- Research
- Open Access
The multivariate slash and skew-slash student t distributions
- Fei Tan^{1}Email author,
- Yuanyuan Tang^{2} and
- Hanxiang Peng^{1}
https://doi.org/10.1186/s40488-015-0025-9
© Tan et al.; licensee Springer. 2015
- Received: 27 February 2014
- Accepted: 22 September 2014
- Published: 24 February 2015
Abstract
In this article, we introduce the multivariate slash and skew-slash t distributions which provide alternative choices in simulating and fitting skewed and heavy tailed data. We study their relationships with other distributions and give the densities, stochastic representations, moments, marginal distributions, distributions of linear combinations and characteristic functions of the random vectors obeying these distributions. We characterize the skew t, the skew-slash normal and the skew-slash t distributions using both the hidden truncation or selective sampling model and the order statistics of the components of a bivariate normal or t variable. Density curves and contour plots are drawn to illustrate the skewness and tail behaviors. Maximum likelihood and Bayesian estimation of the parameters are discussed. The proposed distributions are compared with the skew-slash normal through simulations and applied to fit two real datasets. Our results indicated that the proposed skew-slash t fitting outperformed the skew-slash normal fitting and is a competitive candidate distribution in analyzing skewed and heavy tailed data.
Mathematics Subject Classification Primary 62E10; Secondary 62P10
Keywords
- Heavy tail
- Skew distribution
- Skew-slash distribution
- Slash distribution
- Student t distribution
Introduction
Skewed and heavy tailed data occur frequently in real life and pose challenges to our usual way of thinking. Examples of such data include household incomes, loss data such as crop loss claims and hospital discharge bills, and files transferred through the Internet to name a few. Candidate distributions for simulating and fitting such data are not abundant. One can’t simply take the normal or the t distributions or as such as substitutes. Even though Cauchy distribution can be used to simulate and fit such data, its sharp central peak and the fact that its first moment does not exist narrow its applications. Thus additional distributions are needed to study such skewed and heavy tailed data.
Kafadar (1988) introduced the univariate normal slash distribution as the resulting distribution of the ratio of a standard normal random variable (rv) and an independent uniform rv (hereafter referred to the distribution as the slash normal). Generalizing the standard normal by introducing a tail parameter, the slash normal has heavier tails than the standard normal, hence it could be used to simulate and fit heavy tailed data. Wang and Genton (2006) generalized the univariate slash normal to the multivariate slash normal and investigated its properties. They also defined the multivariate skew-slash normal as the resulting distribution of the ratio of a skewed normal rv and an independent uniform rv (hereafter referred to the distribution as the skew-slash normal). They applied it to fit two real datasets.
In this article, we introduce the slash (student) t distribution and the skew-slash (student) t distribution. They could be used to simulate and fit skewed and heavy tailed data. The slash t distribution generalizes the slash normal distribution of Kafadar (1988) and the multivaraiate slash normal distribution of Wang and Genton (2006), and the skew-slash t distribution generalizes the skew-slash normal distribution of the latter two authors. In the skew-slash t there is one parameter to regulate the skewness of the distribution and another parameter to control the tail behavior. By setting the skewness parameter to zero, the skew-slash t reduces to the slash t. By letting the tail parameter to be infinity, the skew-slash t simplifies to the skew t of Azzalini and Capitanio (2003). As both the slash and skew t take the t and hence the normal as their special cases, so does the skew-slash t. To fit data, one can start with the skew-slash t. If the fitted value of the degrees of freedom is very large, then one takes the simpler skew-slash normal model. This idea of course can be used to perform the hypothesis testing of a skew-slash normal sub-model against a skew-slash t model. We have derived the formulas for the densities, moments, marginal distributions and linear combinations of these distributions. Thus it could be expected that they can be used to analyze skewed and heavy tailed data.
The skew-slash normal and t fitting to the GAD data
SSLT | SSLN | |||
---|---|---|---|---|
MLE | SE | MLE | SE | |
AIC | -242.318 | -240.687 | ||
μ | 3.726 | 0.0070 | 3.748 | 0.036 |
σ ^{2} | 0.002 | 0.0004 | 0.009 | 0.038 |
λ | -2.056 | 0.6190 | -3.199 | 3.204 |
q | 3.812 | 2.1990 | 7.456 | 83.194 |
r | 3.044 | 1.4080 |
Skew-slash t and skew-slash normal comparison
True | Estimated | True | Estimated | |||
---|---|---|---|---|---|---|
SSLT | SSLT | SSLN | SSLN | SSLT | SSLN | |
AIC | 1524.939 | 1543.771 | 1416.485 | 1425.878 | ||
μ _{1} | 0 | -0.160 | -0.467 | 0 | -0.309 | -0.690 |
μ _{2} | 0 | 0.032 | -0.225 | 0 | 0.025 | -0.212 |
\(\sigma _{11}^{2}\) | 1 | 0.827 | 0.809 | 1 | 0.587 | 0.651 |
σ _{12} | 0 | 0.122 | 0.230 | 0 | 0.130 | 0.304 |
\(\sigma _{22}^{2}\) | 1 | 1.154 | 1.527 | 1 | 1.019 | 1.314 |
λ _{1} | -5 | -2.830 | -1.581 | -5 | -1.989 | -0.860 |
λ _{2} | 3 | 1.688 | 1.422 | 3 | 1.363 | 1.127 |
q | 5 | 6.383 | 4.553 | 5 | 6.697 | 5.151 |
r | 8 | 7.610 | 7.506 |
Azzalini and Dalla Valle (1996) introduced the multivariate skew normal distribution that extends the normal distribution with an additional skewness parameter. It provides an alternative modeling distribution to skewed data that are often observed in many areas such as economics, computer science and life sciences. Many authors have investigated skew t distributions, see e.g. Azzalini and Dalla Valle (1996), Gupta (2003), and Sahu et al. (2003). Azzalini and Capitanio (2003) proposed the multivariate skew t distribution by allowing a skewness parameter in a multivariate t distribution.
It is our belief that the proposed slash and skew-slash t distributions throw some additional light on this theory and contribute to the family of candidate distributions for modeling and simulating skewed and heavy tailed data.
The article is organized as follows. In Section 2, we introduce the multivariate slash t distribution, study its relationships with other distributions and derive the density function. We investigate its tractable properties such as heavy tail behavior and closeness of marginal distributions and linear combinations. We give the stochastic representations, moments and characteristic function. We close this section with an example which graphically displays the densities. In Section 3, we define the skew-slash t distribution, derive the densities and characteristic functions, and give the moments and distributions of linear combinations of these distributions. We first define the standard skew-slash distribution in Subsection 3.1 and study their relationships with other distributions. In subsection 3.2, we characterize the skew t, skew-slash normal and skew-slash t distributions using hidden truncation or selective sampling model and the order statistics of the components of a bivariate normal or t variable. In Subsection 3.2, we define the general multivariate skew-slash distribution. An example is presented to illustrate the densities of the proposed distributions. Section 4 covers parameter estimation and statistical inference. Here we briefly discuss the maximum likelihood and Byesian approaches. Section 5 is devoted to simulations as well as applications of the proposed skew-slash t distribution to fit two real datasets. Finally, some concluding remarks are given in Section 6.
The multivariate slash t distribution
In this section, we define the multivarate slash t distribution, derive the density and study its tail behaviors and relationships with other distributions. We give the stochastic representations, moments, and characteristic function and discuss marginal distributions and linear combinations. We close this section with an example.
where |M| denotes the determinant of a square matrix M. If m=0 and R=I _{ k } where I _{ k } denotes the k×k indentity matrix, it is referred to as the standard k-variate t distribution and denoted by t _{ k }(r). We now introduce the k-variate slash t distribution. Write U∼U(0,1) for the rv uniformly distributed over (0,1).
Definition 1.
When m=0 and R=I _{ k }, it is referred to as the standard (k-variate) slash t and denoted by S L T _{ k }(q,r). It can be easily seen that the k-variate slash t distribution generalizes the k-variate t distribution as stated below.
Remark 1.
The limiting distribution of the slash t distribution S L T _{ k }(q,r), as q→∞, is the student t distribution t _{ k }(r).
From this density it immediately follows that the standard k-variate slash t distribution S L T _{ k }(q,r) is symmetric about 0 as the standard k-variate t is so.
This manifests that the standard univariate slash t is also heavy tailed.
Stochastic representations Stochastic Representations not only reveal the relations with other distributions but are very useful, for instance, in calculating moments and random generation. We provide two stochastic representations for the slash t distribution based on the two stochastic representations of the multivariate t distribution. According to Kafadar (1988), a continuous random variable ξ has a slash normal distribution, written ξ∼S L N(q,0,Σ), if it can be expressed as ξ=Z/U ^{1/q } where Z∼N _{ k }(0,Σ) and U∼U(0,1) are independent.
where ξ∼S L N _{ k }(q,0,R), η=m U ^{−1/q }, and both ξ and η are independent of S.
where ξ∼S L N _{ k }(q,0,r I _{ k }), η=m U ^{−1/q }, and ξ,η are independent of V. Here V ^{−1/2} is the inverse of the symmetric square root V ^{1/2} of V, where V has a k-variate Wishart distribution with degrees of freedom r+k−1 and covariance matrix R ^{−1}.
Hence the k-variate slash t has the same correlation matrix R as the k-variate t.
For details, see Kotz and Nadarajah (2004). Based on these formulae we have the following.
Theorem 1.
where c is given in (2.13). Otherwise if q≤p then \({\mathbb {E}}(X_{1}^{p_{1}}\cdots X_{k}^{p_{k}})\) diverges.
PROOF.
Note that the integral \({\int _{0}^{1}} qv^{q-1-p}\,dv\) converges to q/(q−p) if q−p>0 and diverges otherwise. These and (2.13) yield the desired results.
The marginal distributions Since the marginal distributions of a k-variate t are still t, the marginal distributions of a k-variate slash t are slash t.
Theorem 2.
The marginal distributions of a k-variate slash t distribution are still slash t.
PROOF.
where \({\mathbf {m}}=({\mathbf {m}}_{1}^{\top }, {\mathbf {m}}_{2}^{\top })^{\top }\) with \({\mathbf {m}}_{1} \in {\mathbb {R}}^{s}\) and R is partitioned into the 2×2 block matrix with R _{11} being the s×s matrix at the position of (1,1)-block. See pages 15-16 of Kotz and Nadarajah (2004). Combining the last two equalities yields the desired equality.
Linear combinations Since the distribution of a linear function of a k-variate t variable is still t, it immediatly yields the following.
Theorem 3.
Let A be a nonsingular nonrandom matrix. If X∼S L T _{ k }(q,r,m,R), then A X∼S L T _{ k }(q,r,A m,A R A ^{⊤}).
for t in some neighborhood of the origin in which the above integral converges.
Example 1.
The multivariate skew-slash t distributions
In this section, we first recall the skew normal and skew t. In subsection 3.1, we define the standard skew-slash t distribution, study its relationships with other distributions and give the moments and characteristic function. In subsection 3.2, we use hidden truncation or selective sampling model and the order statistics to characterize the skew, slash and skew-slash normal and t distributions. In subsection 3.2, we define the general skew-slash t distribution, study its linear transformation and give an example in the end.
where ϕ _{ k } is the pdf of the k-variate standard normal N _{ k }(0,I) and Φ is the cdf of the standard normal N(0,1). Denote it by S K N _{ k }(λ).
Kotz and Nadarajah (2004) wrote “…that the possibilities of constructing skewed multivariate t distributions are practically limitless". The two authors surveyed the definitions given by Gupta (2003), Sahu et al. (2003), Jones (2002) and Azzalini and Capitanio (2003).
Based on these definitions, we may define the skew-slash t distributions in different ways. In this article, however, we will take the following approach. First, we will define the standard skew-slash t based on the standard skew t, a common special case of Gupta (2003), Azzalini and Capitanio (2003), and others. We then introduce a general skew-slash t distribution by introducing location and scale parameters. Our definition may lose some nice interpretations. But we think that this definition is natural, concise and, in particular, convenient in applications.
where Ψ is the cdf of the univariate standard t distribution with r+k degrees of freedom. Denote this distribution by S K T _{ k }(λ,r,Σ). This form of the density given here is slightly different from that of Gupta (2003). Several constant parameters appeared in his density formual are not explicitly expressed in our form of the density. We have incorporated them in the parameters in the above density. Accordingly parameters of the same names may have different values.
where t _{ k }(t;r) is the density of the standard k-variate distribution t _{ k }(r) with degrees of freedom r. This is a common special case of the skew t shared by Gupta (2003), Azzalini and Capitanio (2003), and others.
3.1 The standard multivariate skew-slash t distribution
We begin with the definition, followed the moments and characteristic function.
Definition 2.
The standard skew-slash t generalizes the proposed standard k-variate slash t, the standard skew t of Gupta (2003) and Azzalini and Capitanio (2003), the standard slash normal of Kafadar (1988), and the standard skew-slash normal of Wang and Genton (2006). This is stated below.
Remark 2.
The limiting distribution of the standard skew-slash t distribution S S L T _{ k }(λ,q,r) is, as q→∞, the standard skew t distribution S K T _{ k }(λ,r). The limiting distribution of S S L T _{ k }(λ,q,r) is, as r→∞, the standard skew-slash normal S S L N _{ k }(λ,q), which includes as special cases the standard skew normal S K N _{ k }(λ) (q=∞) and the standard slash normal S L N _{ k }(q) (λ=0). As λ=0, S S L T _{ k }(λ,q,r) reduces to the slash t distribution S L T _{ k }(q,r).
for some neighborhood of the origin in which the above integral converges, where φ _{ S } is the characteristic function of the standard skew t distribution S∼S K T _{ k }(λ,r).
3.2 Hidden truncation and order-statistics characterization
In this subsection, we characterize the skew t, skew-slash normal and skew-slash t distributions using the hidden truncation or selective sampling model and the order statistics of the components of a bivariate normal or t variable.
Using this we immediately derive the following results.
Proposition 1.
Remark 3.
The density function in (3.21) is the conditional pdf of the k-variate slash t rv X=T/U ^{1/q } given A. It is noteworthy that the hidden truncation model yields the pdf (3.20) and (3.21), the former is proportional to the pdf (3.15) of the skew t distribution and the latter is proportional to the pdf (3.16) of the proposed skew-slash t distribution. For more discussion see e.g. Chapter 6 of Genton (2004) and the references therein. The skew-slash normal of Wang and Genton (2006) is the special case of r=∞.
In their Theorem 1, Arnold and Lin (2004) showed that the order statistics of the components of a random vector from a bivariate normal distribution obey the skew-normal law. Using (3.23) and (3.24) we can show that the order statistics of the components of a random vector from a bivariate t distribution obey the skew t law. Thus we extend their result from the normal to t distribution as stated below.
Proposition 2.
where \(\lambda =\lambda (r,\rho)=\sqrt {({1+1/r})({1-\rho })/({{1+\rho })}}\).
Remark 4.
The density functions in (3.25) and (3.26) reduce to the result (c) of Theorem 1 of Arnold and Lin (2004) when the df r=∞ as \(\lambda (\infty, \rho)=\sqrt {(1-\rho)/(1+\rho)}\) is equal to their skewness parameter γ in the skew-normal distribution.
PROOF OF PROPOSITION 2.
We now apply (3.24), with both f _{1} and f _{2} equal to the pdf of t _{1}(r) and both F _{1} and F _{2} equal to the cdf Ψ(;r+1) of the t distribution t(r+1), to obtain the pdf t _{(2)} given in (3.26), noting in this case \(b(t_{2}, t_{2}; r, \rho)={\lambda t_{2}}/{\sqrt {1+{t_{2}^{2}}/r}}\). Aanloguously we can prove (3.25) in view of the equality \(\bar \Psi (t; r)=\Psi (-r; r)\) by the symmetry of the univariate t distribution. This completes the proof.
As a corollary of Proposition 2, we obtain a characterization of the skew-slash normal and t distributions through the order statistics of the components of a random vector from a bivariate t distribution as stated below.
Corollary 1.
Remark 5.
The density functions in (3.27) and (3.28) are (i) the pdf of the order statistics of the components of the random vector (T _{1},T _{2})/U ^{1/q } from the bivariate slash t distribution t _{2}(r,q), and (ii) reduce to the case of the skew-slash normal of Wang and Genton (2006) when the df r=∞.
PROOF OF COROLLARY 1.
where the independence of T _{1,2} and U is used to claim the first equality while the second equality follows from a change of variables. Similarly (3.28) can be proved and this finishes the proof.
The multivariate skew-slash t distributions We now introduce a general multivariate skew-slash t distribution by incorporating location and scale parameters.
Definition 3.
where Σ ^{1/2} is the the choleski decomposition of the positive definite covariance matrix Σ.
where Q(w;μ,Σ)=(w−μ)^{⊤} Σ ^{−1}(w−μ).
As in the case of the standard skew-slash t, one notices that the skew-slash t generalizes the slash t, the skew t of Azzalini and Capitanio, the slash normal of Kafadar, the skew normal of Azzalini and Dalla Valle and the skew-slash normal of Wang and Genton. This is stated below.
Remark 6.
The limiting distribution of the multivariate skew-slash t distribution S S L T _{ k }(λ,q,r,μ,Σ), as q tends to infinity, is the skew t distribution S K T _{ k }(λ,r,μ,Σ). The limiting distribution of S S L T _{ k }(λ,q,r,μ,Σ) is, as r tends to infinity, the skew-slash normal S S L N _{ k }(λ,q,μ,Σ), which include as special cases the k-variate skew normal S K N _{ k }(λ,μ,Σ) (q=∞) and the k-variate slash normal S L N _{ k }(q;μ,Σ) (λ=0). As λ=0, S S L T _{ k }(λ,q,r,μ,Σ) simplifies to the k-variate slash t distribution S L T _{ k }(q,r,μ,Σ).
Linear combinations Since the distribution of a linear function of a k-variate skew t variable is still skew t (see e.g. Section 5.9 of Kotz and Nadarajah (2004)), it immediatly yields the following result. Note that the relationship between our skewness parameter λ and their α is \({\boldsymbol {\lambda }}=\sqrt {1+k/r}{\boldsymbol {\alpha }}\). Let D=diag(σ _{1,1},…,σ _{ k,k }) denote the diagonal matrix consisting of the diagonal entries of Σ=(σ _{ i,j }) and R=D ^{−1/2} Σ D ^{−⊤/2} be the correlation matrix.
Theorem 4.
To give graphical view of the skewness and tail behaviors of the skew-slash t distributions, we plot the density curves of the univariate standard skew-slash t and contours of the bivariate standard skew-slash t below.
Example 2.
Statistical inference
In this section, we discuss maximum likelihood estimation and the Bayesian method and provide the approximate sampling distribution of the estimates.
where \(\hat {\mathbf {J}}^{jj}\) is the (j,j)- entry of the estimated inverse information matrix \(\hat {\mathbf {J}}^{-1}\).
The numerical value of the MLE \(\hat {\boldsymbol {\theta }}\) can be found by solving the score equation (4.33) using the newton’s method. Alternatively, one can directly search the solution of the maximization problem (4.32), for example, using the subroutine optim in the R package.
As for initial values of the newton’s algorithm, one can use the moment estimates of the parameters or other available consistent estimates. One technical issue here is that the estimate \(\hat \Sigma \) of Σ must be positive definite. What we did in our applications was that we estimated the entries of Σ and then verified the positive definiteness of \(\hat \Sigma \).
The Bayesian approach Given observed data D, the likelihood function L(D|θ) can be obtained from the proposed multivariate slash or skew-slash t distribution with parameter vector θ. The posterior density then satisfies p(θ|D)∝L(D|θ)π(θ), where π(θ) is the joint prior density of θ based on the available prior information on it. We choose a prior density for each component of θ and take the joint prior density of θ to be equal to the product of the marginal prior densities. The resulting full Bayesian model has the hierarchical structure with the conditional density of D|θ and the prior distribution θ∼π(θ) in the proposed model. One can obtain a random sample from the joint posterior density by the Markov Chain Monte Carlo (MCMC) method, and a parametric Baysian analysis of the model can be implemented using the Gibbs sampling method in R or JAGS.
Simulations and applications
where ϕ _{ k } is the density of the k-variate normal distribution N _{ k }(μ,Σ) with mean vector μ and covariance matrix Σ.
5.1 Simulation study
Comparison between the skew-slash t and skew-slash normal To compare the two distributions, we first generated data from the 2-variate skew-slash t then fitted it with both 2-variate skew-slash t and skew-slash normal, and vice versa (i.e. generated data from the 2-variate skew-slash normal then fitted it with the two distributions). Reported in Table 2 are the average AIC values and average MLE’s of the parameters based on the sample size n=250 and repetitions M=200.
Notice that for data generated from both the skew-slash t and skew-slash normal, the average AIC values of the skew-slash t fitting were lower than those of the skew-slash normal, indicating a better overall model fitting of the former to the data than the latter.
Maximum likelihood and Bayesian estimates comparison
True | SLT | SSLT | |||
---|---|---|---|---|---|
Parameter | Value | MLE | Bayes | MLE | Bayes |
Univariate | |||||
λ | 3 | 2.496 | 2.511 | ||
q | 5 | 6.196 | 6.236 | 6.689 | 6.479 |
r | 8 | 8.015 | 7.084 | 7.650 | 7.194 |
Bivariate | |||||
λ _{1} | -5 | -5.543 | -3.288 | ||
λ _{2} | 3 | 3.333 | 1.967 | ||
q | 5 | 6.410 | 4.113 | 5.899 | 4.118 |
r | 8 | 7.480 | 8.468 | 7.726 | 8.609 |
5.2 Applications
Model fitting to the GAD Data Gestational age at delivery (GAD) is a variable widely studied in epidemiology, see, for example, Longnecker et al. (2001). We applied the skew-slash t and skew-slash normal distributions to fit the log transformed GAD of n=100 observations. Figure 1 is the histogram superimposed with the fitted density curves, while Table 1 reports the MLE’s, the standard errors (SE) of the parameter estimates and the AIC. We can see from Figure 1 that the skew-slash t distribution was able to better capture the peak of the histogram, giving a better estimation of the density to the majority of data points. In the mean time, the AIC values in Table 1 indicated that the skew-slash t fitting was better than the skew-slash normal.
Model fitting to the AIS data Azzalini and Dalla Valle (1996) used their skew normal distribution to fit (LBM-lean body mass, BMI-body mass index) pairs of the athletes from Australian Institute of Sport (AIS), where the data of n=202 observations were reported in Cook and Weisberg (1994). Wang and Genton (2006) used their skew-slash normal distribution to re-fit the data. Here we applied the proposed skew-slash t distribution to re-fit the (LBM, BMI) pairs in the AIS data. Before fitting we standardized the variables.
The skew-slash normal and t fitting to the AIS data
SSLT | SSLN | ||
---|---|---|---|
MLE | SE | MLE | |
AIC | 981.018 | 978.426 | |
μ _{1} | -0.210 | 0.151 | -0.210 |
μ _{2} | -0.936 | 0.110 | -0.954 |
\(\sigma _{11}^{2}\) | 0.566 | 0.097 | 0.831 |
σ _{12} | 0.465 | 0.086 | 0.708 |
\(\sigma _{22}^{2}\) | 2.042 | 0.454 | 1.486 |
λ _{1} | -1.745 | 0.531 | -2.225 |
λ _{2} | 3.771 | 1.016 | 2.091 |
q | 121.021 | 660.502 | 22.534 |
r | 25.124 | 17.756 |
In conclusion, the proposed slash and skew-slash t are competitive candidate models for fitting skewed and heavy tailed data. The parameters can be estimated under either the frequentist method or Bayesian paradigm. Although for a particular dataset the skew-slash t may not be the final model, it is a good choice to start with in model selection due to its flexibility and the fact that it takes the skew normal, skew t and hence the usual normal and t as its submodels.
Concluding remarks
In this article, we defined the multivariate slash and skew-slash distributions in a pursuit of providing additional distributions to simulate and fit skewed and heavy tailed data. We investigated the heavy tail behaviors and tractable properties of these distributions which are useful in simulations and applications to real data. We characterized the skew t, the skew-slash normal and the skew-slash t distributions using both the hidden truncation or selective sampling model and the order statistics of the components of a bivariate normal or t variable. We demonstrated that the proposed skew-slash t model takes as sub-models the slash t, the slash normal, the skew-slash normal, the skew normal, the skew t and hence the usual normal and t. This nested property can be used in hypothesis testing.
Our simulations and applications to real data indicated that the proposed skew-slash t fitting outperformed the skew-slash normal fitting. Even though the skew-slash normal contains a tail parameter q, the fitting with it to the GAD data was unsatisfactory as the SE of the MLE of the tail parameter q was large, see Table 1. This suggests that not all heavy-tail properties in data can be explained by the tail parameter q in the slash distributions. Thus it makes sense for us to further search for distributions which can be used to fit heavy tailed data. Our proposed slash and skew-slash t distributions can be considered as an example in this attempt. We complete our remarks by pointing out that the degrees-of-freedom parameter r and the tail parameter q would explain different types of fat tail behaviors existed in data.
Declarations
Acknowledgements
The authors gratefully thank two anonymous reviewers for their suggestions that substantially improved the article.
Authors’ Affiliations
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