APPENDIX
A. CONSTANTS, DISTRIBUTIONS AND ASYMPTOTICS
A.1 Mathematical Constants
This section provides some information on several not well-known mathematical constants that are used throughout this thesis. Detailed knowledge of these constants is not presented here, since these constants are of minor importance to the subject of this thesis.
A.1.1 Euler-Mascheroni Constant
The Euler-Mascheroni constant, γ, sometimes also referred to as the Euler constant, is de- fined as the limit of the sequence
γ = lim
n→∞
N
X
n=1
1
n− ln(N)
! .
The numerical value of the Euler-Mascheroni constant is approximately 0.5772157. This constant is often encountered in number theory. For more information see Weisstein (2005).
A.1.2 Ap´ery’s Constant
Ap´ery’s constant, ζ(3), is defined by ζ(3) = 1
Γ(3) Z ∞
0
u2 eu− 1du.
where ζ(z) is the Riemann zeta function. The numerical approximation of Ap´ery’s constant is equal to 1.2020569. For more information see Weisstein (2006a).
A.2 Statistical Distributions
This section provides additional information on a few distributions not often encountered.
Complete and detailed information is not presented here, since the main interested of this thesis lies elsewhere.
A.2.1 The Extreme Value Distribution
The extreme value distribution, sometimes also called the Gumbel distribution, plays an important role in proportional hazard models. Therefore some of the most important prop- erties of this distribution are given. The cumulative distribution function of the extreme value distribution is
F() = exp
−e−µ(−η)
, (A.1)
A. Constants, Distributions and Asymptotics 36
where ∈ IR, µ > 0 is a scale parameter and η ∈ IR is a location parameter. Alternative spec- ifications are also used in some instances. The case where η = 0 and µ = 1 is the standard extreme value distribution. The probability density function corresponding to (A.1) is
f() = µe−µ(−η)exp
−e−µ(−η) .
The expectation of the extreme value distribution is η +γµ, where γ is the Euler-Mascheroni constant (≈ 0.577) and the variance is6µπ22. The skewness ψ of the extreme value distribution is given by
ψ = 12√ 6ζ(3) π3 ,
where ζ(3) is Ap´ery’s constant (≈ 1.202). The kurtosis excess κ of the extreme value distri- bution is equal to 125. If 1 and 2 are independent and have an extreme value distribution with location parameters η1 and η2 respectively, and the same location parameter µ, then
∗ ≡ 1− 2is logistically distributed with location parameter η∗ ≡ η1− η2and scale parame- ter µ. For more information on the extreme value distributions see Weisstein (2006b), Walck (1996) and Johnson, Kotz, and Balakrishnan (1995).
A.2.2 The Logistic Distribution
The cumulative distribution function of the logistic distribution with a location parameter η and scale parameter µ is
F() = 1
1 + exp (µ(− η)), (A.2)
where ∈ IR. The case where η = 0 and µ = 1 is the standard logistic distribution. The probability density function corresponding to (A.2) is
f() = µexp (µ(− η)) (1 + exp (µ(− η)))2.
The expectation of a logistic distribution is η and the variance is equal to 3µπ22. The skewness ψof the logistic distribution is equal to 0, while the kurtosis excess κ is equal to 65 for this dis- tribution. For more information on the logistic distribution see Weisstein (2003) and Walck (1996).
A.3 Slutsky’s Theorem
Without mentioning, Slutsky’s theorem is often applied in the main text of this thesis. There- fore, Slutsky’s theorem (Slutsky, 1925) is provided in this section without a proof as it ap- pears in Wansbeek and Meijer (2000).
Let{XN} be a sequence of random vector variables which, as N −→ ∞, converges in dis- tribution to the random vector variable X and let {YN} be a sequence of random vector
A. Constants, Distributions and Asymptotics 37
variables which, as N −→ ∞, converges in probability to the constant vector c. Further- more, let f (x, y) be a vector-valued function of its two vector-valued arguments and letC be the set of points (x, y) at which f (x, y) is continuous. Then,
f(XN, YN) −→d f(X, c),
provided that the probability that (X, c)∈ C is 1.
A.4 Central Limit Theorem
Let{XN} be a sequence of random variables which are independently and identically dis- tributed with a finite mean and variance, i.e., E [Xn] = µ < ∞ and Var (Xn) = σ2 <∞ for all n. Now define ¯XN as the average of the sequence of random variables, i.e.,
X¯N = 1 N
N
X
n=1
Xn. Then it holds that
√N
X¯N − µ σ
A
∼ N (0, 1) .
B. MATHEMATICAL DERIVATIONS
This appendix contains mathematical derivations of a selected few equations in this thesis as not to clutter the main text and still give ample information on the derivations involved.
B.1 Derivations for Chapter 4 Equation (4.3)
P{∗n≤ E} = P {− ln Λ0(tn)− Ξnβ ≤ E}
= P{− ln Λ0(tn)≤ E + Ξnβ}
= P{ln Λ0(tn) >−E − Ξnβ}
= P{Λ0(tn) > exp (−E − Ξnβ)}
= Ptn>Λ−10 (exp (−E − Ξnβ))
= 1− F Λ−10 (exp (−E − Ξnβ)) exp(Ξnβ)
= exp −Λ0 Λ−10 (exp (−E − Ξnβ)) exp(Ξnβ)
= exp (− exp (−E − Ξnβ) exp(Ξnβ))
= exp(− exp(−E)).
Equation (4.5)
L(β) = ln
N
Y
n=1
f(tn, ξn, β)
!
=
N
X
n=1
ln f (tn, ξn, β)
=
N
X
n=1
ln (λ(tn, ξn, β)(1− F (tn, ξn, β)))
=
N
X
n=1
ln (λ(tn, ξn, β) exp(−Λ(tn, ξn, β)))
=
N
X
n=1
ln λ(tn, ξn, β)−
N
X
n=1
Λ(tn, ξn, β)
=
N
X
n=1
Ξnβ−
N
X
n=1
tnexp(Ξnβ)
= ι0Ξβ− t0exp(Ξβ).
B. Mathematical Derivations 39
Equation (4.6)
s(β) = ∂L(β)
∂β0
= ∂ι0Ξβ
∂β0 −∂t0exp(Ξβ)
∂β0
= ι0Ξ− t0∆Ξ.
Equation (4.7)
E [s(β)] = Eι0Ξ− t0∆Ξ
= ι0Ξ− E[t]0∆Ξ
= ι0Ξ− (exp(−Ξβ))0∆Ξ
= ι0Ξ− ι0Ξ
= 0.
Equation (4.8)
H(β) = ∂2L(β)
∂β∂β0
= ∂ι0Ξ− t0∆Ξ
∂β
= ∂ ι0Ξ− (exp(Ξβ))0∆∗Ξ
∂β
= −Ξ0∆∆∗Ξ.
Equation (4.9)
I(β) = − (E [H(β)])−1
= EΞ0∆∆∗Ξ−1
= Ξ0∆ E [∆∗] Ξ−1
= Ξ0Ξ−1
. Equation (4.10)
plim
N→∞
ˆb = plim
N→∞
(Ξ0Ξ)−1Ξ0t∗
= plim
N→∞
(Ξ0Ξ)−1Ξ0(Ξb + )
= plim
N→∞
(Ξ0Ξ)−1Ξ0Ξb + (Ξ0Ξ)−1Ξ0
= b + plim
N→∞
Ξ0Ξ N
−1Ξ0 N
!
= b.
B. Mathematical Derivations 40
B.2 Derivations for Chapter 5
Equation (5.4) plim
N→∞
ˆb = plim
N→∞
X0X−1
X0t∗
= plim
N→∞
X0X−1
X0(Xb + u)
= plim
N→∞
X0X−1
X0(Xb + − V b)
= plim
N→∞
X0X−1
X0Xb+ X0X−1
X0− X0X−1
X0V b
= b + plim
N→∞
X0X N
−1X0 N
!
− plim
N→∞
X0X N
−1 X0V N
! b
= b− plim
N→∞
X0X N
−1X0V N
! b
= b− Σ−1X ΣVb.
Equation (5.5)
P{u∗n≤ U} = P {− ln Λ0(tn)− Xnβ ≤ U}
= P{− ln Λ0(tn)≤ U + Xnβ}
= P{ln Λ0(tn) >−U − Xnβ}
= P{Λ0(tn) > exp (−U − Xnβ)}
= Ptn>Λ−10 (exp (−U − Xnβ))
= 1− F Λ−10 (exp (−U − Xnβ))
= exp −Λ0 Λ−10 (exp (−U − Xnβ)) exp(Ξnβ)
= exp (− exp (−U − Xnβ) exp(Ξnβ))
= exp (− exp (Ξnβ− Xnβ− U))
= exp (− exp(−Vnβ− U)) . Equation (5.7a)
Ex3n
= E(ξn+ vn)3
= Eξ3n+ 3ξn2vn+ 3ξnv2n+ vn3
= Eξ3n + E vn3 . Equation (5.7b)
Ex2nt∗n
= E(ξn+ vn)2(b1+ b2ξn+ n)
= Eb1ξn2+ 2b1ξnvn+ b1v2n+ b2ξn3+ 2b2ξn2vn+ b2ξnvn2+ ξ2nn+ 2ξnnvn+ nvn2
= b1σ2ξ+ b1σ2v+ b2Eξn3 .
B. Mathematical Derivations 41
Equation (5.7c) Eh
xnt∗n2i
= E(ξn+ vn)(b1+ b2ξn+ n)2
= Eb21ξn+ 2b1b2ξn2+ 2b1ξnn+ b22ξn3+ 2b2ξn2n+ ξn2n+ b21vn+ 2b1b2ξnvn + E2b1nvn+ b22ξn2vn+ 2b2ξnnvn+ 2nvn
= 2b1b2σ2ξ+ b22Eξn3 . Equation (5.7d)
Eh t∗n3i
= E(b1+ b2ξn+ n)3
= Eb31+ 3b21b2ξn+ 3b21n+ 3b1b22ξn2+ 6b1b2ξnn+ 3b12n+ b32ξn3+ 3b22ξn2n + E3b2ξn2n+ 3n
= b31+ 3b1b22σ2ξ+1
2b1π2+ b32Eξ3n + E 3n . Equation (5.8a)
Ex4n
= Eh
(ξn+ vn)4i
= Eξ4n+ 4ξn3vn+ 6ξn2v2n+ 4ξnv3n+ vn4
= Eξ4n + 6σξ2σv2+ Evn4 . Equation (5.8b)
Ex3nt∗n
= Eh
(ξn+ vn)3(b1+ b2ξn+ n)i
= E
ξ3n+ 3ξn2vn+ 3ξnv2n+ v3n (b1+ b2ξn+ n)
= Eb1ξn3+ 3b1ξn2vn+ 3b1ξnvn2+ b1v3n+ b2ξn4+ 3b2ξn3vn+ 3b2ξn2vn2 + Eb2ξnv3n+ ξn3n+ 3ξn2vnn+ 3ξnvn2n+ v3nn
= b1Eξn3 + b1Evn3 + b2Eξn4 + 3b2σ2ξσ2v. Equation (5.8c)
Ex2nt∗2n
= Eh
(ξn+ vn)2(b1+ b2ξn+ n)2i
= E
ξn2 + 2ξnvn+ v2n
b21+ 2b1b2ξn+ 2b1n+ b22ξn2+ 2b2ξnn+ 2n
= Eb21ξn2+ 2b21ξnvn+ b21v2n+ 2b1b2ξn3+ 4b1b2ξn2vN+ 2b1b2ξnvn2+ 2b1ξn2n
+ E4b1ξnvnn+ 2b1v2nn+ b22ξn4+ 2b22ξn3vN + b22ξ2nv2n+ 2b2ξ3nn
+ E4b2ξ2nvnn+ 2b2ξnvn2n+ ξ2n2n+ 2ξnvn2n+ vn22n
= b21σ2ξ+ b21σ2v+ 2b1b2Eξn3 + b22Eξn4 + b22σξ2σv2+π2σξ2
6 +π2σv2 6 .
B. Mathematical Derivations 42
Equation (5.8d) Eh
xnt∗n3 i
= Eh
(ξn+ vn) (b1+ b2ξn+ n)3i
= Eb31ξn+ 2b21b2ξ2n+ 2b21ξnn+ b1b22ξn3+ 2b1b2ξn2n+ b1ξn2n+ b31vn
+ E2b21b2ξnvn+ 2b21vnn+ b1b22ξn2vn+ 2b1b2ξnvnn+ b1vn2n+ b21b2ξ2n + E2b1b22ξn3+ 2b1b2ξn2n+ b32ξn4+ 2b22ξ3nn+ b2ξn22n+ b21b2ξnvn + E2b1b22ξn2vn+ 2b1b2ξnvnn+ b32ξ3nvn+ 2b22ξ2nvnn+ b2ξnvn2n + Eb21ξnn+ 2b1b2ξn2n+ 2b1ξn2n+ b22ξ3nn+ 2b2ξn22n+ ξn3n+ b21vnn
+ E2b1b2ξnvnn+ 2b1vn2n+ b22ξn2vnn+ 2b2ξnvn2n+ vn3n
= 3b21b2σ2ξ+b2π2σξ2
2 + 3b1b22Eξ3n + b32Eξn4 . Equation (5.8e)
Eh t∗n4
i
= Eh
(b1+ b2ξnn)4i
= Eb41+ 4b31b2ξn+ 4b31n+ 2b21b22ξn2+ 4b21b2ξnn+ 2b212n+ 4b21b22ξn2 + E8b21b2ξnn+ 4b1b32ξ3n+ 8b1b22ξn2n+ 4b1b2ξn2n+ 4b212n+ 4b1b22ξn2n + E8b1b2ξn2n+ 4b13n+ b42ξ4n+ 4b32ξn3n+ 2b22ξn22n+ 4b22ξn22n
+ E4b2ξn3n+ 4n
= b41+ 6b21b22σ2ξ+ b21π2+ b22π2σ2ξ+ 4b1b32Eξn3 + 4b1E3n +b42Eξn4 + E 4n .
Equation (5.9)
ft|ξ(tn|ξn) = λ (tn|ξn) exp (−Λ (tn|ξn))
= exp (Ξnβ) exp
−tneΞnβ
= exp
Ξnβ− tneΞnβ . Equation (5.10)
L(θ) = ln
N
Y
n=1
ft,x,ξ(tn, xn, ξn)
!
=
N
X
n=1
ln (ft,x,ξ(tn, xn, ξn))
=
N
X
n=1
ln
exp
Ξnβ− tneΞnβ−2σ12
v (xn− ξn)2−2σ12
ξ
ξn2
2πq σξ2σv2
=
N
X
n=1
Ξnβ−
N
X
n=1
tnexp (Ξnβ)− 1 2σv2
N
X
n=1
(xn− ξn)2− 1 2σξ2
N
X
n=1
ξn2
−N ln 2πq
σ2ξσ2v
= ι0Ξβ− t0exp (Ξβ)−(x− ξ)0(x− ξ) 2σv2 − ξ0ξ
2σξ2 − N ln 2πq
σξ2σv2 .
B. Mathematical Derivations 43
B.3 Derivations for Chapter 6 Equation (6.2)
P{dn= 1} = P {cn− Ξnb− n>0}
= P{n< cn− Ξnb}
= F(cn− Ξnb)
= exp
−eΞnb−cn . Equation (6.3)
P{dn= 1|Xn} = P {t∗n< cn|Xn}
= P{Xnβ+ u∗n< cn|Xn}
= P{un< cn− Xnβ|Xn}
= Fu(cn− Xnβ) . Equation (6.6a)
E[dn] = P{dn= 1}
= E [1 (kn− n>0)]
= E [1 (n< kn)]
= Ez[E[1 (n< kn)|zn]]
= Ez[P{n< kn|zn}]
= Ez[F(kn)] . Equation (6.6b)
E [ξndn] = σξE [zndn]
= σξE [zn1(zn< ln)]
= σξE[Ez[zn1(zn< ln)|n]]
= σξE[Ez[zn|zn< ln, n] P{zn< ln|n}]
= σξE
−φ(ln) Φ(ln)Φ(ln)
= −σξE[φ(ln)] .
B. Mathematical Derivations 44
Equation (6.6c) Eξn2dn
= σξ2Ez2ndn
= σξ2Ez2n1(zn< ln)
= σξ2EEzzn21(zn< ln)|n
= σξ2EEzzn2|zn< ln, n P {zn< ln|n}
= σξ2E
1 Φ(ln)
Z ln
−∞
y2φ(y)dy
Φ(ln)
= σξ2E
Z ln
−∞
y2φ(y)dy
= σξ2E
"
−y√ 2πe−12y2
ln
−∞
+ Z ln
−∞
√1
2πe−12y2dy
#
= σξ2E[−lnφ(ln)− 0 + Φ(ln)]
= σξ2E[Φ(ln)− lnφ(ln)] . Equation (6.6d)
E [ndn] = E [n1(n< kn)]
= Ez[E[n1(n< kn)|zn]]
= Ez[E[n|n< kn, zn] P{n< kn|zn}]
= Ez
1 F(kn)
Z kn
−∞
uf(u)du
F(kn)
= Ez
Z kn
−∞
uf(u)du
= Ez
knF(kn)− Z ∞
e−kn−γ
e−t t dt
= Ez[knF(kn)− mn] . Equation (6.6e)
E2ndn
= E2n1(n< kn)
= EzE2n1(n< kn)|zn
= EzE2n|n< kn, zn P {n< kn|zn}
= Ez
1 F(kn)
Z kn
−∞
u2f(u)du
F(kn)
= Ez
Z kn
−∞
u2f(u)du
. Equation (6.6f)
E [ξnndn] = σξE [znndn]
= σξE [znn1(zn< ln)]
= σξE[Ez[znn1(zn< ln)|n]]
= σξE[nEz[zn1(zn< ln)|n]]
= −σξE[nφ(ln)] .
