# Lecture 17 – Associated Single Decrement Model

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```					Lecture 17 – Associated Single Decrement Model

Absolute rate of decrement

Discuss its general relationship with probabilities in a multiple
decrement model

Speciﬁc relationship under constant force assumption

Example 3

Uniform distr. assumption under multiple decrement model
setting

AS4422a — Lecture 17                                                       1
Absolute rate of decrement
(j)                               (j)
µx (t): if we pretend that µx (t) is the force of mortality in a single
decrement model
′ (j)          t    (j)
µx (s)ds
t px       =e−   0

′
(j)               ′
t qx    = 1 − t px(j)
′ (j)
t qx      is called absolute rate of decrement.
′ (j)
t qxis interpreted as the net probability of decrement due to
cause j , without competing with other causes
The corresponding model is called the associated single
decrement model.
(j)
In comparison, t qx is the probability of decrement due to cause
j while competing with all other causes.

AS4422a — Lecture 17                                                    2
Basic Relationship

(τ                                       (m)
µx ) (s) = µ(1) (s) + µ(2) (s) + · · · + µx (s)
x          x
t (1)      (2)        (m)
(τ )                 0 [µx (s)+µx (s)+···+µx (s)]ds
t px      =e       −

t       (1)                    t    (2)                       t      (m)
µx (s)ds                       µx (s)ds                       µx       (s)ds
=e       −    0                 ·e   −       0                  ···e−       0

m
′
(j)
=             t px
j=1

(τ )            ′ (j)
=⇒              t px         ≤ t px
(τ ) (j)                      ′ (j)
(j)
=⇒             t px µx (t)             ≤ t px µx (t)
1                                                                   1
(τ ) (j)                    (j)              ′ (j)                   ′ (j)    (j)
=⇒                    t px µx (t)dt           =     qx         ≤ qx           =           t px        µx (t)dt
0                                                                   0

AS4422a — Lecture 17                                                                                                       3
Constant force assumption

On age interval [x, x + 1), assume:

(j)      (j)    (τ         (τ
µx (t) = µx , & µx ) (t) = µx ) , for 0 ≤ t < 1

AS4422a — Lecture 17                                               4
Example 3

(1)     (2)    ′ (1)    ′ (2)
x    qx      qx     qx       qx
66   0.03    0.06             *
67   0.04    0.07    *

′ (2)        ′ (1)
Calculate q66 and q67

AS4422a — Lecture 17                                                 5
Uniform distr. assumption for multiple decrements
MUDD:
(j)         (j)
t qx      = t qx
(τ )
t qx
(τ
= t qx )

AS4422a — Lecture 17                            6

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