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Extensions of true skewness for unimodal distributions

A 2022 paper arXiv:2009.10305v4 introduced the notion of true positive and negative skewness for continuous random variables via Fréchet p-means. In this work, we find novel criteria for true skewness, establish true skewness for the Weibull, Lévy, skew-normal, and chi-squared distributions, and discuss the extension of true skewness to discrete and multivariate settings. Furthermore, some relevant properties of the p-means of random variables are established.

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1. Introduction

A commonly accepted measure for the skewness of a probability distribution is given by

Pearson’s moment coefficient of skewness

, or the standarized third central moment

where has mean

and variance

. Usually, we say that a distribution is positively skewed if and negatively skewed if . It is also expected that positively skewed distributions satisfy the mean-median-mode inequalities

while negatively skewed distributions satisfy the reverse inequalities

However, a distribution with positive moment coefficient of skewness does not always satisfy the mean-median-mode inequalities. For instance, the Weibull distribution with shape parameter has positive moment coefficient of skewness but satisfies the reversed mean-median-mode inequalities, corresponding to negative skew [6]. (For other counterexamples, see [2].) This discrepancy is important when comparing Pearson’s moment coefficient with other measures of skewness, such as Pearson’s first skewness coefficient


and Pearson’s second skewness coefficient


The direction of skewness can therefore be inconsistent between different skewness measures. In 2022, Kovchegov [8] introduced the notion of true positive and negative skewness to unify Pearson’s coefficients in determining the sign of skewness. It relies on the idea that for a positively skewed distributions, the left part of the distribution should stochastically dominate the right part, resulting in a left tail that “spreads shorter” and a right tail that “spreads longer.” We use a class of centroids known as -means. See [8].

Definition 1.1 (Kovchegov [8], 2022).

For and random variable with finite -st moment, the -mean, of is the unique solution of the equation

(1.1)

If moreover has finite -th moment, the notion of -mean in (1.1) is equivalent to the Fréchet -mean defined as

(1.2)

For , the uniqueness of the -mean follows from the fact that

is a strictly increasing continuous function of . Occasionally, we write to emphasize that is a function of .

Notice that identically distributed random variables have the same -means and that and correspond to the median and mean of the distribution, respectively. For a random variable let

be the domain of . Notice that is a real number; it does not have to be an integer. If has a unique mode, then we denote it by . In this case, we include in the set . We omit the subscript when the random variable is unambiguous.

Remark 1.2.

For distributions which are “nice enough,” the domain can be extended in a well-defined way to include values of in the interval . The conditions for which this is permissible is discussed in [8]. In this work, we limit our consideration to .

Definition 1.3 (Kovchegov [8], 2022).

We say a random variable (resp. its distribution and density) witha uniquely defined median is truly positively skewed if is a strictly increasing function of in , provided has non-empty interior. Analogously, is truly negatively skewed if is a strictly decreasing function of in .

Remark 1.4.

It is possible that, for a unimodal distribution, is strictly increasing only on and that there exists such that . Kovchegov [8] differentiates between this case, which is referred to in that work as true positive skewness, and the case where holds for all , which is referred to as the stronger true mode positive skewness. Because we consider only unimodal distributions, for simplicity we take “true positive skewness” to mean true mode positive skewness in the sense of Kovchegov [8], unless explicitly mentioned otherwise.

Consider that has unique mode and median. Naturally, Pearson’s first skewness coefficient is positive if and only if and Pearson’s second skewness coefficient is positive if and only if . Additionally, it was noticed in [8] that Pearson’s moment coefficient of skewness is positive if and only if . Now, true positive skewness guarantees provided finiteness of the corresponding moments. Thus, Pearson’s first and second skewness coefficients and Pearson’s moment coefficient of skewness coincide in sign. Therefore, the direction of skewness is unified across several different criteria under true positive skewness.

Remark 1.5.

An additional advantage of the notion of true positive skewness is that it allows us to characterize the skewness of distributions that have infinite integer moments. Indeed, each of Pearson’s skewness coefficients requires at least a finite mean, which excludes a large class of heavy-tailed distributions from the classical study of skewness.

1.1. Existing criteria for true skewness

In general, demonstrating the true skewness of an arbitrary distribution requires a detailed analysis of the solutions of a class of integral equations (1.1). Such analysis is simplified by introducing arguments from stochastic ordering.

Definition 1.6.

For random variables and , we say that stochastically dominates (resp. the distribution of stochastically dominates the distribution of

) if the cumulative distribution function

of is majorized by the cumulative distribution function of , i.e. if holds for every . We say that the stochastic dominance is strict if there exists a point for which .

If is a continuous random variable with a density function , then we may rewrite (1.1) as an equality of integrals, the quantities of which we define as a normalizing term , i.e.,

(1.3)

where has support on the possibly infinite interval . The following result establishes the relationship between true positive skewness and stochastic dominance of the left and right parts of a distribution.

Theorem 1.7 (Kovchegov [8], 2022).

Let be a continuous random variable supported on with density function . For fixed , if the distribution with density function exhibits strict stochastic dominance over the distribution with density function , then is increasing at .

Remark 1.8.

In particular, Theorem 1.7 explains why stochastic dominance of the density function representing the right part of over the density function representing the left part of for all implies positive Pearson’s moment coefficient of skewness () and positive Pearson’s second skewness coefficient (). Importantly, Theorem 1.7 provides a mathematical justification why skewness to the right, as expressed through stochastic domination, implies true positive skewness, yielding positivity of the above listed Pearson’s coefficients of skewness.

Remark 1.9.

The crux of the proof of Theorem 1.7 is that is increasing if and only if

(1.4)

Therefore, weaker versions of stochastic ordering between the left and right parts actually suffice for Theorem 1.7, in particular the concave ordering (see, e.g., [10, 13]).

We will frequently use the following criterion for true positive skewness.

Lemma 1.10 (Kovchegov [8], 2022).

Let be a continuous random variable supported on the possibly infinite interval with density function . For fixed , suppose there exists such that

  1. [label=()]

  2. ;

  3. for ; and

  4. for .

If , or if and , then is increasing at .

The following is a special case.

Proposition 1.11 (Kovchegov [8], 2022).

If is strictly decreasing on its support, then is truly positively skewed.

The requirement of strict monotonicity in the proposition can be relaxed, which will be necessary when we consider uniform mixtures in a later section.

Proposition 1.12.

If is non-increasing on its support , and there exist any two points , , such that , then is truly positively skewed.

The strict inequalities in Lemma 1.10 can be relaxed in a similar manner.

2. Results

2.1. Properties of -means

In this section, we establish several simple but fundamental properties of -means and their behavior. Here and throughout this work, let denote the -mean of a random variable whenever defined. We use when the random variable in question is unambiguous.

Proposition 2.1.

Let be a random variable supported on the possibly infinite interval . Then for all .

The following fact was used implicitly in [8], but we prove it for completeness.

Proposition 2.2.

The map is continuously differentiable on .

When investigating specific distribution families, we may assume that the scale and location parameters are 1 and 0 respectively unless otherwise noted. This is justified by the following, which implies that true positive skewness is preserved under positive affine transformations.

Proposition 2.3.

For any and , .

Next, we consider the asymptotic behavior of as . This requires to have finite moments of all orders, which clearly holds if has bounded support. We consider only continuous random variables, but analogous results hold in the discrete case.

Proposition 2.4.

Let be a continuous random variable supported on the finite interval . Then as .

Suppose instead has infinite support that is bounded below. We have an analogous result if the support is instead bounded from above.

Proposition 2.5.

Let be a continuous random variable supported on for finite . If has finite moments of all orders and holds for every , then as .

A consequence of Proposition 2.5 is that no distribution with support on the positive half-line is truly negatively skewed. Similarly, no distribution with support on the negative half-line is truly positively skewed.

2.2. Examples of true skewness

True positive skewness has already been shown in [8] for several distributions: exponential, gamma, beta (with the mode in the left half-interval), log-normal, and Pareto. Using Lemma 1.10, we establish true positive skewness of Lévy and chi-squared distributions, and identify the parameter regions for which Weibull and skew-normal distributions are truly skewed. Recall that Lévy distribution has undefined Pearson moment coefficient of skewness because it has no finite integer moments. Yet, as we already mentioned in Remark 1.5, finiteness of integer moments is not required for true skewness. Thus, to the authors’ knowledge, Lévy distribution’s positive skewness is formally established for the first time in this work.

Definition 2.6.

The Lévy distribution with location parameter and scale parameter

is a continuous probability distribution, denoted by

, with the density function

(2.1)
Definition 2.7.

The chi-squared distribution with degrees of freedom is a continuous probability distribution, denoted by , with the density function

(2.2)
Definition 2.8.

The Weibull distribution with shape paramter and scale parameter is a continuous probability distribution, denoted by , with the density function

(2.3)
Definition 2.9.

The skew-normal distribution with shape parameter is a continuous probability distribution, denoted by , with the density function

(2.4)

where and

are respectively the density and distribution functions of the standard normal Gaussian distribution.

A common strategy for showing the true skewness of the preceding distributions employs the following observation.

Lemma 2.10.

Suppose there exists a constant such that, for every , is increasing at whenever . If for some , then is increasing on .

When considering distribution families, we can always assume the location parameter is 0 and the scale parameter is 1, since they do not affect the direction of skewness (see Proposition 2.3).

Theorem 2.11.

The distribution is truly positively skewed.

Theorem 2.12.

The distribution is truly positively skewed.

Theorem 2.13.

The distribution is truly positively skewed if and only if . Moreover, it is never truly negatively skewed.

The Lévy, chi-squared, and Weibull distributions are supported only on the half-line, but similar techniques apply if the distribution is supported on the entire line.

Theorem 2.14.

The distribution is truly positively skewed if , truly negatively skewed if , and symmetric if .

2.3. True skewness under limits in distribution

It is reasonable to conjecture that true skewness is preserved under uniform limits of distribution functions since Lemma 1.10 implies that true skewness is, essentially, a feature of a continuous random variable’s density function. However, one must be somewhat careful: let be a sequence of independent gamma random variables, for some fixed , such that each can be expressed as a sum of i.i.d. exponential random variables. We know from [8] that the

’s are truly positively skewed, but the central limit theorem implies that their limit in distribution is Gaussian and thus symmetric.

Therefore, we introduce the notion of true non-negative skewness to refer to a random variable whose -means are non-decreasing, i.e., , as opposed to the strict increase required by true positive skewness. Notice that truly positively skewed as well as symmetric distributions are truly non-negatively skewed.

Theorem 2.15.

Let be the distribution functions of respectively. Suppose that uniformly and that holds for some and every . If the ’s are truly non-negatively skewed, then is truly non-negatively skewed.

If, moreover, there exists a constant such that holds for every and every , then is truly positively skewed.

The condition that the distribution functions converge uniformly is satisfied often in practice, since uniform convergence follows from pointwise convergence if is continuous (see, e.g., [5, Exercise 3.2.9]), which holds if

has no point masses. Alternatively, uniform convergence of the distribution functions holds if the characteristic functions converge uniformly.

Theorem 2.15 can be used when considering the parameter regions of true skewness for certain distribution families. As an example, consider a sequence of independent Weibull random variables, where ; these are truly positively skewed by Theorem 2.13. It is clear that the distribution functions of the ’s converge uniformly to the distribution function of , and one can show directly that the -th moments of the ’s are uniformly bounded for every (see, e.g., [12, Eq. 2.63d]). Theorem 2.15 then implies that is truly non-negatively skewed.

2.4. Additional criteria for true skewness

In this section, we establish two criteria for true positive skewness, one based upon a stochastic representation and the other based upon numerically verifiable conditions.

Theorem 2.16.

Let be a continuous random variable with density function decreasing on its support. If is convex and strictly increasing on the support of , then is truly positively skewed.

One immediate application of this theorem is when

is exponentially distributed. It is well-known that

and that , so Theorem 2.16 immediately yields the true positive skewness of the and distributions.

Our second criteria for true positive skewness does not rely on the -means of a distribution other than its mode and median, provided that it has a density function supported on the half-line which is twice continuously differentiable. It also does not require knowledge of the density expressed in terms of elementary functions, which has conveniently been provided in each of the specific distributions previously examined; instead, it requires certain bounds on the logarithmic derivative within certain intervals. This theorem may have applications in numerically checking true positive skewness for one-sided stable distributions, for which very little descriptive information is known in general, given specific parameter values.

Theorem 2.17.

Let have support on with continuous unimodal density . Suppose has exactly two positive inflection points such that , and

  1. on ,

  2. on .

If , then is truly positively skewed.

Corollary 2.18.

Let have support on with continuous unimodal density . Suppose has exactly one positive inflection point . If , then is truly positively skewed.

Remark 2.19.

The conditions of Theorem 2.17 can be relaxed, at the potential cost of practical ease. In particular, the following require one to compute , as defined in (3.13).

  1. [label=()]

  2. The condition serves to guarantee that is convex at for all , but the required convexity can certainly be achieved for a weaker lower bound on . Indeed, one can see in the proof in Section 3 that is sufficient; we obtain (3.16) via Lemma 3.4. Note that this is not always a weaker lower bound.

  3. The quantity in conditions (1) and (2) can be replaced by .111 This makes the proof significantly lengthier; in fact, Lemmas 3.2, 3.3, and 3.4 are otherwise unnecessary. For the proof of Theorem 2.17 in Section 3, we present the most general argument. As we show later, , so actually this replacement creates a stronger condition on the lower bound of to the left of the mode and a weaker condition on the lower bound of to the right of the mode. This replacement is useful when the density has a steeper right tail.

  4. Similarly, condition (2) only needs to hold on .

Example 2.20 (Log-logistic distribution).

The log-logistic distribution with shape parameter has density function

If , then strictly decreasing and true positive skewness follows from Proposition 1.11. Suppose . One can verify that is unimodal with mode

median , and inflection points

Straightforward computations show that holds if and only if , and holds if and only if . Moreover, holds if and only if . Therefore, the log-logistic distribution is truly positively skewed if , by Corollary 2.18.

3. Proofs

3.1. Proofs of results for Section 1

Proof of Proposition 1.12.

Clearly is finite, otherwise could not be a non-increasing density function. Notice that for all . Otherwise (1.3) fails to hold since and by assumption.

Now the existence of such that implies that there exists a non-singleton interval in on which . Then strict stochastic dominance of over follows by integrating each density, applying monotonicity of the integral, and using equation (1.3). ∎

3.2. Proofs of results for Section 2.1

Proof of Proposition 2.1.

If , then while , contradicting (1.1). Analogous if . ∎

Proof of Proposition 2.2.

Consider the function given by

Both integrands and are continuously differentiable functions of and within their support and are dominated by an integrable function since . By the Leibniz integral rule, observe that

which is strictly negative and finite for all and . The map is the zero level curve of and so is continuously differentiable by the implicit function theorem. ∎

Proof of Proposition 2.3.

Let . It is easy to see that . For every ,

holds, and the result follows. ∎

Proof of Proposition 2.4.

By Proposition 2.3, it suffices to show that if has support on , then .

Let be given. Suppose for the sake of contradiction that there exists a subsequence such that for all . For , we have

(3.1)

and

(3.2)

From (3.2) and (3.2), we have

as . This contradicts (1.1). Thus, every subsequence has only finitely many points that lie below . We can similarly show that every subsequence has only finitely many points that lie above . It follows by taking that every subsequence converges to , which completes the proof. ∎

Proof of Proposition 2.5.

We may assume . Suppose for the sake of contradiction that there exists such that for all . By Proposition 2.3, we may assume . Note that on , so by the bounded convergence theorem as . On the other hand,

hence (1.1) fails to hold for large , contradiction. ∎

3.3. Proofs of results for Section 2.2

Proof of Lemma 2.10.

Suppose for the sake of contradiction that the set

is non-empty. Since is continuous by Proposition 2.2, then is closed and thus contains its infimum . By construction, holds for every , hence is increasing on and so , contradicting the fact that . ∎

Proof of Theorem 2.11.

By Proposition 2.3, it suffices to consider the distribution. Let be its density function, i.e., as in (2.1).

Fix . To show that is increasing at , it suffices to show that the log density ratio of the left and right parts

for , has exactly one positive critical point. Indeed, observe that as and that . Moreover, (1.3) implies that cannot be non-positive for all . Therefore, if has a single positive critical point, then it must be a maximum, and it follows that the conditions of Lemma 1.10 are satisfied and so is increasing at .

To identify the positive critical points, observe that

so the critical points are solutions to the equation

Further simplification yields a quadratic equation in ,

which has exactly one root in if . This holds for arbitrary , so we may conclude true positive skewness if and hold, by Lemma 2.10. The mode and median of the Lévy distribution are well-known (see, e.g., [11]) and can be computed directly as and , which completes the proof. ∎

Proof of Theorem 2.12.

Fix and let be the density function of , i.e., as in (2.2). Fix . There are two cases.

Case 1: . Here, is decreasing on its domain and so true positive skewness follows from Proposition 1.11 and Theorem 1.7.

Case 2: . As in the proof of Theorem 2.11, to show that is increasing at , it suffices to show that the ratio of the left and right parts

for , has exactly one positive critical point, since as and . Observe that

so the critical points are solutions to the equation , i.e.,

There is exactly one positive critical point when . Since has support on the positive half line, then is always non-negative by Proposition 2.1. Thus is increasing at if . Now by Lemma 2.10, true positive skewness of follows if the median satisfies , but this inequality is well-known (see, e.g., Sen [16, Eq. (1.4)] and the references therein). ∎

Proof of Theorem 2.13.

By Proposition 2.3, it suffices to consider the distribution. Let be its density function, i.e., as in (2.3).

It is well-known (see, e.g., [12]) that the distribution has finite moments of all orders and is unimodal with median and mode given by

(3.3)

Notice that holds if and only if .

Since the Weibull distribution has support on the positive half-line, then the second part of the theorem follows from Proposition 2.5. For the first part of the theorem, the “only if” part follows from (3.3).

It remains to show the “if” part. For , is strictly decreasing, so we are done by Proposition 1.11.

Suppose and fix . As in the proof of Theorem 2.11, to show that is increasing at , it suffices to show that the log density ratio of the left and right parts

for , has exactly one positive critical point, since as and . Observe that

so the critical points are solutions to the equation

(3.4)

Since , the binomial series for and converge, hence

in the notation of the generalized binomial coefficient. Substituting into (3.4) yields the equation

(3.5)

where

(3.6)

We analyze the sign changes of the coefficients in (3.5) to determine the number of its positive roots by splitting into several cases for the value of .

Case 1: . Then are positive and are negative. There are an even number (possibly zero, if ) of negative factors in , so it is positive. Also note that for all , hence (3.6) is negative for all .

For the series expression , we have shown that holds for non-zero even and

holds for odd

. By Descartes’ rule of signs for infinite series, has no positive real roots if and has at most one positive real root if .

Suppose such that has no positive real roots. By extension, has no positive real roots, so has no positive extrema and is strictly monotonic on . Since and , then is strictly negative on , hence for all . By monotonicity of the integral, this contradicts (1.3). Thus holds, and has exactly one positive root in , which implies that is increasing at .

This holds for arbitrary , so is increasing on . Moreover, since holds, then holds for every , and we are done.

Case 2: . The inequality still holds for all , but we now have and . If , then there are an odd number of negative factors in , so the numerator of (3.6) is positive. If , then the numerator is simply .

Thus our coefficients in are positive for even and zero for odd , with . By Descartes’ rule of signs, has at most two positive real roots if and at most one if .

Suppose . If has a single positive root, then . But by continuity this limit tends to . If instead has no positive real roots, then by the same argument above we contradict (1.3). Thus . In this case, if has zero or two positive roots, then again and we reach a contradiction. Hence has exactly one positive real root. We may now conclude in the same fashion as in Case 1.

Case 3: . We again look at the numerator of (3.6). If , the numerator is . If , the numerator is . If , then has positive factors , and an odd number of negative factors . Additionally, when , so the numerator of (3.6) is positive.

Our coefficients in are zero for odd , positive for even , negative for , and positive for . This yields two sign changes and hence at most two positive roots of if . Recall from Case 1 that holds if and only if holds. If the latter holds, then must have exactly one positive root since is negative at the right limit of its domain, and so is increasing at . Now by Lemma 2.10, true positive skewness follows if , but this holds immediately from (3.3) and our assumption .

Case 4: . We can plug directly into (3.4) to obtain the roots

The argument in Case 1 can be adapted to show that holds for every , hence exactly one positive root exists.

Case 5: . Similarly, we plug into (3.4) and obtain the roots

Again, since holds for every , then exactly one positive root exists. ∎

Proof of Theorem 2.14.

Recall that are the density and distribution functions of the standard Gaussian distribution. For simplicity, set

We show true positive skewness for positive shape parameter ; the proof for true negative skewness for is analogous, and clearly the skew-normal is simply the normal distribution, which is symmetric, when . Thus, fix and let be the density function of , i.e., as in (2.4).

Fix . To show that is increasing at , it suffices to show that the log density ratio of the left and right parts