Sankhyā : The Indian Journal of Statistics
2009, Volume 71-A, Part 1, pp. 64-72
c 2009, Indian Statistical Institute
On the characterization of life distributions via percentile
residual lifetimes
Gwo Dong Lin
Academia Sinica, Taipei, Taiwan
Abstract
We revisit characterization results for life distributions in terms of percentile
residual life functions and prove that two suitable percentile residual life functions together are enough to characterize the continuous life distributions.
This extends and modifies the existing results in the literature.
AMS (2000) subject classification. Primary 62E10, 60E05; Secondary 39B22.
Keywords and phrases. Characterization, quantile function, mean residual
lifetime, percentile residual lifetime, Schröder’s functional equation
1
Introduction and main results
Both the mean residual lifetime and percentile residual lifetime are of
interest in reliability theory, biometry, survival analysis and actuarial studies.
Consider a nonnegative random variable X with distribution function
F (x) = P (X ≤ x) for x ≥ 0 and let rF be the right extremity of F , namely,
F (rF ) = 1 and F (x) < 1 for all x < rF (if any). Define the quantile function
of F by F −1 (u) = inf{x : F (x) ≥ u}, u ∈ (0, 1), and let Xt be any random
variable that has the conditional distribution of X − t given that X > t,
denoted Ft (x) = (F (t + x) − F (t))/(1 − F (t)), x ≥ 0. It is well known that
the distribution F is uniquely determined by its mean residual life function
e(t) ≡ E(Xt ) = E(X − t|X > t) on [0, rF ) provided X > 0 and E(X) < ∞;
explicitly,
Z x
e(0)
1
F (x) = 1 −
exp −
dt
for x ∈ [0, rF )
e(x)
0 e(t)
(see for example Meilijson (1972)). In contrast, the distribution F cannot
be characterized by a single 100αth percentile residual life function
qα,F (t) ≡ Ft−1 (α) = F −1 (1 − αF (t)) − t, t ∈ [0, rF ),
(1.1)
Percentile residual lifetimes
65
where α ∈ (0, 1), F = 1 − F and α = 1 − α. For example, all distributions
G in the family
Gα = {G : G (x) = e−x [1 + sin (2πx/log α)] , x ≥ 0,
− 1
|| < 1 + (2π/ log α)2 2 }
(1.2)
share the same (constant) 100αth percentile residual life function qα,G (t) =
− log α, t ≥ 0, with the standard exponential distribution G0 (x) = 1 −
e−x , x ≥ 0. (See Remark B in the Section 3 for construction of the family
Gα .)
In this note we shall prove that two percentile residual life functions
qαi ,F , i = 1, 2, with ratio log α1 / log α2 irrational, together are enough to
characterize the distribution F within the class of continuous distributions of
nonnegative random variables. This extends and modifies the existing results
in the literature (see Joe (1985), Song and Cho (1995) and the references
therein). The main results are stated below, while their proofs are given
in the next section. Finally, we give detailed discussions in the section of
Remarks.
The first result gives necessary and sufficient conditions for two percentile
residual life functions qα,F and qα,G to be equal. For convenience, we denote
u = 1 − u for u ∈ [0, 1).
Theorem 1. Let F and G be two continuous distribution functions on
[0, ∞) and F (0) = G(0) ≡ u0 ∈ [0, 1). Further, let α ∈ (0, 1) be a fixed
real number. Then the percentile residual life functions qα,F = qα,G if and
only if (i) rF = rG ≡ r and (ii) there exist two periodic functions Ki :
[− log u0 , ∞) → [0, ∞), i = 1, 2, with the same period − log α such that
F (x) = G(x)K1 (− log G(x)), x ∈ [0, r), and
G(x) = F (x)K2 (− log F (x)), x ∈ [0, r).
(1.3)
The next theorem claims that two suitable percentile residual life functions qαi ,F , i = 1, 2, together are enough to characterize the continuous life
distribution F .
Theorem 2. Let F and G be two continuous distribution functions on
[0, ∞) and F (0) = G(0) ≡ u0 ∈ [0, 1). Let αi ∈ (0, 1), i = 1, 2, be two real
numbers such that log α1 and log α2 are non-commensurate, namely, the ratio log α1 / log α2 is irrational. Assume further that qαi ,F = qαi ,G for i = 1, 2.
Then F = G.
66
Gwo Dong Lin
One of the equalities in (1.3) is enough to claim qα,F = qα,G provided we
impose a strict monotonicity condition on F or G.
Theorem 3. In addition to the assumptions of Theorem 1, let F be strictly
increasing on [0, rF ). Assume further that rF = rG ≡ r and that there exists
a periodic function K : [− log u0 , ∞) → [0, ∞) with period − log α such that
F (x) = G(x)K(− log G(x)), x ∈ [0, r).
(1.4)
Then qα,F = qα,G .
2
Proofs
To prove the main results, we need the following crucial lemma (see, e.g.,
Hájek et al. (1999), p. 32). For completeness, we give a detailed proof here.
Lemma. Let H be a distribution function on R ≡ (−∞, ∞) and let H
be continuous at H −1 (u∗ ) for some u∗ ∈ (0, 1), where H −1 (u) = inf{z :
H(z) ≥ u}, u ∈ (0, 1), is the quantile function of H. Then H(H −1 (u∗ )) = u∗ .
In particular, if F is continuous on [0, ∞) and F (0) = u0 ∈ [0, 1), then
F (F −1 (u)) = u for all u ∈ (u0 , 1).
Proof. Denote H −1 (u∗ ) = z∗ ∈ R. Let zn > z∗ , n = 1, 2, . . ., and
limn→∞ zn = z∗ . Then H(zn ) ≥ u∗ for n = 1, 2, . . ., by the definition of
quantile function, and hence
H(z∗ ) = lim H(zn ) ≥ u∗ ,
n→∞
(2.1)
in which the first equality is due to the continuity of H at z∗ . On the other
hand, let zn0 < z∗ , n = 1, 2, . . ., and limn→∞ zn0 = z∗ . Then H(zn0 ) < u∗ and
hence
H(z∗ ) = lim H(zn0 ) ≤ u∗ .
n→∞
(2.2)
Combining (2.1) and (2.2), we conclude that H(z∗ ) = u∗ . The proof is
complete.
Proof of Theorem 1. (Necessity) Suppose that qα,F = qα,G . Then, by
definition (1.1), we have rF = rG ≡ r and
F −1 (1 − αF (x)) = G−1 (1 − αG(x)), x ∈ [0, r).
This in turn implies, by Lemma above, that
1 − αF (x) = F [F −1 (1 − αF (x))] = F [G−1 (1 − αG(x))], x ∈ [0, r),
(2.3)
Percentile residual lifetimes
67
because F is continuous on [0, ∞) and 1 − αF (x) ≥ 1 − α u0 > u0 . Equivalently,
αF (x) = F [G−1 (1 − αG(x))], x ∈ [0, r).
(2.4)
Especially, it reduces to α u0 = F (G−1 (1 − α u0 )) when x = 0. Similarly, by
(2.3), we have 1 − αG(x) = G[F −1 (1 − αF (x))], x ∈ [0, r), and hence
αG(x) = G[F −1 (1 − αF (x))], x ∈ [0, r).
(2.5)
Letting x = G−1 (1 − u) in (2.4), we further have
αF (G−1 (1 − u)) = F [G−1 (1 − αu)], u ∈ (0, u0 ),
(2.6)
because G(x) = G(G−1 (1 − u)) = 1 − u for u ∈ (0, u0 ) due to the continuity
of G and the fact that 1 − u > u0 = G(0). Denote by `F (`G , respectively)
the left extremity of F (G, respectively). Namely, `F = inf{x : F (x) >
0, x ≥ 0}. We will prove that `F = `G ≡ ` (say) in two separate cases: (a)
u0 > 0 and (b) u0 = 0.
For Case (a), it is seen that `F = `G = 0. For Case (b), if 0 ≤ x ≤ `G ,
G(x) = 1 and hence F (x) = F (G−1 (1 − α))/α = 1 by (2.4). This means
that 0 ≤ x ≤ `F . Similarly, if 0 ≤ x ≤ `F , then 0 ≤ x ≤ `G due to (2.5).
Therefore, `F = `G for Case (b). Next, we claim that
F [G−1 (G(x))] = F (x) for x ∈ (`, r),
(2.7)
which is in fact a consequence of (2.4) and (2.6) by letting u = G(x) in (2.6).
[Note that if u0 > 0, then equality (2.6) also holds true for u = u0 because
G(G−1 (u0 )) = G(0) = u0 .] Define the function
K1 (x) = F (G−1 (1 − e−x ))ex , x > − log u0 , and K1 (− log u0 ) = 1. (2.8)
Then we have that K1 (x − log α) = K1 (x) for all x > − log u0 , because, by
(2.6) and (2.8),
K1 (x − log α) = F (G−1 (1 − e−x+log α ))ex−log α = F (G−1 (1 − e−x α))ex /α
= αF (G−1 (1 − e−x ))ex /α = F (G−1 (1 − e−x ))ex
= K1 (x), x > − log u0 ,
and that K1 (− log u0 − log α) = F (G−1 (1 − α u0 ))/(α u0 ) = 1 = K1 (− log u0 )
due to (2.4) with x = 0. Namely, K1 is a periodic function with period
68
Gwo Dong Lin
− log α. This in turn implies, by definition (2.8), that for x ∈ (`, r) with
G(x) > u0 ,
F (x)
F [G−1 (G(x))]
,
=
K1 (− log G(x)) =
G(x)
G(x)
in which the last equality is due to (2.7). On the other hand, for x ∈ (`, r)
with G(x) = u0 > 0 (if any, G is constant on the interval [0, x]), F (x) = u0
by (2.4) and hence
K1 (− log G(x)) = K1 (− log u0 ) = 1 =
F (x)
.
G(x)
Finally, for 0 ≤ x ≤ `, the above equality also holds true. Therefore K1
satisfies the first equality in (1.3). Exchange the roles of F and G in (2.8)
and define the function
K2 (x) = G(F −1 (1 − e−x ))ex , x > − log u0 , and K2 (− log u0 ) = 1.
Then along the similar lines as above we can prove that K2 is also a periodic
function with period − log α and that the second equality in (1.3) holds true.
(Sufficiency) Suppose that rF = rG ≡ r and that there exist two periodic
functions Ki , i = 1, 2, with the same period − log α, which together satisfy
the equalities in (1.3). We want to prove that qα,F = qα,G . By letting
x = G−1 (1 − αG(t)) in the first equality in (1.3) and by the assumptions on
the function K1 , we have, for t ∈ [0, r),
F [G−1 (1 − αG(t))] = αG(t)K1 (− log α − log G(t))
= αG(t)K1 (− log G(t)) = αF (t).
Equivalently,
F [G−1 (1 − αG(t))] = 1 − αF (t), t ∈ [0, r).
This implies, by the definition of quantile function, that
F −1 (1 − αF (t)) ≤ G−1 (1 − αG(t)), t ∈ [0, r).
(2.9)
Similarly, by the second equality in (1.3), we have
G−1 (1 − αG(t)) ≤ F −1 (1 − αF (t)), t ∈ [0, r).
Combining (2.9) and (2.10) yields that
F −1 (1 − αF (t)) = G−1 (1 − αG(t)), t ∈ [0, r),
(2.10)
69
Percentile residual lifetimes
and hence qα,F = qα,G by definition (1.1). The proof is complete.
Proof of Theorem 2. Let K1 be the function defined in (2.8). Then by
Theorem 1 and the conditions on F and G, we have that rF = rG ≡ r (say),
that K1 is a periodic function with periods − log αi , i = 1, 2 (see the proof
of Theorem 1; note that we apply the same function K1 to both cases α1
and α2 ), and that
F (x) = G(x)K1 (− log G(x)) for x ∈ [0, r).
Since the ratio log α1 / log α2 of periods is irrational, K1 has to be a constant
function (see, e.g., Olmsted (1959), p. 549) and hence K1 (x) = 1 for all
x ≥ − log u0 because K1 (− log u0 ) = 1. This implies that F (x) = G(x) for
x ∈ [0, r), and hence F = G. The proof is complete.
Proof of Theorem 3. Denote
φ(t) = t + qα,G (t) = G−1 (1 − αG(t)), t ∈ [0, r).
(2.11)
Then G(φ(t)) = 1 − αG(t), t ∈ [0, r), by the conditions on G and the fact
that 1 − αG(t) ≥ 1 − α u0 > u0 . Hence
G(φ(t)) = αG(t), t ∈ [0, r).
(2.12)
Applying (1.4) and (2.12), we have, for t ∈ [0, r),
F (φ(t)) = G(φ(t))K(− log G(φ(t))) = αG(t)K(− log α − log G(t))
= αG(t)K(− log G(t)) = αF (t).
This in turn implies that
φ(t) = F −1 (1 − αF (t)) = t + qα,F (t), t ∈ [0, r),
(2.13)
because F is continuous and strictly increasing on [0, r) and 1 − αF (t) ≥
1 − α u0 > u0 = F (0). Therefore, by (2.11) and (2.13), we conclude that
qα,F = qα,G . The proof is complete.
3
Remarks
(A) In Theorems 1 and 2, we have extended and modified the results of Joe
(1985, Theorems 1 and 2). First, we assume F (0) = G(0) ≡ u0 ∈ [0, 1) instead of u0 = 0 and apply the following identities in the proof: F (F −1 (u)) =
u and G(G−1 (u)) = u for all u ∈ (u0 , 1) (see Lemma above). Note that if
70
Gwo Dong Lin
F (0) = u0 > 0, then F (F −1 (u)) = u0 > u for all u ∈ (0, u0 ). Next, in the
sufficiency part of Theorem 1, we need the second equality in (1.3) to obtain
the conclusion: qα,F = qα,G , because one equality in (1.3) is not enough to
claim the result that F is constant on an interval I if and only if G is constant on I (a key point used in the proofs of Joe (1985)). These facts have
been neglected by Joe (1985), so his proofs are incomplete. Moreover, we
have included all the possible cases of left and right extremities of distributions (see, e.g., (2.7)), while Joe assumed at least rF = rG = ∞ implicitly.
(B) Under stronger conditions on distributions (continuous and strictly increasing), Song and Cho (1995) proved the above Theorem 2 using a different approach, namely, using the general solution of Schröder’s functional
equation given by Gupta and Langford (1984). Again, they assumed that
u0 = 0, `F = `G = 0 and rF = rG = ∞ (see their Lemma on page 334).
Moreover, in the above family Gα of distributions G (see (1.2)), we correct
the example given by Song and Cho (1995). To construct the family Gα ,
we may apply Theorem 3 above. Consider the standard exponential distribution G0 and the function K (x) = 1 + sin (2πx/log α) , x ≥ 0, where
− 1
|| < 1 + (2π/ log α)2 2 . Clearly, G0 (0) = 0, rG0 = ∞, K (0) = 1, and K
is periodic with period − log α. Set G (x) = G0 (x)K (− log G0 (x)), x ≥ 0.
The range of guarantees that the density of G is positive on [0, ∞). Then
by Theorem 3, the distributions G and G0 have the same 100αth percentile
residual life function qα,G0 . Finally, see Arnold and Brockett (1983) for more
characterization results by the properties of percentile residual life function.
(C) In Theorems 1 and 2, it is possible to weaken the condition qα,F (t) =
qα,G (t) for all t ∈ [0, r) (where r = rF = rG and α = α1 , α2 in Theorem 2)
by assuming that qα,F (t) = qα,G (t) for t ∈ I0 , where I0 is some proper subset
of [0, r). Two cases are given below. For convenience, define the function
φα,H (t) = t + qα,H (t) = H −1 (1 − αH(t)) on [0, r), where H = F, G. It is
seen that φα,F = φα,G if and only if qα,F = qα,G .
∞
Case 1. We can take I0 to be the set {ti }∞
i=0 , where t0 = 0 and {ti }i=1 is
any dense subset of (0, r). To see this, recall that in general the quantile
function H −1 is left-continuous on (0, 1). Therefore, the function φα,H on
[0, r) is uniquely determined by the restricted function of φα,H on I0 due to
the continuity of H.
Case 2. Instead of the continuity conditions on F and G, assume that the
functions φα,F and φα,G are real analytic and strictly increasing on [0, r) and
(n)
(n)
that for each n ≥ 1, the nth derivatives φα,F and φα,G are strictly monotone
Percentile residual lifetimes
71
in some interval [0, δn ). Then in this case we can take I0 to be any subset
{ti }∞
i=1 of (0, r) such that limi→∞ ti = 0. To see this, recall that the assumption φα,F (ti ) = φα,G (ti ) for i = 1, 2, . . . implies φα,F = φα,G by Lemma 5 of
Lin and Hu (2008, p. 1154).
Acknowledgements
The author would like to thank Professor J. S. Huang and the anonymous
Referee for helpful suggestions. Especially, the discussion in the Remark
C was suggested by the Referee. This work was partly supported by the
National Science Council of the Republic of China (Taiwan) under grant
NSC 98-2118-M-001-012.
References
Arnold, B. C. and Brockett, P. L. (1983). When does the βth percentile
residual life function determine the distribution ? Operations Research,
31, 391–396.
Gupta, R. C. and Langford, E. S. (1984). On the determination of a distribution by its median residual life function: a functional equation. J.
Appl. Probab., 21, 120–128.
Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests. Academic
Press, New York.
Joe, H. (1985). Characterizations of life distributions from percentile residual lifetimes. Ann. Inst. Statist. Math., 37, 165–172.
Lin, G. D. and Hu, C.-Y. (2008). On characterizations of the logistic distribution. J. Statist. Plann. Inference, 138, 1147-1156.
Meilijson, I. (1972). Limiting properties of the mean residual lifetime function. Ann. Math. Statist., 43, 354–357.
Olmsted, J. M. H. (1959). Real Variables: An Introduction to the Theory
of Functions. The Appleton-Century Mathematics Series, AppletonCentury-Crofts, Inc., New York.
Song, J.-K. and Cho, G.-Y. (1995). A note on percentile residual life.
Sankhyā, Ser. A, 57, 333–335.
72
Gwo Dong Lin
Gwo Dong Lin
Institute of Statistical Science
Academia Sinica
Taipei 11529, Taiwan
Republic of China
E-mail: [email protected]
Paper received November 2007; revised November 2008.