General Mathematics Vol. 18, No. 1 (2010), 71–80
On a general class of modified gamma approximating operators 1 Vasile Mihe¸san
Abstract By using the generalized gamma distribution we shall define the (a,b)
general modified gamma transform Γ α,β,γ , a, b ∈ R from which we obtain as a special case both general modified gamma operators of the first and second kind. We obtain generalization of a several positive linear operator, as a special case of this general gamma oper ators.
2000 Mathematics Subject Classification: 41A36. Key words and phrases: Euler’s gamma distribution, generalized gamma distribution, linear positive operators, generalized modified gamma transform, modified gamma operators of the first and second kind.
1
Received 06 August, 2009
Accepted for publication (in revised form) 19 September, 2009
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V. Mihe¸san
Introduction
In this paper we continue our earlier investigations [5], [6], [7], [8], [9] concerning to use Euler’s gamma distribution for constructing linear positive operators. In probability theory and statistics, the gamma distribution (G) is a two parameters family of continuous probability distribution. The probability density function (p.d.f.) of the gamma distribution can be expressed in terms of the gamma function parametrized in terms of a shape parameter α and an inverse scale parameter β = 1/θ, called a rate parameter (1)
G(t; α, β) =
β α α−1 −βt t e for t > 0 and α, β > 0. Γ(α)
The Weibull distribution (named after Naloddi Weibull) is a continuous probability distribution given by (2)
γ
W (t; β, γ) = γβ γ tγ−1 e−(βt) for t > 0,
where γ > 0 is the shape parameter and β > 0 is a rate parameter. The most general form of the gamma distribution is the generalized gamma distribution (GG). It was introduced by Stacy and Mihran [12], [13], in order to combine the power of two distribution: the gamma distribution (1) and the Weibull distribution (2). The generalized gamma distribution is a three-parameter distribution, with the probability density function (p.d.f.) given by (3)
γβ αγ γα−1 −(βt)γ GG(t; α, β, γ) = t e Γ(α)
On a general class of modified gamma approximating operators
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for t > 0, where α > 0 and γ > 0 are shape parameter and β > 0 is rate parameter. The moments of (3) can be shown to be EGG (tk ) = β k
(4)
Γ (α + k/γ) . Γ(α)
The generalized gamma distribution is a flexible distribution and it includes as special cases several distributions: the exponential distribution, the gamma distribution, the half normal distribution, the Levy distribution, the Weibull distribution and the log-normal distribution in limit case (α tends to infinity). For more details for the generalized gamma distribution see also [1], [2], [3]. The generalized beta distribution (GB) was introduced by J.B. McDonald and Y.J. Xu [4]. It is five-parameter distribution, with the probability density function (p.d.f.) given by q−1
tγp−1 (1 − (1 − c) (t/d)γ ) GB(t; γ, c, d, p, q) = p+q dγp B(p, q) (1 + c (t/d)γ )
(5)
for 0 < tγ < dγ /(1 − c) and zero otherwise, with 0 ≤ c ≤ 1 and γ, d, p, q, positive, γ ∈ R∗ . The moments of (5) can be shown to be [4] k k p + , B (p + k/γ, q) γ γ (6) EGB (tk ) = dk ; c 2 F1 k B(p, q) p+q+ γ where
2 F1
denotes the hypergeometric series which converges for all k if
c < 1, or for kγ < q if c = 1. Substituting k = 0 into (6) verifies that (5) integrates to one.
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V. Mihe¸san The generalized beta distribution (GB) includes the generalized beta of
the first kind (GB1) and the generalized beta of the second kind (GB2), corresponding to c = 0 and c = 1, (see [4]). The generalized gamma is a limiting case of GB, a.e. 1 γ1 (7) GG(t; α, β, γ) = lim GB t; γ, c, d = q , α, q . q→∞ β Hence, the generalized beta includes generalized gamma as a limiting case for all admissible values of c. We obtain by (6) EGG (tk ) = lim EGB (tk )
(8)
q→∞
k
q γ β k B (α + k/γ, q) Γ (α + k/γ) = lim = βk . q→∞ B(α, q) Γ(α) By using the generalized gamma distribution we shall define the general (a,b)
modified gamma transform Γα,β,γ , a, b ∈ R from which we obtain as a special case both general modified gamma operators of the first and second kind. We obtain generalization of a several positive linear operator, as a special case of this general gamma operators.
2
The general modified gamma transform
By using (3) we define the general modified gamma transform of a function f (9)
(a,b) Γα,β,γ f
β αγ =γ Γ(α)
Z
∞ γ
γ
tαγ−1 e−(βt) f (cta e−bt )dt 0 (a,b)
where α, β, γ > 0; a, b ∈ R and f ∈ L1,loc (0, ∞) such that Γα,β,γ |f | < ∞.
On a general class of modified gamma approximating operators
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(a,b)
We determine c ∈ R such that Γα,β,γ e1 = e1 , that is c=
Γ(α)x (β γ + b)α+a/γ · αγ β Γ (α + a/γ)
and we obtain from (9) the (a, b)-general modified gamma operators (a,b)
(10)
(Γα,β,γ f )(x)
β αγ =γ Γ(α)
Z
∞
t
αγ−1 −(βt)γ
e
0
f
(β γ + b)α+a/γ Γ(α) a −btγ · t e x dt. β αγ Γ (α + a/γ)
(a,b)
One observe that Γα,β,γ is a positive linear operator. (a,b)
Theorem 1 The moment of order k of the operator Γα,β,γ has the following value (11)
(a,b)
(Γα,β,γ ek )(x) =
xk (β γ + b)k(α+a/γ) Γ (α + ka/γ) Γk−1 (α) · · . (β γ + kb)α+ka/γ Γk (α + a/γ) β αγ(k−1)
Proof. We have (a,b)
= = = = = =
(Γα,β,γ ek )(x) γ k Z ∞ β αγ (β + b)α+a/γ Γ(α) Γ(α) γα−1 −(βt)γ a −btγ t e · t e x dt γ Γ(α) 0 β αγ Γ(α + a/γ) Z ∞ γ k(α+a/γ) β αγ Γk (α) γ (β + b) γ γ tγα−1 e−(βt) · tka e−kbt xk dt kαγ k Γ(α) 0 β Γ (α + a/γ) Z ∞ αγ γ k(α+a/γ) k k β (β + b) Γ (α)x γ γ γ · · k tγα+ka−1 e−(βt) −kbt dt kαγ Γ(α) β Γ (α + a/γ) 0 Z ∞ αγ γ k(α+a/γ) ka (β + b) Γk (α)xk β γ 1/γ γ γ · · k tγ (α+ γ )−1 e−((β +(kb) )t) dt kαγ Γ(α) β Γ (α + a/γ) 0 γ k(α+a/γ) k−1 (β + b) Γ (α)xk Γ (α + ka/γ) 1 · γ · · β αγ(k−1) Γk (α + a/γ) (β γ + kb)α+ ka γ γ (β γ + b)α+a/γ (β γ + kb)
α+ ka γ
·
Γ (α + ka/γ) Γk−1 (α) xk · . Γk (α + a/γ) β αγ(k−1)
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V. Mihe¸san
Consequently, we obtain (a,b)
(12)
(Γα,β,γ e2 )(x) =
(β γ + b)2(α+a/γ) Γ (α + 2a/γ) Γ(α)xk · 2 · 2a Γ (α + a/γ) β αγ (β γ + 2b)α+ γ
and (a,b)
Γα,β,γ ((t − x)2 ; x)
(13)
2a
(β γ + b)2(α+a/γ) Γ(α)Γ (α + 2a/γ) − β αγ (β γ + 2b)α+ γ Γ2 (α + a/γ) 2 = ·x . β αγ (β γ + 2b)α + 2a/γΓ2 (α + a/γ)
3
The modified gamma first kind operators
If we put in (10) b = 0 we obtain the general modified gamma first kind operators (14)
(a) (Γα,β,γ f )(x)
β αγ =γ Γ(α)
Z
γ = Γ(α)
Z
∞
t
αγ−1 −(βt)γ
e
f
0
Γ(α)(βt)a x dt Γ (α + a/γ)
or equivalent (15)
(a) (Γα,β,γ f )(x)
∞
u
αγ−1 −uγ
0
e
f
Γ(α)ua x Γ (α + a/γ)
du
(a)
where f ∈ L1,loc (0, ∞) such that Γα,β,γ |f | < ∞. (a)
We observe that Γα,β,γ does not depend on β and we may consider β = 1. (a)
Corollary 1 The moment of order k of the operator Γα,β,γ has the following value (16)
(a)
(Γα,β,γ ek )(x) =
Γk−1 (α)Γ (α + ka/γ) k x . Γk (α + a/γ)
On a general class of modified gamma approximating operators
77
Proof. The result follows from (11) for b = 0. Consequently, we obtain Γ(α)Γ (α + 2a/γ) 2 x . Γ2 (α + a/γ)
2 Γ(a) α,γ ((t − x) ) =
(17)
For γ = 1 we obtain the modified gamma first kind operators (see [9]) and for α = 1 we obtain the modified Weibull first kind operators (see [11]). Special cases Case 1. If we consider a = 1 in (15) we obtain the modified gamma first kind operator (18)
γ (Γα,γ f )(x) = Γ(α)
Z
∞
t
αγ−1 −tγ
e
f
0
Γ(α)tx Γ (α + 1/γ)
dt.
Corollary 2 The moment of order k of the operator Γα,γ has the following value (19)
(Γα,γ ek )(x) =
Γk−1 (α)Γ (α + k/γ) k x . Γk (α + 1/γ)
Proof. The result follows from (16) for a = 1. We deduce (20)
(Γα,γ e2 )(x) = Γα,γ ((t − x)2 ; x) =
Γ(α)Γ (α + 2/γ) 2 x , Γ2 (α + 1/γ)
Γ(α)Γ (α + 2/γ) − Γ2 (α + 1/γ) 2 x . Γ2 (α + 1/γ)
If we choose α = n, n ∈ N in (18) then we obtain the generalization of the Post-Wider positive linear operator defined for f ∈ L1,loc (0, ∞) by (see [9]) (21)
γ (Pn,γ f )(x) = Γ(n)
Z
∞
t 0
γn−1 −tγ
e
f
Γ(n) tx dt. Γ (n + 1/γ)
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V. Mihe¸san If we replace α = nx, n ∈ N in (18) then we obtain the generalization of
the Rathore positive linear operator defined for f ∈ L1,loc (0, ∞) by (see [9]) (22)
γ (Rn,γ f )(x) = Γ(nx)
Z
∞
t
γnx−1 −tγ
e
f
0
Γ(nx)tx dt Γ (nx + 1/γ)
Case 2. If we replace a = −1 in (15) we obtain the modified gamma operator (23)
e α,γ f )(x) = γ (Γ Γ(α)
Z
∞
t
αγ−1 −tγ
e
f
0
Γ(α) x · Γ (α − 1/γ) t
dt.
e α,γ has the following Corollary 3 The moment of order k of the operator Γ value (24)
Γk−1 (α)Γ (α − k/γ) k e x , (Γα,γ ek )(x) = Γk (α − 1/γ)
0 ≤ k < αγ.
Proof. The result follows from (16) for a = −1. We obtain e α,γ e2 )(x) = (Γ
Γ(α)Γ (α − 2/γ) 2 x Γ2 (α − 1/γ)
2 e α,γ ((t − x)2 ; x) = Γ(α)Γ (α − 2/γ) − Γ (α − 1/γ) x2 . Γ Γ2 (α − 1/γ)
For α = n + 1, n ∈ N we obtain the generalization of the operator introduced and studied by A. Lupa¸s and M. M¨ uller γ (Gn,γ f )(x) = Γ(n + 1)
Z
∞
t 0
(n+1)γ−1 −tγ
e
f
Γ(n + 1) x · Γ (n + 1 − 1/γ) t
dt.
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On a general class of modified gamma approximating operators
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The modified gamma second kind operators
If we choose in (10) a = 0 then we obtain the modified gamma second kind operators (25)
(b) (Γα,β,γ f )(x)
β αγ =γ Γ(α)
Z
∞
t
αγ−1 −(βt)γ
e
f
0
βγ + b βγ
α
e
−btγ
x dt
(b)
where f ∈ L1,loc (0, ∞) such that Γα,β,γ |f | < ∞. (b)
Corollary 4 The moment of order k of the operator Γα,β,γ has the following value (b)
(26)
(Γα,β,γ ek )(x) =
xk (β γ + b)kα · . (β γ + kb)α β αγ(k−1)
Proof. The result follows from (11) for a = 0. Consequently, we obtain (b)
(27)
(28)
(Γα,β,γ (e2 )(x) = (b)
Γα,β,γ ((t − x)2 ; x) =
(β γ + b)2α x2 · (β γ + 2b)α β αγ
(β γ + b)2α − β αγ (β γ + 2b)α 2 ·x . β αγ (β γ + 2b)α
For γ = 1 we obtain the modified gamma second kind operators (see [9]) and for α = 1 we obtain the modified Weibull second kind operators (see [11]).
References [1] Ali, M.M., Woo, J., Nadorajah, S., Generalized gamma variables with drought application, Journal of the Korean Statistical Society, 37, 2008, 37-45.
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[2] Gomes, O., Combes, C., Dussauchoy, Parameter estimation of the generalized gamma distribution, Mathematics and Computer in Simulations, 2008 (Available online at www.sciencedirect.com). [3] Huang, P.H., Hwang, T.-Y., On new moment estimation of parameters of the generalized gamma distribution using it’s characterization, Taiwanese Journal of Mathematics, Vol. 10, no. 4, 2006, 1083-1093. [4] McDonald, J.B., Xu, Y.J., A generalization of the beta distribution with applications, Journal of Econometrics, 66, 1995, 133-152. [5] Mihe¸san, V., Approximation of continuous functions by means of linear positive operators, Ph. D. Thesis, Babe¸s-Bolyai University, ClujNapoca, 1997. [6] Mihe¸san, V., On the gamma and beta approximation operators, Gen. Math., 6, 1998, 61-64. [7] Mihe¸san, V., On a general class of gamma approximating operators, Studia Univ. Babe¸s-Bolyai, Mathematica, Vol. XLVIII, 4, 2003, 49-54. [8] Mihe¸san, V., Gamma approximating operators, Creat. Math. Inform., vol. 17, 2008, No. 3. [9] Mihe¸san, V., The modified gamma approximating operators, Aut. Comp. Appl. Math., Vol. 17, 2008, No. 1, 21-26. [10] Mihe¸san, V., Weibull approximating operators, Aut. Comp. Appl. Math. (in press). [11] Mihe¸san, V., The modified Weibull approximating operators (in press). [12] Stacy, E.W., A generalization of gamma distribution, Ann. Math. Statist., 33, 1962, 1187-1192. [13] Stacy, E.W., Mihran, G.A., Parameter estimation for a generalized gamma distribution, Technometric, Vol. 7, No. 3, 1965.
Vasile Mihe¸san Technical University of Cluj-Napoca Department of Mathematics 400020 Cluj-Napoca, Romania e-mail:
[email protected] 
