.52	1.94
0.8	4.32	3.05	2.28	6.97	6.67	4.35	2.13
0.9	3.59	2.10	1.33	57.51	58.26	26.36	1.33

The estimator HI.Opt with the optimal choice of

k = n^{2 / 3}

for the Hill estimator has the smallest bias, when

α \leq 0.6

, and MSE, when

α \leq 0.7

. However, B₃ from Taylor’s law of the local semivariance has better performance, especially much smaller bias, than HI.Opt for

α \geq 0.8

. Since HI.Opt tends to overestimate α, especially when

α \geq 0.7

, we defined the estimator MHB3 to be the minimum of B₃ and HI.Opt. This MHB3 not only reduces the bias dramatically but also improves the MSE of B₃ for α close to 1.

The advantages of B₃ and MHB3 gradually vanish when sample size increases because

k = n^{2 / 3}

is an asymptotically optimal choice. However, for sample sizes smaller than 10⁴, B₃ and MHB3 can improve HI.Opt even more. More comparisons are in SI Appendix for sample sizes

n = 10^{2}, 10^{3}

, and 10⁵. On the other hand, although the behavior of B₁, B₂, and B₃ depends on c in

F (c, α)

, one sees similar patterns in bias and MSE. B₃ and MHB3 still have better bias and MSE for

α \geq 0.8

for small sample sizes. More comparisons are in SI Appendix for

F (2, α)

and

F (0.5, α)

.

Tables in SI Appendix also show that both bias and MSE decrease when sample size increases, as expected of consistent estimators and as proved in Corollary 1.

B. Asymptotic Distribution of $N_{n}^{+} / n^{α}$

To illustrate Theorem 9, we generate 10³ independent random samples from

F (1, α)

with sample size

n = 10^{6}

and calculate

N_{n}^{+} / n^{α}

for each random sample. We use the 10³ values of

N_{n}^{+} / n^{α}

to estimate the distribution of

N_{n}^{+} / n^{α}

. To estimate the distribution of

U^{- α} / Γ (1 - α)

, we generate 10³ independent random values

U_{1}, \dots, U_{10^{3}}

from

F (1, α)

and calculate the corresponding

U_{i}^{- α} / Γ (1 - α)

for

i = 1, \dots, 10^{3}

. Then, we use the 10³ values of

U_{i}^{- α} / Γ (1 - α)

to estimate the distribution of

U^{- α} / Γ (1 - α)

. The histograms and quantile–quantile plots of

N_{n}^{+} / n^{α}

and

U^{- α} / Γ (1 - α)

with

α = 0.25

and

α = 0.5

are in Figs. 1 and 2, respectively. The histograms mostly overlap. The P values of the two-sample Kolmogorov–Smirnov (KS) test are 0.1995 and 0.9135, respectively. These observations support the convergence of

N_{n}^{+} / n^{α}

in distribution.

Fig. 1

Histogram and quantile–quantile plot of $N_{n}^{+} / n^{α}$ and $U^{- α} / Γ (1 - α)$ for $α = 0.25$ . The P value of the KS test is 0.1995.

Fig. 2

Histogram and quantile–quantile plot of $N_{n}^{+} / n^{α}$ and $U^{- α} / Γ (1 - α)$ for $α = 0.50$ . The P value of the KS test is 0.9135.

As expected, the speed of convergence of

N_{n}^{+} / n^{α}

in Theorem 9 depends on α. Similarly, the speeds of convergence of the moment ratios in Theorems 3 and 6 also depend on both α and the orders of the moments. We discuss the sample sizes required to see the convergence in distributions in Theorems 3, 6, and 9 in SI Appendix. From our simulation results, smaller α and higher-order moments result in faster convergence in distribution for the ratios of the moments.

Notes

*

F_{α}

-asymptotic scale invariance is defined in SI Appendix, section D.

†

F_{α}

-asymptotic quasiconcavity is defined in SI Appendix, section D.

Data Availability

Computer code has been deposited in GitHub (https://github.com/cftang9/TLHM). Readers can generate the tables and figures using the R code there.

Acknowledgments

J.E.C. thanks Roseanne Benjamin for help during this work. S.C.P.Y. acknowledges financial support from Hong Kong General Research Fund Grants HKGRF-14300717 “New Kinds of Forward-Backward Stochastic Systems with Applications,” HKGRF-14300319 “Shape-Constrained Inference: Testing for Monotonicity,” and HKGRF-14301321 “General Theory for Infinite Dimensional Stochastic Control: Mean Field and Some Classical Problems.”

Supporting Information

Appendix 01 (PDF)

Download
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References

1

R. Carmona, “Heavy tail distributions” in Statistical Analysis of Financial Data in R (Springer, New York, NY, ed. 2, 2014), chap. 2, 69–120.

Crossref

Google Scholar

2

W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, NY, 1971), vol. 2.

Google Scholar

3

S. I. Resnick, Heavy-Tail Phenomena: Probabilistic and Statistical Modeling (Springer Science & Business Media, 2007).

Google Scholar

4

G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, New York, NY, 1994).

Google Scholar

5

J. Nešlehová, P. Embrechts, V. Chavez-Demoulin, Infinite mean models and the LDA for operational risk. J. Oper. Risk 1, 3–25 (2006).

Crossref

Google Scholar

6

M. Campolieti, Heavy-tailed distributions and the distribution of wealth: Evidence from rich lists in Canada, 1999–2017. Physica A 503, 263–272 (2018).

Crossref

Google Scholar

7

C. Schluter, Top incomes, heavy tails, and rank-size regressions. Econometrics 6, 10 (2018).

Crossref

Google Scholar

8

F. M. Scherer, D. Harhoff, J. Kukies, “Uncertainty and the size distribution of rewards from innovation” in Capitalism and Democracy in the 21st Century, D. C. Mueller, U. Cantner, Eds. (Springer, 2001), pp. 181–206.

Crossref

Google Scholar

9

G. Silverberg, B. Verspagen, The size distribution of innovations revisited: An application of extreme value statistics to citation and value measures of patent significance. J. Econom. 139, 318–339 (2007).

Crossref

Google Scholar

10

Y. Cen, “City size distribution, city growth and urbanisation in China,” PhD thesis, University of Birmingham, Birmingham, United Kingdom (2015).

Google Scholar

11

N. Bérubé, M. Sainte-Marie, P. Mongeon, V. Larivière, Words by the tail: Assessing lexical diversity in scholarly titles using frequency-rank distribution tail fits. PLoS One 13, e0197775 (2018).

Crossref

PubMed

Google Scholar

12

P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events: For Insurance and Finance (Springer Science & Business Media, 2013), vol. 33.

Google Scholar

13

R. Ibragimov, D. Jaffee, J. Walden, Nondiversification traps in catastrophe insurance markets. Rev. Financ. Stud. 22, 959–993 (2009).

Crossref

Google Scholar

14

R. Ibragimov, Portfolio diversification and value at risk under thick-tailedness. Quant. Finance 9, 565–580 (2009).

Crossref

Google Scholar

15

M. Brown, J. E. Cohen, V. H. de la Peña, Taylor’s law, via ratios, for some distributions with infinite mean. J. Appl. Probab. 54, 657–669 (2017).

Crossref

Google Scholar

16

L. R. Taylor, Aggregation, variance and the mean. Nature 189, 732–735 (1961).

Crossref

Google Scholar

17

R. A. J. Taylor, Taylor’s Power Law: Order and Pattern in Nature (Elsevier Academic Press, Cambridge, MA, 2019).

Google Scholar

18

Z. Eisler, I. Bartos, J. Kertész, Fluctuation scaling in complex systems: Taylor’s law and beyond. Adv. Phys. 57, 89–142 (2008).

Crossref

Google Scholar

19

P. Berck, J. M. Hihn, Using the semivariance to estimate safety-first rules. Am. J. Agric. Econ. 64, 298–300 (1982).

Crossref

Google Scholar

20

S. A. Bond, S. E. Satchell, Statistical properties of the sample semivariance. Appl. Math. Finance 9, 219–239 (2002).

Crossref

Google Scholar

21

W. W. Hogan, J. M. Warren, Toward the development of an equilibrium capital-market model based on semivariance. J. Financ. Quant. Anal. 9, 1–11 (1974).

Crossref

Google Scholar

22

H. Jin, H. Markowitz, X.Y. Zhou, A note on semivariance. Math. Finance. 16, 53–61 (2006).

Crossref

Google Scholar

23

K. Liagkouras, K. Metaxiotis, The constrained mean-semivariance portfolio optimization problem with the support of a novel multiobjective evolutionary algorithm. J. Softw. Eng. Appl. 6, 22–29 (2013).

Crossref

Google Scholar

24

T. J. Nantell, B. Price, An analytical comparison of variance and semivariance capital market theories. J. Financ. Quant. Anal. 14, 221–242 (1979).

Crossref

Google Scholar

25

R. B. Porter, Semivariance and stochastic dominance: A comparison. Am. Econ. Rev. 64, 200–204 (1974).

Google Scholar

26

C. G. Turvey, G. Nayak, The semivariance-minimizing hedge ratio. J. Agric. Resour. Econ. 28, 100–115 (2003).

Google Scholar

27

C. Z. van de Beek, H. Leijnse, P. J. J. F. Torfs, R. Uijlenhoet, Climatology of daily rainfall semi-variance in The Netherlands. Hydrol. Earth Syst. Sci. 15, 171–183 (2011).

Crossref

Google Scholar

28

A. Giometto, M. Formentin, A. Rinaldo, J. E. Cohen, A. Maritan, Sample and population exponents of generalized Taylor’s law. Proc. Natl. Acad. Sci. U.S.A. 112, 7755–7760 (2015).

Crossref

PubMed

Google Scholar

29

J. E. Cohen, M. Xu, W. S. F. Schuster, Allometric scaling of population variance with mean body size is predicted from Taylor’s law and density-mass allometry. Proc. Natl. Acad. Sci. U.S.A. 109, 15829–15834 (2012).

Crossref

PubMed

Google Scholar

30

J. E. Cohen, M. Xu, W. S. F. Schuster, Stochastic multiplicative population growth predicts and interprets Taylor’s power law of fluctuation scaling. Proc. Biol. Sci. 280, 20122955 (2013).

Crossref

PubMed

Google Scholar

31

W. F. Sharpe, Mutual fund performance. J. Bus. 39, 119–138 (1966).

Crossref

Google Scholar

32

R. E. Miller, A. K. Gehr, Sample size bias and Sharpe’s performance measure: A note. J. Financ. Quant. Anal. 13, 943–946 (1978).

Crossref

Google Scholar

33

M. Eling, S. Farinelli, D. Rossello, L. Tibiletti, One-size or tailor-made performance ratios for ranking hedge funds? J. Deriv. Hedge Funds 16, 267–277 (2011).

Crossref

Google Scholar

34

F. A. Sortino, L. N. Price, Performance measurement in a downside risk framework. J. Invest. 3, 59–64 (1994).

Crossref

Google Scholar

35

T. N. Rollinger, S. T. Hoffman, Sortino: A ‘Sharper’ Ratio (Red Rock Capital, Chicago, IL, 2013).

Google Scholar

36

S. Farinelli, L. Tibiletti, Sharpe thinking in asset ranking with one-sided measures. Eur. J. Oper. Res. 185, 1542–1547 (2008).

Crossref

Google Scholar

37

H. Albrecher, S. A. Ladoucette, J. L. Teugels, Asymptotics of the sample coefficient of variation and the sample dispersion. J. Stat. Plan. Inference 140, 358–368 (2010).

Crossref

Google Scholar

38

J. E. Cohen, R. A. Davis, G. Samorodnitsky, Heavy-tailed distributions, correlations, kurtosis and Taylor’s Law of fluctuation scaling. Proc. R. Soc. Lond. A Math. Phys. Sci. 476, 20200610 (2020).

Google Scholar

39

A. Cascon, C. Keating, W. F. Shadwick, An Introduction to Omega (The Finance Development Centre, Fuqua-Duke University, 2002).

Google Scholar

40

F. A. Sortino, R. Van Der Meer, A. Plantinga, The Dutch triangle. J. Portfol. Manage. 26, 50–57 (1999).

Crossref

Google Scholar

41

J. H. J. Einmahl, The empirical distribution function as a tail estimator. Stat. Neerl. 44, 79–82 (1990).

Crossref

Google Scholar

42

S. J. Wolfe, “On moments of probability distribution functions” in Fractional Calculus and Its Applications, B. Ross, Ed. (Springer-Verlag, Berlin, Germany, 1975), pp. 306–316.

Crossref

Google Scholar

43

A. W. Marshall, I. Olkin, Inequalities: Theory of Majorization and Its Applications (Academic Press, New York, NY, 1979).

Google Scholar

44

B. M. Hill, A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975).

Crossref

Google Scholar

45

P. Hall, On some simple estimates of an exponent of regular variation. J. R. Stat. Soc. B 44, 37–42 (1982).

Google Scholar

46

M. I. Gomes, A. Guillou, Extreme value theory and statistics of univariate extremes: A review. Int. Stat. Rev. 83, 263–292 (2015).

Crossref

Google Scholar

47

H. Bergström, On some expansions of stable distribution functions. Ark. Mat. 2, 375–378 (1952).

Crossref

Google Scholar

Information & Authors

Information

Published in

Proceedings of the National Academy of Sciences

Vol. 118 | No. 46
November 16, 2021

PubMed: 34772810

Classifications

Copyright

Data Availability

Computer code has been deposited in GitHub (https://github.com/cftang9/TLHM). Readers can generate the tables and figures using the R code there.

Submission history

Accepted: September 29, 2021

Published online: November 12, 2021

Published in issue: November 16, 2021

Change history

November 23, 2021: The SI Appendix has been updated.

Keywords

Acknowledgments

J.E.C. thanks Roseanne Benjamin for help during this work. S.C.P.Y. acknowledges financial support from Hong Kong General Research Fund Grants HKGRF-14300717 “New Kinds of Forward-Backward Stochastic Systems with Applications,” HKGRF-14300319 “Shape-Constrained Inference: Testing for Monotonicity,” and HKGRF-14301321 “General Theory for Infinite Dimensional Stochastic Control: Mean Field and Some Classical Problems.”

Notes

Reviewers: S.R., Texas Tech University; and J.R., London School of Economics.

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Authors

Affiliations

Mark Brown^2,1 [email protected]

Department of Statistics, Columbia University, New York, NY 10027;

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Joel E. Cohen^2,1 https://orcid.org/0000-0002-9746-6725 [email protected]

Laboratory of Populations, The Rockefeller University, New York, NY 10065-6399;

Earth Institute, Columbia University, New York, NY 10027;

Department of Statistics, Columbia University, New York, NY 10027;

Department of Statistics, University of Chicago, Chicago, IL 60637;

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Chuan-Fa Tang^2,1 [email protected]

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080;

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Sheung Chi Phillip Yam^2,1 [email protected]

Department of Statistics, Chinese University of Hong Kong, Hong Kong

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Notes

2

To whom correspondence may be addressed. Email: [email protected], [email protected], [email protected], or [email protected].

Author contributions: M.B., J.E.C., C.-F.T., and S.C.P.Y. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

1

M.B., J.E.C., C.-F.T., and S.C.P.Y. contributed equally to this work.

Competing Interests

The authors declare no competing interest.

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Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

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Abstract

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1. Preliminary

2. Background and Prior Results

3. Taylor’s Laws for Semivariances

A. Lower Semivariances and Sortino Ratio

B. Upper Semivariances

4. Fluctuation Scaling for Higher Moments

A. Higher Sample Moments, Skewness, and Kurtosis

B. Central Lower and Local Lower Partial Moments

C. Central Upper Moments and Local Upper Moments

D. Omega Index, Upside Potential Ratio, and Farinelli–Tibiletti Ratio

5. Number of Observations Exceeding Sample Mean of Stable Law

A. Asymptotic Distributions and Moments of Nn+/nα

6. Numerical Experiments

A. Tail Estimators

B. Asymptotic Distribution of Nn+/nα

Notes

Data Availability

Acknowledgments

Supporting Information

References

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Keywords

Acknowledgments

Notes

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A. Asymptotic Distributions and Moments of $N_{n}^{+} / n^{α}$

B. Asymptotic Distribution of $N_{n}^{+} / n^{α}$