- Email: [email protected]
Preservation of classes of life distributions under weighting with a general weight function
Statistics and Probability Letters 78 (2008) 3056–3061
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Preservation of classes of life distributions under weighting with a general weight function Paweł Błażej ∗ University of Wrołcaw, Poland
article
a b s t r a c t
info
Article history: Received 12 April 2006 Received in revised form 20 March 2008 Accepted 1 May 2008 Available online 28 May 2008
In the paper a representation of weighted distributions is used to obtain some results concerning their relations with life distributions. Extensions of the results by Gupta [Gupta, R.C., Keating, J.P., 1986. Relations for the reliability measures under length biased sampling. Scand. J. Statist. 13, 49–56], Jain et al. [Jain, K., Singh, H., Bagai, I., 1989. Relations for the reliability measures of weighted distributions. Comm. Statist. Theory Methods 18, 4393–4412] and Bartoszewicz and Skolimowska [Bartoszewicz, J., Skolimowska, M., 2006. Preservation of classes of life distributions and stochastic orders under weighting. Statist. Probab. Lett. 76, 587–596] are given. © 2008 Elsevier B.V. All rights reserved.
MSC: 60E15 60E07 62N05
1. Preliminaries Let X , Y be nonnegative random variables with distribution functions F , G and let f , g be their density functions if they exist. Denote by F the tail (survival function) of F , by F −1 (u) = inf{x : F (x) ≥ u} the quantile (or inversed) function of F . We use increasing in place of nondecreasing and decreasing in place of nonincreasing. An absolutely continuous distribution F is said to be IFR (DFR) if the hazard rate function of F , rF (x) = f (x)/F (x), is increasing (decreasing). A distribution F is said to be DMRL (IMRL), if the function mX (t ) =
E (X − t |X > t ), 0
t < t ∗; otherwise,
where t ∗ = sup{t : F¯ (t ) > 0}, is decreasing (increasing) in t . A distribution F with F (0) = 0 is said to be IFRA (DFRA) if F −1 (u)/[− log(1 − u)] is decreasing (increasing) in u ∈ (0, 1). A distribution F is said to be NBU (NWU) if F (x + y) ≤ (≥)F (x)F (y) for all x, y, x + y from support of F . It is well known that IFR ⊂ IFRA ⊂ NBU
and
DFR ⊂ DFRA ⊂ NWU
and IFR ⊂ DMRL,
and IMRL ⊂ DFR.
Let w : R → R be a function for which 0 < E [w(X )] < ∞. Then +
Fw (x) =
∗
1
Z
x
E [w(X )] −∞
w(z )dF (z )
Corresponding address: University of Wrocław, Mathematical Institute, Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland. E-mail address: [email protected].
0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.05.028
(1)
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
3057
is a distribution function, called the weighted distribution associated with F . If the distribution F has the density f , then fw (x) =
w(x)f (x) E [w(X )]
is the density of Fw . The idea of weighted distributions is due to Fisher (1934). Rao (1965) defined weighted distributions with a general weight function w . Patil and Rao (1977, 1978), gave some statistical models leading to weighted distributions and applied their results to the analysis of data relating to human populations and ecology. Gupta and Keating (1986) considered relations for reliability measures of the length-biased distributions. Jain et al. (1989) studied relations for reliability measures of weighted distributions. Bartoszewicz and Skolimowska (2006) obtained some results about preservation of classes of life distributions and stochastic orders under weighting. They used a representation of weighted distributions by the Lorenz curve. 2. Results 2.1. New representation of weighted distribution We use a representation of weighted distribution being a composition of two distribution functions. Lemma 1. For any distribution function F and a weight function w , the weighted distribution function Fw is of the form Fw (x) = F ∗ (F (x)), where F ∗ (u) =
u
Z
1 E [w(X )]
w(F −1 (z ))dz ,
u ∈ (0, 1).
0
Proof. Putting z = F −1 (u) in (1), we have Fw (x) =
F (x)
Z
1 E [w(X )]
w(F −1 (u))du =
0
F (x)
Z
f ∗ (u)du,
0
where f ∗ (u) = w(F −1 (u))/E [w(X )] is a density function on (0, 1), i.e. it is the density of F ∗ .
We give some examples of the new representation of weighted distributions. Example 1 (Length-Biased Distributions). Let F be a distribution function and let w(x) = x be the weight function, then Fw is called the length-biased distribution. It is obvious that the function F −1 (u)/E (X ) is a density on (0, 1). Example 2 (Weighted Distributions and the Lorenz Curve). Denote by LX (p) =
p
Z
1 E (X )
F −1 (u)du,
0 ≤ p ≤ 1,
0
the Lorenz curve of X and L¯ X = 1 − LX . It is well known that LX is a distribution function and is convex. Bartoszewicz and Skolimowska (2006) observed that: (a) if w is increasing left continuous, then F w ( x) =
1 E (U )
Z
F (x)
w(F −1 (u))du = LU (F (x)),
0
(b) if w is decreasing left continuous, then F w ( x) =
1 E (U )
Z
F (x)
w(F −1 (u))du = L¯ U (F (x)),
0
where U = w(X ). It is evident that in these cases f ∗ (u) = w(F −1 (u))/E (U ) is a density on (0, 1). Example 3 (Order Statistics). Let X1:n , X2:n , . . . , Xn:n be order statistics in a sample X1 , X2 , . . . , Xn from an absolutely continuous distribution F . It is well known that the density fi:n of Xi:n , i = 1, 2 . . . , n, is of the form fi:n (x) =
n! F i−1 (x)[1 − F (x)]n−i f (x), (i − 1)!(n − i)!
i = 1, 2, . . . , n,
3058
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
and hence this distribution may be considered as a weighted one with the weight function wi (x) = F i−1 (x)[1 − F (x)]n−i . On the other hand, fi:n (x) = f ∗ (F (x))f (x), where f ∗ (u) =
n! ui−1 (1 − u)n−i (i − 1)!(n − i)!
is the density of the beta distribution B(i, n − i + 1). Many authors studied relations between the weighted distribution and the original one in terms of stochastic orders. We give some theorems about relations between these distributions using F ∗ . In our first step we give some definitions. We say that X is stochastically smaller than Y (F ≤st G) if F (x) ≥ G(x). A random variable X is smaller than Y in hazard rate order (F ≤hr G) if G(x)/F (x) is increasing. We say that X is smaller than Y in the reversed hazard order (F ≤rh G) if G(x)/F (x) is increasing. We say that X is smaller than Y in the likelihood ratio order (F ≤lr G) if g (x)/f (x) is increasing. Thus we can formulate the following theorem which is inspired by Theorem 1 in Lehmann and Rojo (1992). Theorem 1. (a) F ≤lr Fw , (Fw ≤lr F ) iff F ∗ is convex (concave) on (0, 1); (b) F ≤hr Fw , (Fw ≤hr F ) iff 1 − F ∗ (1 − u) is anti-star shaped (star shaped); (c) F ≤rh Fw , (Fw ≤rh F ) iff F ∗ is star shaped (anti-star shaped) ; (d) F ≤st Fw , (Fw ≤st F ) iff F ∗ (u) ≤ u (F ∗ (u) ≥ u) on (0, 1). Proof. The proof of (a) follows easily, F ∗ is convex (concave) if and only if f ∗ (u) is increasing (decreasing) which is equivalent to F ≤lr Fw , (Fw ≤lr F ). To prove (b) note that 1 − F ∗ (1 − u) is anti-star shaped (star shaped) if and only if [1 − F ∗ (F (x))]/F (x) is an increasing (decreasing) function of x for any distribution F which is equivalent to F w (x)/F (x) being an increasing (decreasing) function of x. The proof of (c) follows clearly as F ∗ is star shaped (anti-star shaped) if and only if F ∗ (F (x))/F (x) is an increasing (decreasing) function of x for any distribution F which is equivalent to Fw (x)/F (x) being an increasing (decreasing) function. The proof of (d) is obvious. 2.2. Weighted distributions with monotone failure rate We write b F instead of Fw , where b F (x) = F ∗ (F (x)) for some F ∗ (depending on w , which is given). Consider the failure rate of the weighted distribution b F: rb F ( x) =
f ∗ (F (x))f (x) 1 −b F ( x)
.
Gupta and Keating (1986) proved that if F is IFR, then the length-biased distribution b F is also IFR. Jain et al. (1989) showed that if the weight function w is increasing and concave and F is IFR, then b F also is IFR. Let b F be the weighted distribution associated with F . Then t
Z
1 −b F (t ) = exp −
0
t
Z
rb F (x)dx = exp −
0
rb F ( x)
r F ( x)
rF (x)dx .
Putting c (F (x)) = rb F (x)/rF (x), we have F (t )
Z t Z b F (t ) = 1 − exp − c (F (x))rF (x)dx = 1 − exp − 0
0
c (z ) 1−z
dz .
Hence c (z )
u
Z
F ∗ (u) = 1 − exp −
0
1−z
dz ,
where c (z ) is a positive function on (0, 1). Remark 1. If
R1 0
c (z )/(1 − z )dz = +∞, then F ∗ (u) is a distribution function on (0, 1).
It is evident that
Z
f ∗ (u) = exp −
0
u
c (z ) 1−z
dz
c ( u) 1−u
.
The next theorem gives a representation of the density f ∗ for which the weighted distribution preserves the IFR (DFR) property.
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
3059
Theorem 2. If F is an absolutely continuous IFR (DFR) distribution and c (z ) is a increasing (decreasing) positive function on R1 (0, 1) such that 0 c (z )/(1 − z )dz = +∞, then b F (x) = F ∗ (F (x)) is IFR (DFR). Proof. It suffices to make the following observation: rb F (x) =
f ∗ (F (x))f (x) 1 − F ∗ (F (x))
h R F (x)
exp −
=
0
c (z ) dz 1 −z
h R F (x) exp − 0
i
c (F (x)) f 1−F (x)
c (z ) dz 1 −z
(x)
i
= c (F (x))rF (x),
which is increasing (decreasing) because of the hypothesis of the theorem.
Remark 2. It is evident that: (a) every positive constant function can be c (z ); (b) every measurable positive increasing function on (0, 1) can be c (z ); (c) every measurable positive decreasing function g for which g (x) ≥ a > 0 can be c (z ). We consider some examples of the function c (z ). Example 4. Let F be the Weibull distribution F (x) =
0, 1 − exp(−xβ ),
x ≤ 0; x > 0, β > 1.
which is obviously IFR. Let c (u) = [− log(1 − u)]α , where α > 0. Since c (u) is increasing by Theorem 2, we have that b F is IFR. In this case F ( x)
[− log(1 − z )]α dz 1−z 0 Z 1 [− log(v)]α = 1 − exp − dv v 1−F (x) Z − log(1−F (x)) = 1 − exp − yα dy = 1 − exp −
Z b F (x) = 1 − exp −
0
1
α+1
[− log(1 − F (x))]α+1
thus
b F (x) =
0,
x ≤ 0;
1 − exp −
1
α+1
β(α+1)
x
,
x > 0, α > 0, β > 1. 1
is a Weibull distribution with the scale parameter (α + 1) β(α+1) and the shape parameter β(α + 1). On the other hand the distribution b F may be considered as a weighted distribution associated with the Weibull distribution and has the weight function
w(x) = xαβ exp −
1
α+1
xβ(α+1) + xβ .
Notice that this weight function does not satisfy the assumptions of the theorems of Gupta and Keating (1986) and Jain et al. (1989). We will consider some well-known theorems about preservation of the IFR property in terms of c (u) function. Example 5. Gupta and Keating (1986) proved that if F is an IFR distribution, then the length-biased distribution b F is also IFR. In this case we have, F −1 (u)(1 − u) c (u) = R 1 , F −1 (z )dz u hence 1 c (u)
=
1 1−u
R1 u
F −1 (z )dz − F −1 (u) F −1 (u)
+ 1.
3060
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
Thus c (u) is increasing only if: 1
Z
1 1−u
F −1 (z )dz − F −1 (u) = mX (F −1 (u)) u
is decreasing in u ∈ (0, 1) which obviously holds (because F is DMRL), and b F is IFR also by Theorem 2. Example 6. Jain et al. (1989) proved that if F is IFR, w(x) is increasing and concave, then b F is IFR (DFR). The main idea of the proof of Jain et al. (1989) consists of showing that A(x) − w(x) is decreasing in x ∈ (0, ∞), where A(x) = E [w(X )|X > x]. It is evident that in this case
w(F −1 (u))(1 − u) c ( u) = R 1 , w(F −1 (z ))dz u thus c (u) is increasing only if 1
Z
1 1−u
w(F −1 (z ))dz − w(F −1 (u)) u
is decreasing which is equivalent to A(x) − w(x) is decreasing. Remark 3. Jain et al. (1989) also observed that if F is IFR and w(x)F (x) is increasing, then b F is IFR. It is clear that this condition implies that c (u) is an increasing function of u ∈ (0, 1). 2.3. Preservation of IFRA (DFRA) and NBU (NWU) classes under weighting Bartoszewicz and Skolimowska (2006) proved preservation of the IFRA (DFRA) and NBU (NWU) classes under weighting assuming continuity. We give some extension of their results in terms of c (u) function. Theorem 3. (a) If F is IFRA (NBU) and c (u) is increasing then b F is IFRA (NBU); (b) If F is DFRA (NWU) and c (u) is decreasing then b F is DFRA (NWU). Proof. The proof follows by using the fact that rb F (x) = c (F (x))rF (x) and using the method of the proof of Theorem 3 of Bartoszewicz and Skolimowska (2006). 2.4. Preservation of NBU (NWU) classes under weighting Bartoszewicz and Skolimowska (2006) proved preservation of NBU (NWU) classes under weighting assuming a submultiplicative (supermultiplicative) function LU where U = w(X ). We reformulate their theorem using some assumptions on F ∗ but first we prove the following lemma. Lemma 2. Let F ∗ be a continuous distribution function on (0, 1). If Fq∗ /F ∗ is nondecreasing where Fq∗ (u) = F ∗ (qu) for any q ∈ (0, 1), then F ∗ (pq) ≤ F ∗ (p)F ∗ (q), p ∈ [0, 1]. Proof. The proof of the lemma is similar to that of Theorem 6 in Bartoszewicz and Skolimowska (2006). It is obvious that Fq∗ (u) F ∗ ( u)
is decreasing in u ∈ [0, 1/q],
implies F ∗ (pq) F ∗ (p)
≤
F ∗ (q) F ∗ (1)
.
Thus we can formulate a theorem about preservation of the NBU property. Theorem 4. Let X be a nonnegative random variable with the distribution function F . If F is NBU and Fq∗ /F ∗ is decreasing, then
b F is NBU. Proof. The proof is similar to that of Theorem 5 in Bartoszewicz and Skolimowska (2006) by observing that LU = F ∗ with U = w(x).
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
3061
Remark 4. Bartoszewicz and Skolimowska (2006) proved a theorem about preservation of the NWU property in the case when the weight function w is decreasing and such that the Lorenz curve LU is supermultiplicative i.e. LU (pq) ≥ LU (p)LU (q)
(2)
for every p, q ∈ [0, 1]. They observed that condition (2) is satisfied by the hyperbolic Lorenz curve considered by Aggarwal and Singh (1984), see also Arnold (1986): Lθ (p) =
(1 − θ )2 p , ((1 + θ )2 − 4θ p)
θ ∈ (0, 1).
(3)
We give an extension of their observation. Example 7 (Preservation of the NWU Property Under the Special Class of F ∗ ). Let X be a nonnegative random variable with the distribution function F . If F is NWU and
(1 − θ )2 (1 − u) ((1 + θ )2 − 4θ (1 − u))
F ∗ (u) = 1 −
(4)
then b F is NWU. It is easy to see that function (4) is the distribution on (0, 1) for any θ ∈ (0, 1). We obtain 1 −b F (x + t ) = 1 − F ∗ (F (x + t )) =
(1 − θ )2 F (x + t ) ((1 + θ )2 − 4θ F (x + t ))
= Lθ (F (x + t )).
By the definition of the NWU property of F and supermultiplicativity (3) we get Lθ (F (x + t )) ≥ Lθ (F (x)F (t )) ≥ Lθ (F (x))Lθ (F (t ))
= (1 − F ∗ (F (x)))(1 − F ∗ (F (t ))) = (1 − b F (x))(1 − b F (t )), i.e b F is also NWU. Acknowledgments The author is very grateful to the referee for the valuable comments. This research was supported by the Ministry of Higher Education and Science, Poland, Grant N201 046 31/3733. References Aggarwal, V., Singh, R., 1984. On optimum stratification with proportional allocation for a class of Pareto distributions. Commun. Stat. Theory Methods 13, 3107–3116. Arnold, B.C., 1986. A class of hyperbolic Lorenz curves. The Indian J. Stat. 48, B, 427–436. Bartoszewicz, J., Skolimowska, M., 2006. Preservation of classes of life distributions and stochastic orders under weighting. Statist. Probab. Lett. 76, 587–596. Fisher, R.A., 1934. The effects of methods of ascertainment upon the estimation of frequencies. Ann. Eugenics 6, 13–25. Gupta, R.C., Keating, J.P., 1986. Relations for the reliability measures under length biased sampling. Scand. J. Statist. 13, 49–56. Jain, K., Singh, H., Bagai, I., 1989. Relations for the reliability measures of weighted distributions. Commun. Stat. Theory Methods 18, 4393–4412. Lehmann, E.L., Rojo, J., 1992. Invariant directional orderings. Ann. Statist 20 (4), 2100–2110. Patil, G.P., Rao, C.R., 1977. The weighted distributions: A survey and their applications. In: Krishnaiah, P.R. (Ed.), Applications of Statistics. North-Holland Publ. Co, Amsterdam, pp. 383–405. Patil, G.P., Rao, C.R., 1978. Weighted distributions and size biased sampling wit applications to wild-life populations and human families. Biometrica 34, 179–189. Rao, C.R., 1965. Weighted distributions arising out of methods of ascertainment: What population does a sample represent? In: Atkinson, A.C., Fienberg, S.E. (Eds.), Celebration of Statistics: The ISI Centenary Volume. Springer Verlag, New York, pp. 543–569.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Preservation of classes of life distributions under weighting with a general weight function Paweł Błażej ∗ University of Wrołcaw, Poland
article
a b s t r a c t
info
Article history: Received 12 April 2006 Received in revised form 20 March 2008 Accepted 1 May 2008 Available online 28 May 2008
In the paper a representation of weighted distributions is used to obtain some results concerning their relations with life distributions. Extensions of the results by Gupta [Gupta, R.C., Keating, J.P., 1986. Relations for the reliability measures under length biased sampling. Scand. J. Statist. 13, 49–56], Jain et al. [Jain, K., Singh, H., Bagai, I., 1989. Relations for the reliability measures of weighted distributions. Comm. Statist. Theory Methods 18, 4393–4412] and Bartoszewicz and Skolimowska [Bartoszewicz, J., Skolimowska, M., 2006. Preservation of classes of life distributions and stochastic orders under weighting. Statist. Probab. Lett. 76, 587–596] are given. © 2008 Elsevier B.V. All rights reserved.
MSC: 60E15 60E07 62N05
1. Preliminaries Let X , Y be nonnegative random variables with distribution functions F , G and let f , g be their density functions if they exist. Denote by F the tail (survival function) of F , by F −1 (u) = inf{x : F (x) ≥ u} the quantile (or inversed) function of F . We use increasing in place of nondecreasing and decreasing in place of nonincreasing. An absolutely continuous distribution F is said to be IFR (DFR) if the hazard rate function of F , rF (x) = f (x)/F (x), is increasing (decreasing). A distribution F is said to be DMRL (IMRL), if the function mX (t ) =
E (X − t |X > t ), 0
t < t ∗; otherwise,
where t ∗ = sup{t : F¯ (t ) > 0}, is decreasing (increasing) in t . A distribution F with F (0) = 0 is said to be IFRA (DFRA) if F −1 (u)/[− log(1 − u)] is decreasing (increasing) in u ∈ (0, 1). A distribution F is said to be NBU (NWU) if F (x + y) ≤ (≥)F (x)F (y) for all x, y, x + y from support of F . It is well known that IFR ⊂ IFRA ⊂ NBU
and
DFR ⊂ DFRA ⊂ NWU
and IFR ⊂ DMRL,
and IMRL ⊂ DFR.
Let w : R → R be a function for which 0 < E [w(X )] < ∞. Then +
Fw (x) =
∗
1
Z
x
E [w(X )] −∞
w(z )dF (z )
Corresponding address: University of Wrocław, Mathematical Institute, Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland. E-mail address: [email protected].
0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.05.028
(1)
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
3057
is a distribution function, called the weighted distribution associated with F . If the distribution F has the density f , then fw (x) =
w(x)f (x) E [w(X )]
is the density of Fw . The idea of weighted distributions is due to Fisher (1934). Rao (1965) defined weighted distributions with a general weight function w . Patil and Rao (1977, 1978), gave some statistical models leading to weighted distributions and applied their results to the analysis of data relating to human populations and ecology. Gupta and Keating (1986) considered relations for reliability measures of the length-biased distributions. Jain et al. (1989) studied relations for reliability measures of weighted distributions. Bartoszewicz and Skolimowska (2006) obtained some results about preservation of classes of life distributions and stochastic orders under weighting. They used a representation of weighted distributions by the Lorenz curve. 2. Results 2.1. New representation of weighted distribution We use a representation of weighted distribution being a composition of two distribution functions. Lemma 1. For any distribution function F and a weight function w , the weighted distribution function Fw is of the form Fw (x) = F ∗ (F (x)), where F ∗ (u) =
u
Z
1 E [w(X )]
w(F −1 (z ))dz ,
u ∈ (0, 1).
0
Proof. Putting z = F −1 (u) in (1), we have Fw (x) =
F (x)
Z
1 E [w(X )]
w(F −1 (u))du =
0
F (x)
Z
f ∗ (u)du,
0
where f ∗ (u) = w(F −1 (u))/E [w(X )] is a density function on (0, 1), i.e. it is the density of F ∗ .
We give some examples of the new representation of weighted distributions. Example 1 (Length-Biased Distributions). Let F be a distribution function and let w(x) = x be the weight function, then Fw is called the length-biased distribution. It is obvious that the function F −1 (u)/E (X ) is a density on (0, 1). Example 2 (Weighted Distributions and the Lorenz Curve). Denote by LX (p) =
p
Z
1 E (X )
F −1 (u)du,
0 ≤ p ≤ 1,
0
the Lorenz curve of X and L¯ X = 1 − LX . It is well known that LX is a distribution function and is convex. Bartoszewicz and Skolimowska (2006) observed that: (a) if w is increasing left continuous, then F w ( x) =
1 E (U )
Z
F (x)
w(F −1 (u))du = LU (F (x)),
0
(b) if w is decreasing left continuous, then F w ( x) =
1 E (U )
Z
F (x)
w(F −1 (u))du = L¯ U (F (x)),
0
where U = w(X ). It is evident that in these cases f ∗ (u) = w(F −1 (u))/E (U ) is a density on (0, 1). Example 3 (Order Statistics). Let X1:n , X2:n , . . . , Xn:n be order statistics in a sample X1 , X2 , . . . , Xn from an absolutely continuous distribution F . It is well known that the density fi:n of Xi:n , i = 1, 2 . . . , n, is of the form fi:n (x) =
n! F i−1 (x)[1 − F (x)]n−i f (x), (i − 1)!(n − i)!
i = 1, 2, . . . , n,
3058
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
and hence this distribution may be considered as a weighted one with the weight function wi (x) = F i−1 (x)[1 − F (x)]n−i . On the other hand, fi:n (x) = f ∗ (F (x))f (x), where f ∗ (u) =
n! ui−1 (1 − u)n−i (i − 1)!(n − i)!
is the density of the beta distribution B(i, n − i + 1). Many authors studied relations between the weighted distribution and the original one in terms of stochastic orders. We give some theorems about relations between these distributions using F ∗ . In our first step we give some definitions. We say that X is stochastically smaller than Y (F ≤st G) if F (x) ≥ G(x). A random variable X is smaller than Y in hazard rate order (F ≤hr G) if G(x)/F (x) is increasing. We say that X is smaller than Y in the reversed hazard order (F ≤rh G) if G(x)/F (x) is increasing. We say that X is smaller than Y in the likelihood ratio order (F ≤lr G) if g (x)/f (x) is increasing. Thus we can formulate the following theorem which is inspired by Theorem 1 in Lehmann and Rojo (1992). Theorem 1. (a) F ≤lr Fw , (Fw ≤lr F ) iff F ∗ is convex (concave) on (0, 1); (b) F ≤hr Fw , (Fw ≤hr F ) iff 1 − F ∗ (1 − u) is anti-star shaped (star shaped); (c) F ≤rh Fw , (Fw ≤rh F ) iff F ∗ is star shaped (anti-star shaped) ; (d) F ≤st Fw , (Fw ≤st F ) iff F ∗ (u) ≤ u (F ∗ (u) ≥ u) on (0, 1). Proof. The proof of (a) follows easily, F ∗ is convex (concave) if and only if f ∗ (u) is increasing (decreasing) which is equivalent to F ≤lr Fw , (Fw ≤lr F ). To prove (b) note that 1 − F ∗ (1 − u) is anti-star shaped (star shaped) if and only if [1 − F ∗ (F (x))]/F (x) is an increasing (decreasing) function of x for any distribution F which is equivalent to F w (x)/F (x) being an increasing (decreasing) function of x. The proof of (c) follows clearly as F ∗ is star shaped (anti-star shaped) if and only if F ∗ (F (x))/F (x) is an increasing (decreasing) function of x for any distribution F which is equivalent to Fw (x)/F (x) being an increasing (decreasing) function. The proof of (d) is obvious. 2.2. Weighted distributions with monotone failure rate We write b F instead of Fw , where b F (x) = F ∗ (F (x)) for some F ∗ (depending on w , which is given). Consider the failure rate of the weighted distribution b F: rb F ( x) =
f ∗ (F (x))f (x) 1 −b F ( x)
.
Gupta and Keating (1986) proved that if F is IFR, then the length-biased distribution b F is also IFR. Jain et al. (1989) showed that if the weight function w is increasing and concave and F is IFR, then b F also is IFR. Let b F be the weighted distribution associated with F . Then t
Z
1 −b F (t ) = exp −
0
t
Z
rb F (x)dx = exp −
0
rb F ( x)
r F ( x)
rF (x)dx .
Putting c (F (x)) = rb F (x)/rF (x), we have F (t )
Z t Z b F (t ) = 1 − exp − c (F (x))rF (x)dx = 1 − exp − 0
0
c (z ) 1−z
dz .
Hence c (z )
u
Z
F ∗ (u) = 1 − exp −
0
1−z
dz ,
where c (z ) is a positive function on (0, 1). Remark 1. If
R1 0
c (z )/(1 − z )dz = +∞, then F ∗ (u) is a distribution function on (0, 1).
It is evident that
Z
f ∗ (u) = exp −
0
u
c (z ) 1−z
dz
c ( u) 1−u
.
The next theorem gives a representation of the density f ∗ for which the weighted distribution preserves the IFR (DFR) property.
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
3059
Theorem 2. If F is an absolutely continuous IFR (DFR) distribution and c (z ) is a increasing (decreasing) positive function on R1 (0, 1) such that 0 c (z )/(1 − z )dz = +∞, then b F (x) = F ∗ (F (x)) is IFR (DFR). Proof. It suffices to make the following observation: rb F (x) =
f ∗ (F (x))f (x) 1 − F ∗ (F (x))
h R F (x)
exp −
=
0
c (z ) dz 1 −z
h R F (x) exp − 0
i
c (F (x)) f 1−F (x)
c (z ) dz 1 −z
(x)
i
= c (F (x))rF (x),
which is increasing (decreasing) because of the hypothesis of the theorem.
Remark 2. It is evident that: (a) every positive constant function can be c (z ); (b) every measurable positive increasing function on (0, 1) can be c (z ); (c) every measurable positive decreasing function g for which g (x) ≥ a > 0 can be c (z ). We consider some examples of the function c (z ). Example 4. Let F be the Weibull distribution F (x) =
0, 1 − exp(−xβ ),
x ≤ 0; x > 0, β > 1.
which is obviously IFR. Let c (u) = [− log(1 − u)]α , where α > 0. Since c (u) is increasing by Theorem 2, we have that b F is IFR. In this case F ( x)
[− log(1 − z )]α dz 1−z 0 Z 1 [− log(v)]α = 1 − exp − dv v 1−F (x) Z − log(1−F (x)) = 1 − exp − yα dy = 1 − exp −
Z b F (x) = 1 − exp −
0
1
α+1
[− log(1 − F (x))]α+1
thus
b F (x) =
0,
x ≤ 0;
1 − exp −
1
α+1
β(α+1)
x
,
x > 0, α > 0, β > 1. 1
is a Weibull distribution with the scale parameter (α + 1) β(α+1) and the shape parameter β(α + 1). On the other hand the distribution b F may be considered as a weighted distribution associated with the Weibull distribution and has the weight function
w(x) = xαβ exp −
1
α+1
xβ(α+1) + xβ .
Notice that this weight function does not satisfy the assumptions of the theorems of Gupta and Keating (1986) and Jain et al. (1989). We will consider some well-known theorems about preservation of the IFR property in terms of c (u) function. Example 5. Gupta and Keating (1986) proved that if F is an IFR distribution, then the length-biased distribution b F is also IFR. In this case we have, F −1 (u)(1 − u) c (u) = R 1 , F −1 (z )dz u hence 1 c (u)
=
1 1−u
R1 u
F −1 (z )dz − F −1 (u) F −1 (u)
+ 1.
3060
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
Thus c (u) is increasing only if: 1
Z
1 1−u
F −1 (z )dz − F −1 (u) = mX (F −1 (u)) u
is decreasing in u ∈ (0, 1) which obviously holds (because F is DMRL), and b F is IFR also by Theorem 2. Example 6. Jain et al. (1989) proved that if F is IFR, w(x) is increasing and concave, then b F is IFR (DFR). The main idea of the proof of Jain et al. (1989) consists of showing that A(x) − w(x) is decreasing in x ∈ (0, ∞), where A(x) = E [w(X )|X > x]. It is evident that in this case
w(F −1 (u))(1 − u) c ( u) = R 1 , w(F −1 (z ))dz u thus c (u) is increasing only if 1
Z
1 1−u
w(F −1 (z ))dz − w(F −1 (u)) u
is decreasing which is equivalent to A(x) − w(x) is decreasing. Remark 3. Jain et al. (1989) also observed that if F is IFR and w(x)F (x) is increasing, then b F is IFR. It is clear that this condition implies that c (u) is an increasing function of u ∈ (0, 1). 2.3. Preservation of IFRA (DFRA) and NBU (NWU) classes under weighting Bartoszewicz and Skolimowska (2006) proved preservation of the IFRA (DFRA) and NBU (NWU) classes under weighting assuming continuity. We give some extension of their results in terms of c (u) function. Theorem 3. (a) If F is IFRA (NBU) and c (u) is increasing then b F is IFRA (NBU); (b) If F is DFRA (NWU) and c (u) is decreasing then b F is DFRA (NWU). Proof. The proof follows by using the fact that rb F (x) = c (F (x))rF (x) and using the method of the proof of Theorem 3 of Bartoszewicz and Skolimowska (2006). 2.4. Preservation of NBU (NWU) classes under weighting Bartoszewicz and Skolimowska (2006) proved preservation of NBU (NWU) classes under weighting assuming a submultiplicative (supermultiplicative) function LU where U = w(X ). We reformulate their theorem using some assumptions on F ∗ but first we prove the following lemma. Lemma 2. Let F ∗ be a continuous distribution function on (0, 1). If Fq∗ /F ∗ is nondecreasing where Fq∗ (u) = F ∗ (qu) for any q ∈ (0, 1), then F ∗ (pq) ≤ F ∗ (p)F ∗ (q), p ∈ [0, 1]. Proof. The proof of the lemma is similar to that of Theorem 6 in Bartoszewicz and Skolimowska (2006). It is obvious that Fq∗ (u) F ∗ ( u)
is decreasing in u ∈ [0, 1/q],
implies F ∗ (pq) F ∗ (p)
≤
F ∗ (q) F ∗ (1)
.
Thus we can formulate a theorem about preservation of the NBU property. Theorem 4. Let X be a nonnegative random variable with the distribution function F . If F is NBU and Fq∗ /F ∗ is decreasing, then
b F is NBU. Proof. The proof is similar to that of Theorem 5 in Bartoszewicz and Skolimowska (2006) by observing that LU = F ∗ with U = w(x).
P. Błażej / Statistics and Probability Letters 78 (2008) 3056–3061
3061
Remark 4. Bartoszewicz and Skolimowska (2006) proved a theorem about preservation of the NWU property in the case when the weight function w is decreasing and such that the Lorenz curve LU is supermultiplicative i.e. LU (pq) ≥ LU (p)LU (q)
(2)
for every p, q ∈ [0, 1]. They observed that condition (2) is satisfied by the hyperbolic Lorenz curve considered by Aggarwal and Singh (1984), see also Arnold (1986): Lθ (p) =
(1 − θ )2 p , ((1 + θ )2 − 4θ p)
θ ∈ (0, 1).
(3)
We give an extension of their observation. Example 7 (Preservation of the NWU Property Under the Special Class of F ∗ ). Let X be a nonnegative random variable with the distribution function F . If F is NWU and
(1 − θ )2 (1 − u) ((1 + θ )2 − 4θ (1 − u))
F ∗ (u) = 1 −
(4)
then b F is NWU. It is easy to see that function (4) is the distribution on (0, 1) for any θ ∈ (0, 1). We obtain 1 −b F (x + t ) = 1 − F ∗ (F (x + t )) =
(1 − θ )2 F (x + t ) ((1 + θ )2 − 4θ F (x + t ))
= Lθ (F (x + t )).
By the definition of the NWU property of F and supermultiplicativity (3) we get Lθ (F (x + t )) ≥ Lθ (F (x)F (t )) ≥ Lθ (F (x))Lθ (F (t ))
= (1 − F ∗ (F (x)))(1 − F ∗ (F (t ))) = (1 − b F (x))(1 − b F (t )), i.e b F is also NWU. Acknowledgments The author is very grateful to the referee for the valuable comments. This research was supported by the Ministry of Higher Education and Science, Poland, Grant N201 046 31/3733. References Aggarwal, V., Singh, R., 1984. On optimum stratification with proportional allocation for a class of Pareto distributions. Commun. Stat. Theory Methods 13, 3107–3116. Arnold, B.C., 1986. A class of hyperbolic Lorenz curves. The Indian J. Stat. 48, B, 427–436. Bartoszewicz, J., Skolimowska, M., 2006. Preservation of classes of life distributions and stochastic orders under weighting. Statist. Probab. Lett. 76, 587–596. Fisher, R.A., 1934. The effects of methods of ascertainment upon the estimation of frequencies. Ann. Eugenics 6, 13–25. Gupta, R.C., Keating, J.P., 1986. Relations for the reliability measures under length biased sampling. Scand. J. Statist. 13, 49–56. Jain, K., Singh, H., Bagai, I., 1989. Relations for the reliability measures of weighted distributions. Commun. Stat. Theory Methods 18, 4393–4412. Lehmann, E.L., Rojo, J., 1992. Invariant directional orderings. Ann. Statist 20 (4), 2100–2110. Patil, G.P., Rao, C.R., 1977. The weighted distributions: A survey and their applications. In: Krishnaiah, P.R. (Ed.), Applications of Statistics. North-Holland Publ. Co, Amsterdam, pp. 383–405. Patil, G.P., Rao, C.R., 1978. Weighted distributions and size biased sampling wit applications to wild-life populations and human families. Biometrica 34, 179–189. Rao, C.R., 1965. Weighted distributions arising out of methods of ascertainment: What population does a sample represent? In: Atkinson, A.C., Fienberg, S.E. (Eds.), Celebration of Statistics: The ISI Centenary Volume. Springer Verlag, New York, pp. 543–569.