H-Functions and Statistical Distributions

H-Functions and Statistical Distributions

∗

P. N. Rathie†, P.K. Swamee‡, G.G. Matos, M. Coutinho, T.B. Carrijo

Abstract Various new applications of H-function to problems in statistical distributions are given. Normal depth problem in rectangular canal, critical depth problem in a trapezoidal canal and sequent depths problem are treated in Hydraulics. Invertible approximations to chi-square and t distributions are obtained. Approximation to skew-normal is given in terms of H-function. Skew-Levy distribution is defined in terms of H-function. Error graphs are given to show that approximations are very good. Some density function graphs are also given.

1

Introduction

The H-function [see Mathai and Saxena(1978), Srivastava et al(1979), Braaksma(1964)] is defined by · ¯ ¸ ¯ (a , A ), . . . , (an , An ), (an+1 , An+1 ), . . . , (ap , Ap ) m,n Hp,q z ¯¯ 1 1 (1) (b1 , B1 ), . . . , (bm , Bm ), (bm+1 , Bm+1 ), . . . , (bq , Bq ) Qn Qm Z 1 j=1 Γ(1 − aj − Aj s) j=1 Γ(bj + Bj s) Q Q z −s ds = 2πi L qj=m+1 Γ(1 − bj − Bj s) pj=n+1 Γ(aj + Aj s) where (1) i =

√

−1,

(2) z(6= 0) is a complex variable, (3) z s = exp (s (ln |z| + i arg z)), (4) An empty product is interpreted as unity, (5) m, n, p and q are non-negative integres satisfying 0 ≤ n ≤ p, 0 ≤ m ≤ q (both n and m are not zeros), ∗ Invited paper presented at the Annual conference of Bharat Gania Parisad held at Department of Mathematics, Lucknow University, Lucknow, India on December 23, 2008. † Department of Statistics, University of Brasilia, Brasilia, DF, Brazil. [E-mail(Corresponding Author)]: [email protected], [email protected] ‡ Civil Engineering, G 130 Greenword City, Sector 40, Gurgaon Haryana, India 122001. E-mail: [email protected]

1

(6) Aj (j = 1, . . . , p) and Bj (j = 1, . . . , q) are assumed to be positive quantities, (7) aj (j = 1, . . . , p) and bj (j = 1, . . . , q) are complex numbers such that none of the poles of Γ(bj + Bj s) (j = 1, . . . , m) coincide with the poles of Γ(1 − aj − Aj s) (j = 1, . . . , n) i.e. Ak (bh + ν) 6= Bh (ak − λ − 1) for ν, λ = 0, 1, . . ., h = 1, . . . , m, k = 1, . . . , n, (8) The contour L runs from −i∞ to +i∞ such that the poles of Γ(bj + Bj s) (j = 1, . . . , m) lie to the left of L and the poles of Γ(1 − aj − Aj s) (j = 1, . . . , n) lie to the right of L. Such a contour is possible on account of (7) with suitable indentations, if required. In Braaksma (1964), p 278, it has been shown that the H-function makes sense and defines an analytic function of z in the following two cases (i) δ > 0, z 6= 0 where δ=

q X

Bj −

p X

Aj

j=1

j=1

(ii) δ = 0, and 0 < |z| < D−1 where δ=

p Y

A

Aj j /

j=1

q Y

Bj

Bj

j=1

The values of H function does not depend on the choice of L. m Y When the poles of Γ(bj − Bj s) are simple, j=1

µ ¶ bh + v Γ bj − Bj m X ∞ Bh X j6=1h = µ ¶× q Y bh + v h=1 v=0 Γ 1 − bj + Bj Bh j=m+1 µ ¶ n Y bh + v Γ 1 − aj + Aj Bh (−1)v z (bh +v)/Bh j=1 ¶ µ p Y v!Bh bh + v Γ aj − Aj Bh j=n+1 m Y

m,n Hp,q (z)

for z 6= 0 if δ > 0 and for 0 < |z| < D−1 if δ = 0 n Y When the poles of Γ(1 − aj + Aj s) are simple, j=1

2

(2)

µ ¶ 1 − ah + v Γ 1 − a j − Aj m X ∞ Ah X j6=1h µ ¶ × q Y 1 − ah + v h=1 v=0 Γ aj + Aj Ah j=m+1 ¶ µ n Y 1 − ah + v Γ bj + Bj Ah (−1)v z (1−ah +v)/Ah j=1 µ ¶ p Y v!Ah 1 − ah + v Γ 1 − bj − Bj Ah j=n+1 m Y

m,n Hp,q (z) =

(3)

for z 6= 0 if δ < 0 and for |z| > D−1 if δ = 0. Invertible approximations for chi-square and student-t distributions are obtained in terms of H-functions. Approximation to skew-normal is defined through a new density function. Skew-Levy distribution is defined in terms of H-function and three particular cases are given. Error and density function graphs are also given. In certain cases, the H-function can be converted in G-function which can be computed by softwares Mathematica or Maple.

2

Approximation to Chi-Square Distribution

The following approximation to the chi-square distribution is proposed: £ ¡ ¢¤ F (x) = 1 − exp −x1+a b + cxd , x > 0, a, b, c, d > 0,

(4)

where F (x)=probability distribution function; x=generalized Weibull variate; a, b, c, d=unknown constants to be determined. Differentiating Eq.(4), the density function f (x) is £ ¡ ¢¤ f (x) = xa [b(1 + a) + c(1 + a + d)xd ] exp −x1+a b + cxd . 1+a

Taking x

= y, Eq. (4) is rewritten as µ ¶ d 1 1 c y = ln − y 1+ 1+a . b 1−F b

(5)

(6)

Applying Lagrange’s inversion expansion, one gets

y=

· ∞ d X + k + 1) 1 (− cb )k Γ(k 1+a k=0

k!

d Γ(k 1+a + 2)

b

µ ln

1 1−F

¶¸kd/(1+a)+1 , 0 ≤ F ≤ 1.

Rewriting properly Eq.(7), and taking x = y 1/(1+a) , we get

3

(7)

x

½µ ¶ µ ¶ 1 1 = ln × b 1−F 1 " #) 1+a d ¯ ³ c ´ · 1 µ 1 ¶¸ 1+a ¯(0,1+d/(1+a)) 11 H12 ln , ¯(0,1),(−1,d/(1+a)) b b 1−F 0 ≤ F ≤ 1.

2.1

(8)

Moments

The h-th moment is given by E(X h ) = b(1 + a)J(a+h) = c(1 + a + d)J(a+d+h) ,

(9)

where Z

∞

Jα =

xα exp[−x(1+a) (b + cxd )]dx.

(10)

0

Evaluating Eq. (10), one gets

∞ −( 1+α 1+a ) X

b (1 + a)

·

1 + α + (1 + a + d)k (−1)k 1+a

h ik d cb(1+ 1+a )

.

(11)

¯ 1+α · ¸ ¯ b−( 1+a ) 1 1 −(1+d/(1+a)) ¯ (a − α, 1 + d/(1 + a)) Jα = H cb . ¯ (0, 1) (1 + a) 1 1

(12)

Jα =

Γ

¸

k=0

k!

Eq.(11) can be rewritten as

2.2

Numerical Approximation

For a trial set of values of a, b, c and d, the average error Eχ2 between the Chi-Square distribution and the proposed distribution is Eχ2 =

10k ¯ 1 X ¯¯ FEq.(4) (xi ) − Fχ2 (xi )¯ , 10k i=1

(13)

where Fχ2 (xi ) represents the Chi-Square distribution function, FEq.(4) (xi ) the values of the distribution function given by Eq.(4) for xi varied between 0 and k at a regular interval of 0.1 and k is the quantile of the Chi-Square distribution such that P (χ2 < k) = 0.99, given with one decimal place after the comma. The aproximattion with the Chi-Square distribution with 3 and 10 degrees of freedom are given here to explain, but many other cases was given. To the first 4

case, taking a = b = 10−3 , the values of c and d wich give minimum average error Eχ2 = 0.003572 are 0.237 and 0.24, respectively. To the second case, taking a = b = 10−5 , the values of c and d which give minimum average error Eχ2 = 0.012688 are 0.0032 and 1.378. Departure 0.015 0.010 0.005 0.000 5

10

15

20

x

-0.005 -0.010 -0.015

Figure 1: Departure from Chi-Square Distribution with 3 degrees of freedom.

Departure 0.03 0.02 0.01 0.00 5

10

15

20

25

30

x

-0.01 -0.02 -0.03

Figure 2: Departure from Chi-Square Distribution with 10 degrees of freedom.

5

3

Approximation to Student’s t Distribution

The following approximation to the Student’s t distributions is proposed [see Rathie and Swamee (2006)]: F (x) = {exp [−x (a + b|x|p )] + 1}

−1

, x ∈ R, a, b, p > 0.

(14)

where F (x)=probability distribution function; x=generalized logistic variate; a, b, p=unknown constants to be determined. Differentiating Eq.(14), the density function is f (x) =

[a + b(1 + p)|x|p ] exp [−x (a + b|x|p )] 2

{exp [−x (a + b|x|p )] + 1}

.

(15)

Special cases of Eq.(14) are studied in Swamee and Rathie (2007) and Page (1977). Rewritten adequately Eq.(14), we get ¶ µ 1 F b x = ln − xp+1 . (16) a 1−F a Applying Lagrange’s inversion expansion, follows that  ∞ X (− b )k Γ(kp + k + 1) · 1 µ 1 − F ¶¸kp+1   a  ln , F ≤ 0.5  − k! Γ(kp + 2) a F k=0 x= X , · µ ¶¸kp+1 ∞  (− ab )k Γ(kp + k + 1) 1 F   ln , F ≥ 0.5.   k! Γ(kp + 2) a 1−F

(17)

k=0

which can be rewritten as µ ¶ · µ # "µ ¶ · ¶¸ µ ¶¸p ¯ ¯ (0, p + 1)  F b 1 1−F 1  11 ¯  ln H1 2 ln  ¯ (0, 1), (−1, p) ,  a 1−F a a F "µ ¶ · # x = µ ¶· µ ¶¸ µ ¶¸p ¯ ¯ (0, p + 1)  1 F b 1 F  11 ¯  ln H1 2 ln   a ¯ (0, 1), (−1, p) , 1−F a a 1−F

F ≤ 0.5 . F ≥ 0.5. (18)

The results for h-th moments about origin, characteristic function and confidence interval for µ were obtained for generalized logistic distribution (see Rathie and Swamee (2007)).

3.1

Numerical Approximation

√ The value of the Student’s t density function for x =√0 is [β(v/2, 1/2) v]−1 . Putting x = 0 in Eq.(14), we get a = 4[β(v/2, 1/2) v]−1 , where v are the degrees of freedom of Student’s t distribution. For a trial set of values of b and p, the average error Et between the Student’s t distribution and the proposed distribution is 6

Et =

20k+1 X 1 |FEq.14 (xi ) − Ft (xi )| 20k + 1 i=1

(19)

where Ft (xi ) represents the Student’s t distribution function, FEq.14 (xi ) the values of the distribution function given by Eq.(14) for xi varied between −k and k at a regular interval of 0.1 and k is the quantile of the Student’s t distribution such that P (−k < X < k) = 0.98, given with one decimal place after the comma. Similarly to the previous case, only the aproximattion with the Student’s t distribution with 7 and 60 degrees of freedom are given here, but other cases was given. Repeating this process for several sets of values of b and p, in the first case a = 1.539966, b = 0.00848 and p = 0.687 give minimum Et = 0.0002651130. To the other case a = 1.589134, b = 0.0645 and p = 1.915 give minimum Et = 0.000083487. Departure 0.0010

0.0005

0.0000 -6

-4

0

-2

2

4

6

x

-0.0005

-0.0010

Figure 3: Departure from Student’s t Distribution with 7 degrees of freedom.

7

Departure 0.0003 0.0002 0.0001 0.0000 -4

0

-2

2

4

x

-0.0001 -0.0002 -0.0003

Figure 4: Departure from Student’s t Distribution with 60 degrees of freedom.

4

Skew-Normal Approximation

In this section, we will define an invertible approximation to Skew-Normal distribution. To do this we will use the distribution given by Eq.(14) (see Rathie and Swamee (2006)). For the values a=1.59413, b=0.07443 and p=1.939 this distribution is a good approximation to the Normal distribution, with a maximum error of 4.10−4 at x=0 to the probability density function and 7.757 10−5 at x=±2.81 to the cumulative distribution funcion.

4.1

Skew Distributions

We can find skew distributions using the following formula:

h(x) = 2.f (x).G(w(x))

(20)

where f(x) is a symmetric density function, G(x) is a cumulative distribution function of any symmetric density function, and w(x) is an odd function. For the Normal N(0,1) distribution φ(z) , we can make the Skew-Normal distribution (Sn(z)) for w(z)=c.z as follows

Sn(z) = 2.φ(z).Φ(c.z)

(21)

The problem with this distribution is that we can not calculate Φ(c.z) explicitely, so we can not find one compact result for the Skew-Normal. 8

To approximate to the Skew-Normal we are going to consider: f(x)=f(z), G(x)=F(c.z), in Eq.(14), to get:

h(z) = 2.f (z).F (c.z) =

4.2

2.[a + b(1 + p)|z|p ].exp[−z(a + b|z|p )] {exp[−z(a + b|z|p )] + 1}2 .{exp[−cz(a + b|cz|p )] + 1} (22)

Particular case when c=1

Considering c = 1 in Eq.(22) we can find a good approximation to the SkewNormal by:

h(z) = 2.f (z).F (z) =

2.[a + b(1 + p)|z|p ].exp[−z(a + b|z|p )] {exp[−z(a + b|z|p )] + 1}3

(23)

We can easily obtain a cumulative distribuition by making a substitution as follows: H(z) = {exp[−z(a + b|z|p )] + 1}−2

(24)

We can also calculate z in terms of H: √ 1 H b √ ) − z|z|p z = ln( a a 1− H

(25)

Using the Lagranges Inversion Theorem we get: √  X ∞ (− ab )n .Γ(np + n + 1) 1 1 − H np+1   √ − [ ln( )] , H ≤ 0.25   n!Γ(np + 2) a H n=0 √ z= ∞ b n  H  X (− a ) .Γ(np + n + 1) [ 1 ln(  √ )]np+1 , H ≥ 0.25  n!Γ(np + 2) a 1 − H n=0

(26)

or       z=

    

√ 1,1 1 H √ a ln[ 1− H ]H1,2 √ 1,1 1 H √ ln[ ]H1,2 a 1− H

·

·

√ 1− b 1 √ H ))p a ( a ln( H

¯ ¸ ¯ (0, p + 1) ¯ ¯ (0, 1), (−1, p) , H ≤ 0.25

√ b 1 H √ ( ln( ))p a a 1− H

¯ ¸ ¯ (0, p + 1) ¯ ¯ (0, 1), (−1, p) , H ≥ 0.25

9

(27)

4.3

Graphs and comparations

Using the moments we can approximate the mean, variance, kurtosis and skewness and compare the values to the given by the Skew-Normal:

Mean Variance Skewness Kurtosis

Proposed distribution 0.564053 0.680923 0.13204 3.04421

Error 0.000136 0.0007671 0.004908 0.017534

Skew-Normal 0.564189 0.6816901 0.136948 3.061744

We have seen that the proposed equations are good approximation to the Skew-Normal, we can also compare the graphs of the distributions. The last graph shows the error between the proposed distribution and the SkewNormal, the maximum error is 1.54766 10−4 at z=2.81 to the cumulative distribution funcion. SnHzL

0.4

0.3

0.2

0.1

z

Figure 5: Approximation of the Skew-Normal; c=1

10

HHzL - SNHzL 0.00015 0.00010 0.00005 z 0

-1

1

2

3

4

5

-0.00005

Figure 6: Error between H(z) and the Skew-Normal (CDF) c=1

5

SKEW-LEVY DISTRIBUTION

Rathie et al (2006) obtained the following density and distribuition functions for Levy symmetric distribuition,  2

f (x, α, γ, 0, 0) =

1 11  x H12 1/ 1/ 2 π 2 αγ α 4γ /α

 ¯ ¯(1− α1 , α2 ) ¯  ¯(0,1),( 12 ,1) , −∞ < x < ∞

 F (x, α, γ, 0, 0) =

1 + 2

2

x 11  x H12 1/ 1/ 2 2π 2 αγ α 4γ /α

(28)

 ¯ ¯(1− α1 , α2 ) ¯  , −∞ < x < ∞ ¯(0,1),( −1 2 ,1) (29)

The corresponding Skew-Levy density function, for a real x, is defined by 

 ¯ 1 2 ¯ 2 11  x ¯(1− α , α )  × H12 1/ 1/ 2/ ¯(0,1),( 12 ,1) π 2 αγ α 4γ α    ¯ 2 2 ¯ 1 2 wx 11  w x ¯(1− α , α ) 1 +  , H12 1/ 1/ 2 ¯(0,1),( −1 2 2 ,1) 2π 2 αγ α 4γ /α 2

h (x, α, γ, 0, 0, w) =

−∞ < x < ∞

11

(30)

5.1

SPECIAL CASES

For α = 12 , and a real x, Eq.(30) reduces to Skew-Lvy distribution µ ¶ 1 h x, , γ, 0, 0, w 2

=

¸ · 27/2 13 64x2 ¯¯ 14 ,0, −1 4 G × ¯ π 2 γ 2 31 γ 4 0 µ · ¸¶ 1 23/2 wx 14 64w2 x2 ¯¯ 12 , 14 ,0, −1 4 + 2 2 G42 , ¯0, −1 2 π γ γ4 2 −∞ < x < ∞

(31)

For α = 1, Eq.(30) reduces to Skew-Cauchy distribution

· ¸µ · ¸¶ 1 wx 12 w2 x2 ¯¯0, 12 2 11 x2 ¯¯0 G + G h (x, 1, γ, 0, 0, w) = , −∞ < x < ∞ ¯ −1 πγ 11 γ 2 0 2 2πγ 22 γ 2 0, 2 (32) For α = 2, Eq.(30) reduces to Skew-Normal distribution

h (x, 2, γ, 0, 0, ω) =

·

1

G10 01

¸ x2 |0 × 4γ

1 1 π /2 γ /2   · 2 2¯ ¸ 1 ωx ω x ¯2 , 1 + G11 1/ 1/ 12 4γ ¯0, −1 2 2 2 2 2π γ −∞ ≤ x ≤ ∞.

(33)

Density functions given by Eq.(31), Eq.(32) and Eq.(??) are traced in subsequent figures for γ = 1 and w = 1.

12

0.7 0.6 0.5 0.4 0.3 0.2 0.1

-4

0

-2

2

4

6

8

10

Figure 7: Graphic Levy h(x, 12 , 1, 0, 0, 1) 0.35 0.30 0.25 0.20 0.15 0.10 0.05

-4

-2

2

4

6

8

10

Figure 8: Graphic Levy h(x, 1, 1, 0, 0, 1) A detailed study the Skew-Levy density function is on way and will be reported in the near future. 13

0.4

0.3

0.2

0.1

-4

-2

2

4

6

8

10

Figure 9: Graphic Levy h(x, 2, 1, 0, 0, 1)

Acknowledgement: Supported, partially, by financial grant from DPP(2008), Univ. of Brasilia, Brasilia, D.F., Brazil

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References [1] Braaksma, B.L.J. (1964) Asymptotic Expansions and Analytics Continuations for a class of Barnes Integral,Comp. Math. 15, 239-341. [2] Luke, Y. L. (1969) The Special Functions and Their Applications, v.1, Academic Press, New York. [3] Mathai, A.M., Saxena, R.K. (1978) The H-funtion with Applications in Statistics and Other Disciplines, Wiley. [4] Page, E. (1977) Approximations to the cumulative normal function and its inverse for use on a pocket calculator,Applied Statistics, 26(1), 75-76. [5] Srivastava, H.M., Gupta, K.C. and Goyal, S.P. (1982) The H-finctions of one and two Variables with Applications, South Asia Pub. [6] Springer,M.D. (1979) The Algebra of Random Variables, John Wiley, New York. [7] Swamee, P.K. (2002) Near lognormal distribution, J. Hydrol. Eng. ASCE,7(6), 441-444. [8] Rathie, P. N., Swamee, P. K. (2006) On a new invertible generalized logistic distribution approximation to normal distribution, Technical Research Report in Statistics, 07/2006, Department of Statistics, University of Brasilia, Brasilia, Brazil. [9] Rathie,P.N., Dorea,C.C.Y. and Matsushita,R (2006) Lvy distribution, H function and applications to currency data. Proceedings of the Seventh International Conference of the Society for Special Functions and Their Applications Pune Univ., India, February 32-23, 2006(7),17-26. [10] Swamee, P. K., Rathie, P. N. (2007) Invertible alternativces to normal and lognormal distribution, J. Hydrol. Eng. ASCE, March/April, 218-221.

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