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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 732875, 12 pages
The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense
Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan
Received 16 April 2013; Accepted 9 July 2013
Academic Editor: Nicola Guglielmi
Copyright © 2013 Chien-Yu Peng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  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 is given by , where denotes the standard Wiener process; and are the random effects. If the product’s lifetime can be defined as the first-passage time when crosses a predefined threshold level , that is, , then the product’s lifetime on the given and is an inverse Gaussian distribution (denoted by ). Hence, the most common measure associated with a lifetime distribution, the first moment, can be expressed as 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 , O’Hagan and Forster , 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 is said to have the skew-normal distribution and is denoted by if its density function is given by where and are the pdf and the cdf of the standard normal distribution and the parameters , , and regulate the location, scale, and skewness, respectively. A systematic treatment of the skew-normal distribution has been developed by Azzalini . Furthermore, Azzalini and Capitanio  defined a skewed variate as the scale mixture where , is distributed independently of , according to a chi-squared distribution with degrees of freedom , and is denoted by . Note that the skew- distribution approaches the skew-normal distribution as ; that is, , where is the pdf of and is defined in (2). Since the skew-normal, skew-Cauchy, and Student’s -distributions were included in the skew- distribution, they have proved themselves quite adequate for modeling real data sets (refer to ).
In the previously-mentioned lifetime example, we can assume the prior distribution of the random effect to be a skew- distribution for adequately realistic situations. The same approach for the random effects in regression models has been presented in Azzalini and Capitanio . Then, the product’s mean lifetime in (1) turns out to be the FNM of a skew- distribution. However, by the result in Piegorsch and Casella , the FNM of a skew- 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 , 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 to be continuous on the real line except for a finite number of points . Then, we will call the following the principal value for an improper integral that exists if and only if for every , , 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  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 . 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- distribution.
The remainder of this paper is organized as follows. The exact form of the PV-FNM of skew- and generalized Student’s -distributions is derived in Section 2. Section 3 provides some applications of the PV-FNM of skew- and generalized Student’s -distributions. Section 4 contains conclusions.
The PV-FNM of a skew- distribution does exist by giving a scale mixture of a skew-normal distribution. In the following, we start with a simple case of skew- distribution and then increase the complexity of the distributions. Note that the notation 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 , then
In what follows, we deal with the PV-FNM of a skew- distribution which is more complicated in calculation. Note that to avoid “reverse” sum (product) in the following formula, we define and for .
Theorem 2. If , then one obtains for odd values of and for even values of where
Note that if is defined in (2), then the nonexistence of the PV-FNM of pointed out by Peng , where . Unfortunately, the PV-FNM of a skew- distribution, , fails to exist for all when . When , the formula of Theorem 2 reduces to that obtained in the skew-Cauchy case as follows.
Corollary 3. If in Theorem 2, one lets , then and
Corollary 4. If in Theorem 2, one lets , then
As shown before, the formula of the PV-FNM of a skew- distribution becomes complicated and difficult to calculate due to the nonzero location parameter in the integral. However, if in Theorem 2, the complicated formula can be simplified significantly as the following results show.
Corollary 5. If in Theorem 2, one lets , then which is an increasing function in .
Moreover, by using the property of the Gamma function, that is, as , we have the following result immediately for : which coincides with the fact given by Peng .
The PV-FNM of a generalized Student’s -distribution can be obtained as follows. The generalized Student’s -distribution () is defined as a scale mixture of the Kotz distribution and an inverse gamma distribution. That is, where and , and is independent of . The Kotz distribution () introduced in Kotz  is given by where and . The inverse gamma distribution () is given by where and . For , , , and , we obtain the distribution. Setting , , and yields the generalized version of the distributions family defined by Arellano-Valle and Bolfarine . The case and gives the Kotz distribution. Setting , , and yields the normal distribution.
Theorem 6. If , then where .
To show the applicability and effectiveness of the main results, some applications are presented in the following section.
One special property of the Student’s -distribution is now obtained from Lemma 1.
Corollary 7. If in Lemma 1, one lets , then and
It is of interest to note that if , then only exists for all when . That is, Moreover, it is well known that the random variable still follows the Cauchy distribution with location parameter and scale parameter . Therefore, assume that , The moment estimators in the principal value sense can be obtained by (19) and (20) as 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 for all real . 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 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, where a (Gaussian) hypergeometric function is given by
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 then a confluent hypergeometric function can be shown to be a limiting case of a hypergeometric function as follows.
Corollary 9. Consider the following:
Note that we also double-checked the correctness of our results in this study via numerical evaluations with MathCAD mathematical software.
The primary objective considered in this study is to derive the exact form of the PV-FNM of a skew- 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- 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- 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. ). This brings difficulty to inference close to the singularity point. Using the principal value on inferring this issue is a challenge worthy of research.
A. Proof of Lemma 1
Proof. It is well known that . Hence, by Theorem 2 in Peng  and using the PV-FNM of the normal distribution showed in the appendix of Quenouille , we obtain For odd degrees of freedom, making use of formula 2.271 in Gradshteyn and Ryzhik , we have For even degrees of freedom, by using formulas 2.171 and 2.124 in Gradshteyn and Ryzhik , we get 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 , , and , one gets
Proof. By using the change of variable , the binomial theorem, and term-by-term integration, the original integral can be expressed as Now, applying formula 2.729 in Gradshteyn and Ryzhik , the result thus follows.
Lemma 11. For , , , and , , one gets
Proof. Using the change of variable , the binomial theorem, and term-by-term integration, we get where and . By using formulas 1.2.11 and 1.2.10 in Prudnikov et al. , one can obtain 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  as follows. If , then
By using (B.7), the definition in (3), and the conditional expectation, we obtain
Now, can be calculated as (18) from Lemma 1. By Fubini’s theorem, turns out to be
The evaluation of this integral is divided into two parts.
(i) For odd degrees of freedom, by using formulas 2.171 and 2.124 in Gradshteyn and Ryzhik , we get where in which . Therefore, the change of variable implies Then, using formula 1.2.7 in Prudnikov et al. , one can obtain as (8). From Lemma 10, let and ; then can be expressed as (9).
(ii) For even degrees of freedom, making use of formula 2.271 in Gradshteyn and Ryzhik , can be expanded as where Let and in Lemma 11; then can be worked out as (10).
Finally, is easy to calculate as 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 , where is integer, then the PV-FNM of a Kotz distribution is given by where .
Proof. Let and ; then where Then, using binomial theorem and term-by-term integration, we get where Following the same argument in the appendix of Quenouille , can be derived as By the property of a normal distribution, we obtain 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 . Hence, using Lemma 12, we obtain
Now, can be evaluated as
The evaluation of this integral is divided into two parts.
(i) For , making use of formula 2.271 in Gradshteyn and Ryzhik , we have
(ii) For , by using formulas 2.171 , 2.103 in Gradshteyn and Ryzhik  and after some algebraic manipulations, we get respectively. Finally, is easy to evaluate as Consequently, combining (C.8)–(C.15) yields the desired result.
D. Proof of Corollary 8
Proof. Since a Student’s -distribution approaches a normal distribution as , we have , where and . In addition, Quenouille  showed that Hence, by restricting limit to odd degrees in Lemma 1 and using L’Hospital rule, we have where . Comparing (D.1) and (D.2), one can obtain 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, By using formula 9.212 in Gradshteyn and Ryzhik  and comparing (24) in Corollary 8 and (E.1), (27) can be obtained directly.
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.
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