A Note on Stochastic Orders

Transcription

1 Intern. J. Fuzzy Mathematical rchive Vol. 9, No. 2, 2015, ISSN: (P), (online) Published on 8 October International Journal of Note on Stochastic Orders D.Rajan 1 and D.Vijayabalan 2 1 Department of Mathematics, T.B.M.. College, Porayar dan_rajan@rediffmail.com 2 Department of Mathematics, T.B.M.. College, Porayar, Tamil Nadu vijayabalantqb@gmail.com Received 1 September 2015; accepted 1 October 2015 bstract. In this paper, we definestochastic orders of fuzzy random variables with some properties. lso we define the hazard rate ordering of fuzzy random variableswith related properties. Keywords: Fuzzy random variables, fuzzy stochastic order, fuzzy hazard rate order MS Mathematics Subject Classification (2010): Introduction The concept of fuzzy random variable was introduced by Kwakernaak [3] and Puri and Ralescu [7]. fuzzy random variable is just a random variable that takes on values in space of fuzzy sets. The outcomes of Kwakernaak s fuzzy random variables are fuzzy real. Subsets and the extreme points of their -cuts are classical random variables. Fuzzy random variables are Mathematical descriptions for fuzzy stochastic phenomena, but only one time descriptions. Fuzzy random variables generalize random variables and random sets. Kwakernaak [4] introduced the concept of a fuzzy random variables as a function X: Ω F(R) where (Ω,,P) is a probability triple and F(R) denotes the set of all canonical fuzzy numbers. Puri and Ralescu [7] defined the notion of fuzzy random variable as a function X: Ω F(R n ) where (Ω,,P) is a probability space and F(R n ) indicates all functions : R n [0,1]. Stochastic ordering of fuzzy random variables is one of the most applicatble in statistics and probability. Various concept of statistic comparison between random variables have been defined and studied in the literature, because of their usefulness in modeling for reliability economics applications and as mathematical tools for proving important results in applied probability [8]. 2. Preliminaries Definition 2.1. et X be a universal set. Then a fuzzy set ( ) defined by its membership function µ % : X [ 0,1]. 245 {( x x ) x X} = %,µ / of X is Definition 2.2. For each 0 1, the -cut of set of % is denoted by its % = { x X:µ % ( x) }.

2 D.Rajan and D.Vijayabalan Definition 2.3. fuzzy number is a fuzzy set of R such that the following conditions are satisfies: 1. % is normal if there exists X µ x = 1 x such that % ( ) 2. % is called convex if λ + ( λ ) ( x1 x2 ) ( ( x1 ) ( x2 )) µ 1 min µ,µ % % % 3. % is called upper semi continuous with compact support; that is for every > 0, there exists δ>0; x y µ x < µ y + < δ % ( ) % ( ) 4. The -cut of fuzzy number is closed interval denoted by =,, where ( ) { x x } { x ( x) } = inf R; µ and = sup R; µ % %. 5. If % is closed and bounded fuzzy number with, function is strictly increasing on 1,, then % is called canonical fuzzy number. and its membership and strictly decreasing on Definition 2.4. fuzzy random variable is a fuzzy set consisting of a membership function and a basic set of underlying variables. fuzzy random variable X is a map X: Ω F(R) satisfying the following conditions: 1. For each ( 0,1] both X and X ( )( ) ( )( ) defined as { } ( )( ) { ( )( ) } X ω x = inf x R; X ω x and X ω x = sup x R; X ω x are finite real valued random variables defined on such Ω,, that the mathematical expectations EX and EX exist. ω Ω and 0,1, X ω x and X ω x 2. For each ( ] ( )( ) ( )( ) m Definition 2.5. fuzzy set valued mapping X : Ω F0 ( R ) = F0 ( R )... F0 ( R ) represented by X( ω) = ( X( 1, ω)... X( m, ω) ) is called the fuzzy random vector if for each K, 1 K m, X( K,ω) is a fuzzy random variable. Definition 2.6. X( ω) is a fuzzy random variable if and only if X ( ω) = X ( ω ),X ( ω), where X ( ω) and X ( ) for each ( ] ( ω) ( ω) 0,1 and X = X ( 0,1] ω are both random variables Definition 2.7. The cut of the distribution function F of fuzzy random variable is defined by =Min,,max,. Definition 2.8. The membership function of the probability distribution function of, denoted as. It is defined as 246

3 Note on Stochastic Orders = 1. Definition 2.9. The cut of probability density functions, where, is given by =Min,,max,. Definition The membership function of the probability distribution function of, denoted as is given by = Stochastic orders Definition 3.1. If and be fuzzy random variables with fuzzy cumulative distribution function and respectively then X Y. Definition 3.2. If and are two fuzzy random variables then. Definition 3.3. If and are two fuzzy random variables then, for all increasing functions. Example 3.4. If and are two exponential fuzzy random variables with mean and respectively such that <then X Y. Properties of stochastic orders Preposition 3.5. (Ordering of mean values) If X Y and is well defined then Proof:ssume first that and are non negative value fuzzy random variables. Then = > > > > = Generally one can express any fuzzy random variables as the difference of two non negative fuzzy random variables. et = For each, ; 0 = 0; <0 = 0; <0 ; 0 X Yimplies > > > > et 0 and 0 then > > > > This implies Similary we can prove There fore = - 247

4 D.Rajan and D.Vijayabalan - = The above inequality true for all 0,1 This shows that,,, That is. Preposition 3.6. (Closed under transformations by increasing function) If and are two fuzzy random variables and if X Ythen, for any increasing function Proof: Suppose first that X Y, and let be an increasing function et =inf; Then for 0,1, > >= > > > > = > > This shows that Conversely suppose that for all increasing function. For any a et denote the increasing function = 1 ; > 0 ; Then for 0,1, = > > > > This shows thatx Y. 4. Hazard rate order Definition 4.1. et X be a non negative fuzzy random variables with fuzzy distribution function F and fuzzy Probability density function. Then the level fuzzy Hazard rate of X at t 0 is defined as =,,,,, The fuzzy hazard rate of denoted as the membership function of is. defined as = Definition 4.2. et X and Y be two non negative fuzzy random variable with continuous distribution functions and with fuzzy hazard rate functions and respectively. X is smaller than Y in the hazard rate order Denoted as X Y if,,,,, and 248

5 Note on Stochastic Orders max, max, max, max, For each, 0,1 Q where, are the survival and density functions of X respectively and,are the survival and density functions of Y respectively. Property 4.3. If and are fuzzy random variables with continuous distribution functions and if X Y then for all, where is fuzzy probability distribution function of and is fuzzy probability distribution function of. Preposition 4.4. If and are two fuzzy random variables such that X Y then X Y. Proof: If X Y then the survival function of is smaller than that of for each 1 and, 0,1. This Shows that min, min, and max, max, This shows that min >, > min >, >,max >, > max >, > The above establishes that X Y and The proof is complete. Property 4.5. (Closed under increasing function) If X and Y be two non negative fuzzy random variables with have the survival functions and. If X Ythen, where >0 and where >0 Proof: Given X Y min min,min min min,min and max max,max max max,max min min >,min > min >,min > max max >,max > max >,max > min min >,min > min >,min > 249

6 D.Rajan and D.Vijayabalan maxmax >,max >. max >,max > REFERENCES 1. F.iche and D.Dubois, n extension of stochastic dominance to fuzzy random variables, Computational Intelligence for Knowledge-Based System design, ecture Notes in Computer science, 6178 (2010) D.Dubois, inear programming with fuzzy data. In J.C. Bezdek (Ed.), nalysis of fuzzy information Boca Ration: CRC Press (1978) H.Kwakernaak, Fuzzy random variables-i, Definitions and theorems, Information Sciences, 15 (1978) H.Kwakernaak, Fuzzy random variables-ii, lgorithms and examples for the discrete case, Information Sciences, 17 (1979) E..ehmann, Ordered families of distributions, nnals of Mathematical Statistics, 26 (1955) E..Piriyakumar and N.Renganathan, Stochastic ordering of fuzzy random variables, Information and Management Sciences, 12 (2001) M..Puri and D..Ralescu, Fuzzy random variables. Journal of Mathematical nalysis and pplications, 114 (1986) M.Shaked and J.G.Shanthikumar, Stochastic orders, 2 nd, edu. New York: Springer (2007). 9. H.J.Zimmermann, Fuzzy set theory, Wiley interdisciplinary Reviews: Computational Statistics, 2 (2010)