Truncated-Exponential Skew-Symmetric Distributions

Transcription

1 Truncated-Exponential Skew-Symmetric Distributions Saralees Nadarajah, Vahid Nassiri & Adel Mohammadpour First version: 15 December 2009 Research Report No. 19, 2009, Probability and Statistics Group School of Mathematics, The University of Manchester

2 Truncated exponential skew symmetric distributions by Saralees Nadarajah School of Mathematics University of Manchester Manchester M13 9PL, UK Vahid Nassiri and Adel Mohammadpour Department of Statistics Amirkabir University of Technology (Tehran Polytechnic) Tehran 15914, IRAN Abstract: The family of skew distributions introduced by Azzalini and extended by others has received widespread attention. However, it suffers from complicated inference procedures. In this paper, a new family of skew distributions that overcomes the difficulties is introduced. This new family belongs to the exponential family. Many properties of this family are studied, inference procedures developed and simulation studies performed to assess the procedures. Some particular cases of this family, evidence of its flexibility and a real data application are presented. At least ten advantages of the new family over Azzalini s distributions are established. Keywords: Azzalini skew distributions; Estimation; Exponential family; Skewness. 1 Introduction The need for skew distributions arises in every area of the sciences, engineering and medicine. The most common approach for the construction of skew distributions is to introduce skewness into known symmetric distributions. Ferreira and Steel (2006) presented a unified approach for constructing such distributions. Let X be a symmetric random variable about zero with f X ( ) and F X ( ) denoting its probability density function (pdf) and cumulative distribution function (cdf), respectively. Define Y as a new random variable with the pdf f Y (y) = f X (y)ω (F X (y)), y R, (1) where ω( ) is a pdf on the unit interval (0,1). Then Y is said to be a skew version of the symmetric random variable X (Definition 1, Ferreira and Steel, 2006). The unified family in (1) contains many of the known families of skew distributions. If ω( ) is a beta pdf then (1) yields the family of distributions studied by Jones (2004). If ω(x) = 2/( 2 + 1

3 1)f X ( sign(1/2 x) F 1 X (x))/f X(F 1 X (x)), 0 < < then (1) yields the family of distributions studied by Fernández and Steel (1998). The most popular version of (1) are the skew distributions introduced by Azzalini (1985). Take ω(x) = 2F X (F 1 X (x)). Then (1) reduces to f Y (y) = 2f X (y)f X (y), y R, R. (2) We shall refer to distributions with the pdf (2) as Azzalini skew distributions. A particular case of (2) is the class of skew normal distributions obtained by setting f X ( ) = φ( ), the standard normal pdf, and F X ( ) = Φ( ), the standard normal cdf. The family given by (2) and the skew normal class have been studied and extended by many authors, see Azzalini (1986), Azzalini and Dalla Valle (1996), Azzalini and Capitanio (1999), Arnold and Beaver (2000), Pewsey (2000), Loperfido (2001), Arnold and Beaver (2002), Nadarajah and Kotz (2003), Gupta and Gupta (2004), Behboodian et al. (2006), Nadarajah and Kotz (2006), Huang and Chen (2007) and Sharafi and Behboodian (2007). In this paper, we shall focus on the family of Azzalini skew distributions. One of the main difficulties of this family is about making inferences on its skewness parameter. It is known, for example, that the maximum likelihood estimator for for Azzalini skew normal distributions does not always exist, see Pewsey (2000). The aim of this paper is to introduce a new family of distributions as a competitor to (2). This new family belongs to the exponential family, so inferences on the skewness parameter become much easier. Its pdf is a particular case of (1) with ω( ) corresponding to a truncated exponential distribution: this particular choice is made because it yields a natural extension of (2) to an exponential family. We shall refer to the new family as truncated exponential skew symmetric distributions. We establish at least ten advantages of the new family over (2). The contents of the rest of this paper are organized as follows. In Section 2 we introduce the new family of distributions and study its mathematical properties. Section 3 provides inference procedures for maximum likelihood estimation, moments estimation, hypotheses testing and simulation. Some particular cases and evidence of flexibility of the new family are presented in Section 4. A simulation study is performed in Section 5 to compare the performances of the methods of maximum likelihood and moments. A real data application to illustrate the usefulness of the new family is presented in Section 6. Finally, some conclusions including a list of ten advantages of the proposed family are noted in Section 7. 2 Truncated exponential skew symmetric distributions and their properties In this section we introduce the truncated exponential skew symmetric random variable and study its properties. 2

4 Definition 1. A random variable Y has the truncated exponential skew symmetric distribution with parameter, TESS(), if its pdf has the following form: f Y (y;) = 1 exp( ) f X(y)exp { F X (y)}, y R, R, (3) where f X ( ) and F X ( ) are, respectively, the pdf and the cdf of a symmetric random variable X about zero. We shall refer to in (3) as the shape parameter. From (3) an explicit expression for the cdf of Y is obtained as: The inverse cdf is: F 1 Y F Y (y;) = 1 exp { F X(y)}, y R, R. (4) 1 exp( ) (y;) = F 1 ( (1/)ln {1 y (1 exp( ))}), y R, R. (5) X We can use (5) for several purposes, e.g. for finding quantiles or generating random numbers. If P(Y q p ) = p, then the pth quantile, q p, can be obtained using (5). We can also use (5) and the inversion method to generate a random sample, Y 1,...,Y n, from TESS(). Note that (3) is a particular case of (1) for ω(x) = exp( x)/{1 exp( )}, a truncated exponential pdf. By introducing the exponential function and replacing F X (y) by F X (y), we have in (3) a natural extension of (2) to an exponential family. Note also that (3) is symmetric with respect to in the sense that f(y;) = f( y; ). Furthermore, in the limit, as 0, Y TESS() has the same distribution as X. Note that (3) is undefined at = 0, so = 0 should be interpreted as the limit 0. If ± then Y TESS() reduces to degenerate random variables. If then F Y (y) = 0 if F X (y) = 0 and F Y (y) = 1 for all other values of y. If then F Y (y) = 1 if F X (y) = 1 and F Y (y) = 0 for all other values of y. The shape parameter,, in (3) satisfies the first three ordering properties of van Zwet (1964). In particular, if is considered as a function of Y, then: 1) (ay + b) = (Y ) for all a > 0 and b R; 2) (Y ) = 0 for symmetric Y ; 3) ( Y ) = (Y ). The fourth and the last ordering of van Zwet (1964) is: suppose 1 (Y ) 2 (Y ) and let f i, F i and Fi 1 denote the pdf, the cdf and the inverse cdf corresponding to i (Y ), i = 1,2; then 1 (Y ) is said to be smaller than 2 (Y ) in convex order if and only if F2 1 (F 1 (y)) is convex in y or equivalently f 1 (F1 1 (u))/f 2 (F2 1 (u)) is increasing in u [0, 1]. Unfortunately, this ordering property does not appear to be satisfied by (3): see Figure 1 showing the plot of f 1 (F1 1 (u))/f 2 (F2 1 (u)) versus u when X is a standard normal random variable, 1 = 10 and 2 = 5. [Figure 1 about here.] The modes of (3) are the roots of the equation f X (y) f X (y) = f X(y). (6) 3

5 Note that the roots are to the left (respectively, right) of zero as > 0 (respectively, < 0). The root, say y = y 0, corresponds to a maximum if The root corresponds to a minimum if The root corresponds to a point of inflexion if ( ) f X (y 0) f X (y 0 ) f X (y f 2 X 0) < (y 0). (7) f X (y 0 ) ( ) f X (y 0) f X (y 0 ) f X(y f 2 X 0 ) > (y 0). (8) f X (y 0 ) ( ) f X (y 0) f X (y 0 ) f X(y f 2 X 0 ) = (y 0). (9) f X (y 0 ) The mode corresponding to a maximum is unique if the y 0 is such that f X (y) > f2 X (y) for all y < y 0 and f X (y) < f2 X (y) for all y > y 0. The mode corresponding to a minimum is unique if the y 0 is such that f X (y) < f2 X (y) for all y < y 0 and f X (y) > f2 X (y) for all y > y 0. The mode corresponding to a point of inflexion is unique if the y 0 is such that either f X (y) < f2 X (y) for all y y 0 or f X (y) > f2 X (y) for all y y 0. Note that (3) can be multi modal if, for example, f X ( ) is multi modal. Note that the tails of Y TESS() have the same behavior as the tails of X because f Y (y) /{exp() 1}f X (y) as y and f Y (y) /{1 exp( )}f X (y) as y. Also 1 F Y (y) /{exp() 1}{1 F X (y)} as y and F Y (y) /{1 exp( )}F X (y) as y. Let Y TESS() and T = F X (Y ). Then it is straightforward to show that F T (t) = 1 exp( t), t [0,1], R, 1 exp( ) and f T (t) = exp( t) 1 exp( ), t [0,1], R M T (s) = 1 exp(s ) s 1 exp( ), R, where M T (s) is the moment generating function of T. In particular, E(T) = 1 exp( )( + 1), R. (10) {1 exp( )} Note that the pdf of T is the same as the ω( ) chosen to construct (3). 4

6 Entropies are measures of variation of the uncertainty. Let s Y and s X denote Shannon entropies (Shannon, 1951) corresponding to f Y ( ) and f X ( ), respectively. We have by (10): where s Y = ln 1 exp( ) I = + 1 exp( )( + 1) 1 exp( ) ln f X (y)f X (y)exp { F X (y)} dy. By the series expansion for exponential, one can express where I k = I = s X + ( 1) k k I k, k! k=1 ln f X (y)f X (y)f k X(y)dy. I, (11) 1 exp( ) Let r Y (γ) and r X (γ) denote Rényi entropies (Rényi, 1961) corresponding to f Y ( ) and f X ( ), respectively. Similar calculations using the series expansion for exponential show that [ 1 r Y (γ) = 1 γ ln γ ] {1 exp( )} γ f γ X (y)exp { γf X(y)} dy [ 1 = 1 γ ln γ ( γ) k ] {1 exp( )} γ f γ X k! (y)f X k (y)dy k=0 { [ ]} 1 = 1 γ ln γ ( γ) k {1 exp( )} γ exp {(1 γ)r X (γ)} + J k, (12) k! where J k = f γ X (y)f X(y)F k X(y)dy. The Shannon entropy in (11) can also be obtained as the limit of (12) as γ 1. Theorems 1 and 2 consider the moments of Y TESS(). Theorem 1. Let Y TESS(). If E X r, r > 0, exists, then E Y r exists. k=1 Proof. We have E Y r = = = + y r 1 exp( ) f X(y)exp { F X (y)} dy + y r exp { F X (y)} f X (y)dy 1 exp( ) 1 exp( ) E [ X r exp { F X (X)}]. The result follows by noting that E[ X r exp{ F X (X)}] max(1,exp( ))E[ X r ]. 5

7 Theorem 2. Let X k:n denote the kth order statistic for a random sample of size n from f X ( ). Let Y T ESS(). If the conditions of Theorem 1 hold then E (Y r ) = 1 exp( ) k=0 ( ) k (k + 1)! E ( Xk+1:k+1 r ). Proof. By the series expansion for exponential, one can express E (Y r ) = = = 1 exp( ) 1 exp( ) 1 exp( ) y r f X (y)exp { F X (y)} dy ( ) k k=0 k=0 k! y r f X (y)f k X (y)dy ( ) k 1 k! (k + 1) E ( Xk+1:k+1 r ) So, the result follows. Let M k:n (t) = E[exp(tX k:n )] and φ k:n (t) = E[exp(itX k:n )] denote, respectively, the moment generating function (mgf) and the characteristic function (chf) of X k:n, where i = 1. It then follows from Theorem 2 that the mgf and the chf of Y TESS() are E[exp(tY )] = 1 exp( ) k=0 ( ) k (k + 1)! M k+1:k+1(t) and E[exp(itY )] = 1 exp( ) k=0 ( ) k (k + 1)! φ k+1:k+1(t), respectively. Let Y k:n denote the kth order statistic for a random sample of size n from f Y ( ). Write Y = Y () when Y TESS(). Then the pdf and the cdf of Y k:n can be expressed as f Yk:n (y) = n! {1 exp( )} n k 1 (k 1)!(n k)! n k l=0 m=0 ( )( ) k 1 n k ( 1) l+n k m l m l + m + 1 exp { (n k m)} [1 exp { (l + m + 1)}]f Y ((l+m+1)) (y) and F Yk:n (y) = n! {1 exp( )} n k 1 (k 1)!(n k)! n k l=0 m=0 ( )( ) k 1 n k ( 1) l+n k m l m l + m + 1 exp { (n k m)}[1 exp { (l + m + 1)}]F Y ((l+m+1)) (y), 6

8 respectively. Also the rth moment, the mgf and the chf of Y k:n can be expressed as E (Y r k:n ) = n! {1 exp( )} n k 1 (k 1)!(n k)! n k l=0 m=0 ( )( ) k 1 n k ( 1) l+n k m l m l + m + 1 exp { (n k m)} [1 exp { (l + m + 1)}] E (Y r ((l + m + 1))), E [exp (ty k:n )] = n! {1 exp( )} n k 1 (k 1)!(n k)! n k l=0 m=0 ( )( ) k 1 n k ( 1) l+n k m l m l + m + 1 exp { (n k m)}[1 exp { (l + m + 1)}]M Y ((l+m+1)) (t) and E [exp (ity k:n )] = n! {1 exp( )} n k 1 (k 1)!(n k)! n k l=0 m=0 ( )( ) k 1 n k ( 1) l+n k m l m l + m + 1 exp { (n k m)}[1 exp { (l + m + 1)}]φ Y ((l+m+1)) (t), respectively. In particular, the rth L-moment (due to Hoskings (1990)) of Y TESS() can be expressed as r 1 ( )( ) r 1 r 1 + j r = ( 1) r 1 j β j, j j j=0 where β r = (1/(r+1))E(Y r+1:r+1 ). The L moments have several advantages over ordinary moments: for example, they apply for any distribution having finite mean; no higher-order moments need be finite (Hoskings, 1990). 3 Inference In this section we draw some inferences for a truncated exponential skew symmetric random variable with additional location and scale parameters. Definition 2. A random variable Y has the truncated exponential skew symmetric distribution with parameters (, κ, δ), TESS(,κ,δ), if its pdf has the following form: ( ) { ( )} y κ y κ f Y (y;,κ,δ) = δ {1 exp( )} f X exp F X δ δ for y R, R, κ R and δ > 0, where f X ( ) and F X ( ) are, respectively, the pdf and the cdf of a symmetric random variable X about zero. (13) Note that the TESS pdf in (13) can be written as h(y)c()exp{w()t(y)}, where h(y) = f X ((y κ)/δ) 0, c() = /{δ(1 exp( ))} > 0, w() = ( R) and t(y) = F X ((y κ)/δ) (y R). 7

9 So, the pdf belongs to the exponential family with respect to (see Lehmann and Casella (1998), page 23) and n i=1 F X(Z i ), where Z i = (Y i κ)/δ, is a complete sufficient statistic for provided that κ and δ are assumed known. For estimating (, κ, δ), we find their moments and maximum likelihood estimators. Suppose y 1,y 2,...,y n is a random sample from (13). Let m k = (1/n) n j=1 yk j for k = 1,2,3. By equating the theoretical moments of (13) with the sample moments, we obtain the equations: 1 exp( ) k l=0 ( ) k κ k l δ l ( ) m ( ) l (m + 1)! E Xm+1:m+1 l = m k m=0 for k = 1,2,3. The moments estimators are the simultaneous solutions of these three equations. Now consider estimation by the method of maximum likelihood. The likelihood function of the three parameters is ( ) { n n } { L(,κ,δ) = f X (z i ) exp δ(1 exp( )) i=1 } n F X (z i ), where z i = (y i κ)/δ. So, the maximum likelihood estimators are the simultaneous solutions of the equations ( ) 1 n 1 = exp() 1 i=1 n F X (z i ), (14) i=1 and where z i = (y i κ)/δ. n f X (z i) n f X (z i ) = f X (z i ) (15) i=1 i=1 n f X z (z i) n i f X (z i ) = z i f X (z i ) n, (16) i=1 i=1 Theorem 3 shows that (14) always has a root and is unique. The proof of this theorem requires the following lemma which is straightforward. Lemma 1. Consider g(x) = 1/x 1/{exp(x) 1}. Then g(x) is strictly decreasing and g(x) (0, 1). Theorem 3. If κ and δ are assumed known then the maximum likelihood estimator of given by (14) always exists and is unique. Proof. Note that (14) can be re expressed as ng() r(z) = 0, where r(z) = n i=1 F X(z i ). By Lemma 1, ng() is strictly decreasing and lies in (0,n). So, ng() r(z) is strictly decreasing and lies in ( r(z),n r(z)). Note that n r(z) > 0 since r(z) < n. So, the theorem follows by the intermediate value theorem (see Rudin (1976), page 93). 8

10 The elements of the Fisher information matrix of the maximum likelihood estimators can be calculated as: E and E ( ) 2 ln L κ δ ( ) E 2 ln L δ 2 ( ) E 2 ln L 2 = n 2 n exp() {exp() 1} 2, E ( ) [ ] 2 ln L κ 2 = nδ f 2E X (Z) f X (Z) ( ) E 2 ln L = n κ δ E [f X(Z)], ( ) 2 ln L = n δ δ E [Zf X(Z)], ( + n δ 2E f X (Z) f X (Z) ) 2 + n δ 2 E [ f X (Z) ], [ ] ] ( ) = n f δ 2E X (Z) [Z nδ2e f X (Z) + n f 2 f X (Z) f X (Z) δ 2E X Z (Z) + n f X (Z) δ 2 E [f X(Z)] + n [ ] δ 2 E Zf X(Z) [ ] [ = n δ 2 2n δ 2 E Z f X (Z) nδ2 f X (Z) E Z ] ( ) X (Z) + n f X (Z) δ 2 E Z 2 f 2 X (Z) f X (Z) 2f + 2n δ 2 E [Zf X(Z)] + n δ 2 E [ Z 2 f X (Z) ], where Z = (Y κ)/δ and Y TESS(,κ,δ). It follows that the standard error for has a closed form if κ and δ are assumed known. In general, the elements of the Fisher information matrix will have to be computed numerically. If = 0 then the last three elements reduce to ( ) [ ] E 2 ln L κ 2 = nδ f 2E X (Z) f X (Z) and ( + n δ 2 E f X (Z) f X (Z) ) 2 ( ) [ ] [ ] ( ) E 2 ln L = nδ f κ δ 2E X (Z) nδ2e Z f X (Z) + n f X (Z) f X (Z) δ 2 E f 2 X Z (Z) f X (Z) ( ) E 2 ln L δ 2 = n δ 2 2n δ 2 E [ ] Z f X (Z) [Z nδ2e f X (Z) 2f, ] ( X (Z) + n f X (Z) δ 2E Z 2 f X (Z) f X (Z) ) 2, 9

11 where Z = (Y κ)/δ and Y T ESS(0, κ, δ). If in addition X is standard normal then ( ) E 2 ln L κ 2 = n δ 2, and where Z = (Y κ)/δ and Y TESS(0,κ,δ). ( ) E 2 ln L = 2n κ δ δ 2 E [Z] ( ) E 2 ln L δ 2 = n { [ 3E Z 2 ] δ 2 1 }, We noted earlier that T = n i=1 F X(Z i ) is a sufficient statistic for provided that κ and δ are assumed known. It can be noted further that the TESS family has the monotone likelihood ratio (MLR) property with respect to T. So, by using Karlin Rubin s theorem, one can find the uniformly most powerful (UMP) level (size) α test for testing H 0 : 0 versus H 1 : > 0, i.e. if ϕ(x) = { 1, if T t 0, 0, if T < t 0 then ϕ(x) is a UMP size α test provided that κ and δ are known, where P(T t 0 ) = α. By (5), t 0 = (1/)ln{1 (1 α)(1 exp( ))} for the case n = 1. For testing H 0 : F = F 0, where F 0 is a known TESS cdf, against H 1 : F F 0, one can use nonparametric goodness of fit tests such as ones based on the chi-square test or the Kolmogrov Smirnov test. These tests can still be used if F 0 is unknown and estimated from some data. In this case, the critical values for the Kolmogrov Smirnov test can be obtained by simulation. Since the TESS cdf has a closed form, performing these tests is straightforward. A quantile quantile plot or a probability probability plot can also be used as informal checks for H 0 : F = F 0. Finally, consider simulating truncated exponential skew symmetric variates. As mentioned in Section 2, the inversion method can be applied since the inverse cdf of TESS() is given by (5). Another method for simulation is the rejection method. If > 0 then the following scheme holds: 1. simulate X = x from the pdf f X ( ); 2. simulate Y = UMf X (x), where U is a uniform variate on the unit interval [0,1] and M = /{1 exp( )}; 3. accept X = x as a TESS() variate if Y < f Y (x). If Y f Y (x) return to step 2. If < 0 then apply the above scheme with M = /{1 exp()} and take the negatives of the simulated variates. 10

12 4 Some particular cases In this section we study some particular cases of truncated exponential skew symmetric distributions. Here, we consider normal, t and Cauchy cases (Cauchy is a particular case of t and normal is a limiting case of t). As Nadarajah and Kotz (2006) did, some other distributions such as Laplace, logistic and uniform can also be studied. If f X ( ) = φ( ) and F X ( ) = Φ( ), in Definition 1, then (3) yields the pdf: f Y (y) = φ(y)exp { Φ(y)}, y R, R. (17) 1 exp( ) The corresponding Azzalini s distribution has the pdf f Y (y) = 2φ(y)Φ(y), y R, R. (18) We shall refer to (17) as the truncated exponential skew normal distribution with parameter, ES normal(). It follows by Theorem 1 that E Y r exists for all r > 0. So, by Theorem 2 and the results in Nadarajah (2007a), E (Y r ) = 2 r/2 π {1 exp( )} k=0 ( ) k 2 k k! k p = 0 p + r even ( F (p) p + r + 1 A ; 1 2 2,..., 1 2 ; 3 2,..., 3 ) 2 ; 1,..., 1, ( ) ( ) k p + r + 1 π p/2 2 p Γ p 2 where F (n) A ( ) denotes the Lauricella function of type A (Exton, 1978) defined by = F (n) A (a,b 1,...,b n ;c 1,...,c n ;x 1,...,x n ) (a) m1 + +m n (b 1 ) m1 (b n ) mn x m1 1 x mn n (c 1 ) m1 (c n ) mn m 1! m n!, m 1 =0 m n=0 where (f) k = f(f + 1) (f + k 1) denotes the ascending factorial with the convention (f) 0 = 1. One can show using equations (6) (9) that (17) has a unique mode corresponding to a maximum at y 0, the unique root of φ(y) + y = 0. If f X ( ) and F X ( ) in Definition 1 are the Student s t pdf and the Student s t cdf with ν degrees of freedom, respectively, then (3) yields the pdf: ( ) (1+ν)/2 1 f Y (y) = 1 + y2 exp { F X (y)}, 1 exp( ) νb(ν/2,1/2) ν y R, R. (19) where B(a,b) = Γ(a)Γ(b)/Γ(a + b) is the beta function. The corresponding Azzalini s distribution has the pdf f Y (y) = ( ) (1+ν)/ y2 F X (y), y R, R. (20) νb(ν/2,1/2) ν 11

13 For general ν, the cdf term, F X (y), takes the form F X (y) = 1 ( yγ ((ν + 1)/2) 1 + 2F 1 2 πνγ (ν/2) 2, ν ; 3 ) 2 ; y2, (21) ν where 2 F 1 ( ) denotes the Gauss hypergeometric function defined by 2F 1 (a,b;c;x) = If ν is an integer then one can simplify (21) to: ( ) y π arctan ν + 1 2π F X (y) = ) π ν/2 i=1 ( B i 1 2, 1 2 k=0 (ν 1)/2 i=1 (a) k (b) k (c) k x k k!. ( B i, 1 ) ν i 1/2 y 2 (ν + y 2 ) i, for ν odd, ν i 1 y (ν + y 2 ) i 1/2, for ν even, see Nadarajah and Kotz (2003). We shall refer to (19) as the truncated exponential skew t(ν) distribution with ν degrees of freedom and parameter, ES t(, ν). It follows from Theorem 1 that E Y r exists for r < ν. So, if r < ν then by Theorem 2 and the results in Nadarajah (2007b), E (Y r ) = 1 exp( ) k=0 ( ) k (k + 1)! {A(r,k + 1,k + 1) + ( 1)r A(r,k + 1,1)}, where A(k,n,r) = n!ν k/2 r 1 n r ( )( ) r 1 n r 2 n ( 1) q 2 p+q (r 1)!(n r)! p q p=0 q=0 ( ν k B 1 p q (1/2,ν/2)B, k + p + q + 1 ) 2 2 ( (k ) ( + p + q + 1 F1:1 1:2 : 1 ν 2 2, 1 ) ( ;...; 1 ν 2 2, 1 ) ; 2 ( ) ( ) ( ) ) ν + p + q : ;... ; ;1,...,1, where FC:D A:B ( ) denotes the generalized Kampé de Fériet function (Exton, 1978) defined by = F A:B C:D ((a) : (b 1);... ;(b n );(c) : (d 1 );... ;(d n );x 1,...,x n ) ((a)) m1 + +m n ((b 1 )) m1 ((b n )) mn x m1 1 x mn n ((c)) m1 + +m n ((d 1 )) m1 ((d n )) mn m 1! m n!, m 1 =0 m n=0 where a = (a 1,a 2,...,a A ), b i = (b i,1,b i,2,...,b i,b ) for i = 1,2,...,n, c = (c 1,c 2,...,c C ), d i = (d i,1,d i,2,...,d i,d ) for i = 1,2,...,n, and ((f)) k = ((f 1,f 2,...,f p )) k = (f 1 ) k (f 2 ) k (f p ) k denotes the product of ascending factorials. The mode of (19) is the root of the equation y(1+y 2 /ν) (ν 1)/2 = ν/{(1 + ν)b(ν/2,1/2)}. It is well known that the asymptotic distribution of Student s t as 12

14 ν is standard normal, see e.g. Johnson et al. (1995, page 363). A similar result can be stated for ES t(,ν): if Z ES normal(), Y ES t(,ν) and ν then Z and Y are identically distributed. A particular case of (19) with interesting properties is: { 1 f Y (y) = 1 exp( ) π(1 + y 2 ) exp ( π arctan(y) )}, y R, R. (22) We shall refer to (22) as the truncated exponential skew Cauchy distribution, ES Cauchy. Note that (22) is a particular case of the Pearson type IV distribution, so the ES Cauchy is a Pearson type IV distribution. Equations (4) and (5) for the ES Cauchy have closed forms. ES Cauchy has the heaviest tails within the class of distributions ES t(,ν) if ν is limited to be an integer. It follows by Theorem 1 that E Y r does not exist for all r 1. Furthermore, one can show using equations (6) (9) that (22) has a unique mode corresponding to a maximum at y 0 = /(2π). [Figures 2 to 5 about here.] Figure 2 to 5 show how the skewness and kurtosis measures of (19) compare to those of (20) for ν = 5,10,20, and = 100, 99,...,99,100 (note that (19) and (20) reduce to (17) and (18), respectively, in the case ν = ). The formulas used to compute the skewness and kurtosis measures are: Skewness(Y ) = E ( Y 3) 3E(Y )E ( Y 2) + 2E 3 (Y ) { E ( Y 2 ) E 2 (Y ) } 3/2 and Kurtosis(Y ) = E ( Y 4) 4E(Y )E ( Y 3) + 6E ( Y 2) E 2 (Y ) 3E 4 (Y ) { E ( Y 2 ) E 2 (Y ) } 2. It is evident from the figures that the truncated exponential skew distributions take a wider range of values for both skewness and kurtosis. The two curves for skewness in Figures 4 and 5 appear to approach the same limit as ± : this was verified by redrawing the figures over a wider range of values. The same comment applies with respect to the two curves for kurtosis in Figure 5. The gain in terms of skewness and kurtosis appears to be greater for the distributions with heavier tails. In the figures, the truncated exponential skew t(5) distribution achieves the greatest gain. This is interesting because heavy tail distributions are becoming increasing popular models for real world data because their tails are more realistic than the normal tails. 5 Simulation study In this section, we compare the performances of the methods of moments and maximum likelihood presented in Section 3. For this purpose, we generated samples of size n = 100 from (13) for 13

15 = 1,1,2,5, κ = 1,0,1,5 and δ = 1,2,5,10 when X is standard normal and Student s t with ν = 5, 10, 20. For each sample, we computed the moments and maximum likelihood estimates following the procedures described in Section 3. We repeated this process 100 times and computed the average of the estimates (AE) and the mean squared error (MSE). The results are reported in Tables 1 to 4. [Table 1 to 4 about here.] It is clear that maximum likelihood performs consistently better than the moments methods for all values of, κ, δ, for all four distributions and with respect to the AE and MSE. This is expected of course. The observations were similar for other values of, κ, δ. 6 Application The aim of this section is to illustrate the usefulness of (3) over (2) using some real data sets. We use the annual maximum daily rainfall data for the years from 1907 to 2000 for fourteen locations in west central Florida: Clermont, Brooksville, Orlando, Bartow, Avon Park, Arcadia, Kissimmee, Inverness, Plant City, Tarpon Springs, Tampa International Airport, St Leo, Gainesville, and Ocala. The data were obtained from the Department of Meteorology in Tallahassee, Florida. By definition annual maximum daily rainfall is non negative: to have (2) and (3) as possible models and to avoid computational difficulties, we define standardized annual maximum rainfall = (annual maximum rainfall - m)/s, where m and s are the observed mean and standard deviation, respectively. We would like to emphasize that the aim here is not to provide a complete statistical modeling or inferences for the data sets involved. We refer the readers to Nadarajah (2005) for a comprehensive analysis of the data sets used. [Figures 6 to 11 about here.] We fitted location scale variations of Azzalini skew normal and the truncated exponential skew normal distributions to the standardized annual maximum rainfall from each of the fourteen locations. The maximum likelihood procedure described by equations (14) (16) was used. The results for four of the locations are: Clermont with sample size n = 94: (17) yielded ln L = 99.0 with = (42.488), κ = (0.533) and δ = (0.178); (18) yielded ln L = with = (13.484), κ = (0.072) and δ = (0.118). Avon Park with sample size n = 94: (17) yielded ln L = 93.0 with = (42.082), κ = (0.586) and δ = (0.196); (18) yielded ln L = with = (8.083), κ = (0.054) and δ = (0.117). 14

16 Gainesville with sample size n = 94: (17) yielded ln L = with = (24.686), κ = (0.567) and δ = (0.192); (18) yielded ln L = with = (12.449), κ = (0.045) and δ = (0.117). Ocala with sample size n = 94: (17) yielded lnl = 97.6 with = (33.526), κ = (0.582) and δ = (0.197); (18) yielded ln L = with = (82.923), κ = (0.039) and δ = (0.119). The numbers within brackets are the standard errors computed by inverting the expected information matrix, see Section 3. The two fitted models given by (17) and (18) are not nested. So, their comparison should be based on criteria such as the Akaike s information criterion or the Bayesian information criterion. However, the two models have the same number of parameters. In this case, these criteria reduce to the usual likelihood ratio test. Comparing the likelihood values, we see that (17) provides a significantly better fit than (18) for each of the four locations. The results were the same for other locations. The locations Clermont, Avon Park, Gainesville and Ocala are illustrated because they showed the most significant improvements. The conclusion based on the likelihood values can be verified by means of probability probability plots and density plots. A probability probability plot consists of plots of the observed probabilities against the probabilities predicted by the fitted model. For example, for the model given by (17), [1 exp{ Φ((x (j) κ)/ δ)}]/[1 exp( )] was plotted versus (j 0.375)/(n+0.25), j = 1,2,...,n (as recommended by Blom (1958) and Chambers et al. (1983)), where x (j) are the sorted values of the observed standardized annual maximum rainfall. The probability probability plots for the two fitted models and for the four locations are shown in Figures 6 to 9. We can see that the model given by (17) has the points much closer to the diagonal line for each location. A density plot compares the fitted pdfs of the models with the empirical histogram of the observed data. The density plots for the four locations are shown in Figures 10 and 11. Again the fitted pdfs for (17) appear to capture the general pattern of the empirical histograms much better. 7 Conclusions We have introduced a new family of skew distributions as a competitor to the well known Azzalini skew distributions. We have studied various mathematical properties and developed procedures for estimation, hypotheses testing and simulation. We have assessed the performances of the estimation procedures by simulation. We have also studied three particular members of the family and their flexibility and illustrated a real data application. The new family of distributions has several 15

17 advantages over Azzalini skew distributions. Some of these are: 1) it belongs to the exponential family; 2) has closed form expressions for pdf, cdf and quantiles; 3) moments, mgf and the chf can be expressed as a series expansion of those of the original symmetric distribution (see Theorem 2); 4) moments, mgf and the chf of order statistics follow directly from those of the original sample; 5) exhibits the same tail behaviors as those of the original symmetric distribution; 6) the maximum likelihood estimator for the shape parameter always exists and is unique (see Theorem 3); 7) the standard error for the shape parameter has a closed form; 8) admits a uniformly most powerful test for hypotheses about the shape parameter; 9) admits wider range of values for skewness and kurtosis and so more flexibility especially when the tails are heavier; 10) provides better fits to real data sets. The work of this paper can be extended in several directions. One is to compare the truncated exponential skew t distributions with other skew t distributions proposed in the literature (for example, those by Azzalini and Capitanio (2003), Jones and Faddy (2003), Ma and Genton (2004) and Azzalini and Genton (2008)). Another is to perform a simulation study to explore the properties of the maximum likelihood and moments estimates for small, medium and large sample sizes. A third direction is to construct multivariate generalizations of the truncated exponential skew symmetric distributions. These issues are beyond the scope of the present investigation, but we may consider some of these in a future paper. References Arnold, B. C., Beaver, R. J., The skew-cauchy distribution. Statistics and Probability Letters 49, Arnold, B. C., Beaver, R. J., Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion). Test 11, Azzalini, A., A class of distributions include the normal ones. Scandinavian Journal of Statistics 12, Azzalini, A., Further results on a class of distributions which includes the normal ones. Statistica XLVI, Azzalini, A., Capitanio, A., Statistical applications of the multivariate skew normal distribution. Journal of the Royal Statistical Society, B, 61, Azzalini, A., Capitanio, A., Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. Journal of the Royal Statistical Society, B, 65,

18 Azzalini, A., Dalla Valle, A., The multivariate skew-normal distribution. Biometrika 83, Azzalini, A., Genton, M. G., Robust likelihood methods based on the skew-t and related distributions. International Statistical Review Behboodian, J., Jamalizadeh, A., Balakrishnan, N., A new class of skew-cauchy distributions. Statistics and Probability Letters 76, Blom, G., Statistical Estimates and Transformed Beta-variables. John Wiley and Sons, New York. Chambers, J., Cleveland, W., Kleiner, B., Tukey, P., Graphical Methods for Data Analysis. Chapman and Hall, London. Exton, H., Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Halsted Press, New York. Fernández, C., Steel, M. F. J., On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association 93, Ferreira, J. T. A. S., Steel, M. F. J., A constructive representation of univariate skewed distributions. Journal of the American Statistical Association 101, Gupta, R. C., Gupta, R. D., Generalized skew normal model. Test 13, Hoskings, J. R. M., L-moments: analysis and estimation of distribution using linear combinations of order statistics. Journal of the Royal Statistical Society, B, 52, Huang, W. J., Chen H. Y., Generalized skew Cauchy distribution. Statistics and Probability Letters 77, Johnson, N. L., Kotz, S., Balakrishnan, N., Continuous Univariate Distributions, Volume 2, second edition. John Wiley and Sons, New York. Jones, M. C., Families of distributions arising from distributions of order statistics (with discussion). Test 13, Jones, M. C., Faddy, M. J A skew extension of the t distribution, with applications. Journal of the Royal Statistical Society, B, 65, Lehmann, E. L., Casella, G., Theory of Point Estimation, second edition. Springer-Verlag, New York. Loperfido, N., Quadratic forms of skew-normal random vectors. Statistics and Probability Letters 54,

19 Ma, Y., Genton, M. G., Flexible class of skew-symmetric distributions. Scandinavian Journal of Statistics 31, Nadarajah, S., Extremes of daily rainfall in west central Florida. Climatic Change 69, Nadarajah, S., 2007a. Explicit expressions for moments of normal order statistics. Demonstratio Mathematica 40, Nadarajah, S., 2007b. Explicit expressions for moments of t order statistics. Comptes Rendus Mathematique 345, Nadarajah, S., Kotz, S., Skewed distributions generated by the normal kernel. Statistics and Probability Letters 65, Nadarajah, S., Kotz, S., Skew distributions generated from different families. Acta Applicandae Mathematica 91, Pewsey, A., Problems of inference for Azzalini s skew-normal distribution. Journal of Applied Statistics 27, Rényi, A., On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, pp University of California Press, Berkeley. Rudin, W., Principles of Mathematical Analysis, third edition. McGraw-Hill Book Company, New York. Shannon, C. E., Prediction and entropy of printed English. The Bell System Technical Journal 30, Sharafi, M., Behboodian, J., The Balakrishnan skewnormal density. Statistical Papers 49, van Zwet, W. R., Convex transformations of random variables. Volume 7 of Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam. 18

20 Table 1. Comparison of maximum likelihood versus moments estimation for the truncated exponential skew t(5) distribution. κ δ Maximum likelihood Moments AE( ) AE( κ) AE( δ) MSE( ) MSE( κ) MSE( δ) AE( ) AE( κ) AE( δ) MSE( ) MSE( κ) MSE( δ)

21

22 Table 2. Comparison of maximum likelihood versus moments estimation for the truncated exponential skew t(10) distribution. κ δ Maximum likelihood Moments AE( ) AE( κ) AE( δ) MSE( ) MSE( κ) MSE( δ) AE( ) AE( κ) AE( δ) MSE( ) MSE( κ) MSE( δ)

23

24 Table 3. Comparison of maximum likelihood versus moments estimation for the truncated exponential skew t(20) distribution. κ δ Maximum likelihood Moments AE( ) AE( κ) AE( δ) MSE( ) MSE( κ) MSE( δ) AE( ) AE( κ) AE( δ) MSE( ) MSE( κ) MSE( δ)

25