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Inter-order relations between moments of a Student t distribution, with an application to L_p-quantiles

This paper introduces inter-order formulas for partial and complete moments of a Student t distribution with n degrees of freedom. We show how the partial moment of order n - j about any real value m can be expressed in terms of the partial moment of order j - 1 for j in {1,…, n }. Closed form expressions for the complete moments are also established. We then focus on L_p-quantiles, which represent a class of generalized quantiles defined through an asymmetric p-power loss function. Based on the results obtained, we also show that for a Student t distribution the L_n-j+1-quantile and the L_j-quantile coincide at any confidence level τ in (0, 1).

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12/03/2019

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1 Introduction

Partial moments play an important role in several areas of statistics and applied mathematics, including reliability modeling (lata1983moments), insurance pricing (Denuit02) and financial risk theory (harlow1989asset). bawa1975optimal was among the first ones to introduce partial moments in financial economics, ANTHONISZ20122122 and yao2021general subsequently investigated the use of (lower) partial moments in the context of asset pricing and portfolio optimization, respectively. unser2000lower examined investors’ risk perception in financial decision making using lower partial moments as a measure of perceived risk, while antle2010asymmetry

considered partial moments as a flexible way to estimate the effects of biophysical and management processes on agricultural output distributions. From a theoretical standpoint, their properties and recursive relationships have been investigated in, for instance,

winkler, where several methods for determining partial moments are discussed or in abraham who characterized different continuous distributions belonging to the exponential and Pearson families based on partial moments. In this work, we are concerned with partial and complete moments of the Student distribution with arbitrary degrees of freedom . In both frequentist and Bayesian paradigms, the Student

has thoroughly pervaded the statistical literature in those situations where heavy tails or outliers arise as a robust alternative to the normal distribution (

jonsson2011heavy). These characteristics make it suitable for the analysis of data in the economic sciences and risk analysis; see, for instance, lange1989 and mcneil2015. In particular, results on the moments of a Student distribution with degrees of freedom have been obtained in winkler where it was shown that the -th order partial moment about the origin can be expressed in terms of the -th order partial moment, provided that . More recently, BernardiBPStudent presented a recurrence formula by substituting repeatedly the partial moment of order in the characterization of the -th order partial moment illustrated in TableofInt07

. In the literature, general expressions for central moments of skewed, truncated or folded Student

distributions have also been derived by nadarajah2004skewed and kim2008moments, for example. However, relationships for partial and complete higher-order moments about any arbitrary point of a Student distribution have not yet been determined.
This paper contributes to the existing literature in two ways. First, we show that for a Student distribution with degrees of freedom, the partial moment of order about any point can be expressed in terms of the partial moment of order times a multiplicative constant that depends on and . In addition, closed form expressions for the complete moments are also derived in terms of the Gamma and the hypergeometric functions. The proofs of these results involve concepts from combinatorial analysis as well as exploit fundamental properties of known special functions. To the best of our knowledge, this is the first time that inter-order relationships for partial and complete moments of the Student distribution are investigated, giving a simple way for calculating higher order moments from those of lower order.

The second contribution of the paper establishes inter-order relations of equality between -quantiles of the Student distribution. In the literature, -quantiles, introduced by Chen96 as a natural extension of quantiles, are an important class of statistical functionals defined as the minimizers of an expected asymmetric -power loss function. Indeed, for and for , they correspond respectively to quantiles and expectiles, which have been proposed by NeweyP87, based on asymmetric least-squares estimation as a “quantile-like” generalization of the mean. Basic properties of the -quantiles have been given by Chen96 and, in the context of generalized quantiles, Remillard95 provided sufficient conditions under which -quantiles (quantiles) and -quantiles (expectiles) coincide. More generally, both quantiles and expectiles, and in turn -quantiles, can be embedded in the wider class of M-quantiles of BrecklingChambers88 which extend the ideas of M-estimation of huber1964robust

by introducing asymmetric influence functions to model the entire distribution of a random variable. Generalized quantile models have been implemented in a broad range of applications, such as multilevel modeling (

alfo2021m and merlo2021quantile), nonparametric regression (Pratesi09

), multivariate analysis (

merlo2022marginal), economics and finance (Gneiting11, Bellinietal14 and daouia2019extreme). In the latter context, -quantiles have received particular consideration as potential competitors to the most used risk measures in banking and insurance, namely Value at Risk and Expected Shortfall. Indeed, besides being elicitable (Lambertetal08), they possess several interesting properties in terms of risk measures (see for instance bellini2012isotonicity and Bellinietal14). Moreover, when , -quantiles are the only risk measure that is both coherent (Artzner99) and elicitable (please see also Bellinietal14 and ziegel2016coherence for a detailed analysis of expectiles as a risk measure).

In this paper, by exploiting the results derived on partial moments we show that for a Student distribution with degrees of freedom, the -quantile and the -quantile coincide for any (and the same holds for any affine transformation). This result generalizes the one in Koenker93 who showed that the Student distribution with 2 degrees of freedom, or any affine transformation thereof, is the unique distribution where -quantiles and -quantiles match each other, and in BernardiBPStudent who proved that the -quantiles and -quantiles coincide for any .

With this paper, we give new insights on the properties of the Student distribution and, at the same time, we contribute in the context of risk management to the framework of belliniDB17, li2022pelve and fiori2022generalized for comparing general pairs of risk measures and determining when these coincide.

The rest of the paper is organized as follows. Section 2 introduces the notation and the mathematical background that will be used in our analysis. In Section 3 we present the first important result of the paper deriving inter-order formulas for partial and complete moments of arbitrary order for a Student distribution. Finally, in Section 4 we introduce the -quantiles and provide a symmetry characterization of -quantiles of the Student distribution. The proofs of the main results are collected in the Appendix A Appendix.

2 Mathematical background

We start by introducing some basic mathematical notation. Let denote the set of positive integers, and the positive real line; will always be an element of . The binomial coefficient if defined by for , where is the factorial with the convention . We denote and the positive and negative part of

, respectively. Given an atomless probability space

, represents the space of random variables with finite -moment, . and denote, respectively, the space of measurable and bounded random variables. Given a continuous real-valued random variable , we denote and its density and distribution functions. To improve readability the subscript may be omitted when no confusion is likely to arise. For , the upper and lower partial moments of order about any point are denoted as follows

(1)

and

(2)

In this paper we make extensive use of some special functions: the Gamma, the Beta and the hypergeometric functions. These functions are widely discussed in every book on special functions. Among many others we cite abramowitz1972handbook, Askey75, temme1996special and mathai2008special. Here, we briefly introduce the hypergeometric function with its most relevant properties.

To this end, we first recall the notion of Pochhammer’s symbol:

with the convention . Note that for , one has , where denotes the Gamma function of the positive real number . For , it is immediate to verify

Then, the hypergeometric function is defined by the series

(3)

with . The radius of convergence of the series is the unity, but it may be extended under weak conditions. For instance, when the analytic continuation of the hypergeometric function can be obtained when (or ) and , see temme1996special. The series terminates if either or is a nonpositive integer, in which case the function reduces to a polynomial and the convergence is guaranteed for any . Indeed, when or is equal to , with , the function is a polynomial of degree in , that is:

The hypergeometric function satisfies a great variety of relations. First we observe that it is symmetric in and , giving . Another useful property which will be often used throughout the paper is the following Euler’s transformation (temme1996special, p. 110). For (or equivalently from the symmetry in with respect to and ) and :

(4)

where denotes the argument of the real number .

The last property of the hypergeometric function that we need is:

(5)

for , and the Gamma functions well defined (temme1996special, p. 120). In this contribution we will briefly encounter the generalized hypergeometric function defined as

The radius of convergence of this series is unity for , while it is for . The parameters can be commuted as well as the parameters . Furthermore if for some and , then

3 Relations between moments of the Student distribution

In this section we present innovative results for the partial and complete moments of the Student distribution. Let be a random variable with Student distribution with degrees of freedom, i.e., , with density function given by

In order to introduce our results, we recall the expressions for the moments of centered around zero. Specifically, the raw moments are well defined for any order and can be calculated as

(6)

The raw moments of odd order are null because the density function is an even function.

In the following, a closed form expression for the complete moments of order centered in is derived. This result can be found in the working paper by kirkby19 based on the unpublished note by winkelbauer12. Here, we derive the same formulas with a different proof.

Proposition 3.1.

Let be a random variable with Student distribution with degrees of freedom. For and :

(7)

where is the hypergeometric function defined in (3).

Proof.

See the Appendix A.1. ∎

Remark 3.2.

Note that , therefore Proposition 3.1 for returns the expression for the raw moments of order provided in (6). Moreover, for both odd and even the first term of the hypergeometric function is a nonpositive integer, implying that the function is a polynomial and the convergence is guaranteed for any and .

A useful property of upper and lower partial moments of the Student distribution that will be needed for the main theorems can be proved by the following lemma.

Lemma 3.3.

Let be a random variable with Student distribution with degrees of freedom and let :

  1. For the following equation holds:

    (8)
  2. For

    (9)
Proof.

For the proof see Appendix A.2. ∎

Thanks to Lemma 3.3, in the following theorem we derive an interesting link between the central moments of order with that of order for any .

Theorem 3.4.

Let be a random variable with Student distribution with degrees of freedom. For and , the following equation for the central moments about the point holds:

  1. For odd:

    (10)
  2. For even:

    (11)
Proof.

For the proof see Appendix A.3. ∎

Theorem 3.4 provides an original inter-order relation between moments of the Student distribution.

Remark 3.5.

It is worth noting that Theorem 3.4 also establishes an expression for the raw moments of . Indeed, for , we have that

(12)

In the next results, we show that a similar relation holds also for the upper and lower partial moments.

Theorem 3.6.

Let be a random variable with Student distribution with degrees of freedom. For and , the following equation for the upper partial moments holds:

(13)
Proof.

For the result follows from Lemma 3.3(1). For , we recall that the upper partial moment of order can be decomposed as:

The result follows from Lemma 3.3(2) together with Theorem 3.4 (1) and (2) for the case odd and even, respectively. ∎

Corollary 3.7.

Let be a random variable with Student distribution with degrees of freedom. For and , the following equation for the lower partial moments holds:

(14)
Proof.

The proof follows from Theorem 3.4 and 3.6, using the equality:

From another perspective, Theorem 3.6 and Corollary 3.7 assert that for a Student distribution with degrees of freedom the ratio between the upper (lower) partial moment of order and that of order is a deterministic function of . Figure 1 depicts the function for various values of and . As it can be observed, this is an even convex function of when the exponent is positive and concave otherwise. We can also see that in both cases the global minimum (maximum) is at . Evidently, if and is odd, this is a constant function equal to 1.

Figure 1: From top to bottom and from left to right the function for degrees of freedom and .

4 On the -quantiles for the Student distribution

In this section we formally introduce the -quantiles and present our second major contribution on identity relations between the -quantiles of a Student distribution. Specifically, consider the asymmetric power loss function , , where . For , the -quantile of a random variable at level , denoted by , is defined as:

(15)

provided the expectation exists. For , corresponds to the quantile at level of and (15) has a unique solution only if the distribution of is strictly increasing in a neighborhood of . When , corresponds to the level expectile introduced by NeweyP87. Expectiles have gained major attention in quantitative risk management as an important alternative to the well known VaR and ES risk measures. See for instance Emmer16 for a comparison of these three risk measures, Delbaen and the reference therein for a characterization of expectiles as the unique example of a coherent elicitable risk measure and belliniDB17 for an empirical analysis of expectiles. Quantiles and expectiles with correspond respectively to the median and the mean of . For , -quantiles can be defined as the unique solution of the following first order condition:

(16)

The main advantage of using (16) is that it is well defined for random variables with -th finite moments, while (15) requires also the -th moment to be finite.

In what follows, we show that for a random variable having a Student distribution with degrees of freedom the -quantile and the -quantile coincide for any level .

Theorem 4.1.

Let be a Student random variable with degrees of freedom. For the following equation holds:

(17)
Proof.

For there is nothing to prove. For and any , the first order condition (16) says that is the unique solution to

(18)

where in the second step we used Theorem 3.6 and Corollary 3.7. However, (18) is satisfied if and only if , from which we obtain the desired result . ∎

From Theorem 4.1 follows an important property of the -quantiles of a Student distribution. Specifically, the -quantile not only minimizes the asymmetric power loss function of order but it also minimizes the one of order for and for any . Their symmetry characterization is reflected by the fact that, for a given number of degrees of freedom , the -quantile (quantile) and -quantile coincide, the -quantile (expectile) tallies with the -quantile, and so forth for every pair of - and -quantiles.

In the next corollary we extend this result to any affine transformation of the Student distribution.

Corollary 4.2.

Let be any affine transformation of a random variable with Student distribution with degrees of freedom, that is . For any ,

Proof.

The proof follows immediately by noting that for any and . ∎

Theorem 4.1 and Corollary 4.2 extend the work of Koenker92 who originally showed that the class of distributions for which the -quantiles and -quantiles coincide, corresponds to a rescaled Student distribution with 2 degrees of freedom. Moreover, in the special case , the equality in Theorem 4.1 reduces to the expression derived in BernardiBPStudent, which shows that the -quantile and the quantile coincide for any level . With our results we further generalize these works by providing a characterization of the symmetry of the -quantiles for the Student distribution with degrees of freedom for any .

Figure 2 illustrates the behavior of the -quantile for of a Student distribution with (left) and (right) degrees of freedom with respect to . It is worth noticing that the -quantiles are monotonically increasing, with a symmetry point for due to the symmetry of the Student centered around zero. Moreover, by looking at the plot on the left, it is possible to see that (green) coincides with (red) for all . Similarly, in the right-hand picture we have that (green) is equal to (pink) and (blue) coincides with (red) for all .

Figure 2: -quantiles for of a Student distribution with (left) and (right) degrees of freedom with respect to . In both plots, the bi-colored dashed lines indicate that the -quantile coincides with the -quantile for and for all .

Appendix A Appendix

a.1 Proof Proposition 3.1

Proposition 3.1 Let be a random variable with Student distribution with degrees of freedom. For and :

(19)

where is the hypergeometric function defined in (3).

Proof.

By using the binomial expansion and the linearity of the expectation we can decompose the term , as:

Note that for and all the moments are finite. We first assume that is even. The term is not zero only for even, therefore the index is changed to obtain:

One can now substitute the values of from (6) and express the binomial coefficient in terms of the Gamma function to get:

(20)

where in the last equation we multiplied and divided by . One can now apply the Legendre duplication formula for the Gamma function, , to the following terms:

By substituting the above formulas in (20), recalling the definition of the Pochhammer symbol and that , one gets

Since the first term of the hypergeometric function is a nonpositive integer, the convergence of the function is guaranteed for any value of and .
Consider now the case odd. The term is not zero only for odd, therefore the index is changed to obtain:

One can now substitute the values of from (6) and express the binomial coefficient in terms of the Gamma function to get:

(21)

where in the last equation we multiplied and divided by . Again, we apply the Legendre duplication formula for the Gamma function to the following terms:

By substituting the above formulas in (21), one gets

where in the last step we multiplied and divided by . Recalling that , we find:

as desired. Since the first term of the hypergeometric function is a nonpositive integer, the convergence of the function is guaranteed for any value of and . ∎

a.2 Proof of Lemma 3.3

Lemma 3.3 Let be a random variable with Student distribution with degrees of freedom and let :

  1. For the following equation holds:

    (22)
  2. For

    (23)
Proof.

We first note that from (1) and (2), the upper and lower partial moments of order are

  1. For , formula 3.254(2) in zwillinger2015table implies:

    (24)

    where . It follows that for