Volume 2010, Article ID 290124,10pages doi:10.1155/2010/290124
Research Article
Moment Estimation Inequalities Based on
g
λ
Random Variable on Sugeno Measure Space
Jingfeng Tian,
1Zhiming Zhang,
2and Dazeng Tian
31College of Science and Technology, North China Electric Power University, Baoding,
Hebei Province 071051, China
2College of Mathematics and Computer Sciences, Hebei University, Baoding, Hebei Province 071002, China 3College of Physics Science and Technology, Hebei University, Baoding, Hebei Province 071002, China
Correspondence should be addressed to Jingfeng Tian,tianjfhxm [email protected]
Received 8 October 2009; Accepted 12 December 2009
Academic Editor: Andrei Volodin
Copyrightq2010 Jingfeng Tian et al. 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 definitions and properties of moment ofgλrandom variable are provided on Sugeno measure
space. Then some important moment estimation inequalities based ongλ random variable are
presented and proven.
1. Introduction
In 1974, the Japanese scholar Sugeno1presented a kind of typical nonadditive measure, Sugeno measure, which is an important generalization of probability measure2–6. As we all know, the definitions and properties of moment of random variable play an important role in probability theory7–9. Likewise, they are also very important for Sugeno measure. In this paper we present the analogous definitions and properties based ongλrandom variable on Sugeno measure space. Then some important moment estimation inequalities based ongλ random variable are presented and proven.
2. Preliminaries
Let us recall some definitions and facts from5.
if and only if there exists
λ∈
− 1
supμ,∞
∪ {0} 2.1
such that
μ ∞
i1 Ei
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 λ
∞
i1
1λ·μEi−1
, asλ /0,
∞
i1
μEi, asλ0,
2.2
for any disjoint sequence{Ei}of sets inζwhose union is also inζ.
Definition 2.2. LetFbe aσ-algebra of subsets ofX. Andμis called Sugeno measure onFif and only if it satisfies theσ-λrule andμX 1. Usually, Sugeno measure onFis denoted by gλ.
We call the tripleX,F, gλa Sugeno measure space, denoted bygλspace, whereλ ∈ −1,∞. In the following, our discussion will be restricted to this space.
Theorem 2.3. For allE, F ∈ F, E⊂Fimply thatgλE≤gλF(monotonicity).
Theorem 2.4. Letgλbe a Sugeno measure onF. Then, for anyE∈ FandF∈ F,
gλE∪F gλE gλF−gλE∩F λgλEgλF 1λgλE∩F ,
gλE−F gλE−gλE∩F 1λgλE∩F ,
gλEc 1−gλE 1λgλE.
2.3
In order to present the analogous definitions and properties based on gλ random variable on Sugeno measure space, we recall some definitions and facts from10.
Definition 2.5. Letξbe a function mapping fromX,F, gλto real lineR. Thenξis called agλ random variable.
Definition 2.6. Letξbe agλrandom variable. Then the distribution function ofξis defined by
Definition 2.7. LetFgλxbe the distribution function ofgλrandom variableξ. Thenξis called
continuousgλrandom variable if there exists a nonnegative real valued functionfgλxsuch
that
Fgλx
x
−∞fgλtdt, ∀x∈R 2.5
is valid. The functionfgλxis called a density function ofξ.
In the following, our discussion will be restricted to the continuous gλ random variable.
Definition 2.8. Let Fgλx be the distribution function of gλ random variable ξ. If
∞
−∞|x|dFgλx < ∞, then we call
∞
−∞xdFgλxan expected value of gλ random variable ξ,
denoted byEgλξ.
Theorem 2.9. Letξ,ηbegλrandom variables; let C and D be constants. Then
Egλ
CξDηCEgλξ DEgλ
η. 2.6
Definition 2.10. Letξbe agλrandom variable. IfEgλ{ξ−Egλξ
2}exists, thenE
gλ{ξ−Egλξ
2}
is called the variance ofξ, denoted byDgλξ.
3. Moment Estimation Inequalities Based on
g
λRandom Variable
We begin this section with a short lemmasee11, which will be useful in the sequel.
Lemma 3.1. Letξbe agλrandom variable whose Sugeno density functionfgλ exists. If the Lebesgue
integral
∞
0
gλ{ξ≥r}dr− 0
−∞gλ{ξ≤r}drλ
∞
0
gλ{ξ≥r} ·gλ{ξ≤r}dr 3.1
is finite, then
Egλξ
∞
0
gλ{ξ≥r}dr− 0
−∞gλ{ξ≤r}drλ
∞
0
Theorem 3.2. Letξbe a nonnegativegλrandom variable. Whenλ≥0, the inequality
∞
i1
gλ{ξ≥i} ≤Egλξ≤1λ
1
∞
i1
gλ{ξ≥i}
3.3
is valid; whenλ <0, the inequality
1λ
∞
i1
gλ{ξ≥i} ≤Egλξ≤1
∞
i1
gλ{ξ≥i} 3.4
holds true.
Proof. IWhenλ≥0, sincegλ{ξ≥r}is a monotone decreasing function ofr,we have
Egλξ
∞
0
gλ{ξ≥r}drλ ∞
0
gλ{ξ≥r} ·gλ{ξ≤r}dr
≥
∞
0
gλ{ξ≥r}dr
∞ i1
i
i−1
gλ{ξ≥r}dr
≥∞ i1
i
i−1
gλ{ξ≥i}dr
∞ i1
gλ{ξ≥i},
Egλξ
∞
0
gλ{ξ≥r}drλ ∞
0
gλ{ξ≥r} ·gλ{ξ≤r}dr
≤
∞
0
gλ{ξ≥r}drλ ∞
0
gλ{ξ≥r}dr
1λ ∞
0
gλ{ξ≥r}dr
1λ
∞
i1 i
i−1
gλ{ξ≥r}dr
≤1λ
∞
i1 i
i−1
gλ{ξ≥i−1}dr
1λ
1
∞
i1
gλ{ξ≥i}
.
IIWhenλ <0, owing to the monotonicity ofgλ{ξ≥r}we also have
Egλξ
∞
0
gλ{ξ≥r}drλ ∞
0
gλ{ξ≥r} ·gλ{ξ≤r}dr
≥
∞
0
gλ{ξ≥r}drλ ∞
0
gλ{ξ≥r}dr
1λ ∞
0
gλ{ξ≥r}dr
1λ
∞
i1 i
i−1
gλ{ξ≥r}dr
≥1λ
∞
i1 i
i−1
gλ{ξ≥i}dr
1λ
∞
i1
gλ{ξ≥i},
Egλξ
∞
0
gλ{ξ≥r}drλ ∞
0
gλ{ξ≥r} ·gλ{ξ≤r}dr
≤
∞
0
gλ{ξ≥r}dr
∞ i1
i
i−1
gλ{ξ≥r}dr
≤∞ i1
i
i−1
gλ{ξ≥i−1}dr
1
∞
i1
gλ{ξ≥i}.
3.6
Definition 3.3. Letξbe agλrandom variable andka positive number. Then1the expected value Egλξk is called the kth moment, 2 the expected value Egλ|ξ|k is called the
kth absolute moment, 3 the expected value Egλ{ξ −Egλξ
k} is called the kth central moment, and 4 the expected valueEgλ{|ξ−Egλξ|
k} is called the kth absolute central moment.
Theorem 3.4. Letξbe a nonnegativegλrandom variable andka positive number. Then
Egλ
ξkk
∞
0 rk−1g
λ{ξ≥r}drkλ ∞
0 rk−1g
Proof. FromLemma 3.1, we infer
Egλ
ξk
∞
0 gλ
ξk≥xdxλ ∞
0 gλ
ξk≥x·gλ
ξk≤xdx
∞
0
gλ{ξ≥r}drkλ ∞
0
gλ{ξ≥r} ·gλ{ξ≤r}drk
k ∞
0 rk−1g
λ{ξ≥r}drkλ ∞
0 rk−1g
λ{ξ≥r} ·gλ{ξ≤r}dr.
3.8
Similar to the case of credibility theory12, we have the following: Theorems3.5,3.6, and3.7.
Theorem 3.5. Letξbe agλrandom variable that takes values inm, nand has expected valueEgλξ,
and letfxbe a convex function onm, n. Then
Egλ
fξ≤ n−Egλξ
n−m fm
Egλξ−m
n−m fn. 3.9
Theorem 3.6. Letξbe agλrandom variable that takes values inm, nand has expected valueEgλξ.
Then
Dgλξ≤
Egλξ−m
n−Egλξ
. 3.10
Theorem 3.7. Letξ be agλ random variable that takes values inm, nand has expected valueμ.
Then, for any positive integerk,
Egλ
|ξ|k≤ n−μ n−m|m|
kμ−m n−m|n|
k,
Egλ
ξ−μk≤ n−μ
n−mμ−m
kμ−m n−mn−μ
k .
3.11
Theorem 3.8. Letξbe agλrandom variable andt >0.ThenEgλ|ξ|
t<∞if and only if∞
i1gλ{|ξ|> i1/t}<∞.
Proof. Fromgλ{|ξ|t≥i}gλ{|ξ| ≥i1/t}andTheorem 3.2, the conclusion is valid.
Theorem 3.9. Letξ be agλrandom variable andt > 0. IfEgλ|ξ|t < ∞,thenlimx→ ∞xtgλ{|ξ| ≥
x} 0.Conversely, if there exists one positive numbertsuch thatlimx→ ∞xtgλ{|ξ| ≥ x} 0, then Egλ|ξ|
Proof. 1Whenλ≥0, we have
Egλ
|ξ|t ∞
0 gλ
|ξ|t≥rdrλ ∞
0 gλ
|ξ|t≥r·gλ
|ξ|t≤rdr
≥
∞
0 gλ
|ξ|t≥rdr.
3.12
SinceEgλ|ξ|t<∞,we obtain
∞
0 gλ{|ξ|t≥r}dr <∞.Consequently,
lim x→ ∞
∞
xt/2
gλ
|ξ|t≥rdr 0. 3.13
Since
∞
xt/2
gλ
|ξ|t≥rdr ≥ xt
xt/2
gλ
|ξ|t≥rdr≥ 1 2x
tg
λ{|ξ| ≥x}, 3.14
we have
lim x→ ∞x
tg
λ{|ξ| ≥x}0. 3.15
2Whenλ <0, we have
Egλ
|ξ|t ∞
0 gλ
|ξ|t≥rdrλ ∞
0 gλ
|ξ|t≥r·gλ
|ξ|t≤rdr
≥
∞
0 gλ
|ξ|t≥rdrλ ∞
0 gλ
|ξ|t≥rdr
1λ ∞
0 gλ
|ξ|t≥rdr.
3.16
Since
Egλ
|ξ|t<∞, 3.17
we obtain
1λ ∞
0 gλ
|ξ|t≥rdr <∞. 3.18
Consequently,
lim x→ ∞1λ
∞
xt/2
gλ
Since
1λ ∞
xt/2
gλ
|ξ|t≥rdr≥1λ xt
xt/2
gλ
|ξ|t≥rdr≥ 1
21λx tg
λ{|ξ| ≥x}, 3.20
we have
lim x→ ∞x
tg
λ{|ξ| ≥x}0. 3.21
Conversely, if limx→ ∞xtgλ{|ξ| ≥ x} 0,then there exists one numberlsuch thatxtgλ{|ξ| ≥ x} ≤1, for allx≥l.
3Whenλ≥0, for anys,where 0≤s < t, we have
Egλ
|ξ|s ∞
0 gλ
|ξ|s≥rdrλ ∞
0 gλ
|ξ|s≥r·gλ
|ξ|s≤rdr
≤
∞
0 gλ
|ξ|s≥rdrλ ∞
0 gλ
|ξ|s≥rdr
1λ ∞
0 gλ
|ξ|s≥rdr
1λ l
0 gλ
|ξ|s≥rdr ∞
l gλ
|ξ|s≥rdr
1λ l
0 gλ
|ξ|s≥rdr ∞
l
srs−1gλ{|ξ| ≥r}dr
≤1λ l
0 gλ
|ξ|s≥rdrs ∞
l
rs−t−1dr
≤1λ l
0 gλ
|ξ|s≥rdrs ∞
0
rs−t−1dr
.
3.22
Since0∞rpdr <∞for anyp <−1,we have
Egλ
|ξ|s≤1λ l
0 gλ
|ξ|s≥rdrs ∞
0
rs−t−1dr
4Whenλ <0, for anys,where 0≤s < t, we have
Egλ
|ξ|s ∞
0 gλ
|ξ|s≥rdrλ ∞
0 gλ
|ξ|s≥r·gλ
|ξ|s≤rdr
≤
∞
0 gλ
|ξ|s≥rdr
l
0 gλ
|ξ|s≥rdr ∞
l gλ
|ξ|s≥rdr
l
0 gλ
|ξ|s≥rdr ∞
l
srs−1gλ{|ξ| ≥r}dr
≤
l
0 gλ
|ξ|s≥rdrs ∞
l
rs−t−1dr
≤
l
0 gλ
|ξ|s≥rdrs ∞
0
rs−t−1dr.
3.24
Since0∞rpdr <∞for anyp <−1,we have
Egλ
|ξ|s≤ l
0 gλ
|ξ|s≥rdrs ∞
0
rs−t−1dr <∞. 3.25
Acknowledgment
This work was supported by the NNSF of Chinano. 60773062, the NSF of Hebei Province of China no. 2008000633, the foundation of North China Electric Power Universityno. 200911033, the KSRP of Department of Education of Hebei Province of Chinano. 2005001D, and the KSTRP of Ministry of Education of Chinano. 206012.
References
1 M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. dissertation, Tokyo Institute of Technology, 1974.
2 A. Basile, “Sequential compactness for sets of Sugeno fuzzy measures,”Fuzzy Sets and Systems, vol. 21, no. 2, pp. 243–247, 1987.
3 M. Berres, “λ-additive measures on measure spaces,”Fuzzy Sets and Systems, vol. 27, no. 2, pp. 159– 169, 1988.
4 T.-Y. Chen, J.-C. Wang, and G.-H. Tzeng, “Identification of general fuzzy measures by genetic algorithms based on partial information,”IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 30, no. 4, pp. 517–528, 2000.
5 Z. Y. Wang and G. J. Klir,Fuzzy Measure Theory, Plenum Press, New York, NY, USA, 1992.
6 S. T. Wierzchon, “An algorithm for identification of fuzzy measure,”Fuzzy Sets and Systems, vol. 9, no. 1, pp. 69–78, 1983.
8 S. H. Sung, “Moment inequalities and complete moment convergence,”Journal of Inequalities and Applications, vol. 2009, Article ID 271265, 14 pages, 2009.
9 Q.-P. Zang, “A limit theorem for the moment of self-normalized sums,”Journal of Inequalities and Applications, vol. 2009, Article ID 957056, 10 pages, 2009.
10 M. Ha, Y. Li, J. Li, and D. Tian, “The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space,”Science in China. Series F, vol. 49, no. 3, pp. 372–385, 2006.
11 M. Ha, H. Zhang, W. Pedrycz, and H. Xing, “The expected value models on Sugeno measure space,”
International Journal of Approximate Reasoning, vol. 50, no. 7, pp. 1022–1035, 2009.
