Moment Estimation Inequalities Based on Random Variable on Sugeno Measure Space

Abstract

The definitions and properties of moment of random variable are provided on Sugeno measure space. Then some important moment estimation inequalities based on random variable are presented and proven.

1. Introduction

In 1974, the Japanese scholar Sugeno [1] presented a kind of typical nonadditive measure, Sugeno measure, which is an important generalization of probability measure [2–6]. As we all know, the definitions and properties of moment of random variable play an important role in probability theory [7–9]. Likewise, they are also very important for Sugeno measure. In this paper we present the analogous definitions and properties based on random variable on Sugeno measure space. Then some important moment estimation inequalities based on random variable are presented and proven.

2. Preliminaries

Let us recall some definitions and facts from [5].

Definition 2.1.

Let be a nonempty set, let be a nonempty class of subsets of , and let be a nonnegative real valued set function defined on . Therefore satisfies the - rule (on ) if and only if there exists

(2.1)

such that

(2.2)

for any disjoint sequence of sets in whose union is also in .

Definition 2.2.

Let be a -algebra of subsets of . And is called Sugeno measure on if and only if it satisfies the - rule and . Usually, Sugeno measure on is denoted by .

We call the triple a Sugeno measure space, denoted by space, where . In the following, our discussion will be restricted to this space.

Theorem 2.3.

For all imply that   (monotonicity).

Theorem 2.4.

Let be a Sugeno measure on . Then, for any and ,

(2.3)

In order to present the analogous definitions and properties based on random variable on Sugeno measure space, we recall some definitions and facts from [10].

Definition 2.5.

Let be a function mapping from to real line . Then is called a random variable.

Definition 2.6.

Let be a random variable. Then the distribution function of is defined by

(2.4)

Definition 2.7.

Let be the distribution function of random variable . Then is called continuous random variable if there exists a nonnegative real valued function such that

(2.5)

is valid. The function is called a density function of .

In the following, our discussion will be restricted to the continuous random variable.

Definition 2.8.

Let be the distribution function of random variable . If , then we call an expected value of random variable , denoted by .

Theorem 2.9.

Let , be random variables; let C and D be constants. Then

(2.6)

Definition 2.10.

Let be a random variable. If exists, then is called the variance of , denoted by .

3. Moment Estimation Inequalities Based on Random Variable

We begin this section with a short lemma (see [11]), which will be useful in the sequel.

Lemma 3.1.

Let be a random variable whose Sugeno density function exists. If the Lebesgue integral

(3.1)

is finite, then

(3.2)

Theorem 3.2.

Let be a nonnegative random variable. When , the inequality

(3.3)

is valid; when , the inequality

(3.4)

holds true.

Proof.

• (I) When , since is a monotone decreasing function of we have

  

  (3.5)

• (II) When , owing to the monotonicity of we also have

(3.6)

Definition 3.3.

Let be a random variable and a positive number. Then the expected value is called the th moment, the expected value is called the th absolute moment, the expected value is called the th central moment, and the expected value is called the th absolute central moment.

Theorem 3.4.

Let be a nonnegative random variable and a positive number. Then

(3.7)

Proof.

From Lemma 3.1, we infer

(3.8)

Similar to the case of credibility theory [12], we have the following: Theorems 3.5, 3.6, and 3.7.

Theorem 3.5.

Let be a random variable that takes values in and has expected value , and let be a convex function on . Then

(3.9)

Theorem 3.6.

Let be a random variable that takes values in and has expected value . Then

(3.10)

Theorem 3.7.

Let be a random variable that takes values in and has expected value . Then, for any positive integer ,

(3.11)

Theorem 3.8.

Let be a random variable and Then if and only if .

Proof.

From and Theorem 3.2, the conclusion is valid.

Theorem 3.9.

Let be a random variable and . If then Conversely, if there exists one positive number such that , then for any where

Proof.

• (1) When , we have

  

  (3.12)

Since we obtain Consequently,

(3.13)

Since

(3.14)

we have

(3.15)

• (2) When , we have

(3.16)

Since

(3.17)

we obtain

(3.18)

Consequently,

(3.19)

Since

(3.20)

we have

(3.21)

Conversely, if then there exists one number such that , for all

• (3) When , for any where , we have

(3.22)

Since for any we have

(3.23)

• (4) When , for any where , we have

  

  (3.24)

Since for any we have

(3.25)