Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution

In this paper, we compute doubly truncated moments for the selection elliptical (SE) class of distributions, which includes some multivariate asymmetric versions of well-known elliptical distributions, such as, the normal, Student's t, slash, among others. We address the moments for doubly truncated members of this family, establishing neat formulation for high order moments as well as for its first two moments. We establish sufficient and necessary conditions for the existence of these truncated moments. Further, we propose optimized methods able to deal with extreme setting of the parameters, partitions with almost zero volume or no truncation which are validated with a brief numerical study. Finally, we present some results useful in interval censoring models. All results has been particularized to the unified skew-t (SUT) distribution, a complex multivariate asymmetric heavy-tailed distribution which includes the extended skew-t (EST), extended skew-normal (ESN), skew-t (ST) and skew-normal (SN) distributions as particular and limiting cases.

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1 Introduction

Truncated moments have been a topic of high interest in the statistical literature, whose possible applications are wide, from simple to complex statistical models as survival analysis, censored data models, and in the most varied areas of applications such as agronomy, insurance, finance, biology, among others. These areas have data whose inherent characteristics lead to the use of methods that involve these truncated moments, such as restricted responses to a certain interval, partial information such as censoring (which may be left, right or interval), missing, among others. The need to have more flexible models that incorporate features such as asymmetry and robustness, has led to the exploration of this area in last years. From the first two one-sided truncated moments for the normal distribution, useful in Tobin’s model (

[1]), its evolution led to its extension to the multivariate case ([2]), double truncation ([3]), heavy tails when considering the Student’s bivariate case in [4], and finally the first two moments for the multivariate Student’s case in [5]. Besides the interval-type truncation in cases before, [6] considers an interesting non-centered ellipsoid elliptical truncation of the form on well known distributions as the multivariate normal, Student’s , and generalized hyperbolic distribution. On the other hand, [7] recently proposed a recursive approach that allows calculating arbitrary product moments for the normal multivariate case. Based on the latter, [8]

proposes the calculation of doubly truncated moments for the normal mean-variance mixture distributions (

[9]) which includes several well-known complex asymmetric multivariate distributions as the generalized hyperbolic distribution ([10]).

Unlike [8], in this paper we focus our efforts to the general class of asymmetric distributions called the multivariate elliptical selection family. This large family of distributions includes complex multivariate asymmetric versions of well-known elliptical distributions as the normal, Student’s , exponential power, hyperbolic, slash, Pearson type II, contaminated normal, among others. We go further in details for the unified skew- (SUT) distribution, a complex multivariate asymmetric heavy-tailed distribution which includes the extended skew- (EST) distribution ([11]), the skew- (ST) distribution ([12]) and naturally, as limiting cases, its analogous normal and skew-normal (SN) distributions when .

The rest of the paper is organized as follows. In Section 2 we present some preliminaries results, most of them being definitions of the class of distributions and its special cases of interest along the manuscript. Section 3, the addresses the moments for the doubly truncated selection elliptical distributions. Further, we establish formulas for high order moments as well as its first two moments. We present a methodology to deal with some limiting cases and a discussion when a non-truncated partition exists. In addition, we establish sufficient and necessary conditions for the existence of these truncated moments. Section 4 bases results from Section 3 to the SUT case. In Section 5, a brief numerical study is presented in order to validate the methodology. In Section 6, we present some Lemmas and Corollaries related to conditional expectations which are useful in censored modeling. An application of selection elliptical truncated moments on tail conditional expectation is presented in Section 7. Finally, the paper closes with some conclusions and direction for future research.

2 Preliminaries

2.1 Selection distributions

First, we start our exposition defining a selection distribution as in [13].

Definition 1 (selection distribution).

Let and

be two random vectors, and denote by

a measurable subset of . We define a selection distribution as the conditional distribution of given , that is, as the distribution of . We say that a random vector has a selection distribution if .

We use the notation with parameters depending on the characteristics of , , and . Furthermore, for

having a probability density function (pdf)

say, then has a pdf given by

(1)

Since selection distribution depends on the subset , particular cases are obtained. One of the most important case is when the selection subset has the form

(2)

In particular, when , the distribution of is called to be a simple selection distribution.

In this work, we are mainly interested in the case where has a joint density following an arbitrary symmetric multivariate distribution . For , this setting leads to a -variate random vector following a skewed version of , which its pdf can be computed in a simpler manner as

(3)

2.2 Selection elliptical (SE) distributions

A quite popular family of selection distributions arises when and have a joint multivariate elliptically contoured distribution, as follows:

(4)

where and are location vectors, , , and are dispersion matrices, and, in addition to these parameters, is a density generator function. We denote the selection distribution resulting from (4) by . They typically result in skew-elliptical distributions, except for two cases: and (for more details, see [13]). Given that the elliptical family of distributions is closed under marginalization and conditioning, the distribution of and are also elliptical, where their respective pdfs are given by

(5)
(6)

with induced conditional generator

with

. These last equations imply that the selection elliptical distributions are also closed under marginalization and conditioning. Furthermore, it is well-know that the SE family is closed under linear transformations. For

and being a matrix of rank and a vector, respectively, it holds that the linear transformation , where is an acronym that stands for identically distributed, and then

(7)

Notice from Equation (3), that alternatively we can write

(8)

2.3 Particular cases for the SE distribution

Some particular cases, useful for our purposes, are detailed next. For further details, we refer to [13].

Unified-skew elliptical (SUE) distribution

Let . is said to follow the unified skew-elliptical distribution introduced by [14] when the truncation subset . From (8), it follows that

(9)

where and

denote the cumulative distribution function (cdf) of the

. Note that the density in (9) extends the family of skew elliptical distributions proposed by [15] (see also, [12]), which consider and .

Scale-mixture of unified-skew normal (SMSUN) distribution

Let

being a nonnegative random variable with cdf

. For a generator function , several skewed and thick-tailed distributions can be obtained from different specifications of the weight function and . It is said that follows a SMSUN distribution, if its probability density function (pdf) takes the general form

(10)

where represents the cdf of a -variate normal distribution with mean vector and variance-covariance matrix . Here follow a unified skew-normal (SUN) distribution, where we write .

  • Unified skew-normal (SUN) distribution

    Setting as a degenerated r.v. in 1 () and , then , , for which . Then, follow a SUN distribution, that is, , with pdf as

    (11)
  • Unified skew- (SUT) distribution

    For and weight function , we obtain and hence (10) becomes

    (12)

    where represents the cdf of a -variate Student’s distribution with location vector , scale matrix

    and degrees of freedom

    . For with pdf as in (12) is said to follow a SUT distribution, which is denoted by and was introduced by [14]. It is well-know that (12) reduces to a SUN pdf (11) as and to an unified skew-Cauchy (SUC) distribution, when .

    Furthermore, using the following parametrization:

    (13)

    where , with being the square root matrix of such that , we use the notation , to stand for a -variate EST distribution with location parameter , positive-definite scale matrix , shape matrix parameter , extension vector parameter and positive-definite correlation matrix . The pdf is now simplified to

    (14)

    with and being the Mahalanobis distance. The pdf in (14) is equivalent to the one found in [11], with a different parametrization. Although the unified skew- distribution above is appealing from a theoretical point of view, the particular case, when , leads to simpler but flexible enough distribution of interest for practical purposes.

    Extended skew- (EST) distribution

    For , we have that , and , hence (14) reduces to the pdf of a EST distribution, denoted by , that is,

    (15)

    with .Here, is a shape parameter which regulates the skewness of , and is a scalar. Location and scale parameters and remains as before. Here, we write Notice that, . Besides, it is straightforward to see that

    where corresponds to the pdf of a multivariate Student’s distribution with location parameter , scale parameter and degrees of freedom . On the other hand, when , we retrieve the skew- distribution say, which density function is given by

    (16)

    that is, . Further properties were studied in [11], but with a slightly different parametrization.

    Figure 1: Densities for particular cases of a truncated SUT distribution. Normal cases at left column (normal, SN and ESN from top to bottom) and Student’s- cases at right (Student’s , ST and EST from top to bottom).

    Six different densities for special cases of the truncated SUT distribution are shown in Figure 1. Symmetrical cases normal and Student’s are shown at first row (), skew cases: skew-normal (SN) and ST at second row () and extended skew cases: extended skew-normal (ESN) and EST at the third row. Location vector and scale matrix remains fixed for all cases.

  • Others unified skewed distributions

    Others unified members are given by different combinations of the weight function and the mixture cdf . For instance, we obtain an unified skew-slash distribution when and ; an unified skew-contaminated-normal distribution when and is a discrete r.v. with probability mass function (pmf) , with being the identity function. Besides, [15] mentions some other distributions as the skew-logistic, skew-stable, skew-exponential power, skew-Pearson type II and finite mixture of skew-normal distribution. It is worth mentioning that even though [15] works with a subclass of the SMSUN, when and , unified versions of these are readily computed by considering the same respective weight function and mixture distribution .

3 On moments of the doubly truncated selection elliptical distribution

Let with pdf as defined in (8) and let also be a Borel set in . We say that a random vector has a truncated selection elliptical (TSE) distribution on when . In this case, the pdf of is given by

where is the indicator function of the set . We use the notation . If has the form

(17)

we say that the distribution of is doubly truncated distribution and we use the notation , where and , where and values may be infinite, by convention. Analogously we define and . Thus, we say that the distribution of is truncated from below and truncated from above, respectively. For convenience, we also use the notation with the last parameter indicating the truncation interval. Analogously, we do denote to refer to a -variate (doubly) truncated elliptical (TE) distribution on . Some characterizations of the doubly TE have been recently discussed in [16].

3.1 Moments of a TSE distribution

For two -dimensional vectors and , let stand for , that is, we use a pointwise notation. Next, we present a formulation to compute arbitrary product moments of a TSLCT-EC distribution.

Theorem 1 (moments of a TSE).

Let as defined in (44). Let be a truncation subset of the form . For , then can be computed as

(18)

with , and , where , with , for .

Proof.

Since , the proof is direct by noting that

Corollary 1 (first two moments of a TSE).

Under the same conditions of Theorem 1, let and , both partitioned as

respectively. Then, the first two moments of are given by

(19)
(20)

where and .

For the particular truncation subset as in (2), Theorem 1 and Corollary 1 hold considering and . Notice that, Theorem 1 and Corollary 1 state that we are able to compute any arbitrary moment of , that is, a TSE distribution just using an unique corresponding moment of a doubly TE distribution .

This is highly convenient since doubly truncated moments for some members of the elliptical family of distributions are already available in the literature and statistical softwares. In particular for the truncated multivariate normal and Student’s-t we have the R packages TTmoment, tmvtnorm and MomTrunc.

3.2 Dealing with limiting and extreme cases

Consider and as in Theorem 1 with truncation subset . As , we have that . Besides, as , we have that and consequently . Thus, for containing high negative values small enough, sometimes we are not able to compute due to computation precision, mainly when we work with distributions with lighter tails densities. For instance, for a normal univariate case, for in R software. The next proposition helps us to circumvent this problem.

Proposition 1 (limiting case of a SE).

As , i.e., , then

(21)
Proof.

Let and . As , we have that , and , hence becomes degenerated on . From Definition 1, , and by the conditional distribution in Equation (6), it is straightforward to show that . Evaluating we achieve (21) concluding the proof. ∎

3.3 Approximating the mean and variance-covariance of a TE distribution for extreme cases

While using the relation (19) and (20), we may face numerical problems trying to compute and for extreme settings of and . Usually, it occurs when because the probability density is far from the integration region

. It is worth mentioning that, for these cases, it is not even possible to estimate the moments generating Monte Carlo (MC) samples via rejection sample due to the high rejection ratio when subsetting to a small integration region. Other methods as Gibbs sampling are preferable under this situation.

Hence, we present correction method in order to approximate the mean and the variance-covariance of a multivariate TE distribution even when the numerical precision of the software is a limitation.

3.3.1 Dealing with out-of-bounds limits

Consider to be partitioned as such that , , where . Also, consider , , and partitioned as before. Suppose that we are not able to compute , because there exists a partition of of dimension that is out-of-bounds, that is . Notice that this happens because Besides, we suppose that . Since the limits of are out-of-bounds (and ), we have two possible cases: or . For convenience, let and . For the first case, as , we have that and . Analogously, we have that and as . Hence, is degenerated on and then . Given that and , it follows that

(22)

with and being the mean and variance-covariance matrix of a -variate TE distribution.

In the event that there are double infinite limits, we can part the vector as well, in order to avoid unnecessary calculation of these integrals.

3.3.2 Dealing with double infinite limits

Now, consider to be partitioned such that the upper and lower truncation limits associated with are both infinite, but at least one of the truncation limits associated with is finite. Then be the number of pairs in that are both infinite, that is, and , by complement. Since and , it follows that and . Let and . Hence, it follows that , that is

(23)

On the other hand, we have that , and , with being a constant depending of the conditional generating function . Finally,

(24)

where and are the mean vector and variance-covariance matrix of a TE distribution, so we can use (19) and (20) as well.

Remark 1.

Note that does not follow a non-truncated elliptical distribution, that is, even though . This occurs due to . In general, the marginal distributions of a TE distribution are not TE, however this holds for due to the particular case and .

Particular cases

Notice that the constant will vary depending of the elliptical distribution we are using. For instance, if then it follows that and . In this case, it takes the form , which is given by

(25)

where denotes the integral

(26)

that is,