Chapter 4
Location-Scale-Based Parametric Distributions
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
4-1
Copyright 1998-2008 W. Q. Meeker and L. A. Escobar.
Based on the authors’ text Statistical Methods for Reliability
Data, John Wiley & Sons Inc. 1998.
December 14, 2015
8h 8min
Motivation for Parametric Models
• Complements nonparametric techniques.
• Parametric models can be described concisely with just a
few parameters, instead of having to report an entire curve.
• It is possible to use a parametric model to extrapolate (in
time) to the lower or upper tail of a distribution.
• Parametric models provide smooth estimates of failure-time
distributions.
4-3
In practice it is often useful to compare various parametric
and nonparametric analyses of a data set.
Functions of the Parameters-Continued
E(T ) =
Z ∞
0
tf (t; θ ) dt =
Z ∞
0
[1 − F (t; θ )] dt.
• The mean time to failure, MTTF, of T (also known as
expectation of T )
R
If 0∞ tf (t; θ ) dt = ∞, we say that the mean of T does not
exist.
Z ∞
SD(T )
.
E(T )
Var(T ).
q0
[t − E(T )]2f (t; θ ) dt
• The variance (or the second central moment) of T and the
standard deviation
Var(T ) =
SD(T ) =
• Coefficient of variation γ2
γ2 =
4-5
Chapter 4
Location-Scale-Based Parametric Distributions
Objectives
• Explain importance of parametric models in the analysis of
reliability data.
• Define important functions of model parameter that are of
interest in reliability studies.
• Introduce the location-scale and log-location-scale families
of distributions.
• Describe the properties of the exponential distribution.
4-4
4-2
• Describe the Weibull and lognormal distributions and the
related underlying location-scale distributions.
Functions of the Parameters
• Cumulative distribution function (cdf) of T
F (t; θ ) = Pr(T ≤ t), t > 0.
f (t; θ )
, t > 0.
1 − F (t; θ )
F (tp; θ ) ≥ p.
• The p quantile of T is the smallest value tp such that
• Hazard function of T
h(t; θ ) =
Location-Scale Distributions
Y belongs to the location-scale family of distributions if the
cdf of Y can be expressed as
y−µ
, −∞ < y < ∞
F (y; µ, σ) = Pr(Y ≤ y) = Φ
σ
where −∞ < µ < ∞ is a location parameter and σ > 0 is a
scale parameter.
Φ is the cdf of Y when µ = 0 and σ = 1 and Φ does not
depend on any unknown parameters.
Note: The distribution of Z = (Y − µ)/σ does not depend
on any unknown parameters.
4-6
Importance of Location-Scale Distributions
Importance due to:
• Most widely used statistical distributions are either members of this class or closely related to this class of distributions: exponential, normal, Weibull, lognormal, loglogistic,
logistic, and extreme value distributions.
• Methods of inference, statistical theory, and computer software generated for the general family can be applied to this
large, important class of models.
t > γ,
4-7
• Theory for location-scale distributions is relatively simple.
Exponential Distribution
For T ∼ EXP(θ, γ),
t−γ
F (t; θ, γ) = 1 − exp −
θ
1
t−γ
f (t; θ, γ) =
exp −
θ
θ
1
f (t; θ, γ)
= ,
h(t; θ, γ) =
1 − F (t; θ, γ)
θ
where θ > 0 is a scale parameter and γ is both a location and
a threshold parameter. When γ = 0 one gets the well-known
one-parameter exponential distribution.
E(T ) = γ + θ,
5
6
6
7
7
Var(T ) = θ 2.
f(t)
0.0
0.4
0.8
1.2
3
4
σ
5
5
5
µ
6
0.3
0.5
0.8
t
5
7
Probability Density Function
Examples of Normal Distributions
4
t
t
5
Hazard Function
4
4-9
Quantiles: tp = γ − θ log(1 − p).
Moments: For integer m > 0, E[(T − γ)m] = m! θm. Then
3
Cumulative Distribution Function
1
F(t) .5
0
20
15
h(t) 10
5
0
3
4 - 11
3.0
1.5
2.0
2.0
0.5
0.0
1.0
θ
0
0
0
γ
2.0
0.5
1.0
2.0
t
4-8
3.0
Probability Density Function
Examples of Exponential Distributions
2.0
Cumulative Distribution Function
1
t
f(t) 1.0
1.0
F(t) .5
0.0
t
Hazard Function
0.0
2.0
1.5
1.0
1.0
3.0
0
h(t)
0.5
0.0
Motivation for the Exponential Distribution
• Simplest distribution used in the analysis of reliability data.
• Has the important characteristic that its hazard function is
constant (does not depend on time t).
• Popular distribution for some kinds of electronic components (e.g., capacitors or robust, high-quality integrated
circuits).
• This distribution would not be appropriate for a population
of electronic components having failure-causing quality-defects.
4 - 10
• Might be useful to describe failure times for components
that exhibit physical wearout only after expected technological life of the system in which the component would be
installed.
−∞ < y < ∞.
Normal (Gaussian) Distribution
For Y ∼ NOR(µ, σ)
1
φnor
σ
y−µ
F (y; µ, σ) = Φnor
σ
y−µ
,
σ
f (y; µ, σ) =
√
Rz
where φnor (z) = (1/ 2π) exp(−z 2/2) and Φnor (z) = −∞
φnor (w)dw
are pdf and cdf for a standardized normal (µ = 0, σ = 1).
−∞ < µ < ∞ is a location parameter; σ > 0 is a scale parameter.
Var(Y ) = σ 2.
−1
−1
Quantiles: yp = µ + σΦnor
(p) where Φnor
(p) is the p quantile
for a standardized normal.
Moments: For integer m > 0, E[(Y − µ)m] = 0 if m is odd,
and E[(Y − µ)m] = (m)!σ m/[2m/2 (m/2)!] if m is even. Thus
E(Y ) = µ,
4 - 12
0.0
t
2.0
2.0
3.0
3.0
f(t)
0.0
0.4
0.8
1.2
0.0
1.0
σ
0
0
0
µ
2.0
0.3
0.5
0.8
t
4 - 13
3.0
Probability Density Function
Examples of Lognormal Distributions
1.0
1.0
Hazard Function
t
Cumulative Distribution Function
1
2
3
4
0
F(t) .5
h(t)
1
0
0.0
Motivation for Lognormal Distribution
• The lognormal distribution is a common model for failure
times.
• It can be justified for a random variable that arises from the
product of a number of identically distributed independent
positive random quantities.
• It has been suggested as an appropriate model for failure
time caused by a degradation process with combinations of
random rates that combine multiplicatively.
• Widely used to describe time to fracture from fatigue crack
growth in metals.
• Useful in modeling failure time of a population electronic
components with a decreasing hazard function (due to a
small proportion of defects in the population).
4 - 15
• Useful for describing the failure-time distribution of certain
degradation processes.
Smallest Extreme Value Distribution
For Y ∼ SEV(µ, σ),
y−µ
F (y; µ, σ) = Φsev
σ
1
y−µ
f (y; µ, σ) =
φsev
σ
σ 1
y−µ
h(y; µ, σ) =
exp
, −∞ < y < ∞.
σ
σ
Φsev (z) = 1 − exp[− exp(z)], φsev (z) = exp[z − exp(z)] are cdf
and pdf for standardized SEV (µ = 0, σ = 1). −∞ < µ < ∞
is a location parameter and σ > 0 is a scale parameter.
−1
Quantiles: yp = µ + Φsev
(p)σ = µ + log [− log(1 − p)] σ.
Mean and Variance: E(Y ) = µ − σγ, Var(Y ) = σ 2π 2/6,
where γ ≈ .5772, π ≈ 3.1416.
4 - 17
Lognormal Distribution
"
#
t > 0.
If T ∼ LOGNOR(µ, σ) then log(T ) ∼ NOR(µ, σ) with
log(t) − µ
F (t; µ, σ) = Φnor
σ
"
#
1
log(t) − µ
φnor
,
σt
σ
f (t; µ, σ) =
φnor and Φnor are pdf and cdf for a standardized normal.
exp(µ) is a scale parameter; σ > 0 is a shape parameter.
h
4 - 14
t
t
45
50
55
55
Hazard Function
40
t
1.0
1.5
1.5
60
2.0
2.0
f(t)
0.02
35
40
µ
55
σ
50
50
50
50
5
6
7
t
45
4 - 16
60
Probability Density Function
30
0.5
η
1.5
β
1
1
1
t
1.0
2.0
Probability Density Function
0.0
0.8
1.0
1.5
i
exp(σ 2) − 1 .
Quantiles: t = exp µ + σΦ−1 (p) , where Φ−1 (p) is the p
p
nor
nor
quantile for a standardized normal.
Moments: For integer m > 0, E(T m) = exp mµ + m2σ 2/2 .
E(T ) = exp µ + σ 2/2 , Var(T ) = exp 2µ + σ 2
50
0.06
Examples of Smallest Extreme Value Distributions
45
Cumulative Distribution Function
1
40
f(t) 0.04
35
F(t) .5
30
0.0
1.5
1.0
35
0.5
t
1.0
Hazard Function
0.5
2.5
2.0
1.5
1.0
0.5
0.0
Examples of Weibull Distributions
60
0
h(t)
0.5
0.0
30
0.0
Cumulative Distribution Function
1
2
3
4
0
F(t) .5
h(t)
1
0
0.0
4 - 18
!β−1
!β−1


Weibull Distribution
t
η
,
t>0
t
exp −
η
Common Parameterization:
β
f (t) =
η
t
η
!β 

t
F (t) = Pr(T ≤ t) = 1 − exp −
η
β
η
h(t) =

!β 
β > 0 is shape parameter; η > 0 is scale parameter.
0
Z ∞
!
!
!#
4 - 19
wκ−1 exp(−w)dw is the gamma function.
2
1
Var(T ) = η 2 Γ 1 +
− Γ2 1 +
β
β
"
Quantiles: tp = η [− log(1 − p)]1/β .
Moments: For integer m > 0, E(T m) = η mΓ(1+m/β). Then
Γ(κ) =
1
,
E(T ) = ηΓ 1 +
β
where
Note: When β = 1 then T ∼ EXP(η).
Motivation for the Weibull Distribution
• The theory of extreme values shows that the Weibull distribution can be used to model the minimum of a large
number of independent positive random variables from a
certain class of distributions.
◮ Failure of the weakest link in a chain with many links
with failure mechanisms (e.g., creep or fatigue) in each
link acting approximately independent.
◮ Failure of a system with a large number of components in
series and with approximately independent failure mechanisms in each component.
4 - 21
• The more common justification for its use is empirical: the
Weibull distribution can be used to model failure-time data
with a decreasing or an increasing hazard function.
−∞ < y < ∞.
Largest Extreme Value Distribution
When Y ∼ LEV(µ, σ),
y−µ
F (y; µ, σ) = Φlev
σ
1
y−µ
f (y; µ, σ) =
φlev
σ
σ
exp − y−µ
o,
n
h
σ i
−1
σ exp exp − y−µ
σ
h(y; µ, σ) =
where Φlev (z) = exp[− exp(−z)] and φlev (z) = exp[−z−exp(−z)]
are the cdf and pdf for a standardized LEV (µ = 0, σ = 1)
distribution.
−∞ < µ < ∞ is a location parameter and σ > 0 is a scale
parameter.
4 - 23
Alternative Weibull Parameterization

!β−1
t
η
!β

"
=
 = Φsev
!β 

exp −
t
η
h
#
i
log(t) − µ
1
φsev
σt
σ
"
log(t) − µ
σ
Note: If T ∼ WEIB(µ, σ) then Y = log(T ) ∼ SEV(µ, σ).
t
η
For T ∼ WEIB(µ, σ) then
β
η
F (t; µ, σ) = 1 − exp −
f (t; µ, σ) =
Φsev (z) = 1 − exp[− exp(z)].
φsev (z) = exp[z − exp(z)]
where σ = 1/β, µ = log(η), and
Quantiles:
−1 (p)
tp = η [− log(1 − p)]1/β = exp µ + σΦsev
#
4 - 20
−1
where Φsev
(p) is the p quantile for a standardized SEV (i.e.,
µ = 0, σ = 1).
40
0.06
30
0.02
0
10
σ
10
10
10
µ
30
5
6
7
t
20
40
Probability Density Function
Examples of Largest Extreme Value Distributions
30
Cumulative Distribution Function
1
t
f(t) 0.04
20
F(t) .5
10
0.0
0
t
20
Hazard Function
10
40
0
0.20
0.15
h(t)0.10
0.05
0.0
0
4 - 22
Largest Extreme Value Distribution - Continued
Quantiles: yp = µ − σ log [− log(p)] .
Mean and Variance: E(Y ) = µ + σγ, Var(Y ) = σ 2π 2/6,
where γ ≈ .5772, π ≈ 3.1416.
Notes:
• The hazard is increasing but is bounded in the sense that
limy→∞ h(y; µ, σ) = 1/σ.
• If Y ∼ LEV(µ, σ) then −Y ∼ SEV(−µ, σ).
4 - 24
5
5
15
t
15
20
20
25
25
0.25
0.20
0.15
f(t)
0.10
0.05
0.0
5
10
µ
20
σ
15
15
15
1
2
3
t
15
4 - 25
25
Probability Density Function
Examples of Logistic Distributions
10
Hazard Function
10
Cumulative Distribution Function
1
0
F(t) .5
h(t)
1.0
0.8
0.6
0.4
0.2
0.0
t
Logistic Distribution-Continued
p
−1
(p) = µ + σ log 1−p
Quantiles: yp = µ + σΦlogis
, where
−1
Φlogis
(p) = log[p/(1 − p)] is the p quantile for a standardized
logistic distribution.
4 - 27
Moments: For integer m > 0, E[(Y − µ)m] = 0 if m is odd,
h
i
P∞
m if m
and E[(Y − µ)m] = 2σ m (m!) 1 − (1/2)m−1
i=1 (1/i)
is even. Thus
E(Y ) = µ,
σ 2π 2
Var(Y ) =
.
3
Loglogistic Distribution
"
#
t > 0.
If Y ∼ LOGIS(µ, σ) then T = exp(Y ) ∼ LOGLOGIS(µ, σ)
with
log(t) − µ
F (t; µ, σ) = Φlogis
σ
"
#
1
log(t) − µ
f (t; µ, σ) =
φlogis
σt
σ
"
#
1
log(t) − µ
Φlogis
,
σt
σ
h(t; µ, σ) =
exp(µ) is a scale parameter; σ > 0 is a shape parameter.
Φlogis and φlogis are cdf and pdf for a LOGIS(0, 1).
4 - 29
Logistic Distribution
For Y ∼ LOGIS(µ, σ),
y−µ
F (y; µ, σ) = Φlogis
σ
1
y−µ
f (y; µ, σ) =
φlogis
σ
σ 1
y−µ
h(y; µ, σ) =
Φlogis
, −∞ < y < ∞.
σ
σ
−∞ < µ < ∞ is a location parameter; σ > 0 is a scale parameter.
φlogis(z) =
Φlogis(z) =
t
2
3
3
4
4
exp(z)
[1 + exp(z)]2
exp(z)
.
1 + exp(z)
f(t)
0.0
0.4
0.8
1.2
0
1
σ
0
0
0
µ
3
0.2
0.4
0.6
t
2
4 - 28
4
Probability Density Function
Examples of Loglogistic Distributions
1
t
2
Hazard Function
1
4 - 26
φlogis and Φlogis are pdf and cdf for a standardized logistic
distribution defined by
0
Cumulative Distribution Function
1
1.0
2.0
3.0
0
F(t) .5
h(t)
0.0
0
Loglogistic Distribution-Continued
h
i
−1
Quantiles: tp = exp µ + σΦlogis
(p) = exp(µ) [p/(1 − p)]σ .
Moments: For integer m > 0,
E(T m) = exp(mµ) Γ(1 + mσ) Γ(1 − mσ).
The m moment is not finite when mσ ≥ 1.
For σ < 1,
E(T ) = exp(µ) Γ(1 + σ) Γ(1 − σ),
and for σ < 1/2,
h
i
Var(T ) = exp(2µ) Γ(1 + 2σ) Γ(1 − 2σ) − Γ2(1 + σ) Γ2(1 − σ) .
4 - 30
Other Topics in Chapter 4
Pseudorandom number generation.
4 - 31
Topics in Chapter 5
• Parametric models with threshold parameters.
• Important distributions used in reliability that can not be
translated into location-scale distributions: gamma, generalized gamma, etc.
• Finite (discrete) mixture distributions
P
i ξi = 1
F (t; θ ) = ξ1F1(t; θ 1) + · · · + ξk Fk (t; θ k )
where ξi ≥ 0, and
• Compound (continuous) mixture distributions.
4 - 32
If failure-times of units in a population are EXP(η) with
1/η ∼ GAM(θ, κ), then the unconditional failure time, T , of
a unit selected at random from the population has a Pareto
distribution of the form
1
F (t; θ, κ) = 1 −
, t > 0.
(1 + θt)κ