- Email: [email protected]
Log-concavity for Bernstein-type operators using stochastic orders
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Electronic Notes in Discrete Mathematics 43 (2013) 27–30 www.elsevier.com/locate/endm
Log-concavity for Bernstein-type operators using stochastic orders F. G. Bad´ıa 1,2 Departamento de M´etodos Estad´ısticos Universidad de Zaragoza Zaragoza, Spain
C. Sang¨ uesa 3 Departamento de M´etodos Estad´ısticos Universidad de Zaragoza Zaragoza, Spain
Abstract This paper aims to study the preservation of log-concavity for Bernstein-type operators. In particular, attention is focused on positive linear operators, defined on the positive semi-axis, admitting a probabilistic representation in terms of a process with independent increments. This class includes classical Gamma, Sz´asz and Sz´asz-Durrmeyer operators. As a main tool in our results we use stochastic orders techniques. Our results include, as a particular case, the log-concavity of certain functions related to the gamma incomplete function. Keywords: log-concavity, Bernstein-type operator, Sz´asz operator, Gamma operator, Stochastic order, Gamma incomplete function.
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.005
28
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F.G. Badía, C. Sangüesa / Electronic Notes in Discrete Mathematics 43 (2013) 27–30
Introduction
We consider positive linear operators, defined on the positive semi-axis, admitting a probabilistic representation in terms of Levy processes (that is, a process with independent and stationary increments). Our aim is to prove the preservation of log-concavity under these operators by using basic characteristics of the underlying processes, together with some results coming from the well-developed theory of stochastic orders in the field of probability theory (cf. [10]). Log-concave functions (or Polya frequency functions of order 2, see [5, Ch. 7]) are a class of functions arising in many different fields such as economics (see, e.g., [2]) statistics (see, e.g., [11]) or applied probability (see, e.g., [9]). Recall that a log-concave function on an interval I ⊆ R is a function f : I → [0, ∞] verifying for all x, y ∈ I and 0 ≤ α ≤ 1 that (1) f (αx + (1 − α)y) ≥ f (x)α f (y)1−α (or equivalently log f is concave on the support of f ). Probabilistic methods play a significant role in the shape-preservation and approximation properties of positive linear operators, specially since the late years of last century (see, for instance, [3] and the references therein). Some classical operators included in this study are the following. Sz´asz operator: ∞ k −tr (tr)k (2) e . f Lr (f, t) = r k! k=0 Gamma-type operator (3)
∞ tr−1 u u e−u du. Gr (f, t) = f r Γ(tr) 0
We consider also Sz´asz-Durrmeyer type variants of the Sz´asz operator defined in the following way. Let X1 , X2 , . . . be a sequence of independent and identically distributed non negative random variables. The operators Mr and Mr∗ are defined as follows:
∞ k+1 X (tr)k i i=1 (4) Mr (f, t) = e−tr , E f r k! k=0 1
This work has been supported by research projects MTM2012-36603-C02-02. The first and second authors acknowledge the support of DGA S11 and E64, respectively. 2 Email: [email protected] 3 Email: [email protected]
F.G. Badía, C. Sangüesa / Electronic Notes in Discrete Mathematics 43 (2013) 27–30
29
k X (tr)k i i=1 (5) Mr∗ (f, t) = e−tr , E f r k! k=0 ∞
where E denotes mathematical expectation. When the Xi are exponential random variables with mean 1, the previous operators were introduced in [7] as a Durrmeyer-type modification of the Sz´asz operator. All the aforementioned operators (and some other positive linear operators) admit the following probabilistic representation. Let (X(t), t ≥ 0) be a stochastic process. Consider a operator of the form: (6) T f (t) = Ef (X(t)), t ≥ 0, f ∈ T , in which T is the set of measurable functions f : [0, ∞) → R such that E|f |(X(t)) < ∞, t ≥ 0. In order to build a sequence of smooth functions Tr f (t) approximating f as r tends to infinity one considers:
X(tr) Tr f (t) = E f (7) , r > 0. r In particular, if the underlying process satisfies E[X(t)] = t (in other words, the operator is centered) and belongs to the class of Levy processes (independent and stationary increments, see [8], for instance), we can guarantee that (8)
lim Tr f (t) = f (t),
r→∞
t≥0
for all bounded and continuous f (this is a consequence of the strong law of large numbers for Levy processes [8, Thm, 36.5, p.246]). Sz´asz operator (2) and Gamma type operator (3) are in this setting. In fact (2) can be written as in (7), in which (X(t), t ≥ 0) is a standard Poisson process. In a similar way (3) can be written as in (7), by considering (X(t), t ≥ 0) a standard gamma process. Sz´asz-Durrmeyer operator M ∗ is represented in terms of (X(t), t ≥ 0), a compound Poisson process. On the other hand, its analogous operator M is represented in terms of (X(t), t ≥ 0), a slight modification of the compound Poisson process. It is interesting to point out that the compound Poisson process is of great interest in actuarial sciences, in order to model the losses of an insurance company as time evolves. In particular, when the summands are gamma distributions, we obtain the class of Tweedie’s distributions (see [4, p. 317], and the references therein). In this work we will give general conditions for operators of the form (7) in order to preserve log-concavity. As a consequence of these results we obtain the preservation of the log-concavity for Szasz and Gamma-type operators. We will also show the preservation of log-concavity for Szasz-Durrmeyer oper-
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F.G. Badía, C. Sangüesa / Electronic Notes in Discrete Mathematics 43 (2013) 27–30
ators, under some specific conditions of the sequence X1 , X2 , . . . defining the summands in these operators. These conditions have to do with properties coming from the field of reliability theory [6], such as increasing failure rate or decreasing reversed failure rate. Finally, as an immediate consequence of the preservation of the log-concavity for the gamma-type operator, we will give some results concerning the log-concavity for some well-known special functions. In particular, the normalized gamma incomplete function on an arbitrary interval is log-concave with respect to the shape parameter. This property is known for the upper and lower normalized incomplete gamma function (see [1], and the references therein).
References [1] Alzer, H. and Baricz, A., Functional inequalities for the incomplete gamma function, J. Math. Anal. Appl. 385 (2012), 167-178. [2] An, M.Y., Logconcavity versus logconvexity: A complete characterization, J. Econom. Theory 80 (1998), 350-369. [3] de la Cal, J. and Carcamo, J., On the approximation of convex functions by Bernstein-type operators, J. Math. Anal. Appl. 334 (2007), 1106-1115. [4] Kaas, R., Goovaerts, M., Dhaene, J. & Denuit, M. “Modern Actuarial Risk Theory,” 2nd Ed., Springer, Heidelberg, 2009. [5] Karlin, S., “Total Positivity,” Vol. I, University Press, Stanford, 1968. [6] Marshall, A.W. and Olkin, I., “Life Distributions,” Springer, New York, 2007. [7] Mazhar, S. M. and Totik, V., Approximation by modified Sz´ asz operators, Acta Sci. Math. (Szeged) 49 (1985), 257-269. [8] Sato, K.I. “L´evy Processes and Infinitely Divisible Distributions,” Cambridge University Press, 1999. [9] Sengupta, D. and Nanda, A.K., Log-concave and concave distributions in reliability, Naval Res. Logist. 46 (1999), 419-433. [10] Shaked, M. and Shanthikumar, J.G., “Stochastic Orders,” Springer, New York, 2007. [11] Walther, G., Inference and modelling with logconcave distributions, Statist. Sci. 24 (2009), 319-327.
Electronic Notes in Discrete Mathematics 43 (2013) 27–30 www.elsevier.com/locate/endm
Log-concavity for Bernstein-type operators using stochastic orders F. G. Bad´ıa 1,2 Departamento de M´etodos Estad´ısticos Universidad de Zaragoza Zaragoza, Spain
C. Sang¨ uesa 3 Departamento de M´etodos Estad´ısticos Universidad de Zaragoza Zaragoza, Spain
Abstract This paper aims to study the preservation of log-concavity for Bernstein-type operators. In particular, attention is focused on positive linear operators, defined on the positive semi-axis, admitting a probabilistic representation in terms of a process with independent increments. This class includes classical Gamma, Sz´asz and Sz´asz-Durrmeyer operators. As a main tool in our results we use stochastic orders techniques. Our results include, as a particular case, the log-concavity of certain functions related to the gamma incomplete function. Keywords: log-concavity, Bernstein-type operator, Sz´asz operator, Gamma operator, Stochastic order, Gamma incomplete function.
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.005
28
1
F.G. Badía, C. Sangüesa / Electronic Notes in Discrete Mathematics 43 (2013) 27–30
Introduction
We consider positive linear operators, defined on the positive semi-axis, admitting a probabilistic representation in terms of Levy processes (that is, a process with independent and stationary increments). Our aim is to prove the preservation of log-concavity under these operators by using basic characteristics of the underlying processes, together with some results coming from the well-developed theory of stochastic orders in the field of probability theory (cf. [10]). Log-concave functions (or Polya frequency functions of order 2, see [5, Ch. 7]) are a class of functions arising in many different fields such as economics (see, e.g., [2]) statistics (see, e.g., [11]) or applied probability (see, e.g., [9]). Recall that a log-concave function on an interval I ⊆ R is a function f : I → [0, ∞] verifying for all x, y ∈ I and 0 ≤ α ≤ 1 that (1) f (αx + (1 − α)y) ≥ f (x)α f (y)1−α (or equivalently log f is concave on the support of f ). Probabilistic methods play a significant role in the shape-preservation and approximation properties of positive linear operators, specially since the late years of last century (see, for instance, [3] and the references therein). Some classical operators included in this study are the following. Sz´asz operator: ∞ k −tr (tr)k (2) e . f Lr (f, t) = r k! k=0 Gamma-type operator (3)
∞ tr−1 u u e−u du. Gr (f, t) = f r Γ(tr) 0
We consider also Sz´asz-Durrmeyer type variants of the Sz´asz operator defined in the following way. Let X1 , X2 , . . . be a sequence of independent and identically distributed non negative random variables. The operators Mr and Mr∗ are defined as follows:
∞ k+1 X (tr)k i i=1 (4) Mr (f, t) = e−tr , E f r k! k=0 1
This work has been supported by research projects MTM2012-36603-C02-02. The first and second authors acknowledge the support of DGA S11 and E64, respectively. 2 Email: [email protected] 3 Email: [email protected]
F.G. Badía, C. Sangüesa / Electronic Notes in Discrete Mathematics 43 (2013) 27–30
29
k X (tr)k i i=1 (5) Mr∗ (f, t) = e−tr , E f r k! k=0 ∞
where E denotes mathematical expectation. When the Xi are exponential random variables with mean 1, the previous operators were introduced in [7] as a Durrmeyer-type modification of the Sz´asz operator. All the aforementioned operators (and some other positive linear operators) admit the following probabilistic representation. Let (X(t), t ≥ 0) be a stochastic process. Consider a operator of the form: (6) T f (t) = Ef (X(t)), t ≥ 0, f ∈ T , in which T is the set of measurable functions f : [0, ∞) → R such that E|f |(X(t)) < ∞, t ≥ 0. In order to build a sequence of smooth functions Tr f (t) approximating f as r tends to infinity one considers:
X(tr) Tr f (t) = E f (7) , r > 0. r In particular, if the underlying process satisfies E[X(t)] = t (in other words, the operator is centered) and belongs to the class of Levy processes (independent and stationary increments, see [8], for instance), we can guarantee that (8)
lim Tr f (t) = f (t),
r→∞
t≥0
for all bounded and continuous f (this is a consequence of the strong law of large numbers for Levy processes [8, Thm, 36.5, p.246]). Sz´asz operator (2) and Gamma type operator (3) are in this setting. In fact (2) can be written as in (7), in which (X(t), t ≥ 0) is a standard Poisson process. In a similar way (3) can be written as in (7), by considering (X(t), t ≥ 0) a standard gamma process. Sz´asz-Durrmeyer operator M ∗ is represented in terms of (X(t), t ≥ 0), a compound Poisson process. On the other hand, its analogous operator M is represented in terms of (X(t), t ≥ 0), a slight modification of the compound Poisson process. It is interesting to point out that the compound Poisson process is of great interest in actuarial sciences, in order to model the losses of an insurance company as time evolves. In particular, when the summands are gamma distributions, we obtain the class of Tweedie’s distributions (see [4, p. 317], and the references therein). In this work we will give general conditions for operators of the form (7) in order to preserve log-concavity. As a consequence of these results we obtain the preservation of the log-concavity for Szasz and Gamma-type operators. We will also show the preservation of log-concavity for Szasz-Durrmeyer oper-
30
F.G. Badía, C. Sangüesa / Electronic Notes in Discrete Mathematics 43 (2013) 27–30
ators, under some specific conditions of the sequence X1 , X2 , . . . defining the summands in these operators. These conditions have to do with properties coming from the field of reliability theory [6], such as increasing failure rate or decreasing reversed failure rate. Finally, as an immediate consequence of the preservation of the log-concavity for the gamma-type operator, we will give some results concerning the log-concavity for some well-known special functions. In particular, the normalized gamma incomplete function on an arbitrary interval is log-concave with respect to the shape parameter. This property is known for the upper and lower normalized incomplete gamma function (see [1], and the references therein).
References [1] Alzer, H. and Baricz, A., Functional inequalities for the incomplete gamma function, J. Math. Anal. Appl. 385 (2012), 167-178. [2] An, M.Y., Logconcavity versus logconvexity: A complete characterization, J. Econom. Theory 80 (1998), 350-369. [3] de la Cal, J. and Carcamo, J., On the approximation of convex functions by Bernstein-type operators, J. Math. Anal. Appl. 334 (2007), 1106-1115. [4] Kaas, R., Goovaerts, M., Dhaene, J. & Denuit, M. “Modern Actuarial Risk Theory,” 2nd Ed., Springer, Heidelberg, 2009. [5] Karlin, S., “Total Positivity,” Vol. I, University Press, Stanford, 1968. [6] Marshall, A.W. and Olkin, I., “Life Distributions,” Springer, New York, 2007. [7] Mazhar, S. M. and Totik, V., Approximation by modified Sz´ asz operators, Acta Sci. Math. (Szeged) 49 (1985), 257-269. [8] Sato, K.I. “L´evy Processes and Infinitely Divisible Distributions,” Cambridge University Press, 1999. [9] Sengupta, D. and Nanda, A.K., Log-concave and concave distributions in reliability, Naval Res. Logist. 46 (1999), 419-433. [10] Shaked, M. and Shanthikumar, J.G., “Stochastic Orders,” Springer, New York, 2007. [11] Walther, G., Inference and modelling with logconcave distributions, Statist. Sci. 24 (2009), 319-327.