The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense

Abstract

The (Cauchy) principal value is a method for assigning values to certain improper integrals which would otherwise be undefined. Using the principal value sense, this study derives an explicit expression of the first negative moment of skew-t and generalized Student's t-distributions for practical applications. Some applications obtained from the FNM of skew-t and generalized Student's t-distributions are also discussed.

1. Introduction

The first negative moment (FNM) of a continuous density function defined on the real axis has important applications that can arise in many practical situations. For example, Peng [1] proposed a degradation model based on incorporating random effects into a Wiener process to predict the mean lifetime of highly reliable products as follows. The degradation model 𝐿(𝑡) is given by 𝐿(𝑡)=𝜂𝑡+𝜎_𝑊𝑊(𝑡), where 𝑊(⋅) denotes the standard Wiener process; 𝜂 and 𝜎_𝑊 are the random effects. If the product’s lifetime 𝑇 can be defined as the first-passage time when 𝐿(𝑡) crosses a predefined threshold level 𝜔, that is, 𝑇=inf{𝑡∣𝐿(𝑡)≥𝜔}, then the product’s lifetime on the given 𝜂 and 𝜎_𝑊 is an inverse Gaussian distribution (denoted by 𝑇∣𝜂,𝜎_𝑊∼ℐ𝒢(𝜔/𝜂,𝜔²/𝜎²_𝑊)). Hence, the most common measure associated with a lifetime distribution, the first moment, can be expressed as 𝐸(𝑇)=𝐸𝐸𝑇∣𝜂,𝜎_𝑊1=𝜔𝐸𝜂. (1) The mean lifetime of the products turns out to be the FNM of a prior distribution defined on the real line. An introduction to a Bayesian framework can be found in Berger [2], O’Hagan and Forster [3], and the references given therein. For example, the inverse prediction (or calibration problem) in simple linear regression and the coefficient of variation for normal distributions are commonly used in the fields of econometrics, engineering, physics, and biological sciences (see [4–6]. Thus, existence of the FNM for a distribution defined on the real line and the corresponding evaluation are practical issues in fundamental statistics.

There is a general tendency towards more flexible distributions to reduce unrealistic assumptions, like normality as a prior distribution in the statistical literature. Therefore, the skew-symmetric distribution is an alternative approach which extends the symmetric distributions by allowing a shape parameter to control skewness. A random variable 𝑌 is said to have the skew-normal distribution and is denoted by 𝑌∼𝒮𝒩(𝜇,𝜎²,𝛼) if its density function is given by 𝑓_𝑌𝑦=2𝜎𝜙𝑦−𝜇𝜎Φ𝛼𝑦−𝜇𝜎, (2) where 𝜙 and Φ are the pdf and the cdf of the standard normal distribution and the parameters 𝜇, 𝜎, and 𝛼 regulate the location, scale, and skewness, respectively. A systematic treatment of the skew-normal distribution has been developed by Azzalini [7]. Furthermore, Azzalini and Capitanio [8] defined a skewed 𝑡 variate as the scale mixture 𝑋=𝑆⁻¹𝑌^∗+𝜇, (3) where 𝑌^∗∼𝒮𝒩(0,𝜎²,𝛼), 𝜈𝑆² is distributed independently of 𝑌^∗, according to a chi-squared distribution with degrees of freedom 𝜈, and is denoted by 𝑋∼𝒮𝑡(𝜇,𝜎,𝛼,𝜈). Note that the skew-𝑡 distribution approaches the skew-normal distribution as 𝜈→∞; that is, lim_𝜈→∞𝑓_𝑋(𝑥)=𝑓_𝑌(𝑥), where 𝑓_𝑋 is the pdf of 𝑋 and 𝑓_𝑌 is defined in (2). Since the skew-normal, skew-Cauchy, and Student’s 𝑡-distributions were included in the skew-𝑡 distribution, they have proved themselves quite adequate for modeling real data sets (refer to [9]).

In the previously-mentioned lifetime example, we can assume the prior distribution of the random effect 𝜂 to be a skew-𝑡 distribution for adequately realistic situations. The same approach for the random effects in regression models has been presented in Azzalini and Capitanio [8]. Then, the product’s mean lifetime in (1) turns out to be the FNM of a skew-𝑡 distribution. However, by the result in Piegorsch and Casella [10], the FNM of a skew-𝑡 distribution does not exist in the usual sense since the value of the density function at origin is larger than zero. Hence, through different viewpoints of the integral of FNM, the concept of the (Cauchy) principal value, widely used in the probability theory such as the weak law of large numbers [11], can be used to avoid the nonexistence of the FNM in the usual sense. From the analytic and practical viewpoint, one can comprehend at least the FNM in the principal value sense (PV-FNM) if that integral does not converge in the usual sense.

We allow for a finite number of discontinuities on the real axis by requiring ℎ to be continuous on the real line except for a finite number of points 𝑥₁<⋯<𝑥_𝑛. Then, we will call the following the principal value for an improper integral that exists if and only if for every 𝜖_𝑖>0, 𝑖=0,…,𝑛, lim_{𝜖0,…,𝜖𝑛→0}^{𝑥1−𝜖1}_−1/𝜖0+^{𝑥2−𝜖2}_𝑥1+𝜖1+⋯+^{𝑥𝑛−𝜖𝑛}_{𝑥𝑛−1+𝜖𝑛−1}+^1/𝜖0_{𝑥𝑛+𝜖𝑛}ℎ(𝑥)d𝑥 (4) exists and is finite. The improper integral of the FNM in the principal value sense (PV-FNM) can exist even when it does not exist in the usual sense. Based on the principal value sense, Peng [1] gave alternative sufficient conditions for the existence and finiteness of the FNM in which the mild conditions are easy to hold for the most commonly used distributions defined on the real line. The PV-FNM of a skew-normal distribution had also been derived by Peng [1]. Furthermore, an explicit solution is highly desirable and attractive when numerical methods are computationally laborious in evaluating an improper integral. Closed form formula not only avoids time-consuming algorithms but also verifies the numerical evaluation. The aim of this study is to obtain an explicit expression of the PV-FNM corresponding to (3) for practical applications and to provide some interesting results from the PV-FNM of a skew-𝑡 distribution.

The remainder of this paper is organized as follows. The exact form of the PV-FNM of skew-𝑡 and generalized Student’s 𝑡-distributions is derived in Section 2. Section 3 provides some applications of the PV-FNM of skew-𝑡 and generalized Student’s 𝑡-distributions. Section 4 contains conclusions.

2. Preliminaries

The PV-FNM of a skew-𝑡 distribution does exist by giving a scale mixture of a skew-normal distribution. In the following, we start with a simple case of skew-𝑡 distribution and then increase the complexity of the distributions. Note that the notation PVE(⋅) can be used for indicating the expectation in the principal value sense. All derivations are given in the Appendices A, B, C, D, and E.

Lemma 1. If 𝑋∼𝑡(𝜇,𝜎,𝜈)≡𝒮𝑡(𝜇,𝜎,0,𝜈), then 1PVE𝑋=𝜎²𝜈𝜇²+𝜎²𝜈^(𝜈+1)/2×∑^(𝜈−1)/2_𝑘=0𝜈−12𝑘×𝜇^2𝑘+1(2𝑘+1)𝜈^𝑘𝜎^2𝑘+1,𝑖𝑓𝜈𝑖𝑠𝑜𝑑𝑑,𝜈!{(𝜈/2)!}²𝜎2^𝜈𝜈𝜇²+𝜎²𝜈^𝜈/2+1×𝜇∑^𝜈/2_𝑘=12^2𝑘−1(𝑘−1)!𝑘!()!𝜇2𝑘1+²𝜎²𝜈^𝑘+𝜇²+𝜎²𝜈×log𝜇²+𝜎²𝜈+𝜇√𝜎²𝜈,𝑖𝑓𝜈𝑖𝑠𝑒𝑣𝑒𝑛. (5)

In what follows, we deal with the PV-FNM of a skew-𝑡 distribution which is more complicated in calculation. Note that to avoid “reverse” sum (product) in the following formula, we define ∑^𝑛2_𝑑=𝑛1𝑐_𝑑=0 and ∏^𝑛2_𝑑=𝑛1𝑐_𝑑=1 for 𝑛₂<𝑛₁.

Theorem 2. If 𝑋∼𝒮𝑡(𝜇,𝜎,𝛼,𝜈), then one obtains for odd values of 𝜈1PVE𝑋=𝜎²𝜈𝜇²+𝜎²𝜈^(𝜈+1)/2×^(𝜈−1)/2_𝑘=0𝜈−12𝑘𝜇^2𝑘+1(2𝑘+1)𝜈^𝑘𝜎^2𝑘+1−𝛼(𝜈+1)Γ((𝜈+1)/2)𝜎²𝜈^𝜈/2√()×2√𝜋Γ𝜈/21+𝛼²𝜈+2^(𝜈−1)/2_𝑘=0∏^𝑘_𝑚=0(𝜈−2𝑚+2)∏^𝑘_𝑚=0()𝐼𝜈−2𝑚+1²¹(𝑘)+∏^(𝜈−1)/2_𝑛=0(𝜈−2𝑛)∏^(𝜈−1)/2_𝑛=0()𝐼𝜈−2𝑛+1²²+2√𝜋𝜎²𝜈Γ((𝜈+1)/2)Γ()𝜎𝜈/2²𝜈𝜇²+𝜎²𝜈^(𝜈+1)/2×log𝛼+√1+𝛼², (6) and for even values of 𝜈1PVE𝑋=𝜈𝜇𝜈+1𝜇²+𝜎²𝜈×^(𝜈/2)−1_𝑘=0∏^𝑘_𝑚=0(𝜈−2𝑚+1)∏^𝑘_𝑚=0()𝜎𝜈−2𝑚²𝜈𝜇²+𝜎²𝜈^𝑘+∏^𝜈/2−1_𝑛=0(𝜈−2𝑛−1)∏^𝜈/2−1_𝑛=0()×𝜈−2𝑛𝜈𝜎²𝜎²𝜈𝜇²+𝜎²𝜈^(𝜈+1)/2×log𝜇²+𝜎²𝜈+𝜇√𝜎²𝜈+2Γ((𝜈+1)/2)𝜎²𝜈^𝜈/2√()×√𝜋Γ𝜈/2log𝛼+1+𝛼²𝜇²+𝜎²𝜈^(𝜈+1)/2()×−𝛼𝜈+1_{𝜈/2𝑘=0𝑘𝜈/2}1+𝛼^2𝑘+1/2𝐼2𝑘+1²³(𝑘), (7) where 𝐼₂₁(𝑘)=𝛼^{𝜈−2𝑘−1}21+𝛼²𝜇²+𝜎²𝜈^(𝜈+1)/2×^𝑘_𝑖=1𝜈−12−𝑖𝜈−12×𝜇−𝑘₂+𝜎²𝜈^−𝑖−1+𝛼²𝜇²+𝜎²𝜈^−𝑖𝑖1+𝛼²𝜇²+𝜎²𝜈^−𝑖+^{((𝜈−1)/2)−𝑘}_𝑗=1𝜈−12𝑘×𝜎−𝑗₂𝜈^−𝑗−𝜇²+𝜎²𝜈^−𝑗𝑗𝛼^2𝑗1+𝛼²𝜇²+𝜎²𝜈^−𝑗+𝜈−12𝜈−12−𝑘×log1+1+𝛼²𝜇²𝜎²𝜈,𝐼₂₂=1𝜇²+𝜎²𝜈^(𝜈+1)/2×^(𝜈−1)/2_𝑖=0𝜈−12𝑖(−1)^𝑖𝛼^2𝑖×2𝑖+1_{2𝑖+1𝑗=1}(−1)^𝑗+11+𝛼²𝜇²+𝜎²𝜈^{𝑗/2−𝑖−1}2𝑖−𝑗+2𝜇^{𝑗−2𝑖−2}1+𝛼^{2(𝑗−1)/2}𝜇+2²1+𝛼²𝜇²+𝜎²𝜈^𝑖+1/2×log1+𝛼²𝜇²+𝜎²𝜈+𝜇√1+𝛼²√𝜎²𝜈+11+𝛼^2𝑖+1/2𝜎log²𝜈1+𝛼²𝜇²+𝜎²𝜈,𝐼₂₃(𝑘)=^𝑘_𝑖=0𝑘𝑖𝜇²+𝜎²𝜈^𝑘−𝑖(−1)^𝑖×^𝑘−𝑖_𝑗=1∏^𝑗_𝑚=1(𝜈+2𝑘−2𝑖−2𝑚+3)∏^𝑗_𝑚=1()×2𝑘−2𝑖−2𝑚+11+𝛼²𝜇²+𝜎²𝜈^−𝜈/2𝜎²𝜈^{𝑖+𝑗−𝑘−1/2}−𝜇²+𝜎²𝜈^{𝑖+𝑗−𝑘−(𝜈+1)/2}×(𝜈+2𝑘−2𝑖+1)𝛼^2−2𝑗×(1+𝛼²)(𝜇²+𝜎²𝜈)^𝑗−1+∏^𝑘−𝑖_𝑛=1(𝜈+2𝑛−1)∏^𝑘−𝑖_𝑛=1()×𝛼2𝑛−1_{2(𝑘−𝑖)}×𝜈+1_𝜈/2𝑙=1∏^𝑙_𝑚=1(𝜈−2𝑚+3)∏^𝑙_𝑚=1()×1𝜈−2𝑚+2(1+𝛼²)(𝜇²+𝜎²𝜈)^{𝑙+𝑘−𝑖}×1𝜇²+𝜎²𝜈^{(𝜈+1)/2−𝑙}−√𝜎²𝜈(1+𝛼²)𝜇²+𝜎²𝜈^{𝜈/2+1−𝑙}+∏^𝜈/2_𝑛=1(𝜈−2𝑛+1)∏^𝜈/2_𝑛=1()×𝛼𝜈−2𝑛+2_{2𝑘−2𝑖−1}1+𝛼²𝜇²+𝜎²𝜈^{(𝜈+1)/2+𝑘−𝑖}×log𝛼+√1+𝛼²×1+𝛼²𝜇²+𝜎²𝜈×𝛼√𝜎²𝜈+1+𝛼²×𝜇²+𝜎²𝜈^1/2−1. (8)(9)(10)

Note that if 𝑌 is defined in (2), then the nonexistence of the PV-FNM of PVE(𝑌^−𝑟) pointed out by Peng [1], where 𝑟=2,3,…. Unfortunately, the PV-FNM of a skew-𝑡 distribution, PVE(𝑋^−𝑟), fails to exist for all (𝜇,𝜎,𝛼,𝜈)∈ℝ×ℝ₊×ℝ×ℝ₊ when 𝑟=2,3,…. When 𝜈=1, the formula of Theorem 2 reduces to that obtained in the skew-Cauchy case as follows.

Corollary 3. If in Theorem 2, one lets 𝜈=1, then 𝑋∼𝒮𝒞(𝜇,𝜎,𝛼)≡𝒮𝑡(𝜇,𝜎,𝛼,1) and 1PVE𝑋=𝜇𝜇²+𝜎²+2𝜋𝜎𝜇²+𝜎²log𝛼+√1+𝛼²−2𝛼𝜋𝜇𝜇²+𝜎²×log1+1+𝛼²𝜇²/𝜎²+√1+𝛼²𝜇/𝜎1+1+𝛼²𝜇²/𝜎². (11)

Corollary 4. If in Theorem 2, one lets 𝜈=2, then 1PVE𝑋=𝜇𝜇²+2𝜎²−𝛼𝜇²2𝜎²1+𝛼²𝜇²+2𝜎²1+𝛼²𝜇²+2𝜎²+2𝜎²𝜇²+2𝜎^23/2×log𝜇²+2𝜎²×+𝜇𝛼+1+𝛼²1+𝜇²/2𝜎²×1+𝛼²𝜇²+2𝜎²⁻¹. (12)

As shown before, the formula of the PV-FNM of a skew-𝑡 distribution becomes complicated and difficult to calculate due to the nonzero location parameter in the integral. However, if 𝜇=0 in Theorem 2, the complicated formula can be simplified significantly as the following results show.

Corollary 5. If in Theorem 2, one lets 𝜇=0, then 1PVE𝑋=2√𝜋𝜎²Γ((𝜈+1)/2)√()√𝜈Γ𝜈/2log𝛼+1+𝛼², (13) which is an increasing function in 𝜈.

Moreover, by using the property of the Gamma function, that is, Γ((𝜈+1)/2)/(√𝜈Γ(𝜈/2))→1/√2 as 𝜈→∞, we have the following result immediately for 𝜇=0: 1PVE𝑋=2𝜋𝜎²log𝛼+√1+𝛼²as𝜈→∞, (14) which coincides with the fact given by Peng [1].

The PV-FNM of a generalized Student’s 𝑡-distribution can be obtained as follows. The generalized Student’s 𝑡-distribution (𝑌∼𝒢𝒯(𝜇,𝜎²,𝑁,𝑟,𝛼,𝛽)) is defined as a scale mixture of the Kotz distribution and an inverse gamma distribution. That is, 𝑌=𝜇+√𝛼𝜎²𝑉𝑍, (15) where 𝑍∼𝒦(0,1,𝑁,𝑟) and 𝑉∼ℐ𝑛𝑣𝒢(𝛼,𝛽), and 𝑍 is independent of 𝑉. The Kotz distribution (𝑋∼𝒦(𝜇,𝜎²,𝑁,𝑟)) introduced in Kotz [12] is given by 𝑓(𝑥)=𝑟^𝑁𝑥−𝜇/𝜎^2(𝑁−1)(𝑁)𝜎−𝑟𝜋Γexp𝑥−𝜇²𝜎², (16) where 𝑟>0 and 𝑁>0. The inverse gamma distribution (𝑉∼ℐ𝑛𝑣𝒢(𝛼,𝛽)) is given by 𝑓(𝑣)=1Γ(𝛼)𝛽𝛽𝑣^𝛼+1exp−𝛽𝑣, (17) where 𝛼>0 and 𝛽>0. For 𝑁=1, 𝑟=1/2, 𝛼=𝜈/2, and 𝛽=1, we obtain the 𝑡 distribution. Setting 𝑁=𝑟=1, 𝛼𝛽=𝜅, and 𝛼=𝜈/2 yields the generalized version of the 𝑡 distributions family defined by Arellano-Valle and Bolfarine [13]. The case 𝛽=1 and 𝛼→∞ gives the Kotz distribution. Setting 𝑁=𝑟=1, 𝛽=1, and 𝛼→∞ yields the normal distribution.

Theorem 6. If 𝑌∼𝒢𝒯(𝜇,𝜎²,𝑁,𝑟,𝛼,𝛽), then 1PVE𝑌=√𝜋𝑟^𝑁−1Γ(𝛼)Γ()×𝑁−1/2_{2𝑁−2𝑖=1𝑖−1𝑗=0}𝑖𝑗×()2𝑁−2𝑖−1−1^{2𝑁−2−𝑖}𝜇^{2𝑁−𝑗−3}𝐸𝑍^∗𝑗×Γ𝑁+𝛼−𝑗/2−1𝛼𝛽𝜎^{2𝑁−𝑗/2−1}+Γ(𝑁+𝛼)2√𝜋𝛽^𝛼𝜇^2𝑁−2Γ()Γ(𝛼)×𝑟𝑁−1/2𝛼𝜎^2𝑁−1/2×𝛼𝜎²𝛼𝛽𝜎²+𝑟𝜇^{2𝑁+𝛼−1/2}⋅∑^{𝑁+𝛼−3/2}_𝑘=0^{𝑘𝑁+𝛼−3/2}×2𝑘+1𝑟𝜇²𝛼𝛽𝜎^2𝑘+1/2,1𝑖𝑓𝑁+𝛼−2{2(!∈ℕ,𝑁+𝛼−1)}{(𝑁+𝛼−1)!}²𝛼𝜎²4𝛼𝛽𝜎²+𝑟𝜇^{2𝑁+𝛼−1}×𝑟𝛼𝜎²𝜇²𝛼𝛽𝜎²+𝑟𝜇²×∑^{𝑁+𝛼−1}_𝑘=12^2𝑘−1(𝑘−1)!𝑘!(2𝑘)!1+𝑟𝜇²𝛼𝛽𝜎^2𝑘+𝛼𝜎²𝛼𝛽𝜎²+𝑟𝜇²×log1+𝑟𝜇²𝛼𝛽𝜎²+𝑟𝜇²𝛼𝛽𝜎²,𝑖𝑓𝑁+𝛼−1∈ℕ, (18) where 𝑍^∗∼𝒩(0,(2𝑟)⁻¹).

To show the applicability and effectiveness of the main results, some applications are presented in the following section.

3. Applications

One special property of the Student’s 𝑡-distribution is now obtained from Lemma 1.

Corollary 7. If in Lemma 1, one lets 𝜈=1, then 𝑋∼𝒞(𝜇,𝜎)≡𝑡(𝜇,𝜎,1) and 1PVE𝑋=𝜇𝜇²+𝜎². (19)

It is of interest to note that if 𝑋∼𝒞(𝜇,𝜎), then PVE(𝑋^𝑟) only exists for all (𝜇,𝜎)∈ℝ×ℝ₊ when 𝑟=±1. That is, PVE(𝑋)=𝜇. (20) Moreover, it is well known that the random variable 1/𝑋 still follows the Cauchy distribution with location parameter 𝜇/(𝜇²+𝜎²) and scale parameter 𝜎/(𝜇²+𝜎²). Therefore, assume that 𝑋₁,…,𝑋∼_𝑛^i.i.d.𝒞(𝜇,𝜎), 𝑀₋₁=1𝑛^𝑛_𝑖=1𝑋_𝑖⁻¹,𝑀₁=1𝑛^𝑛_𝑖=1𝑋_𝑖. (21) The moment estimators in the principal value sense can be obtained by (19) and (20) as 𝜇=𝑀₁,𝜎²=𝑀₁𝑀₋₁1−𝑀₁𝑀₋₁. (22) This result provides one kind of initial value to compute the maximum likelihood estimators for the Cauchy distribution.

Another application of Lemma 1 is a representation of Dawson’s integral, which is defined as 𝒟(𝑧)=exp−𝑧^2𝑧₀exp𝑡²d𝑡 (23) for all real 𝑧. Dawson’s integral comes up in the theory of propagation of electromagnetic waves along the earth's surface. A precise computation is usually necessary for Dawson’s integral in mathematical physics. Thus, how to represent this special function 𝒟(𝑧) becomes a more practical issue. The representation of Dawson's integral and a related result are investigated in the following corollaries.

Corollary 8. Dawson’s integral can be represented as the limit of a Gaussian hypergeometric function. That is, 𝒟(𝑧)=𝑧exp−𝑧²lim_2𝑛→∞𝐹₁123,−𝑛;2;−2𝑧²,2𝑛+1 (24) where a (Gaussian) hypergeometric function is given by ₂𝐹₁(𝑎,𝑏;𝑐;𝑥)=^∞_𝑘=0Γ(𝑎+𝑘)Γ(𝑎)Γ(𝑏+𝑘)Γ(𝑏)Γ(𝑐)Γ()𝑥𝑐+𝑘^𝑘.𝑘! (25)

The accuracy of this representation is omitted here as it is not essential for this work. Moreover, if a (Kummerian) confluent hypergeometric function is defined as ₁𝐹₁(𝑎;𝑏;𝑥)=^∞_𝑘=0Γ(𝑎+𝑘)Γ(𝑎)Γ(𝑏)Γ()𝑥𝑏+𝑘^𝑘,𝑘! (26) then a confluent hypergeometric function can be shown to be a limiting case of a hypergeometric function as follows.

Corollary 9. Consider the following: lim_2𝑛→∞𝐹₁123,−𝑛;2;−2𝑧²=2𝑛+1¹𝐹₁12;32;𝑧². (27)

Note that we also double-checked the correctness of our results in this study via numerical evaluations with MathCAD mathematical software.

4. Conclusion

The primary objective considered in this study is to derive the exact form of the PV-FNM of a skew-𝑡 distribution. In addition to our attempt, the exact form of the PV-FNM of other distributions will be of great interest. For this objective, the principal value sense is used for the improper integral with singular points, so the PV-FNM can exist and be finite if that integral cannot converge in the usual sense. Hence, an explicit expression for the PV-FNM of a univariate skew-𝑡 distribution is obtained. The complex formula also has direct bearing on computations in theoretical studies. Some applications discovered from the PV-FNM of a skew-𝑡 distribution are discussed.

Many other interesting distributions and FNM-related issues are worthy of further investigation. For example, it is well known that the skew-normal distribution presents singular Fisher information matrix (see Gómez et al. [14]). This brings difficulty to inference close to the singularity point. Using the principal value on inferring this issue is a challenge worthy of research.

Appendices

A. Proof of Lemma 1

Proof. It is well known that 𝑋∣𝑆²∼𝒩(𝜇,𝜎²/𝑆²). Hence, by Theorem 2 in Peng [1] and using the PV-FNM of the normal distribution showed in the appendix of Quenouille [15], we obtain 1PVE𝑋1=PVEPVE𝑋∣𝑆²=𝜈^𝜈/2+1𝜎^𝜈𝜇₀d𝑦𝜇²+𝜎²𝜈−𝑦^2𝜈/2+1. (A.1) For odd degrees of freedom, making use of formula 2.271 (6) in Gradshteyn and Ryzhik [16], we have ^𝜇₀d𝑦𝜇²+𝜎²𝜈−𝑦^2𝜈/2+1=1𝜇²+𝜎²𝜈^(𝜈+1)/2×^(𝜈−1)/2_𝑘=0𝜈−12𝑘𝜇^2𝑘+1(2𝑘+1)𝜎²𝜈^𝑘+1/2. (A.2) For even degrees of freedom, by using formulas 2.171 (4) and 2.124 (1) in Gradshteyn and Ryzhik [16], we get ^𝜇₀d𝑦𝜇²+𝜎²𝜈−𝑦^2𝜈/2+1=𝜇×𝜈+1_{(𝜈/2)−1𝑘=0𝑘𝑚=0}()×𝜈−2𝑚+1_𝑘𝑚=0()×𝜇𝜈−2𝑚₂+𝜎²𝜈^𝑘+1𝜎²𝜈^{𝜈/2−𝑘−1}+∏^𝜈/2−1_𝑛=0(𝜈−2𝑛−1)∏^𝜈/2−1_𝑛=0(𝜈−2𝑛)𝜇²+𝜎²𝜈^𝜈/2𝜇0d𝑦𝜇²+𝜎²𝜈−𝑦²_𝜇0,d𝑦𝜇²+𝜎²𝜈−𝑦²=1𝜇²+𝜎²𝜈log𝜇²+𝜎²𝜈+𝜇√𝜎²𝜈, (A.3)(A.4) respectively. By combining (A.2), (A.3), and (A.4) and some algebraic manipulations, the result then follows.

B. Proof of Theorem 2

To prove Theorem 2, we need the following auxiliary results.

Lemma 10. For 𝑝>𝜇², 𝛼, 𝜇∈ℝ and 𝑟=0,1,2,…, one gets ^𝜇₀1𝑝+𝛼²𝑦^2𝑟+3/2×log𝑝+𝛼²𝑦²+𝑦1+𝛼²𝑝+𝛼²𝑦²𝑝+𝛼²𝑦²−𝑦1+𝛼²𝑝+𝛼²𝑦²=1d𝑦𝑝^𝑟𝑟+1_𝑖=0𝑟𝑖(−1)^𝑖𝛼^2𝑖×2𝑖+1_{2𝑖+1𝑗=1}(−1)^𝑗+1𝛼²𝜇²+𝑝^{𝑗/2−𝑖−1}2𝑖−𝑗+2𝜇^{𝑗−2𝑖−2}1+𝛼^{2(𝑗−1)/2}𝜇+2²𝛼²𝜇²+𝑝^𝑖+1/2×log𝛼²𝜇²+𝑝+𝜇√1+𝛼²𝑝−𝜇²+11+𝛼^2𝑖+1/2log𝑝−𝜇²𝛼²𝜇².+𝑝 (B.1)

Proof. By using the change of variable 𝑡=𝑦/𝑝+𝛼²𝑦², the binomial theorem, and term-by-term integration, the original integral can be expressed as 1𝑝^𝑟𝑟+1_𝑖=0𝑟𝑖(−1)^𝑖𝛼^{2𝑖𝜇/𝑝+𝛼2𝜇2}₀𝑡^2𝑖√log1+1+𝛼²𝑡1−√1+𝛼²𝑡d𝑡. (B.2) Now, applying formula 2.729 (1) in Gradshteyn and Ryzhik [16], the result thus follows.

Lemma 11. For 𝑝>𝜇², 𝛼, 𝜇∈ℝ, and 𝑠, 𝑘=0,1,2,…, one gets ^𝜇₀𝑦^2𝑘+1𝑝+𝛼²𝑦^2𝑠+1𝑝−𝑦^2𝑘+1/2=d𝑦_𝑘𝑖=0𝑘𝑖𝑝^𝑘−𝑖(−1)^𝑖×^𝑘−𝑖_𝑗=1∏^𝑗_𝑚=1(2𝑠+2𝑘−2𝑖−2𝑚+3)∏^𝑗_𝑚=1()×𝛼2𝑘−2𝑖−2𝑚+1₂𝜇²+𝑝^−𝑠𝑝−𝜇^{2𝑖+𝑗−𝑘−1/2}−𝑝^{𝑖+𝑗−𝑘−𝑠−1/2}×(2𝑠+2𝑘−2𝑖+1)×𝛼^2−2𝑗1+𝛼²𝑝^𝑗−1+∏^𝑘−𝑖_𝑛=1(2𝑠+2𝑛−1)∏^𝑘−𝑖_𝑛=1()×𝛼2𝑛−1_{2(𝑘−𝑖)}2𝑠+1^𝑠_𝑙=1∏^𝑙_𝑚=1(2𝑠−2𝑚+3)∏^𝑙_𝑚=1()×12𝑠−2𝑚+2(1+𝛼²)𝑝^{𝑙+𝑘−𝑖}×1𝑝^{𝑠+1/2−𝑙}−𝑝−𝜇²𝛼²𝜇²+𝑝^{𝑠+1−𝑙}+∏^𝑠_𝑛=1(2𝑠−2𝑛+1)∏^𝑠_𝑛=1()×𝛼2𝑠−2𝑛+2_{2𝑘−2𝑖−1}(1+𝛼²)𝑝^{𝑠+𝑘+1/2−𝑖}√×log𝛼+1+𝛼²𝛼²𝜇²+𝑝𝛼𝑝−𝜇²+1+𝛼²𝑝. (B.3)

Proof. Using the change of variable 𝑡=𝑝−𝑦², the binomial theorem, and term-by-term integration, we get ^𝜇₀𝑦^2𝑘+1𝑝+𝛼²𝑦^2𝑠+1𝑝−𝑦^2𝑘+1/2=()d𝑦−1^𝑠+1𝛼^𝑘2𝑠+2_𝑖=0𝑘𝑖𝑝^𝑘−𝑖(−1)^𝑖×_𝒜d𝑡𝑡^{2(𝑘−𝑖)}𝑡²−𝑞^𝑠+1, (B.4) where 𝑞=𝑝(1+𝛼²)/𝛼² and 𝒜={𝑡∣𝑝−𝜇²≤𝑡≤√𝑝}. By using formulas 1.2.11 (2) and 1.2.10 (13) in Prudnikov et al. [17], one can obtain _𝒜d𝑡𝑡^{2(𝑘−𝑖)}𝑡²−𝑞^𝑠+1=(−1)^𝑠+1𝛼^2𝑠+2×2𝑠+2𝑘−2𝑖+1_{𝑘−𝑖𝑗=1}∏^𝑗_𝑚=1(2𝑠+2𝑘−2𝑖−2𝑚+3)∏^𝑗_𝑚=1()×𝛼2𝑘−2𝑖−2𝑚+1₂𝜇²+𝑝^−𝑠𝑝−𝜇^{2𝑖+𝑗−𝑘−1/2}−𝑝^{𝑖+𝑗−𝑘−𝑠−1/2}𝛼^2−2𝑗(1+𝛼²)𝑝^𝑗+∏^𝑘−𝑖_𝑛=1(2𝑠+2𝑛−1)∏^𝑘−𝑖_𝑛=1()𝛼2𝑛−11+𝛼²𝑝^{2(𝑘−𝑖)𝑘−𝑖𝒜}d𝑡𝑡²−𝑞^𝑠+1_𝒜,d𝑡𝑡²−𝑞^𝑠+1=(−1)^𝑠+1×2𝑠+1_𝑠𝑙=1∏^𝑙_𝑚=1(2𝑠−2𝑚+3)∏^𝑙_𝑚=1()×𝛼2𝑠−2𝑚+2_2𝑠+2(1+𝛼²)𝑝^𝑙×1𝑝^{𝑠+1/2−𝑙}−𝑝−𝜇²𝛼²𝜇²+𝑝^{𝑠+1−𝑙}+∏^𝑠_𝑛=1(2𝑠−2𝑛+1)∏^𝑠_𝑛=1()×()2𝑠−2𝑛+2−1^𝑠+1𝛼^2𝑠+1(1+𝛼²)𝑝^𝑠+1/2√×log𝛼+1+𝛼²𝛼²𝜇²+𝑝𝛼𝑝−𝜇²+1+𝛼²𝑝, (B.5)(B.6) respectively. Substituting (B.5) and (B.6) into (B.4) yields the desired result.

Now, we return to the proof of Theorem 2.

Proof. First, we represent the expression (8) of the PV-FNM of a skew-normal distribution in Peng [1] as follows. If 𝑌∼𝒮𝒩(𝜇,𝜎²,𝛼), then 1PVE𝑌=1𝜎^2𝜇₀𝑥exp²−𝜇²2𝜎²−𝛼d𝑥21+𝛼²𝜎³√𝜋exp−𝜇²2𝜎²×^𝜇₀exp−𝛼²𝑥²2𝜎^2𝑥₀exp1+𝛼²𝑦²2𝜎²+d𝑦d𝑥2𝜋𝜎²exp−𝜇²2𝜎²log𝛼+√1+𝛼². (B.7) By using (B.7), the definition in (3), and the conditional expectation, we obtain 1PVE𝑋=𝐼₁−𝐼₂+2𝜋𝜎²log𝛼+√1+𝛼²𝐼₃, (B.8) where 𝐼₁=PVE^𝜇₀𝑆²𝜎²𝑥exp²−𝜇²2𝜎²𝑆²,𝐼d𝑥₂=𝛼21+𝛼²𝜎³√𝜋×PVE_𝜇0𝑦0𝑆³−𝜇×exp²+𝛼²𝑦²−1+𝛼²𝑥²2𝜎²𝑆²,𝐼d𝑥d𝑦₃=PVE𝑆exp−𝜇²2𝜎²𝑆². (B.9) Now, 𝐼₁ can be calculated as (18) from Lemma 1. By Fubini’s theorem, 𝐼₂ turns out to be 𝐼₂=2𝛼√1+𝛼²(𝜈+1)Γ((𝜈+1)/2)𝜎²𝜈^𝜈/2√()𝐼𝜋Γ𝜈/2^∗2, (B.10) where 𝐼^∗₂=^𝜇₀^𝑦₀d𝑥d𝑦𝜇²+𝜎²𝜈+𝛼²𝑦²−(1+𝛼²)𝑥^2(𝜈+3)/2. (B.11) The evaluation of this integral is divided into two parts.
(i) For odd degrees of freedom, by using formulas 2.171 (4) and 2.124 (1) in Gradshteyn and Ryzhik [16], we get 𝐼^∗₂=1×𝜈+2_{(𝜈−1)/2𝑘=0}∏^𝑘_𝑚=0(𝜈−2𝑚+2)∏^𝑘_𝑚=0()𝐼𝜈−2𝑚+1²¹(𝑘)+∏^(𝜈−1)/2_𝑛=0(𝜈−2𝑛)∏^(𝜈−1)/2_𝑛=0()𝐼𝜈−2𝑛+12√1+²²𝛼², (B.12) where 𝐼₂₁(𝑘)=^𝜇₀𝑦𝑝+𝛼²𝑦^2𝑘+1𝑝−𝑦^{2(𝜈+1)/2−𝑘}𝐼d𝑦,₂₂=^𝜇₀1𝑝+𝛼²𝑦^2𝜈/2+1×log𝑝+𝛼²𝑦²+𝑦1+𝛼²𝑝+𝛼²𝑦²𝑝+𝛼²𝑦²−𝑦1+𝛼²𝑝+𝛼²𝑦²d𝑦, (B.13) in which 𝑝=𝜇²+𝜎²𝜈. Therefore, the change of variable 𝑡=𝑦² implies 𝐼₂₁(𝑘)=(−1)^{((𝜈+1)/2)−𝑘}2𝛼^2𝑘+2𝜇2₀𝑝𝑡+𝛼^2−𝑘−1𝑡−𝑝^{𝑘−(𝜈+1)/2}d𝑡. (B.14) Then, using formula 1.2.7 (6) in Prudnikov et al. [17], one can obtain 𝐼₂₁(𝑘) as (8). From Lemma 10, let 𝑝=𝜇²+𝜎²𝜈 and 𝑟=(𝜈−1)/2; then 𝐼₂₂ can be expressed as (9).
(ii) For even degrees of freedom, making use of formula 2.271 (6) in Gradshteyn and Ryzhik [16], 𝐼^∗₂ can be expanded as 𝐼^∗₂=^𝜈/2_𝑘=0^𝑘𝜈/21+𝛼^2𝑘𝐼2𝑘+1²³(𝑘), (B.15) where 𝐼₂₃(𝑘)=^𝜇₀𝑦^2𝑘+1𝜇²+𝜎²𝜈+𝛼²𝑦^2𝜈/2+1𝜇²+𝜎²𝜈−𝑦^2𝑘+1/2d𝑦. (B.16) Let 𝑝=𝜇²+𝜎²𝜈 and 𝑠=𝜈/2 in Lemma 11; then 𝐼₂₃(𝑘) can be worked out as (10).
Finally, 𝐼₃ is easy to calculate as 𝐼₃=2𝜈Γ((𝜈+1)/2)Γ()𝜎𝜈/2²𝜈𝜇²+𝜎²𝜈^(𝜈+1)/2. (B.17) Consequently, combining (18), (8), (9), (10), and (B.17) yields the desired result.

C. Proof of Theorem 6

We need the following lemma.

Lemma 12. Letting 𝑌∼𝒦(𝜇,𝜎²,𝑁,𝑟), where 𝑁 is integer, then the PV-FNM of a Kotz distribution is given by 1PVE𝑌=𝜇𝜎^2𝑁−2⋅2√𝜋𝑟^𝑁𝜎²Γ()×𝑁−1/2_𝜇0𝑟𝑥exp²−𝜇²𝜎²+√d𝑥𝜋𝑟^𝑁−1()×𝜎Γ𝑁−1/2_{2𝑁−2𝑖=1𝑖−1𝑗=0}𝑖𝑗×()2𝑁−2𝑖−1−1^{2𝑁−2−𝑖}⋅𝜇𝜎^{2𝑁−𝑗−3}𝐸𝑍^∗𝑗, (C.1) where 𝑍^∗∼𝒩(0,(2𝑟)⁻¹).

Proof. Let 𝑥=𝑦/𝜎 and 𝑎=𝜇/𝜎; then 1PVE𝑌=𝑟^𝑁−1/2()𝐼(𝑎),𝜎Γ𝑁−1/2 (C.2) where 𝐼(𝑎)=^∞_−∞1𝑥(𝑥−𝑎)^2𝑁−2exp−𝑟(𝑥−𝑎)²d𝑥. (C.3) Then, using binomial theorem and term-by-term integration, we get 𝐼(𝑎)=𝑎^2𝑁−2𝐼₁(𝑎)+^2𝑁−2_𝑖=1𝑖()2𝑁−2−𝑎^{2𝑁−2−𝑖}𝐼₂(𝑎), (C.4) where 𝐼₁(𝑎)=^∞_−∞1𝑥exp−𝑟(𝑥−𝑎)²𝐼d𝑥,₂(𝑎)=^∞_−∞𝑥^𝑖−1exp−𝑟(𝑥−𝑎)²d𝑥. (C.5) Following the same argument in the appendix of Quenouille [15], 𝐼₁(𝑎) can be derived as 𝐼₁(𝑎)=2√𝜋𝑟^𝑎₀exp𝑟𝑥²−𝑎²d𝑥. (C.6) By the property of a normal distribution, we obtain 𝐼₂(𝑎)=𝜋𝑟𝐸𝑍^∗+𝑎^𝑖−1=𝜋𝑟^𝑖−1_𝑗=0𝑗𝑖−1𝜇𝜎^{𝑖−𝑗−1}𝐸𝑍^∗𝑗. (C.7) Substituting (C.6) and (C.7) into (C.2) yields the desired result.

Now, we return to the proof of Theorem 6.

Proof. It is well known that 𝑌∣𝑉∼𝒦(𝜇,𝛼𝜎²𝑉,𝑁,𝑟). Hence, using Lemma 12, we obtain 1PVE𝑌1=PVEPVE𝑌=2√∣𝑉𝜋𝑟^𝑁𝛽^𝛼𝜇^2𝑁−2Γ(𝑁−1/2)Γ(𝛼)𝛼𝜎^2𝑁𝐼₁+√𝜋𝑟^𝑁−1(𝛼)Γ()×𝛽Γ𝑁−1/2_{2𝑁−2𝑖=1𝑖−1𝑗=0}𝑖𝑗×()2𝑁−2𝑖−1−1^{2𝑁−2−𝑖}𝜇^{2𝑁−𝑗−3}𝐸𝑍^∗𝑗𝐼₂, (C.8) where 𝐼₁=^𝜇₀^∞₀𝑣^{−(𝑁+𝛼+1)}−𝑟𝑦×exp𝛽−²−𝜇²𝛼𝜎²1𝑣𝐼d𝑣d𝑥,₂=𝛽^𝛼+1√𝛼𝜎^{22𝑁−𝑗−2}×^∞₀𝑣^{−(2𝑁−𝑗−2)/2−𝛼−1}−𝛽exp𝑣d𝑣. (C.9)(C.10) Now, 𝐼₁ can be evaluated as 𝐼₁=Γ(𝑁+𝛼)𝛼𝜎²𝑟×^{𝑟𝜇2/𝛼𝜎2}₀d𝑦𝛽+𝑟𝜇²/𝛼𝜎²−𝑦^2𝑁+𝛼. (C.11) The evaluation of this integral is divided into two parts.
(i) For 𝑁+𝛼−1/2∈ℕ, making use of formula 2.271 (6) in Gradshteyn and Ryzhik [16], we have ^{𝑟𝜇2/𝛼𝜎2}₀d𝑦𝛽+𝑟𝜇²/𝛼𝜎²−𝑦^2𝑁+𝛼=𝛼𝜎²𝛼𝛽𝜎²+𝑟𝜇^{2𝑁+𝛼−1/2}⋅^{𝑁+𝛼−3/2}_𝑘=0^{𝑘𝑁+𝛼−3/2}2𝑘+1𝑟𝜇²𝛼𝛽𝜎^2𝑘+1/2. (C.12)
(ii) For 𝑁+𝛼−1∈ℕ, by using formulas 2.171 (4), 2.103 (4) in Gradshteyn and Ryzhik [16] and after some algebraic manipulations, we get ^{𝑟𝜇2/𝛼𝜎2}₀d𝑦𝛽+(𝑟𝜇²/𝛼𝜎²)−𝑦^2𝑁+𝛼={2(𝑁+𝛼−1)}!{(𝑁+𝛼−1)!}²𝛼𝜎²4𝛼𝛽𝜎²+𝑟𝜇^{2𝑁+𝛼−1}×𝑟𝛼𝜎²𝜇²𝛼𝛽𝜎²+𝑟𝜇²×^{𝑁+𝛼−1}_𝑘=12^2𝑘−1(𝑘−1)!𝑘!()!×2𝑘1+𝑟𝜇²𝛼𝛽𝜎^2𝑘+^{𝑟𝜇2/𝛼𝜎2}₀d𝑦𝛽+𝑟𝜇²/𝛼𝜎²−𝑦²_{𝑟𝜇2/𝛼𝜎20},d𝑦𝛽+𝑟𝜇²/𝛼𝜎²−𝑦²=𝛼𝜎²𝛼𝛽𝜎²+𝑟𝜇²log1+𝑟𝜇²𝛼𝛽𝜎²+𝑟𝜇²𝛼𝛽𝜎², (C.13)(C.14) respectively. Finally, 𝐼₂ is easy to evaluate as 𝐼₂=𝛽^{𝑗/2−𝑁+2}Γ𝑁+𝛼−𝑗/2−1𝛼𝜎^{2𝑁−𝑗/2−1}. (C.15) Consequently, combining (C.8)–(C.15) yields the desired result.

D. Proof of Corollary 8

Proof. Since a Student’s 𝑡-distribution approaches a normal distribution as 𝜈→∞, we have PVE(1/𝑍)=lim_𝜈→∞PVE(1/𝑋), where 𝑍∼𝒩(𝜇,𝜎²) and 𝑋∼𝑡(𝜇,𝜎²,𝜈). In addition, Quenouille [15] showed that 1PVE𝑍=√2𝜎𝒟𝜇√.2𝜎 (D.1) Hence, by restricting limit to odd degrees in Lemma 1 and using L’Hospital rule, we have lim_𝜈→∞1PVE𝑋_𝜈isodd−𝜇=exp²2𝜎²×lim^𝑛_{𝑛→∞𝑘=0}𝑛𝑘𝜇^2𝑘+1(2𝑘+1)(2𝑛+1)^𝑘𝜎^2𝑘+1, (D.2) where 𝜈=2𝑛+1. Comparing (D.1) and (D.2), one can obtain exp𝑡^𝑧₀²d𝑡=lim^𝑛_{𝑛→∞𝑘=0}𝑛𝑘22𝑛+1^𝑘𝑧^2𝑘+12𝑘+1=𝑧lim_2𝑛→∞𝐹₁123,−𝑛;2;−2𝑧².2𝑛+1 (D.3) Thus, substituting (D.3) into (23) yields the desired result (24).

E. Proof of Corollary 9

Proof. It is well known that Dawson’s integral is a special case of the confluent hypergeometric function. That is, 𝒟(𝑧)=𝑧₁𝐹₁31;2;−𝑧². (E.1) By using formula 9.212 (1) in Gradshteyn and Ryzhik [16] and comparing (24) in Corollary 8 and (E.1), (27) can be obtained directly.

Acknowledgments

This work was partially supported by the National Science Council (Grant no: NSC-102-2623-E-001-001-ET) of Taiwan, Republic of China. The author is grateful to the editor, associate editor, and referees for their help with this paper.