Generalized aging intensity functions

Reliability Engineering and System Safety 178 (2018) 198–208

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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

Generalized aging intensity functions

T

Magdalena Szymkowiak Institute of Automation and Robotics, Poznan University of Technology, Pl. M. Skłodowskiej-Curie 5, Poznań60-965, Poland

A R T I C LE I N FO

A B S T R A C T

Keywords: Generalized failure rate Generalized aging intensity Characterization Generalized aging intensity order

A family of generalized aging intensity functions is introduced and studied. The functions characterize lifetime distributions of univariate positive absolutely continuous random variables. Further on, the generalized aging intensity orders are defined and analyzed.

1. Introduction In the paper we study properties of positive unbounded and absolutely continuous random variables with distribution functions F and dF (x ) corresponding density functions f (x ) = dx positive on (0, +∞) . In the reliability theory these variables are mainly used to describe elements and systems life. A classic notion of the lifetime analysis is the failure rate function of F (known also as the hazard rate function) which is defined as

f (x ) d ln[1 − F (x )] =− . 1 − F (x ) dx

rF (x ) =

(1)

Other related and popular notions are the cumulative failure rate function (called often shortly hazard function)

∫0

RF (x ) =

x

rF (t )dt = −ln[1 − F (x )],

(2)

and the average failure rate

HF (x ) =

1 x

∫0

x

rF (t )dt =

1 1 RF (x ) = − ln[1 − F (x )]. x x

(3)

[14] introduced the aging intensity of F defined by

LF (x ) =

rF (x ) . HF (x )

(4)

Note that the aging intensity function can be also determined by means of the following formulae:

L (x ) =

x rF (x ) = RF (x )

d ln[1 − F (x )] dx 1 ln[1 − F (x )] x

=

−x f (x ) . [1 − F (x )] ln[1 − F (x )]

According to the definition of survival function F (x ) = 1 − F (x ), the failure rate (1) can be interpreted as the local infinitesimal conditional

E-mail address: [email protected]. https://doi.org/10.1016/j.ress.2018.06.012 Received 26 February 2018; Received in revised form 29 May 2018; Accepted 17 June 2018 Available online 21 June 2018 0951-8320/ © 2018 Elsevier Ltd. All rights reserved.

probability of an instantaneous failure occurring immediately after the time point x given that the unit has survived until x. The average failure rate (3) can be treated as a global baseline failure rate. Therefore the aging intensity (4) is defined as the ratio of the instantaneous failure rate rF to the average failure rate HF and expresses the average aging behavior of the item. It describes the aging property quantitatively: the larger the aging intensity, the stronger the tendency of aging (see [14]). Moreover, aging intensity LF (x ) =

d RF (x ) dx 1 RF (x ) x

can be treated as the

elasticity E RF (x ) of the nondecreasing positive cumulative failure rate function (2). The elasticity is an important economic notion, and its thorough discussion can be found in [30]. If function g is differentiable at x and g(x) ≠ 0, the elasticity of g at x is defined as Eg (x ) =

d g (x ) dx . 1 g (x ) x

It

measures the percentage the function g changes when x changes by a small amount. It can be also treated as the relative accuracy of approximating value of cumulative function g(x) with use of its derivative d g (x ) . Accordingly, LF (x ) = E RF (x ) measures the percentage the cudx mulative failure rate function changes (increases) when time x changes (increases) by a small amount. It is well known (see, for example [2]) that the failure rate r of an absolutely continuous random variable X with support (0, +∞) uniquely determines its distribution function F as follows:

Fr (x ) = 1 − exp ⎜⎛− ⎝

∫0

x

r (t )dt ⎟⎞ for x ∈ (0, +∞). ⎠

The necessary and sufficient conditions on r for being the failure rate of a distribution function is that r is nonnegative, and has infinite integral over (0, +∞) . Similarly, the cumulative failure rate function R and the average failure rate H uniquely determine their distribution function F through the following relationships:

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

FR (x ) = 1 − exp(−R (x )) for x ∈ (0, +∞), FH (x ) = 1 − exp(−xH (x )) for x ∈ (0, +∞).

Wα (x ) =

For this purpose, R(x) has to increase from 0 at 0 to + ∞ at + ∞, and clearly xH(x) has to share the property. Contrary to the unique characterization of the distribution through the failure rate, the cumulative failure rate function and the average failure rate, the aging intensity L characterizes the family of distributions depending on parameter 0 < k < +∞.

b

L (t ) dt < +∞ = t

∫0

a

L (t ) dt = t

∫a

+∞

(6)

(see [24]). For negative and positive α, the distribution functions represent Pareto and power random variables, respectively, up to linear transformations. Case α = 0 corresponds to the standard exponential distribution introduced above, and represents the limit of Wα as α → 0. The quantile function of Wα is equal to

Theorem 1. (see [31]) Let L: (0, +∞) → (0, +∞) satisfy the following conditions:

∫a

x > 0, α < 0, ⎧ 1 ⎪1 − (1 − αx ) α for ⎧ 1 ⎨ 0 < x < α , α > 0, ⎩ ⎨ ⎪1 − exp(−x ) for x > 0, α=0 ⎩

1

Wα−1 (x ) =

L (t ) dt t

⎧ α [1 − (1 − x )α ] for 0 < x < 1, α ≠ 0, ⎨− ln(1 − x ) for 0 < x < 1, α = 0. ⎩

(7)

Let F be a distribution function of the univariate absolutely continuous dF (x ) random variable X with its density function f (x ) = dx and support (0, +∞) . Then

for all 0 < a < b < +∞. Then L is an aging intensity function for the family of absolute continuous random variables with support (0, +∞) and their distribution functions given by the formula

1

FL, k (x ) = 1 − exp ⎜⎛−k exp ⎛ ⎝ ⎝

∫a

x

L (t ) ⎞ ⎞ dt ⎟ for t ⎠⎠

RWα, F (x ) = (Wα−1 ∘F )(x ) =

x ∈ (0, +∞)

(8)

and for every k ∈ (0, +∞) and for some arbitrarily chosen a ∈ (0, +∞).

satisfying

Note that fixing k we determine value of FL, k at a, namely FL, k (a) = 1 − exp(−k ) . Although in [31] a different parametrization of the family of distributions was presented we claim that the above one is more universal and meaningful in a general context developed here. In Section 2, we introduce a new family of generalized aging intensity functions, including the above one as a special case, and describe some properties of them. We show in Section 3 that these generalized aging intensities characterize lifetime distributions. It occurs that some generalized aging intensity functions uniquely characterize single distributions, and the others characterize families of distributions dependent on scaling parameters, as in Theorem 1. Some exemplary characterizations are presented in Section 4. In Section 5, we define stochastic orders based on generalized aging intensities, and prove some relations between them. Finally, in Section 6 we indicate applicability of α-generalized aging intensity function for identification of various compound parametric models of lifetime analysis.

= 0, and (9)

are the Wα-generalized cumulative failure function, and the Wα-generalized failure rate, respectively. They are further simply called the αgeneralized cumulative failure function, and the α-generalized failure rate, and denoted by Rα, F and rα, F, respectively. Analysis of α-generalized failure rates for various α provides more information about variability of lifetime distribution functions. The simplest and more natural one is 1-generalized failure rate which coincides with the density function. The increasing and decreasing density function gives the most rough illustration of the aging tendency of the life random variable at various time moments. Another widely acceptable and more subtle device is the standard 0-generalized failure rate which compares variability of the instantaneous failing tendency expressed by the density value f(x) at time x with the cumulative failure probability 1 − F (x ) at some moment after x. For instance, a decreasing failure rate of some life distribution on some time interval means that the density f(x) decreases faster than the cumulative survival function 1 − F (x ) there. If rF(x) is increasing in some time period, then f(x) decreases slower (it may even increase) than 1 − F (x ) . Studying various α-generalized failure rates enables us to obtain deeper comparisons of variability rates of the density and survival functions. For instance, decrease of the α-generalized failure rate with α < 0 is a sharper condition than the standard decreasing failure rate property, and gives more detailed information about the relations between the density and survival functions. The classes of distributions with monotone α-generalized failure rates were considered by [7–9]. Further, the α-generalized average failure rate is equal to

Observe that RF (x ) = (W0−1 ∘F )(x ), where W0 (x ) = 1 − exp(−x ), x > 0, is the standard exponential distribution function, and consequently

d(W0−1 ∘F )(x ) dRF (x ) = . dx dx

[3–5] proposed and studied a generalization of this concept and defined the G-generalized failure rate function for an arbitrary strictly increasing distribution function G with the density g. Under the assumptions, the G-generalized cumulative failure function and the Ggeneralized failure rate are defined by

Hα, F (x ) =

RG, F (x ) = (G−1∘F )(x ), dRG, F (x ) f (x ) d(G−1∘F )(x ) , rG, F (x ) = = = dx dx g [(G−1∘F )(x )]

1 1 R α, F (x ) = (Wα−1 ∘F )(x ) x x 1

=

respectively (see also [27], [11], [10] for further developments). Accordingly, we define the G-generalized aging intensity as

x rG, F (x ) x f (x ) LG, F (x ) = = . RG, F (x ) g ((G−1∘F )(x )) (G−1∘F )(x )

(Wα−1 ∘F )(0)

d(Wα−1 ∘F )(x ) rWα, F (x ) = = [1 − F (x )]α − 1f (x ) for x > 0 dx

2. Generalized aging intensity

rF (x ) =

⎧ α [1 − [1 − F (x )]α ] for x > 0, α ≠ 0, ⎨− ln[1 − F (x )] for x > 0, α = 0, ⎩

⎧ αx [1 − [1 − F (x )]α ] for x > 0, α ≠ 0, ⎨− ⎩

1 x

ln(1 − F (x ))

for x > 0, α = 0.

(10)

Finally, the α-generalized aging intensity being the special case of the G-generalized aging intensity (5) can be determined by the formula

(5)

Lα, F (x ) =

In the paper, we restrict ourselves to the analysis of G-generalized aging intensity functions for G belonging to the parametric family of generalized Pareto distributions for which we obtain most intuitive interpretations. We say that Xα follows a generalized Pareto distribution with parameter α ∈  if its distribution function is expressed as

=

199

rα, F (x ) = Hα, F (x )

d (Wα−1 ∘F )(x ) dx 1 (Wα−1 ∘F )(x ) x

⎧ αx [1 − F (x )]

α − 1f (x ) 1 − [1 − F (x )]α

for x > 0, α ≠ 0

⎨− ⎩

xf (x ) [1 − F (x )]ln[1 − F (x )]

for x > 0, α = 0.

(11)

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

The α-generalized aging intensity describes the relation between the current values of the α-generalized failure rates and those in the past. Large values of Lα, F(x) show how much the actual α-generalized failure rate exceeds the average of the rates observed so far. This reveals a weaker property than increasing α-generalized failure rate which asserts that the actual failure rate is greater than all the previous ones, and allows to describe more general and flexible models. We also note that the α-generalized aging intensity can be treated as the elasticity function of the α-generalized cumulative failure rate function, Lα, F (x ) = E Rα, F (x ) . It measures the percentage the α-generalized cumulative failure rate function increases when time x increases by a small amount. Below we present interpretations of α-generalized aging intensities for particular α. Obviously, the 0-generalized aging intensity function coincides with the standard aging intensity (4) of [14]. Similarly, the 0generalized failure rate, cumulative failure rate and average failure rate are identical with their classic counterparts (1), (2) and (3), respectively. For α = 1, the generalized failure rate (9) is just equal to the density function f, and the 1-generalized aging intensity xf (x ) L1, F (x ) = F (x ) = EF (x ) can be treated as the elasticity function of the distribution function F. If α = n is a positive integer, then

Ln, F (x ) =

rate

LORF ∘ exp (x ) =

which evaluated at ln x amounts to

LORF ∘ exp (ln x ) =

Observe that the denominator is the distribution function (we denote it by F1: n) of the sample minimum of n independent random variables with the common distribution function F. It is also the distribution function of the series system composed of n identical items working independently. The respective density function is f1: n (x ) = n [1 − F (x )]n − 1f (x ) . Consequently, Ln, F has can be interpreted as the elasticity function of the distribution function F1: n of the minimum of the sample of size n with the parent distribution function F. [29] (see also [26], [16] and [15]) proposed a notion of fractional order statistics with non-integer sample size α > 0. In particular, the distribution function of the sample minimum in the model is F1: α (x ) = 1 − [1 − F (x )]α . Accordingly,

α > 0,

R α, F (x ) =

(15)

∫0

x

rα, F (t )dt >

∫0

x

rβ, F (t )dt = Rβ, F (x ).

Dividing the above expressions by x, we trivially deduce the same relation for the α-generalized average failure rate (10). Similar monotonicity properties are shared by the α-generalized aging intensities, but verification of the claim needs more elaborate arguments. Proposition 1. For a given distribution function F with the positive density function on (0, +∞), the α-generalized aging intensity (11) is decreasing in α ∈  at every x ∈ (0, +∞) .

(12)

Proof. For some 0 < c < 1 define the function

is the elasticity function of the sample minimum distribution function based on possibly non-integer sample size. For α = −β < 0, a more intuitive representation of the α-generalized aging intensity (11) is

βxf (x ) L−β, F (x ) = . [1 − F (x )][1 − [1 − F (x )]β ]

x f (x ) = L−1, F (x ). F (x )[1 − F (x )]

Condition that the above function is constant and equal to some γ > 0, say, characterizes the family of the log-logistic distribution functions Fγ , λ (x ) = 1 − [1 + (λx ) γ ]−1 , x > 0, with fixed shape parameter γ and arbitrary scale λ > 0 (see [13,32,33]). We could easily extend our investigations to the absolutely continuous lifetime distribution functions satisfying 0 < F(x) < 1 for all x > 0. Then we would admit that the density function, and so the Gand α-generalized aging intensity functions take on zero values on some subintervals of (0, +∞) except for some neighborhoods of 0 and + ∞. However, from the practical point of view such a generalization sounds artificially: we admit that an item fails either at arbitrarily early stage of life or lives arbitrarily long with positive probabilities, but we exclude the possibility that it can fail in some fixed time intervals distant from 0 and infinity. This is the reason for resigning of the generalization, and assuming throughout the paper that the density and aging intensities are just positive on (0, +∞) . It is obvious that the α-generalized failure rate (9) is decreasing with respect to α ∈  for every x such that 0 < F(x) < 1 and f(x) > 0. This implies an analogous conclusion for the α-generalized cumulative failure rate function (8), because for α < β yields

nx [1 − F (x )]n − 1f (x ) . 1 − [1 − F (x )]n

αx [1 − F (x )]α − 1f (x ) , Lα, F (x ) = 1 − [1 − F (x )]α

f (exp(x )) exp(x ) , F (exp(x ))(1 − F (exp(x )))

hc (α ) =

αc α , 1 − cα

α ∈ ∖ {0}.

It has derivative

dhc (α ) c α (1 − c α + ln c α ) = . dα (1 − c α )2

(13)

For α = −1, it simplifies to

L−1, F (x ) =

This is negative, because c α ∈ (0, +∞) ∖ {1}, and the function 0 < t ↦ 1 − t + ln t is negative for all 1 ≠ t > 0. To check the last claim it suffices to note that this function is strictly concave, and has a single zero at t = 1. Therefore the original function hc is decreasing on either of the half-lines (−∞, 0) and (0, +∞) . Define

x f (x ) F (x )[1 − F (x )]

which has some connection with the log-odds rate. We recall that the log-odds function and the log-odds rate of the distribution function F are defined by

F (x ) , 1 − F (x ) f (x ) dLOF (x ) = . LORF (x ) = dx F (x )(1 − F (x ))

hc (0) = lim hc (α ) = lim

LOF (x ) = ln

α→0

α→0 1

αc α c α + αc α ln c 1 = lim =− . α→0 − cα −c α ln c ln c

Under the above extension, the function hc is decreasing in α on the whole real axis. Fixing c = 1 − F (x ) and multiplying h1 − F (x ) (α ) by the positive

(14)

The latter has an interesting representation f (x ) LORF (x ) = rF (x ) + r˘F (x ), where r˘F (x ) = F (x ) is the reversed hazard rate function which can be interpreted as the instantaneous failure rate occurring immediately before the time point x (the failure occurs just before the time point x, given that the unit has not survived until the time point x). If a positive random variable X has the distribution function F, then ln X has the distribution function F○exp , and log-odds

factor

xf (x ) , 1 − F (x )

we conclude that α−1

⎧ αx [1 − F (x )] αf (x ) for α ≠ 0, xf (x ) 1 − [1 − F (x )] h1 − F (x ) (α ) = xf ( x ) 1 − F (x ) ⎨− for α = 0, ⎩ [1 − F (x )]ln[1 − F (x )] which is just Lα, F(x), is decreasing with respect to α ∈  . 200

□

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

Proposition 2. We have Lα, Wβ (x ) > 1 for α < β, and Lα, Wβ (x ) < 1 for α > β. Also, Lα, Wα (x ) = 1 for all α ∈  .

Using Proposition 2 and Lemma 1, we can provide another proof of Proposition 1. Namely, if α < β, then Lα, Wβ ((W β−1 ∘F )(x )) > 1 by Proposition 2, and Lα, F(x) > Lβ, F(x) by Lemma 1. For α > β, relation Lα, Wβ ((W β−1 ∘F )(x )) < 1 implies Lα, F(x) < Lβ, F(x).

Proof. We study first 0 ≠ α ≠ β ≠ 0. By the formulae (6) and (7), α

1 − (1 − βx ) β α

(Wα−1 ∘Wβ )(x ) = derivatives

d(Wα−1 ∘Wβ )(x ) dx

on the support of Wβ. The composition has α −1

= (1 − βx ) β

d2 (Wα−1 ∘Wβ )(x ) dx 2

Proposition 3. Function Lα, Wβ (x ) for 0 < x < +∞ and some α ≠ β, (see (16)–(18)) is increasing when α < β, and decreasing for α > β. Proof. We first focus on the case 0 ≠ α ≠ β ≠ 0 for which the formula (16) holds. It has the derivative

, α −2

= (β − α )(1 − βx ) β

dLα, Wβ (x )

.

dx

This means that Wα−1 ∘Wβ is strictly convex for β > α, and strictly concave for β < α. In the first case,

(Wα−1 ∘Wβ )(x ) − (Wα−1 ∘Wβ )(0) x

(Wα−1 ∘Wβ )(x )

=

x

<

d(Wα−1 ∘Wβ )(x ) dx

(19) where parameter γ = is introduced for the sake of simplicity of notation. The fraction is always positive, because for every β ≠ 0 and x from the support of Wβ we have 1 ≠ 1 − βx > 0 . The sign of (19) is identical with that of

,

Mβ, γ (x ) = −β [γ (γ − 1) − γ 2 (1 − βx ) + γ (1 − βx ) γ ]. Observe that Mβ, γ (0) = 0, and

dMβ, γ (x )

d(Wα−1 ∘W0 )(x ) = exp(−αx ), dx

dx

γ(x)

Resulting convexity for α < 0 and concavity for α > 0 of the composition Wα−1 ∘W0 implies that Lα, W0 is greater and less than 1, respectively. Further, we can observe that convexity and concavity of Wα−1 ∘W0 implies concavity and convexity of its inverse W0−1 ∘Wα , respectively. This means that L0, Wα < 1 for α < 0 and L0, Wα > 1 for α > 0. Finally, identity Lα, Wα = 1 is an immediate consequence of the trivial equality (Wα−1 ∘Wα )(x ) = x . □

1 − (1 −

α βx ) β

,

L0, Wβ (x ) =

−βx , β ≠ 0. (1 − βx )ln(1 − βx )

dL0, Wβ (x )

(18)

Lα, F (x ) = Lβ, F (x ) Lα, Wβ ((W β−1 ∘F )(x )).

dx

=

as

d(W β−1 ∘F )(x )

dx

dx

we get

,

x

d(Wα−1 ∘F )(x ) dx d

=

d

(W β−1 ∘F )(x ) dx (Wα−1 ∘Wβ )((W β−1 ∘F )(x )) x dx (W β−1 ∘F )(x ) (Wα−1 ∘Wβ )((W β−1 ∘F )(x ))

dx

, Mβ,

α β

ln(1 − βx ) + 1 − (1 − βx ) . (1 − βx )2 ln2 (1 − βx )

(20)

exp(−αx ) [1 − exp(−αx )]2

(21)

is positive for all x > 0, whereas function

In Figs. 1 and 2, the α-generalized aging intensities Lα, Wβ are plotted, for β equal to 0 and to − 1, respectively, and different α. The presented functions have the properties proved in Propositions 1, 2 and 3: the αgeneralized aging intensity is decreasing in α ∈  ; Lα, Wβ (x ) > 1 for α < β, Lα, Wβ (x ) < 1 for α > β and Lα, Wα (x ) = 1 for all α ∈  ; Lα, Wβ (x ) is increasing in x when α < β, and decreasing for α > β.

Combining these formulae, we obtain

(Wα−1 ∘F )(x )

dMβ, γ (x )

Nα (x ) = α − α 2x − α exp(−αx ) vanishes at 0, Nα (0) = 0 . Moreover, dNα (x ) dN (x ) = α 2 [exp( −αx ) − 1] > 0 for x ∈ (0, +∞) if α < 0 and dαx < 0 dx for x ∈ (0, +∞) if α > 0. The properties are shared by (21). This implies that (17) is increasing if α < β = 0 and decreasing if α > β = 0 . □

(Wα−1 ∘Wβ )((W β−1 ∘F )(x )) (W β−1 ∘F )(x ) (Wα−1 ∘F )(x ) . = x x (W β−1 ∘F )(x )

Lα, F (x ) =

=β

The fraction

d(Wα−1 ∘Wβ )((Wα−1 ∘F )(x ))

< 0 when α > β. For

dLα, W0 (x ) exp(−αx ) [α − α 2x − α exp(−αx )]. = dx [1 − exp(−αx )]2

Lemma 1. For any α, β ∈  and F, we have

d(Wα−1 ∘F )(x )

dx

− 1 > 0,

For all possible choices of β and x, expression t = 1 − βx takes on all positive values except for 1. Recalling arguments of the proof of Proposition 1, we notice that the numerator in (20) is negative, and the sign of the derivative (20) is opposite to that of β. This means that (18) is increasing for β < 0 and decreasing for β > 0, as desired. It remains to analyze the case β = 0 ≠ α . Differentiating (17), we obtain

(17)

(Wα−1 ∘Wβ ∘W β−1 ∘F )(x ),

dLα, Wβ (x )

α β

are positive when γ − 1 = − 1 < 0, which again γ(x), and dx gives α < β. They are all negative for α > β. This completes the proof for 0 ≠ α ≠ β ≠ 0. For α = 0 ≠ β , function (18) has derivative

For completeness, we remind here that Lα, Wα (x ) = 1 for all α ∈  .

(Wα−1 ∘F )(x )

> 0, and so Mβ,

> 0 for all x > 0 when γ − 1 =

dLα, Wβ (x )

(16)

α ≠ 0,

dx

1

0 ≠ α ≠ β ≠ 0,

αx exp(−αx ) , 1 − exp(−αx )

dx

i.e., when α < β. By similar arguments,

dx

Lα, W0 (x ) =

Proof. Representing

> 0, and finally

dLα, Wβ (x )

dMβ, γ (x )

β > 0 and 0 < x < β , we have 0 < 1 − βx < 1. Accordingly,

Using the formulae of the above proof we easily obtain

Lα, Wβ (x ) =

= β 2γ 2 [(1 − βx ) γ − 1 − 1].

If β < 0, then inequality 1 − βx > 1 implies

d2 (Wα−1 ∘W0)(x ) = −α exp(−αx ). dx 2

αx (1 −

(1 − βx ) γ − 2 , [1 − (1 − βx ) γ ]2

α β

and so Lα, Wβ (x ) > 1. For β < α we similarly obtain the reversed inequality. 1 − exp(−αx ) with Suppose now that α ≠ 0, and take (Wα−1 ∘W0)(x ) = α derivatives

α −1 βx ) β

= −β [γ (γ − 1) − γ 2 (1 − βx ) + γ (1 − βx ) γ ]

3. Characterizations through α-generalized aging intensity

(W β−1 ∘F )(x )

If α < 0 then the α-generalized aging intensity characterizes a family of distributions depending on parameter 0 < k < +∞, as it was for α = 0 (see Theorem 1).

= Lα, Wβ ((W β−1 ∘F )(x )) Lβ, F (x ). □ 201

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

(

a L (t )

)

so that k1 = k exp − ∫a 1 t dt runs over (0, +∞) as k does so. A proper choice of the lower integral limit a provides the simpler representation of family (23). We illustrate the idea in the examples of Section 4. Note that choosing k we determine value of (23) at point a getting 1 F (a) = Wα (k ) = 1 − (1 − kα ) α . Remark 2. Function ζL (x ) = exp

(∫

a

x L (t ) dt t

)

is a strictly increasing

transformation of the support (0, +∞) of each Wα with α ≤ 0 onto itself. Its inverse ηL (x ) = ζ L−1 (x ) is increasing mapping transforming (0, +∞) onto (0, +∞), as well. If Xα has a generalized Pareto distribution function Wα, then Fα, L, 1 is the distribution function of η (X ) ηL(Xα). Scale transformations L α for 0 < k < +∞ generate all the k distributions with α-generalized aging intensity L. Proof. Assume first that L satisfying (22) is the α-generalized aging intensity of a distribution function with support (0, +∞) . By (11) we obtain d

(Wα−1 ∘F )(t ) L (t ) = dt −1 t (Wα ∘F )(t )

Fig. 1. Lα, W0 .

for t ∈ (0, +∞).

(24)

Integrating the above over arbitrarily chosen interval (a, x ) ⊂ (0, +∞), we get

∫a

x

L (t ) dt = t

∫a

d x dt

(Wα−1 ∘F )(t )

(Wα−1 ∘F )(t )

dt = ln

(Wα−1 ∘F )(x ) . (Wα−1 ∘F )(a)

(25)

Therefore

(Wα−1 ∘F )(x ) = exp ⎛ (Wα−1 ∘F )(a) ⎝

∫a

x

L (t ) ⎞ dt , t ⎠

(Wα−1 ∘F )(x ) = (Wα−1 ∘F )(a)exp ⎛ ⎝

∫a

x

L (t ) ⎞ dt , t ⎠

(26)

and finally

F (x ) = Wα ⎛ (Wα−1 ∘F )(a)exp ⎛ ⎝ ⎝ ⎜

∫a

(Wα−1 ∘F )(x ) = (Wα−1 ∘F )(b) exp ⎛ ⎝ Theorem 2. Function L: (0, +∞) → (0, +∞) 0 < a < b < +∞ relations b

L (t ) dt < +∞ = t

∫0

a

L (t ) dt = t

∫a

+∞

satisfying

L (t ) dt t

for

all

⎜

∫a

x

(Wα−1 ∘F )(x ) = (Wα−1 ∘F )(a)exp ⎛ ⎝

(22)

L (t ) ⎞ ⎞ dt = 1 − ⎡1 − kα exp ⎛ ⎢ t ⎠⎠ ⎝ ⎣ ⎟

(Wα−1 ∘F )(x ) = (Wα−1 ∘F )(a)exp ⎛ ⎝

1

∫a

x

L (t ) ⎞ ⎤α dt t ⎠⎥ ⎦

Remark 1. We strongly emphasize that distribution functions (23) depend on single parameter k, and choice of a is immaterial here. Replacing a by a1 ∈ (0, +∞), we obtain x L (t )

t

a1 L (t ) x L (t ) dt ⎞ ⎞⎟ = Wα ⎜⎛k exp ⎛− dt ⎞ exp ⎛ dt ⎞ ⎞⎟ a t t ⎠⎠ ⎝ ⎠ ⎝ a ⎠⎠ ⎝ x L (t ) = Wα ⎜⎛k1 exp ⎛ dt ⎞ ⎞⎟ t ⎝ a ⎠⎠ ⎝

∫

x

L (t ) ⎞ dt . t ⎠

∫a

b

L (t ) ⎞ dt t ⎠

∫a

b

L (t ) dt + t

∫b

x

L (t ) ⎞ dt t ⎠

which again leads to (26) for originally chosen a. The differential equation (24) has a family of solutions dependent on the boundary conditions represented by the values of Wα−1 ∘F at a. We now determine the values k = (Wα−1 ∘F )(a) which correspond to the solutions F being absolutely continuous distribution functions supported on (0, +∞) . By assumption, F(a) for any 0 < a < +∞ may take on an arbitrary value in (0,1), and, in consequence, every k ∈ (0, +∞) may become the value of (Wα−1 ∘F )(a) . We finally note that if α < 0, then the formula (23) determines absolute continuous distribution functions with support (0, +∞) for every 0 < k < +∞, since

for arbitrarily fixed 0 < a < +∞ and every 0 < k < +∞.

∫

∫b

and so

(23)

Wα ⎛⎜k exp ⎛ ⎝ a1 ⎝

⎟

However, applying (26) for x = b, we get

is for every α < 0 the α-generalized aging intensity for the family of absolute continuous distribution functions with support (0, +∞) given by the formula

Fα, L, k (x ) = Wα ⎛k exp ⎛ ⎝ ⎝

L (t ) ⎞ ⎞ dt . t ⎠⎠

We show that the representation does not depend on the choice of a. Indeed, for b ≠ a we similarly obtain

Fig. 2. Lα, W−1.

∫a

x

∫

∫

202

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

(

(

dWα k exp ∫a dFα, L, k (x ) = dx dx

x L (t ) dt t

) ) k exp ⎛ ⎝

∫a

x

α ≤ 0 and for none α > 0, or for every α > 0 and none α ≤ 0, or it is not the aging intensity for any real α. For instance, function L (x ) = xg (x ), x > 0, where a positive function g has a finite integral over (0, +∞) cannot be the α-generalized aging intensity function for any α ∈  and any lifetime distribution function.

L (t ) ⎞ L (x ) dt t ⎠ x

> 0 for x ∈ (0, +∞), lim Fα, L, k (x ) = Wα ⎛k exp ⎛− ⎝ ⎝ ⎜

x → 0+

lim Fα, L, k (x ) = Wα ⎛k exp ⎛ ⎝ ⎝ ⎜

x →+∞

∫0

∫a

L (t ) ⎞ ⎞ dt = Wα (0) = 0, t ⎠⎠

a

⎟

L (t ) ⎞ ⎞ dt = Wα (+∞) = 1. t ⎠⎠

+∞

4. Examples of characterization through α-generalized aging intensity

⎟

□

We first determine distribution functions which have α-generalized aging intensities Lα, F identical with those of the selected lifetime distribution function F for various α ≤ 0. Then

Observe that the above proof can be literally extended to the case α = 0 which justifies Theorem 1. On the contrary, if α > 0 then αgeneralized aging intensity L uniquely characterizes a single absolutely continuous distribution with support (0, +∞) . Theorem 3. Function 0 < a < +∞ relations

∫a

+∞

L (t ) dt < +∞ = t

∫0

L: (0, +∞) → (0, +∞)

satisfying

for

L (t ) dt t

a

∫a

all

∫x

(27)

= 1 − ⎡1 − exp ⎛− ⎢ ⎝ ⎣

∫x

Fα, Lα, F , k (x ) = Wα (k (Wα−1 ∘F )(x ))

L (t ) ⎞ ⎞ dt ⎟ t ⎠⎠

+∞

+∞

(

1 α

1 α

L (t ) ⎞ ⎤ dt . t ⎠⎥ ⎦

(

+∞ L (t ) dt t

exp − ∫x dx

(28)

) ) × 1 exp ⎛− α

=

⎝

∫x

L (t ) ⎞ L (x ) dt t ⎠ x +∞ L (t ) dt t

Moreover, since limx → 0+ ∫x we also have

= +∞ and limx →+∞ ∫x

1 lim Fα, L (x ) = lim+Wα ⎛ exp ⎛− x → 0+ x→0 ⎝ ⎝α 1 lim Fα, L (x ) = lim Wα ⎛ exp ⎛− x →+∞ x →+∞ ⎝ ⎝α

+∞

∫x

⎜

⎜

∫x

1 = = (Wα−1 ∘F )(a)exp ⎛ α ⎝

∫a

=0

F−1, L−1, F , k (x ) =

L (t ) ⎞ ⎞ dt = Wα (0) = 0, t ⎠⎠

1 exp ⎛− α ⎝

∫a

+∞

+∞

∫a

+∞

< 0,

for x

kF (x ) , (k − 1) F (x ) + 1

which is similar to the family determined by Marshall–Olkin transformation (see [19]). Observe that if k = 1 in the formula (29), we recover the baseline distribution function. If we take F = Wα , for any α ≤ 0, we have

L (t ) ⎞ ⎞ 1 dt = Wα ⎛ ⎞ = 1. t ⎝α⎠ ⎠⎠ ⎟

+∞

1 − exp(−kx ), α = 0, Fα, Lα, Wα , k (x ) = Wα (kx ) = ⎧ ⎨1 − (1 − αkx ) α1 , α < 0, ⎩

L (t ) ⎞ dt t ⎠

L (t ) ⎞ dt t ⎠

L (t ) ⎞ ⎞ dt ⎟ t ⎠⎠

for arbitrary 0 < a < +∞ which leads to (28) with a replaced by x.

x > 0, k > 0,

which means that all the distribution functions sharing the α-generalized aging intensity with Wα for some α ≤ 0 are the scale transformations of the original Wα. Many generalizations of standard life distribution models have been proposed in reliability literature to help reliability practitioners providing a good fitting of data sets. Some of them are characterized by means of α-generalized aging intensity functions. They appear in the following part of the paper, and are gathered in Table 1 for easy reference.

and

1 F (a) = Wα ⎜⎛ exp ⎛− ⎝ ⎝α

= 0,

⎟

so that

(Wα−1 ∘F )(a) =

α < 0,

The 0-generalized aging intensity L0, F characterizes the proportional hazard family with frailty parameter k > 0 and the baseline distribution function F (see, e.g., [20], p. 233 and [17]). Transformation (29) for α < 0 is not known in literature. For α = −1 it simplifies to

If F is a distribution function with α-generalized aging intensity L, then by arguments of the proof of Theorem 2 it satisfies (26) for arbitrary 0 < a < +∞. Then for x → +∞ we obtain

Wα−1 (1)

⎧1 ⎪ ⎨1 ⎪ ⎩ > 0 and k > 0.

α = 0,

(29) +∞

> 0forx ∈ (0, +∞). +∞ L (t ) dt t

k ⎧1 − [1 − F (x )] , ⎨1 − (1 − k (1 − (1 − F (x ))α )) α1 , ⎩ α − [1 − F (x )]k , 1 − F (x ) − , α 1 {k + (1 − k )[1 − F (x )]−α }− α

=

Proof. We note that for α > 0 the formula (28) determines an absolutely continuous distribution function with support (0, +∞) . Firstly,

dWα dFα, L (x ) = dx

Lα, F (t ) dt = ln(Wα−1 ∘F )(x ) − ln(Wα−1 ∘F )(a) t

(see (25)). The simplest form of the formula (23) is obtained when dependence on a is hidden, i.e., for a = (F −1∘Wα )(1) for which the last terms in the right-hand side of the above equation vanishes. Then we obtain

is for every α > 0 the α-generalized aging intensity for the unique absolute continuous distribution function with support (0, +∞) given by the formula

1 Fα, L (x ) = Wα ⎜⎛ exp ⎛− ⎝ ⎝α

x

□

Example 1. Suppose that F (x ) = W0 (x ) = 1 − exp(−x ) is the standard exponential distribution function with scale parameter λ = 1. We easily verify that L0, W0 (x ) = 1 satisfies (22). By the above considerations, this 0-generalized aging intensity function generates the family of exponential distributions with scale parameters λ = k for 0 < k < +∞. For α ≠ 0 we have (17). Since

As we see, the elasticity function EF(x), being 1-generalized aging intensity, uniquely determines unbounded lifetime distribution functions. Moreover, the elasticity of the sample minimum E F1: α (x ), even for non-integer sample size, uniquely characterizes the parent distributions as well. Due to (22) and (27), a function L: (0, +∞) → [0, +∞) may be either α-generalized aging intensity of some distributions for every 203

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

Table 1 Life distributions.

is (18). The antiderivative of

∫a

x

L0, Wβ (t ) t

is

−β dt = ln(W0−1 ∘Wβ )(x ) − ln(W0−1 ∘Wβ )(a) (1 − βt )ln(1 − βt ) −ln(1 − βx ) −ln(1 − βa) . = ln − ln β β

Distribution name

Distribution function

Notation

References

Exponential Two-parameter Weibull Modified Weibull Inverse two-parameter Weibull Lomax

1 − exp(−λx ) 1 − exp(−(λx ) γ )

Exp(λ) W2(γ, λ)

[20] [20]

1 − exp(−(λx ) γ exp(δx )) 1 − exp(−(λx )−γ )

MW(γ, λ, δ) invW2(γ, λ)

[18] [20]

1 − [1 + λx ]−ξ

LO(λ, ξ)

[20]

Log-logistic

1 − [1 + (λx ) γ ]−1

LLog(γ, λ)

[20]

Modified log-logistic

1 − [1 + (λx ) γ exp(δx )]−1

MLLog(γ, λ, δ)

F0, L0, Wβ , k (x ) = W0 (k (W0−1 ∘Wβ )(x )) = 1 − (1 − βx ) β ,

This is the family of distribution functions of Lomax random variables k with scale parameter λ = −β and frailty parameters ξ = − β , for 0 < k < +∞. The respective density functions are

Power Lomax

1 − (1 + (λx ) γ )−ξ

POLO(γ, λ, ξ)

[25]

Modified power Lomax

1 − (1 + (λx ) γ exp(δx ))−ξ

Half-Cauchy

2 π

MPOLO(γ, λ, ξ, δ) HC(λ)

[1]

arctan(λx )

( )

1 1 Therefore (22) are satisfied. For a = β ⎡1 − exp − β ⎤, the formula (23) ⎣ ⎦ takes on the form k

k −1

f0, L0, W , k (x ) = k (1 − βx ) β β

∫a

x

Lα, W0 (t ) dt = t

∫a

x

α exp(−αt ) dt = ln(Wα−1 ∘W0)(x ) 1 − exp(−αt )

∫a

x

Lα, Wβ (t ) t

dt = ln(Wα−1 ∘Wβ )(x ) − ln(Wα−1 ∘Wβ )(a) α

= ln

(22) and (27) are satisfied for α < 0 and α > 0, respectively. In the 1 former case we plug a = (W0−1 ∘Wα )(1) = − α ln(1 − α ) into (23), and finally obtain

α

1 − (1 − βx ) β 1 − (1 − βa) β . − ln α α

(31)

β

1 If α < 0, Lα, Wβ satisfies (22). With a = β ⎡1 − (1 − α ) α ⎤, we rewrite (23) ⎣ ⎦ as

Fα, Lα, W0, k (x ) = Wα (k (Wα−1 ∘W0)(x )) 1

x > 0.

For 0 ≠ α ≠ β, we have (16), and so

− ln(Wα−1 ∘W0)(a) 1 − exp(−αx ) 1 − exp(−αa) , = ln − ln α α

= 1 − [1 − k (1 − exp(−αx ))]α ,

,

x > 0.

α

(

)

1 α

Fα, Lα, Wβ , k (x ) = Wα (k (Wα−1 ∘Wβ )(x )) = 1 − ⎡1 − k 1 − (1 − βx ) β ⎤ , ⎣ ⎦

x > 0,

x > 0,

which have the density functions

(comp. (29)). They have the density functions 1

fα, Lα, W , k (x ) = k exp( −αx )[1 − k (1 − exp(−αx ))] α − 1 , 0

x > 0.

α −1

fα, Lα, W , k (x ) = k (1 − βx ) β

(30)

β

(

α

)

1 α −1

⎡1 − k 1 − (1 − βx ) β ⎤ ⎦ ⎣

,

x > 0. (32)

The various density functions (30) of the distributions characterized through the same α-generalized aging intensity L−1, W0 are plotted in Fig. 3. By Theorem 3, the only distribution function with the α-generalized aging intensity (17) for any α > 0 is the standard exponential distribution function.

The exemplary density functions (32) of the distributions characterized through α-generalized aging intensity L−2, W−1 are plotted in Fig. 4. For α > 0, (31) implies that the relations (27) hold. By Theorem 2, the only distribution function with the α-generalized aging intensity (16) is Wβ itself.

1

2

Example 2. Put F (x ) = Wβ (x ) = 1 − (1 − βx ) β for some β < 0 being the distribution function of the Lomax distribution with scale parameter 1 λ = −β and frailty parameter ξ = − β . Its 0-generalized aging intensity

Example 3. Let F (x ) = π arctan x for x > 0 be the distribution function of the half-Cauchy distribution function with scale parameter λ = 1. For α = 0, we have

Fig. 3. Density functions (30) of distributions characterized through L−1, W0 .

Fig. 4. Density functions (32) of distributions characterized through L−2, W−1. 204

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

L0, F (x ) = −

2x π (1 + x 2) 1 −

(

1 2 π

) (

arctan x ln 1 −

2 π

arctan x

)

indefinite integral A ln x + Bx + C , and satisfies the assumptions of Theorem 2. It does not obey (27), and cannot be the α-generalized aging intensity for any α > 0. If α ≤ 0, then the following classes of lifetime distribution functions

. (33)

Equation

∫a

x

L0, F (t ) dt = ln(W0−1 ∘F )(x ) − ln(W0−1 ∘F )(a) t 2 = ln ⎡−ln ⎛1 − arctan x ⎞ ⎤ π ⎝ ⎠⎦ ⎣

Fα, L, k (x ) = Wα (kx A exp(Bx )) =

implies that relations (22) hold. Therefore 0-generalized aging intensity π (33) with a = tan 2 (1 − exp(−1)) characterize the family of the distribution functions

)

k 2 F0, L0, F , k (x ) = W0 (k (W0−1 ∘F )(x )) = 1 − ⎛1 − arctan x ⎞ , π ⎝ ⎠

x > 0.

For α ≠ 0 yields

(

)

2

α−1

1 − π arctan x 2αx Lα, F (x ) = − π (1 + x 2) 1 − 1 − 2 arctan x π

(

)

α

.

Example 6. Take now L (x ) = Ax −B for all x > 0 and some A, B > 0. We L (x ) A have ∫ x dx = − B x −B + C which implies that the relations (27) hold.

(34)

Since

∫a

x

+∞ L (x )

⎟

⎜

1

α A Fα, L (x ) = 1 − ⎡1 − exp ⎛− x −B ⎞ ⎤ , ⎝ B ⎠⎦ ⎣

⎟

relations (22) and (27) hold under assumptions α < 0 and α > 0, respectively. In the latter case, α-generalized aging intensity (34) uniquely determines the half-Cauchy distribution. In the former, (23) with

a = tan

(

π 2

1

x > 0.

This is an element of the proportional hazard family with frailty parameter 1 and baseline inverse Weibull distribution with shape α

)

1 − (1 − α ) α ⎤ characterize the family ⎡ ⎣ ⎦

Fα, Lα, F , k (x ) = Wα (k

A

dx = B x −B , and by (28) L is the αIn particular, we have ∫x x generalized aging intensity for every fixed α > 0 of the unique distribution function

α Lα, F (t ) 1 2 dt = ln ⎛ ⎡1 − ⎛1 − arctan x ⎞ ⎤ ⎞ t π ⎝ ⎠ ⎦⎠ ⎝α ⎣ α 1 2 − ln ⎛ ⎡1 − ⎛1 − arctan a ⎞ ⎤ ⎞, π ⎝ ⎠ ⎦⎠ ⎝α ⎣ ⎜

x > 0,

have the linear α-generalized aging intensity functions. In order to get these representations, we used (23) with 0 < a < 1 being the unique B solution to the equation ln x = − A x . It was shown before (see, e.g., [31]) that the linear 0-generalized aging intensity function determines the scale family of modified Weibull distributions with parameters 1 γ = A, λ = k A for 0 < k < +∞, and δ = B, defined in [18]. For any α < 0, the linear α-generalized aging intensity characterizes the family of distributions that we call the modified power Lomax 1 distributions with parameters γ = A, λ = (kα ) A , for 0 < k < +∞, 1 ξ = − α and δ = B . In the case α = −1, the linear α-generalized aging intensity characterizes the family of distributions called here the 1 modified log-logistic distributions with parameters γ = A, λ = k A for 0 < k < +∞, and δ = B .

2 − ln ⎡−ln ⎛1 − arctan a ⎞ ⎤ π ⎝ ⎠⎦ ⎣

(

A α = 0, ⎧1 − exp(−kx exp(Bx )), ⎨1 − (1 − kαx A exp(Bx )) α1 , α < 0, ⎩

parameter γ = B and scale parameter λ = 233, and [17]).

(Wα−1 ∘F )(x ))

1 A −B B

()

(see, e.g., [20], p.

1

α α 2 = 1 − ⎡1 − k + k ⎛1 − arctan x ⎞ ⎤ , π ⎝ ⎠ ⎦ ⎣

x > 0.

5. α-generalized aging intensity order

For α = −1, we simply have

k arctan x F−1, L−1, F , k (x ) = (k − 1)arctan x +

π 2

,

[21] introduced the aging intensity order in the following way. For the random variables X, Y with the common support (0, +∞), and the distribution functions F and G, respectively, we say that X succeeds Y in the aging intensity order and write X ≥ AIY, if LF(x) ≤ LG(x) for all x ∈ (0, +∞) . It means that if a random variable has a smaller aging intensity function than another then it is bigger (better) in the aging intensity AI order, i.e., it has a weaker tendency of aging. Analogously, the α-generalized aging intensity order can be defined for every α ∈  . For the absolutely continuous random variables X, Y having the distribution functions F and G, respectively, both supported on (0, +∞) we say that X is greater in the α-generalized aging intensity order for some α ∈ , and write X ≥ αAIY, if Lα, F(x) ≤ Lα, G(x) for all x ∈ (0, +∞) . The family of α-generalized aging intensity orders contains some orders known in literature as special cases. It is obvious that the standard aging intensity order of [21] coincides with our 0-generalized aging intensity order. The reversed hazard rate order is based on the notion of reversed hazard rate function f (x ) r˘F (x ) = F (x ) for a lifetime distribution function F with the density f (see, e.g., [12]). We say that an absolutely continuous lifetime random variable X with the distribution function F succeeds another absolutely continuous lifetime random variable Y with the distribution function G, and write X ≥ rhY if r˘F (x ) ≥ r˘G (x ) for every x > 0. We recall its apparent similarity to the more popular hazard rate order X ≥ hrY determined by the relation rF(x) ≤ rG(x), x > 0, (see (1)). Both imply the standard stochastic ordering X ≥ stY defined as F(x) ≤ G(x), x > 0 (this is applied for comparing general random variables as well, and preservation of the inequality on the whole real axis is required then). Further properties of

x > 0.

Now we characterize lifetime distribution functions by means of some simple α- generalized aging intensity functions. Example 4. Suppose that L (x ) = A > 0 for x > 0. Function L (x ) has the x indefinite integral A ln x + C . Therefore it satisfies (22), and not (27). x Choosing a = 1 so that ∫a dt = A ln x , for α ≤ 0 we obtain

Fα, L, k (x ) = Wα (kx A) =

A ⎧1 − exp(−kx ), α = 0, ⎨1 − (1 − kαx A) α1 , α < 0, ⎩

x > 0.

Some special cases of the result were known in literature. A constant 0generalized aging intensity function characterizes the scale family of Weibull distributions with shape parameter γ = A and scale parameters 1 λ = k A for 0 < k < +∞ (see, e.g., [31]). In the case α = −1, constant function (15) characterizes the scale family of log-logistic distributions 1 with shape parameter γ = A and scale parameters λ = k A for 0 < k < +∞ (see [32,33]). For any α < 0, constant α-generalized aging intensity characterizes the scale family of power Lomax distributions with shape parameter γ = A, scale parameters 1 1 λ = (−kα ) A for 0 < k < +∞, and frailty parameter ξ = − α . Example 5. Now we assume that the generalized aging intensity is linear L (x ) = A + Bx for some A, B > 0. Then the function L (x ) has the x

205

Reliability Engineering and System Safety 178 (2018) 198–208

M. Szymkowiak

LF , α (x ) LG, α (x ) f (x ) = ≤ [1 − F (x )][1 − [1 − F (x )]−α ] x x g (x ) , = [1 − G (x )][1 − [1 − G (x )]−α ]

the orders are presented in [28]. It is easy to verify that

g (x ) f (x ) ≥ r˘G (x ) = G (x ) F (x ) xg (x ) xf (x ) ⇔ L1, F (x ) = ≥ L1, G (x ) = ⇔ X ≤1AI Y . G (x ) F (x )

X ≥rh Y ⇔ r˘F (x ) =

Letting α → −∞, we obtain

rF (x ) =

In a similar way we conclude that the orderings X1: α ≥ rhY1: α and X ≤ αAIY are equivalent for every positive, non-necessarily integer α. [23] determined the LOR order based on the log-odds rate function (14). Relation X ≥ LORY for the lifetime random variables X and Y with the respective distribution functions F and G, and the density functions f and g, positive on (0, +∞), is defined by the inequality LORF(x) ≤ LORG(x) valid for all positive x. Observe that

g (x ) f (x ) , ≤ rG (x ) = 1 − G (x ) 1 − F (x )

x > 0.

(ii) Under the above notation, suppose now that β > 0. By (12), for every α > β and x > 0 yields α−1

Lα, F (x ) f (x ) 1 − [1 − G (x )]α 1 − F (x ) ⎞ =⎛ 1 ( ) (x ) 1 − [1 − F (x )]α Lα, G (x ) G x g − ⎝ ⎠ if G (x ) < F (x ), ⎧ 0, ⎪ f (x ) , if G (x ) = F (x ), as α → +∞. → ⎨ g (x ) ⎪+∞, if G (x ) > F (x ), ⎩

1 ≤

f (x ) ≥ LORG (x ) F (x )[1 − F (x )] g (x ) = G (x )[1 − G (x )] xf (x ) ⇔ L−1, F (x ) = ≥ L−1, G (x ) F (x )[1 − F (x )] xg (x ) = ⇔ X ≥−1AI Y . G (x )[1 − G (x )]

X ≥LOR Y ⇔ LORF (x ) =

⎜

⎟

The limit is actually greater than or equal to 1 if for every x > 0 either G(x) > F(x) or G (x ) = F (x ) and g(x) ≤ f(x). It suffices to conclude that X ≥ stY, but we actually show that the equality of the distribution functions at some point necessarily implies the same for the density functions there. Indeed, if G (x ) = F (x ) and g(x) < f(x) then G < F on L some right neighborhood of x, and Lα, F → 0 there which contradicts our α, G assumption for sufficiently large α. □

Now we prove a lemma useful in comparisons of α-generalized aging intensity orders for various α. Lemma 2. Assume that absolutely continuous distribution functions F and G supported on (0, +∞) satisfy F ≤ G on (0, +∞) .

In view of the above proposition, the hazard rate order and the stochastic order can be treated as the (−∞) and + ∞-generalized aging intensity orders, respectively. Hence Proposition 4(i) and (ii) can be extended to − ∞ ≤ α ≤ β, and β ≤ α ≤ +∞, respectively. The corresponding (−∞) -generalized aging intensity function L−∞, F (x ) = xrF (x ) uniquely characterizes distribution function F unlike its counterparts for all negative α, though. Observe that L−∞, F does not satisfy assumptions of Theorems 2 and 3, because a +∞ ∫0 rF (x ) dx < +∞ = ∫a rF (x ) dx for every 0 < a < +∞. Defining L+∞, F does not make any sense, because lim α →+∞Lα, F (x ) = 0 for every x > 0.

(i) If Lβ, F(x) ≤ Lβ, G(x) for some 0 < x < +∞, then for all α < β we have Lα, F(x) ≤ Lα, G(x). (ii) If Lβ, F(x) ≥ Lβ, G(x) for some 0 < x < +∞, then for all α > β we have Lα, F(x) ≥ Lα, G(x). Proof. (i) For x satisfying Lβ, F(x) ≤ Lβ, G(x) we also have (W β−1 ∘F )(x ) ≤ (W β−1 ∘G )(x ), by assumption and increasing monotonicity of W β−1. By Proposition 3, for every α < β we have

Corollary 1. Relation X ≥ rhY implies X1: α ≥ rhY1:

Lα, Wβ ((W β−1 ∘F )(x )) ≤ Lα, Wβ ((W β−1 ∘G )(x )).

α

for every α > 1.

Proof. The assumption implies X ≥ stY, and is equivalent to X ≤ 1AIY. By Proposition 4(ii), we also have X ≤ αAIY for every α > 1 which is just the claim. □

Applying Lemma 1, we obtain

Lα, F (x ) = Lβ, F (x ) Lα, Wβ ((W β−1 ∘F )(x )) ≤ Lβ, G (x ) Lα, Wβ ((W β−1 ∘G )(x ))

The result for the sample minima of the samples with integer sizes was presented in Theorem 3.2. by [22].

= Lα, G (x ). The proof of part (ii) is analogous.

x > 0.

□

Corollary 2.

Observe that strict inequalities in the assumption imply strict inequalities in the statements. An immediate consequence of Lemma 2 is the following.

(i) If X ≥ stY and X ≥ LORY, then X ≥ αAIY for all α ∈ [−∞, −1]. (ii) If X ≥ stY and X ≤ LORY, then X ≤ αAIY for all α ∈ [−1, +∞].

Proposition 4. Assume that X ≥ stY.

These are the consequences of equivalence of LOR and -1AI orders, and Proposition 4(i) and (ii), respectively, together with Proposition 5. Corollary 2 is a generalization of Proposition 2.18 in [23] which asserts that condition X ≥ stY combined with either X ≥ LORY or X ≤ LORY imply that X ≥ hrY and X ≥ rhY, respectively. They are just the conclusions of Corollary 2(i) and (ii) with α = −∞ and α = 1, respectively. Further on, we establish α-generalized aging intensity orderings within some parametric classes of life distributions. We use the notation consistent with that of Table 1. First we recall two results of [6] based on the 0AI order. They showed that if X1, X2 have the two-parameter Weibull distributions W2(γi, λi) for i = 1, 2, respectively, and γ1 ≤ γ2, then X1 ≥ 0AIX2. Also, if X1, X2 have the modified Weibull distributions MW(γi, λi, δi) for i = 1, 2, and γ1 ≤ γ2 with δ1 ≤ δ2, then X1 ≥ 0AIX2. The statements are based on the fact that the classic and modified Weibull distributions are characterized by the constant and linear 0generalized aging intensity functions, respectively (see Examples 1 and 5). Below we present analogous results for the α-generalized aging

(i) If X ≥ βAIY for some β ∈ , then for all α < β we have X ≥ αAIY. (ii) If X ≤ βAIY for some β ∈ , then for all α > β we have X ≤ αAIY. Next we present the relationships between αAI orders and the hazard rate order, and the standard stochastic order. Proposition 5. (i) If there exists β ∈  such that for all α < β we have X ≥ αAIY, then X ≥ hrY. (ii) If there exists β ∈  such that for all α > β we have X ≤ αAIY, then X ≥ stY. Proof. (i) Let F, G and f, g denote the distribution and density functions of X and Y, respectively. We may assume that β < 0. Then by (13), for every α < β and x > 0 we have 206

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M. Szymkowiak

orders with negative α. Proposition 6. For any α < 0, if two random variables X1, X2 have the 1 power Lomax distributions POLO (γi, λi , − α ) for i = 1, 2 with γ1 ≤ γ2, then X1 ≥ αAI X2. In the special case α = −1, we get the following. Corollary 3. If Xi have the log-logistic distributions LLog(γi, λi) for i = 1, 2 with γ1 ≤ γ2, then X1 ≥−1AI X2 . Proposition 7. For any α < 0, if Xi have the modified power Lomax 1 distributions MPOLO (γi, λi , − α , δi ) for i = 1, 2 with γ1 ≤ γ2 and δ1 ≤ δ2, then X1 ≥ αAIX2. Corollary 4. If Xi have the modified log-logistic distributions MLLog(γi, λi, δi) for i = 1, 2 with γ1 ≤ γ2, and δ1 ≤ δ2, then X1 ≥−1AI X2 . Propositions 6 and 7 can be easily deduced from Examples 4 and 5. The statements of Corollaries 3 and 4 can be expressed in terms of the log-odds ratio order as well. 6. Analysis of α-generalized aging intensity through data

Fig. 5. Estimated L−̂ 1, F (x ) and regression line for grouped data from Example 7.

We present an application of the α-generalized aging intensity functions for identifying parametric families of lifetime distributions from random data. The method is especially useful in the problems when the density and distribution functions have complicated forms, but under a proper choice of α the respective aging intensity has relatively easy representation. Due to (11), a natural estimate of α-generalized aging intensity is

Table 3 Parameters of MLLog(γ, λ, δ).

Theoretical parameters Estimated parameters

γ

λ

2 γ̂

0.8 λ̂

1.9521

0.8340

δ 1

δ̂ 0.9430

α − 1

⎧ αx [1 − F (x )] fα (x ) , α ≠ 0, ⎪ 1 − [1 − F (x )] Lα̂ , F (x ) = ⎨ xf (x ) ⎪− [1 − F (x )]ln[1 − F (x )] , α = 0, ⎩

for some unknown positive γ, λ, and δ (see Table 1).

By Example 5, its − 1-generalized aging intensity function is L−1, Fγ , λ, δ (x ) = γ + δx , x > 0. For verifying our claim, we check whether the estimate of the respective aging intensity accurately approximates a linear function. We solve the problem under an additional restriction that we can only use grouped data for the inference. This means that instead of disposing precise values of observations, we merely know the numbers of observations which fall down into fixed classes of partition of the positive half-axis. In our study, we generate N independent random variables X1 , …, XN with MLLog(γ, λ, δ) lifetime distribution by means of the following procedure. Using function rnd of MATLAB, we generate standard uniform random variables U1, …, UN . Then applying the inverse transform technique with we obtain Fγ , λ (x ) = 1 − [1 + (λx ) γ ]−1 ,

Table 2 Generated grouped data and respective values of estimators.

Yi = Fγ−, λ1 (1 − Ui ) = λ U − 1 , i = 1, …, N , with log-logistic distribui tion LLog(γ, λ). Next, solving numerically with use of MATLAB the

f (x ) is a nonparametric density estimator, and where   (x ) = ∫ x  F f (t )dt is the corresponding estimator of the distribution 0 function. The 0-generalized aging intensity estimators were presented and studied in [31]. Example 7. Suppose that our aim is to test if a random samples X1 , …, XN comes from the modified log-logistic lifetime distribution function

Fγ , λ, δ (x ) = 1 − [1 + (λx ) γ exp(δx )]−1 ,

x > 0,

1

class

x ∈ [xj , xj + 1)

nj

 f (x )

 (xj ) F

L−̂ 1, F (xj )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0-0.21 0.21-0.42 0.42- 0.63 0.63-0.84 0.84-1.05 1.05-1.26 1.26-1.47 1.47-1.68 1.68-1.89 1.89-2.10 2.10-2.31 2.31-2.52 2.52-2.73 2.73-2.94 2.94-3.15 3.15-3.36 3.36-3.57 3.57-3.78 3.78-3.99 3.99-4.20

44 112 190 171 152 109 95 39 33 16 14 7 5 6 1 1 2 1 1 1

0.2095 0.5333 0.9048 0.8143 0.7238 0.5190 0.4524 0.1857 0.1571 0.0762 0.0667 0.0333 0.0238 0.0286 0.0048 0.0048 0.0095 0.0048 0.0048 0.0048

0.0220 0.1000 0.2510 0.4315 0.5930 0.7235 0.8255 0.8925 0.9285 0.9530 0.9680 0.9785 0.9845 0.9900 0.9935 0.9945 0.9960 0.9975 0.9985 0.9995

1.0225 1.8667 2.5266 2.4398 2.8340 2.9968 4.2867 3.0487 4.2252 3.3935 4.7456 3.8265 4.0957 8.1818 2.2454 2.8338 8.2831 7.0175 12.3519 39.0195

(

)

1

Lambert equations Xi exp

1 γ

( X ) = Y with respect to X , we get random δ γ

i

i

i

variables with desired distribution MLLog(γ, λ, δ). Finally, using function histogram we group the obtained data into k classes [x j , x j + 1) = [x j , x j + Δx ), j = 1, …, k , of length Δx. The frequency of the j-th class is denoted by nj = nj (X1, …, XN ) . Under the assumptions, a natural estimate of the density function is the histogram-type estimator

 f (x ) =

nj N Δx

,

for x j ≤ x < x j + 1,

j = 1, …, k .

The respective distribution function and − 1-generalized aging intensity function estimates have the forms j−1 (x − x j ) ⎞  (x ) = 1 ⎛ ∑ ni + nj F , ⎜ Δx ⎟ N i=1 ⎠ ⎝

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M. Szymkowiak

L−̂ 1, F (x ) =

x f (x )  (x )(1 − F  (x )) F

Acknowledgments

x nj N Δx

=

The author is grateful to three anonymous reviewers for their useful comments and suggestions to the original version of this paper.

,

j−1 k (Δx ∑i = 1 ni + nj (x − x j )) ⎜⎛Δx ∑i = j ni − nj (x − x j ) ⎟⎞ ⎝ ⎠

References

where x ∈ [x j , x j + 1) and j = 1, …, k . For the exemplary generated data we put γ = 2, λ = 0.8 and δ = 1, sample size N = 1000, number of classes k = 20 with length Δx = 0.21.  f , distribution function F The values of estimates of density function  and − 1-generalized aging intensity function L−̂ 1, F at the middle points 1 x j = 2 (x j + x j + 1) of the classes are given in Table 2 (note that the density estimate is constant over the whole class). A cursory look at the graph of L−̂ 1, F (x ) after removing few outlying values at the right end on Fig. 5 allows us to confirm the hypothesis that the parent − 1-generalized aging intensity function is linearly increasing. As a consequence, we may accept the assumption that the random sample has some modified log-logistic distribution. If we need more formal arguments, the following statistical procedure should be performed. First, using the regression estimation method, we calculate the estimates of the slope and the intercept which for our data amount to δ ̂ = 0.9430 and γ ̂ = 1.9521. Then we plug them into the log-likelihood function, and determine λ ̂ minimizing it. The problem resolves into finding the solution to the equation nj k N = 2 which is here equal to λ ̂ = 0.8309 . In Table 3, ∑ j=1 −γ ̂ (λ ̂ xj )

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exp(−δ ̂ xj ) + 1

the theoretical parameters of MLLog(γ, λ, δ) and their estimators are compared. We see that the estimators based on the empirical generalized aging intensity are quite accurate despite of the significant loss of statistical information caused by grouping. Finally, for checking if our data really fit the modified log logistic lifetime distribution we use the most adequate for grouped data the chisquare goodness-of-fit test (available in MATLAB function chi2gof). Combining some classes with low frequencies, we determine statistics χ 2 = 13.2394 with ν = 10 degrees of freedom. Moreover, we determine p-value p = 0.2106. It means that at significance levels α < 0.2106 the hypothesis that the considered data follow the modified log logistic distribution should not be rejected. 7. Conclusions In this paper, the α-generalized aging intensity functions of an univariate positive absolutely continuous random variable were defined. They provide tools for a deeper analysis of variability of lifetime distributions, comparing the instantaneous values of the α-generalized failure rates and those observed in the past. They can also be treated as the elasticity functions of the α-generalized cumulative failure rates which measure the percentage these functions increase when time increases by a small amount. Some properties of the α-generalized aging intensity functions were presented. Moreover, they were used for characterization of continuous random variables. It was proved that for each positive α this characterization is unique, but for the other α each generalized aging intensity function characterizes a scale family of distributions. Stochastic orders based on the α-generalized aging intensities, and their connections with some other orders known earlier in literature were also discussed. Finally, to demonstrate applicability of the α-generalized aging intensity in the reliability practice, the analysis of some random data was performed. The recognition of the shape of some α-aging intensity function estimate admits a simple identification of data lifetime distribution.

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