Metodoloski zvezki, Vol. 11, No. 1, 2014, 21-29
Optimal Unbiased Estimates of P{X < Y} for Some Families of Distributions
Marko Obradovic, Milan Jovanovic, Bojana Milosevic1
Abstract
In reliability theory, one of the main problems is estimating parameter R = P{X <Y}. In this paper we shall present UMVUEs for R in different cases i.e. for different distributions of X and Y. Some of them are already existing and some are original.
1 Introduction
In reliability theory the main parameter is the reliability of a system, and its estimation is one of the main goals. The system fails if the applied stress X is greater than strength Y, so R is a measure of system performance. In most cases this parameter is given as R = P{X < Y}, although for some discrete cases the expression R = P{X < Y} is also considered.
The problem was first introduced by Birnbaum (1956). Since then numerous papers have been published. Most of results are presented in (Kotz et al., 2003). The vast majority of papers presuppose independence of stress and strength variables, as well as that they come from the same family of, in most cases continuous, distributions. There exists a wide range of applications in engineering, military, medicine and psychology.
The unbiasedness of an estimator is a desired property especially when dealing with relatively small sample sizes, where we cannot count on asymptotic unbiasedness. Since in many cases most popular estimators are biased, it is often important to find the unique minimum variance unbiased estimator (UMVUE).
1.1 UMVUE of R
Let X = (Xi,..., Xni) and Y = (Yi,..., Yn2) be the samples from the distributions of random variables X and Y. Then, using the following theorem we can construct UMVUEs.
Theorem 1 If V (X, Y) is any unbiased estimator of parameter 9 and T is a complete sufficient statistic for 9, then E (V (X, Y)|T) is the UMVUE of 9.
1 Faculty of Mathematics, University of Belgrade, Studenski trg 16, Belgrade, Serbia; email addresses: marcone@matf.bg.ac.rs, mjovanovic@matf.bg.ac.rs, bojana@matf.bg.ac.rs
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Marko Obradovic, Milan Jovanovic and Bojana Milosevic
This theorem is the combination of Rao-Blackwell and Lehmann-Scheffe theorems. Their proofs could be found in (Hogg et al., 2005).
However, in continuous case, the use of this theorem might not be technically convenient. Therefore, the following theorems were proposed to help deriving the UMVUEs (Kotz et al., 2003).
Theorem 2 Let 90 e 6 be an arbitrary value of 9 and let T be a complete sufficient statistic for 9. Denote by geo (t) and geo (t\Xi = xi,...,Xk = xk ,Yi = yi,...,Yk = yk) the pdf of T and the conditional pdf of T for given Xj = xj ,Yj = yj, j = l,...,k, respectively. Then the U MVU E of joint pdf fe (xi, ...,xk ,y1, ...,yk) is of the form
f(x i, ...,xk ,yi, ...,yk) =
nk , ,	(t\Xi = xi,...,Xk = xk,Yi = yi,...,Yk = yk)
j=i feo(xj ,yj >-geOxk-.
Theorem 3 The UMVUE of R is
R = J JI(x< y)f(x, y)dxdy,
where f is given in theorem 2 for k = 1.
2 Existing results
In this section we present a brief summary of existing results obtained for UMVUEs of R for some distributions.
• Exponential distribution
Let X and Y be independent exponentially distributed random variables with densities
fX(x; a) = ae-ax, x > 0, fY(y; $)= fie-l3y,y > 0, where a and $ are unknown positive parameters. The complete sufficient statistics
ni	n-2
for a and $ are Tx = E Xj and Ty = E Yj.
j=i j=i
The UMVUE of R was derived by Tong (1974; 1977), and it is given by
eWr(ni-ijr;su (ty)m, if ty < tx
R = { i=°
n^i(-i)i r(Sjrjn:-i) (ty) i, if ty >tx . i=0 v '
Optimal Unbiased Estimates of P{X < Y}
23
Normal distribution
Let X and Y be normally distributed independent random variables with densities
1	(x —Mi)2
fx(x; ^i, ai) = ,-^e	, x E R,

fY (y; ^2,a2) =
(y-M2)
2

nal
e 2ct22 , y e R,
where al, and al are unknown parameters. The complete sufficient statistics for O^i, al, m, al) are (X, SX, Y, Si).
The UMVUE of R was derived by Downtown (1973) and it is given by
R
b 11 B( 1
2' 2 / \ 2 2
1
q n i—4	Q -/¿2—4
(1 - ul)(1 - vl)dudv,
where B(a,b) is the beta function and
n = i(u,v) E [-1,1] x [-1,1]|- uSx 1) +vsy ^1) +(Y - XX) > 0
v/nr
v7^
Gamma distribution
Let X and Y be independent gamma distributed random variables with densities
fx(x; ai,ai) fY (y; al,al)
xai-i	x
r(ai)aa:
e CTi, x > 0,
y«2-i
e CT2 , y > 0,
r(«l)a2a2
where ai and al are known integer values and ai and al are unknown positive
ni
parameters. The complete sufficient statistics for ai and al are Tx = E Xj and
j=i
«2
Ty = e Yj. j=i
The UMVUE of R was derived by Constantine et al. (1986), and it is given by
R
(ra2-i)«2-i
1 - E
k=0
B(ai+a2+k,(ni - i)ai) B(ai,(ni — i)ai)B(a2,(n2 — i)a2) \ «2+k
y. A(n2-i)«2-i\ (-i)k (Tx\ 1 x V k J «2 +k \Ty J
if Ty < Tx
(ni- i)ai-i i- i-
B(«2+"i+k,(n2-i)«2)
X
B(a2,(n2-i)a2)B(ai,(ni-i)ai) ai+k
«i-M (-i)~ I Ty 1
k
k=0
/(ni-i)ai-i\ (— ^
n (—/^tyAa
) ai+k yTx )
if Ty > T-
x.
1
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Marko Obradovic, Milan Jovanovic and Bojana Milosevic
Gompertz distribution
Let X and Y be independent Gompertz distributed random variables with densities
-A1(ecx-1)
/X(x; c, Ai) = Aiecxe c ,x> 0,
-A2(ecy-1)
/y(y;c, A2) = A2ecye c ,y> 0,
where c is a known positive value and A1 and A2 are unknown positive parameters. The complete sufficient statisitcs for A1 and A2 are
1 «1	1 «2
Wx = - ^(ecX> - 1), Wy = - ^(ecyj - 1). C j=i C j=i
The UMVUE of R was derived by Saracoglu et al. (2009) and it is given by 1 - zW P^+g-k) (WY, if Wx <Wy
R _ /	fc=0
eV^r(rari)l)r(«2+k) (WY)k , if Wx > Wy. fc=0
Generalized Pareto distribution
Let X and Y be independent random variables from generalized Pareto distribution with densities
/x(x; ai, A) = aiA(1 + Ax)-(ai+1), x > 0, /y(y; a2,A) = a2A(1 + Ay)-(a2+1), y > 0,
where A is known positive value and ai and a2 are unknown positive
parameters. The complete sufficient statistics for parameters a1 and a2 are
«1 «2 Tx = £ ln(1 + Xj) and Ty = £ ln(1 + Yj). j=i j=i
The UMVUE of R was derived by Rezaei et al. (2010), and it is given by
«2 — 1 / \ k 1 - Z-(-1)k r(nr1+k)rC2-k) TY , if Tx < Ty
R = ( k=0
eW P(rar1(-n1)r(:2+k) (TY) k , if Tx > Ty. fc=0
Poisson distribution
Let X and Y be independent Poisson distributed random variables with mass functions
e-A1 A®
P{X = x; Ai} =-¡-1, x = 0,1,...,
x!
e-A2
P{Y = y; A2} =	, y = 0,1,...,
where A1 and A2 are unknown positive parameters. The complete sufficient statistics for A1
«1 «2 and A2 are Tx = £ Xj and Ty = £ Yj.
j=i j=i
Optimal Unbiased Estimates of P{X < Y}
25
The UMVUE of R was derived by Belyaev and Lumelskii (1988) and it is given by
R=......1CX) ^ t1 - £ ) ^
Negative binomial distribution
Let X and Y be indepent random variables from negative binomial distributions with mass functions
P{X = x;mi,pi} =	- ^pf(1 -pi)m1, x = 0,1,...,
P {Y = y; m2,p2} = ^ +y - ^ py2(1 - p2)m2, y = 0,1,...,
where m1 and m2 are known integer values and pi and p2 are unknown probabilities. The
ni	n2
complete sufficient statistics for pi and p2 are Tx = Xj and Tv = Yj.
j=i j=i
The UMVUE of R was derived by Ivshin and Lumelskii (1995) and it is given by
min{Tx,Ty-i} Ty (m1+x-i\(TX-x+m1(n1-i)-i\ (m2+y-i\(Ty-y+m,2(n2-i)-i\ n =	V^	V^ I x A	Tx-x	) V y A	Ty-y	)
R =	2-^t	(m1«1+Tx-i\	(m2U2+Ty-i\
x=0 y=x+i	V TX J	V Ty J
3 New results
In this section we shall derive the UMVUE of R for some new distributions. The first model is where stress and strength both have Weibull distribution with known but different shape parameter and unknown rate parameters. As a special case we present the model where stress has exponential and strength has Rayleigh distribution. An example with real data for Weibull model is presented. In the second model, both stress and strength have logarithmic distribution with unknown parameters.
3.1 Weibull model
Let X and Y be independent random variables from Weibull distribution with densities
fx(x; ai,ai) = ana?1 x21-ie-(a1x)a1, x > 0, fv(y; a2, a2) = «2a2a2ya2-ie-(CT2y)a2, y > 0.
The Weibull distribution is one of the most used distribution in modeling life data. Many researchers have studied the reliability of Weibull model. Most of them did not consider unbiased estimators (e.g. Kundu and Gupta, 2006), and recently the case with common known shape parameter a has been studied in (Amiri et al., 2013).
We consider the case where shape parameters ai and a2 are known positive values, while rate parameters ai and a2 are unknown positive parameters.
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Marko Obradovic, Milan Jovanovic and Bojana Milosevic
«1
The complete sufficient statistics for parameters ai and a2 are TX = E X®1 and
j=i
«2
TY = E Yj*2. Since Xai and Ya2 have exponential distributions with rate parameters j=i j
a^1 and a^2, both statistics have gamma distribution, i.e. TX has r(n1, ) and TY has
k
r(n2,a-a2). Similarly, for k < min(ni,n2), Tx — E Xf has r(n - k,a-ai) and
j=i
k
TY — E Y"2 has r(n2 — k,a-a2). Using this and transformation of random variables
j=i
«i
(Xi,...,Xk , E Xf) to (Xi,...,Xk ,Tx ) we get, for ai = 1,
j=k+i
(tx — E xf )ni-k-1 , ai^ fc
g(tx|Xi = xi,...,Xk = Xk) =-rj(ni — k)-e j=i I{tx xJ1 }.
Using theorem 2, we get that
k (tx — E xj1 )ni-k-ir(ni) k /(xi,...,xk) = ak n x"i-i-t«=-ir(n — k)-1 {tx > £ x? }.
For k =1 we obtain that
r«l)ni-2
(t _ x°l )r
/(x) = ai(ni - 1)xai-1 ( X,- )wi-i I{tx > xai}.
(tx )n
Analogously we get that
^ (tv _ y/«2 )n2 2
f(y) = «2(^2 — 1)ya2-i ( Yy«2-i I{t- > ya2}.
i i
Denote M = min{t^^1, t-2 }. Using the independence of samples and the theorem 3, we obtain
R = J J I{x < y}/(x)/(y)dxdy 0 0
i
M	ty2
= / "'"i " 1)("2	- x")n'-2dx j a«r-HY - y )-2dy
0	x
0
M
f ai("i — 1)xai-i
-itn2-i-(tX
X tY
t„i-it„2-i (tx — xai )ni-2(tY — xa2 )n2-idx. tt
Optimal Unbiased Estimates of P{X < Y}
27
Now applying the binomial formula we obtain that the UMVUE of R is
ao s
ni — 2 n2 -1
R
sp sp ( —1)r+sai(ni —1) /ni —2\/n2 —TXl.1 ' ' «i(r+1)+«2s V r / V s
r=0 s=0 ni — 2 n2 — 1
^i (r + i)
v^	( —1)r+s«i(ni —1) /ni — 2)(n2 — ^ TY
^ ^ ai(r+1)+a2s V r / V r=0 s=0
2
T r+i
if Tai < Ty2
i i if tj^1 > T^2
(3.1)
3.1.1 Exponential-Rayleigh model
As a special case of Weibull model we have a model where X has exponential and Y has Rayleigh distribution with densities
fx(x; a) = ae—ax, x > 0,
fy (y; /3) = 2P2ye
2ye—^2y2
y > 0,
where a and 3 are unknown positive parameters. The complete sufficient statistics for a
ni	n2
and 3 are Tx = E Xj and Ty = E Y^2.
j=1 j=1 The UMVUE of R is
E2	1 (nir—2)(n2s—1)(TX)s, if TX < VT^
R
r=0 s=0 ni —2 n2 —1
e (—;rr+)+i;1) (nir—2)(n2s—1)(^tty)r+1, if tx > vTy.
r=0 s=0
3.1.2 Numerical example
Here we present an example with real data. We wanted to compare daily wind speeds in Rotterdam and Eindhoven. We obtained two samples of 30 randomly chosen daily wind speeds (in 0.1 m/s) from the period of April 1st 2010 to April 1st 2014 taken from the website of Royal Netherlands Meteorological Institute. The first sample is from Rotterdam and the second one is from Eindhoven:
Rotterdam (X): 48, 15, 27, 18, 40, 26, 84, 19, 35, 32, 55, 29, 45, 51, 47, 66, 38, 13, 39, 28, 50, 36, 15, 74, 53, 85, 18, 58, 18, 48.
Eindhoven (Y): 44, 25, 43, 35, 20, 59, 25, 38, 26, 15, 37, 16, 35, 17, 34, 27, 40, 37, 33, 17, 51, 50, 33, 52, 25, 21, 34, 39, 23, 60.
It is well known that wind speed follows Weibull distribution. To check this we used Kolmogorov-Smirnov test. Since this test requires that the parameters may not be estimated from the testing sample, we estimated them beforehand using some other larger samples from the same populations. We got that X follows Weibull distribution with shape parameter a = 2.8 and rate parameter a = 1/47 (Kolmogorov-Smirnov test statistics is 0.157 and the p-value is greater than 0.1), while Y follows Weibull distribution with shape parameter a = 2.6 and rate parameter a = 1/41 (Kolmogorov-Smirnov test statistics is 0.158 and the p-value is greater than 0.1).
Finally, using (3.1) we estimated the probability that the daily wind speed is lower in Rotterdam than in Eindhoven to be f =0.32.
i
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Marko Obradovic, Milan Jovanovic and Bojana Milosevic
3.2 Logarithmic distribution
Let X and Y be independent random variables from logarithmic distribution with mass functions
— 1
P {X = x; p} = —-r-, x =1, 2,...
ln(1 — p) x
-1 qy
P {Y== Rr—^ 7 •»=1-2-
where p and q are unknown probabilities.
The logarithmic distribution has application in biology and ecology. It is often used for modeling data linked to the number of species.
ni	«
The complete sufficient statistics for p and q are Tx = Xj and Ty = Y.
j=i j=i The sum of n independent random variables with logarithmic distributions with the same parameter p has Stirling distribution of the first kind SDFK(n,p) (Johnson et al., 2005), so Tx has SDFK(n1,p) and Ty has SDFK(n2, q) with the following mass functions
P{Tx = x; ni,p} = x!(— ln(1 — p))«, x = ni,ni + 1,...,
P {Ty = y; n2,q} = y!(— ln(1 — q))«, y = ^ + 1 —
where s(x, n) is Stirling number of the first kind. An unbiased estimator for R is I{Xi < Yi}. Since
E(I{Xi < Yi}|Tx = tx,Ty = ty) = P{X' <rYi=X =>ry = tY}
P {Tx = tx, T Y = ty }
M ty —«2 +i	«1	«2
E E P{Xi = x}P{Yi = y}P{ E Xfc = tx — x}P{E Y = ty — y}
x=i y=x+i	fc=2	l=2
=	P {Tx = tx }P {Ty = ty }
= Mty—tx!ty!|s(tx — x,ni — 1)||s(ty — y,n — 1)|
y=X+ nin2 (tx — x)!(tY — y)!xy|s(tx, ni) ||s(tY ,n2)|,
where M = min{tx — ni + 1, tY — n2}, using theorem 1 we get that the UMVUE of R is
min{Tx —ni + i,Ty—«2} TY—n2+i
Tx!Ty!|s(Tx - x,ni - 1)||s(TY - y,n - 1)| x=1	y=x+1 n1n2(TX - x)!(TY - y)!xy|s(TX,n1)||s(TY,n2)|
R = £ £
4 Conclusion
In this paper we considered the unbiased estimation of the probability P{X < Y} when X and Y are two independent random variables. Some known results of UMVUEs for R for some distributions were listed. Two new cases were presented, namely Weibull model with known but different shape parameters and unknown rate parameters and Logarithmic model with unknown parameters. An example using real data was provided.
Optimal Unbiased Estimates of P{X < Y}
29
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