CS2 - Part 1&2

Question 1

Explain what it means for a Markov chain to be periodic with period d

Answer 1

• A state in a Markov chain is periodic with period d>1 if a return to that state is possible only in a number of steps that is a multiple of d
• A Markov chain has period d if all the states in the chain have period d
• If a Markov chain is irreducible, all states have the same periodicity

Question 2

Irreducible Markov chain

Answer 2

• A Markov chain is said to be irreducible if any state j can be reached from anz other state i

Question 3

Mathematical definition of the Markov property

Answer 3

Question 4

Condition for unique stationary distribution

Answer 4

• Finite number of states (so it has at least one stationary distribution)
• Irreducible (so it has a unique stationary distribution)

Question 5

Stationarity of stochastic process

Answer 5

If the statistical properties of a process do not vary over time, the process is stationary.

Question 6

Stationarity

Answer 6

If the statistical properties of a process do not vary over time, the process is stationary.

Question 7

Weak stationarity

Answer 7

E(Xt) and var(Xt) are constant

cov(Xt1, Xt2) depends only on the lag t2 - t1

Question 8

White noise

Answer 8

White noise is a stochastic process that consists of a set of independent and identically distributed random variables. The random variables can be either discrete or continuous and the time set can be either discrete or continuous. White noise processes are stationary and have the Markov property

Question 9

Poisson process

Answer 9

Question 10

Chapman-Kolmogorov equations

Answer 10

Question 11

Survival probability

Answer 11

Question 12

Joint distribution of waiting time Vi and Death indicator Di

Answer 12

Question 13

Maximum likelihood estimate of µ

Answer 13

Question 14

Poisson model mortality

Answer 14

Question 15

Maximum likelihood estimator Possion model

Answer 15

Question 16

Distribution of the waiting times between consecutive events of a Poisson process Nt ~ Poi (lambda*t)

Answer 16

Wt~Exp(lambda)

Question 17

Chi-squared test to test goodness of fit of probability distribution

Answer 17

X^2 = (A-E)^2 / E

degrees of freedom: n - p

where

• n = number of categories
• p = 1 + number of variables in the probability distribution

e.g.

• Exponential distribution: n = 2 (lambda)
• Normal distribution: n = 3 (mean and std)

Question 18

Occupancy probability for Markov jump processes (time-homogeneous and -inhomogeneous)

Answer 18

Question 19

Integrated form of the Kolmogorov backward equation (time-inhomogeneous)

check Chapter 5 Section 7 for explanation

Answer 19

Question 20

Integrated form of the Kolmogorov forward equation (time-inhomogeneous)

check Chapter 5 Section 8 for explanation

Answer 20

Question 21

Kolmogorov equations (forward and backward)

time-inhomogeneous

Answer 21

Question 22

Number of possible triplets in a Markov jump process

Answer 22

For each state X, the number of possible triplets with X as second state is

number of ways into X * number of ways out of X

Question 23

Fx(t) as a function of F(t) and S(t)

Answer 23

(F(x+t) - F(x)) / S(x)

Question 24

Sx(t) as a function of F(t) and S(t)

Answer 24

Sx(t) = S(x+t) / S(x)

Question 25

Central rate of mortality

Answer 25

Question 26

Expected future lifetime after age x

Answer 26

Question 27

Curtate future lifetime random variable

Answer 27

Question 28

Curtate expectation of life

Answer 28

Question 29

Relationship between complete and curtate expectations of life

Answer 29

Question 30

variances of the complete and curtate future lifetimes

Answer 30

Question 31

Integral expression for tqx

Answer 31

Question 32

Integral expression for tpx

Answer 32

Question 33

4 relationships between F(t), S(t), f(t), mu\_t

Answer 33

Question 34

Kaplan-Meier estimate of the survival function

Answer 34

Question 35

Nelson-Aalen estimate of the survival function

Answer 35

Question 36

Parametric estimation of the survival function. General likelihood function

Answer 36

Question 37

Inequality between Kaplan-Meier and Nelson-Aalen survival functions

Answer 37

Question 38

Simple random walk

Answer 38

Question 39

Answer 39

Question 40

mu expressed with survival probability S(t)

Answer 40

Question 41

Definition of Markov jump process

Answer 41

A Markov jumnp process is a stochastic process with a continuous time set and a discrete state space that satisfies the Markov property

Question 42

Maximum likelihood estimates of transition rates in time-homogeneous Markov jump process

Answer 42

Question 43

Principal difficulties in fitting a Markov jump process model with time-inhomogeneous rates and how to overcomet these

Answer 43

* Divide time interval into subintervals, assume that the transition rates are constant over each subinterval and estimate the transition rates for each subinterval. * However, estimates would be based on muchs smaller amount of data compared to the time-homogeneous case, and would be less reliable * Alternatively, we could select an appropriate funcational form for the transition probabilities and use the data to estimate the relevant parameters. This is only possible if we have an idea of what kind of formula would be appropriate.

Question 44

Examples of right censoring

Answer 44

Data are right censored if the censoring mechanism cuts short observations in progress. Example is the ending of a mortality investigation before all the lives being observed have died. * Life insurance policyholders surrender their policies * Active lives of a pension scheme retire * Endowment assurance policies mature

Question 45

Examples of left censoring

Answer 45

Data are left censored if the censoring mechanism prevents us from knowing when entry into the state that we wish to observe took place. * When estimating functions of exact age and we don't know the exact date of birth * When estimating functions of exact policy duration and we don't know the exact date of policy entry * When estimating functions of the duration since onset of sckness and we don't know the exact date of becoming sick

Question 46

Example of interval censoring

Answer 46

Data are interval censored if the observational plan only allows us to say that an event of interest fell within some interval of time. * When we only know the calendar year of withdrawal * when estimating functions of exact age and we only know that deaths were aged 'x nearest birthday' at the date of death

Question 47

Random censoring

Answer 47

* Individuals may leave the observation by a means other than death, and the time of leaving is not known in advance * Special case of right censoring Examples: * Life insurance withdrawals * Emigration from a population * Members of a company pension scheme may leave voluntarily when they move to another employer

Question 48

Type 1 censoring

Answer 48

* If the censoring times are known in advance then the mechanism is called 'Type I censoring'. * Degenerate case of random censoring * Special case of right censoring Examples * Estimating functions of exact age and we stop following individuals once they have reached their 60th birthday * When lives retire from a pension scheme at normal retirement age

Question 49

Type II censoring

Answer 49

* Observation is continued until a predetermined number of deaths has occurred. * Number of events of interest is non-random Example * When a medical trial is ended after 100 lives on a particular course of treatment have died.

Question 50

Non-informative censoring

Answer 50

* Censoring is non-informative if it gives no information about the lifetimes Ti * In the case of random censoring, the independence of each pair Ti, Ci is sufficient to ensure that the censoring in non-informative. * Informative censoring is more difficult to analyse Example * Non-informative censoring occurs if at any given time, lives are equally likely to be censored, regardless of their subsequent force of mortality. This means that we cannot tell anything about a person's mortality after the date of the consoring event from the fact that they have been censored.

Question 51

Example of Informative censoring

Answer 51

* Lives that are in better health may be more likely to surrender their policies than those in a poor state of health. Lives that are censoired are therefore likely to have lighter mortality than those that remain in the investigation.

Question 52

Answer 52

Question 53

Proportional hazards models: Fully parametric models

Answer 53

Fully parametric models assume a lifetime distribution based on a statistical distribution whose parameters must then be determined. Commonly used distributions include: * Exponential distribution (constant hazard) * Weibull distribution (monotonic hazard) * Gombertz-Makeham formula (exponential hazard) * Log-logistic distribution ('humped' hazard)

Question 54

Answer 54

Question 55

Graduation by comparison to standard table: Considerations to take into account

Answer 55

Table must satisfy following criteria: * it must be _available for all classes of lives_, e.g. maile and females * it must _relate to a similar class of lives_, e.g. assurances and not annuitites * it must be a _'benchmark' table_, i.e. generally acceptable to all other actuaries * it should be _up-to-date_, i.e. relate to fairly recent experience * it must _cover the age range_ for which rates are required * **In addition, it should have the _correct pattern_ of rates by age (not necessarily the correct level of rates)** * **It should _not have any special features_ that are unlikely to be present in the experience being graduated.**

Question 56

var(X+Y)

Answer 56

var(X)+var(Y)+ 2 cov(X,Y)

Question 57

cov(X+Z, Y)

Answer 57

cov(X,Y) + cov(Z,Y)

Question 58

cov(a\*X, b\*Y)

Answer 58

a\*b\*cov(X, Y)

Question 59

Mean residual live

Answer 59

Question 60

Answer 60

Question 61

Time series: Exponential smoothing

Answer 61

Question 62

Answer 62

Question 63

Individual risk model -\> Formula and assumptions

Answer 63

Question 64

Maximum likelihood estimator of markov chain transition probabilities including 95% confidence interval

Answer 64

Question 65

Define compound poisson process

Answer 65