A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 152, 2000
G r a ż y n a T r zp i o t *
MULTIVALUED STOP-LOSS STOCHASTIC DOM INANCE TEST
Abstract. Stochastic Dominance tests can be employed to assist decision-makers in ordering uncertain alternatives. Thes tests require specification o f alternatives probability distributions and the assumption of the utility function of the decision-maker. With these assumptions, decision alternatives can be partitioned into classes by stochastic dominance or inverse stochastic dominance (stop-loss dominance). This paper notices procedures to identify this class of alternatives in case of multivalued probability distributions.
1. INTRODUCTION
S tochastic D om inance tests can be em ployed to assist d ecision-m akers in o rd e rin g u n ce rtain altern ativ e s. T hese tests re q u ire specificatio n o f alte rn a tiv e p ro b a b ility d istrib u tio n s an d th e a ssu m p tio n o f th e utility fu n ctio n o f the decision-m aker. W ith these assu m p tio ns decision altern ativ es can be p artitio n ed in to class by stochastic do m in an ce o r inverse sto chastic d o m in an ce (stop-loss dom inan ce) depending on describing m odel fo r gains o r fo r looses. T h is p a p e r n otices p ro c ed u re s to identify th is class o f altern atives in case o f m ultivalued probability d istrib ution s.
2. MULTIVALUED STOCHASTIC DOMINANCE
L et F and G be the cum ulative distrib u tio n s o f tw o d istinct u n ce rtain alternatives X and Y. X dom inates Y by first, second and th ird sto ch astic d o m in an ce (F S D , SSD , T S D ) if and only if
H j( x ) = F (x ) - G (x) < 0 for all x e [ a , b] ( X F S D Y) (1)
H z( x ) = $ H 1( y ) d y š O fo r all x e [ a , b] ( X S S D Y) (2) a
x
Я3(х) = $H2(y)dy < 0 for all x e [a, h], and E(F(x)) ^ E ( G ( x ) ) ( X T S D Y ) (3) T h e relatio n sh ip between the three stochastic d om inan ce rules can be sum m arised by the follow ing diagram : F S D => SSD => T S D , which m eans th at dom inance by FSD implies dom inance by SSD and dom inance by SSD in tu rn implies dom inance by TS D . F o r p ro o f o f FSD and SSD sec H a d a r and R u s s e l l [1969], H a n o c h and L e v y [1969] and R o t h s c h i l d and S t i g - l i t z [1970]. T h e criterion for T S D was suggested by W h i t m o r e [1970].
W hen we have am biguities in probabilities and outcom es, we have no single-valued d istrib u tio n , such a situation can be represented by a set o f probability distributions. Each family has two extrem e probability distributions the scalar o utcom e space X .
Definition 1. Low er prob ability d istributio ns for all values x (e X , are d en o ted by
P*(xd = I p(Áj) (4)
A ccording to this definition wc have: £ p * (x ,) = 1. /
Definition 2. U pper probability d istrib u tio n s for all values x , e X , are den o ted by
P*(x i) = I P(Aj) (5)
J. Jti - m ax {y. y e A j }
N ow we also have: £ Х ( х (-) = 1. i
In case o f the p oint values o f ra n d o m variables bo th d istrib u tio n s (low er a n d u p p e r p ro b a b ility d is trib u tio n s ) are ex actly th e sam e: P * ( x i) = P*(xi) = p(x;) and we have probability distributions in classical sense.
Exam ple 1. W e determ ine lower and upper p ro bability d istrib u tio n s for ran d o m variable X, whose outcom es are m ultivalued, included in som e intervals:
[2, 4] 13. 4] 14, 5] [5, 6]
P(Aj) 0.2 0.5 0.2 0.1
A ccording to the D efinitions 1 and 2 we have low er and u p p er p ro b a b ility d istrib u tio n s for random variable X:
x) 2 3 4 5 6
P'(Xj) 0.2 0.5 0.2 0.1
-P'(Xj) - - 0.7 0.2 0.1
W hen we have lower and upper probab ility d istrib u tio n s fo r ra n d o m variables, w hose outcom es are m ultivalued, we can ra n k such a sto ch astic alternative.
Let tw o distinct uncertain m ultivalued alternatives X and Y have low er cum ulative d istrib u tio n s F * ( x ) and G * ( x ) respectively, u pp er cum ulative d istrib u tio n s F*(x) and Gr(x), for x e [ a , b] respectively, th en we have m ultivalued first, second and third stochastic d om inan ce if and only if
I I l (x) = F> (x) - G*(x) < 0 , for all x e [a, b], (X F S D Y) (6)
X
H2(x) = \ I I l(y)dy < 0, for all x e [ a , b], (X S S D Y ) (7) a
x
H3(x) = j H 2(y)dy 4 o, lo r all x £ [a, b], (X T S D Y)
and F ( F . (x)) > E(G*(x)) (8)
F o r p ro o f see L a n g e w i s c h and C h o o b i n e h [1996].
Exam ple 2. ( T r z p i o t , 1998) Let us take a ra n d o m v ariab le С an d D w hose outcom es are m ultivalued, included in som e intervals as follows:
[0, 1] [1, 2] [2, 3] [3, 4]
p(C) - 0.2 0.4 0.4
/>(!>) 0.3 0.15 0.55
-W e determ ine low er and upper probability distributions fo r ra n d o m variables С and D as follows:
xi 0 1 2 3 4
C: p*(Xj) - 0.2 0.4 0.4
-C: p'(Xj) - - 0.2 0.4 0.4
D: />.(*,) 0.3 0.15 0.55 -
-D: p'(xj) - 0.3 0.15 0.55
-N o w we can receive the values o f lower an d upp er cum ulative d istrib u tio n s (see Fig. 1). XJ ( - oo, 0] (0, 1] (1, 2] (2, 3] (3, 4] (4. oo) C . ( Xj) 0 0 0.2 0.6 1 1 C'(xj) 0 0 0 0.2 0.6 1 D. (x, ) 0 0.3 0.45 1 1 1 D\ x j ) 0 0 0.3 0.45 1 1
Fig. 2. Upper and lower integrals of cumulated distributions of С and D
It is easy to check, th a t th e D efinition 1 an d the D efin ition 2 are n o t tru e (see F ig. 1 and. Fig. 2).
Fig. 3. Upper and lower double integrals of cumulated distributions of С and D
In this exam ple we can establish a th ird degree m ultivalued stoch astic dom inance: С T S D D (see Fig. 3.).
3. MULTIVALUED INVERSE STOCHASTIC DOMINANCE
D en o tin g the density function by / , we have the cum ulative d istrib u tio n F and inverse d istrib u tio n F by the follow ing ( G o o v a e r t s , 1984).
J f ( x ) d x = J d F ( x ) = J dF(x) = 1 (9)
—00 — oo —00
F ( x ) = \ - F ( x ) (10)
Let F and G be the inverse cum ulative d istrib u tio n s o f tw o distinct uncertain alternatives X and У X dom inates У by first, second and third inverse stochastic dom in an ce (F IS D , SISD , T IS D ) if and only if
H v(x) = F (x ) - G (x) ^ 0 for all x e [ a , b] ( X F IS D У ) (11)
H 2(x) = j H ^ d y ^ O for all x e [ a , b] ( X SISD У ) (12)
X
Ь
H 3(x) = j í í 2( y ) d y > 0 for all x e [ a , h]
X
and E(F( x)) > E(G(x)) (X T IS D Y ) (13) F o r p ro o f sec G o o v a e r t s [1984]. F irst degree stop-loss (inverse) dom inance coincides with first degree stochastic dom inance. T h e л-th degree stop-loss dom in an ce im plies higher degree stop-loss dom inance.
D enoting the density function by f , we have the lower inverse p ro bability d istrib u tio n s F » ( x ) an d upper inverse probability d istrib u tio n s F*(x) by the following:
F * ( x ) = l - F . ( x ) (14)
F*(x) = 1 — F*(x) (15)
Let tw o distinct uncertain m ultivalued alternatives X and Y have low er inverse p ro b a b ility d is trib u tio n s F * ( x ) an d G * (x ) resp ectiv ely , u p p e r inverse p ro b ab ility d istrib u tio n s F*(x) and G*(x), for x e [ a , b] respectively, then we have m ultivalued first, second and third inverse stochastic dom inance if and only if
H 2( x) = \ H t ( y ) d y ^ 0 for all x e [ a , h] (X S IS D Y ) (17)
X
b
Н ъ(х) = ß 2(y)dy < 0 for all x e [ a , h] (X T IS D Y )
X
and E ( F ( x ) ) > E ( G ( x ) ) (18)
T h e p ro o f proceeds p arallel to provided by G o o v a e r t s [1984] and L a n g e w i s c h a nd C h o o b i n c h [1996].
Exam ple 3. Let us tak e ran d o m variables A and В w hose ou tco m es are m ultivalued, included in som e intervals as follows:
4 H , -5] [-5, -2] [-5, -11 К -31 l - i . oi
P(A) - - - 0.5 0.5
P(B) 0.25 0.6 0.15 -
-W e determ ine lower and upper probability d istrib ution s for ran do m variables A an d В as follows: XJ -6 -5 -4 -3 -2 -1 0 A: p. ( xj ) - - 0.5 - - 0.5 -A: p'(xj) - - - 0.5 - 0.5 B: p. (xj) 0.25 0.75 - - - - -В p'(xj) - 0.25 - - 0.6 0.15
-N ow we can receive the values o f low er and u pp er cum ulative d istrib u tio n s.
XJ (-CO, -6] (-6. -5] (-5, -4] H . -3J (-3, -2] (-2, -1] (-1. 0] (0, oo)
A.(Xj) 0 0 0 0.5 0.5 0.5 1 • 1
A \ x j ) 0 0 0 0 0.5 0.5 0.5 1
B . ( x j ) 0 0.25 1 1 1 1 1 I
Fig. 4. Upper and lower inverse cumulated distributions of A and В
It is easy to check th a t th e D efinition 1 is n o t true. W e determ in e low er an d u p p er inverse cum ulative distrib u tio n s for ra n d o m variables A and В as follow s (Fig. 4): XJ (-00, -6] H , -5] (-5, -41 H . -31 (-3, -2] (-2, -1] (-1. 0] (0, oo) A .(xj) 1 1 1 0.5 0.5 0.5 0 0 A' ( xj ) 1 1 1 1 0.5 0.5 0.5 0 B. ( xj ) 1 0.75 0 0 0 0 0 0 B'(xj) 1 1 0.75 0.75 0.75 0.15 0 0
Fig. 6. Upper and lower integrals of inverse cumulated distributions of A and В
W e can observe th a t the form u las (16) an d (17) are n o t tru e (Fig. 4, 5). In this exam ple we can establish a th ird degree m ultivalued sto ch astic dom inance: A T I S D B (see Fig. 6).
4. CONCLUSIONS
S top-loss do m in an ce ran k in g is an extension o f stop-loss o rd erin g in analo gy to the stochastic dom inan ce. W e can apply tw o different ra n k in g by stochastic d om inance o r stop-loss d om inan ce which d epends on o u r goals, w hether we look for a good criterion fo r m axim ising gains o r for m inim ising losses. Som etim es we need b o th and this is a case o f o p e ra tio n research problem . T h e c u rren t use o f stochastic d om in an ce assum es th a t prob ability distributions are know n an d unique. In em pirical study specifying uniq ue p ro b ab ility m ay be unjustifiable, so we need m ultivalued view o f the p roblem . Sim ple exam ple is the co n tin u o u s o bserv atio n o f assets, each day we have price from m in to m ax price o f the day. W h en we w a n t to ra n k these assets we should apply m ultivalued stoch astic d o m in an ce or m ultivalued inverse (stop-loss) stochastic d om in ance d ep en din g on o u r prob lem . T h e extended m ultivalued sto chastic d om in an ce to m u ltiv alued inverse (stop-loss) stochastic d om inance, established here, pro v id e a v alu able techniqu e to p are do w n th e n u m b er o f alternatives.
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