CS2 - Part 1&2 Flashcards
Explain what it means for a Markov chain to be periodic with period d
- 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
Irreducible Markov chain
- A Markov chain is said to be irreducible if any state j can be reached from anz other state i
Mathematical definition of the Markov property
Condition for unique stationary distribution
- Finite number of states (so it has at least one stationary distribution)
- Irreducible (so it has a unique stationary distribution)
Stationarity of stochastic process
If the statistical properties of a process do not vary over time, the process is stationary.
Stationarity
If the statistical properties of a process do not vary over time, the process is stationary.
Weak stationarity
E(Xt) and var(Xt) are constant
cov(Xt1, Xt2) depends only on the lag t2 - t1
White noise
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
Poisson process
Chapman-Kolmogorov equations
Survival probability
Joint distribution of waiting time Vi and Death indicator Di
Maximum likelihood estimate of µ
Poisson model mortality
Maximum likelihood estimator Possion model
Distribution of the waiting times between consecutive events of a Poisson process Nt ~ Poi (lambda*t)
Wt~Exp(lambda)
Chi-squared test to test goodness of fit of probability distribution
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)
Occupancy probability for Markov jump processes (time-homogeneous and -inhomogeneous)
Integrated form of the Kolmogorov backward equation (time-inhomogeneous)
check Chapter 5 Section 7 for explanation
Integrated form of the Kolmogorov forward equation (time-inhomogeneous)
check Chapter 5 Section 8 for explanation
Kolmogorov equations (forward and backward)
time-inhomogeneous
Number of possible triplets in a Markov jump process
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
Fx(t) as a function of F(t) and S(t)
(F(x+t) - F(x)) / S(x)
Sx(t) as a function of F(t) and S(t)
Sx(t) = S(x+t) / S(x)
Central rate of mortality
Expected future lifetime after age x
Curtate future lifetime random variable
Curtate expectation of life
Relationship between complete and curtate expectations of life
variances of the complete and curtate future lifetimes
Integral expression for tqx
Integral expression for tpx
4 relationships between
F(t), S(t), f(t), mu_t
Kaplan-Meier estimate of the survival function
Nelson-Aalen estimate of the survival function
Parametric estimation of the survival function. General likelihood function
Inequality between Kaplan-Meier and Nelson-Aalen survival functions
Simple random walk
mu expressed with survival probability S(t)
Definition of Markov jump process
A Markov jumnp process is a stochastic process with a continuous time set and a discrete state space that satisfies the Markov property
Maximum likelihood estimates of transition rates in time-homogeneous Markov jump process
Principal difficulties in fitting a Markov jump process model with time-inhomogeneous rates and how to overcomet these
- 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.
Examples of right censoring
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
Examples of left censoring
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
Example of interval censoring
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
Random censoring
- 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
Type 1 censoring
- 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
Type II censoring
- 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.
Non-informative censoring
- 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.
Example of Informative censoring
- 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.
Proportional hazards models: Fully parametric models
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)
Graduation by comparison to standard table: Considerations to take into account
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.
var(X+Y)
var(X)+var(Y)+ 2 cov(X,Y)
cov(X+Z, Y)
cov(X,Y) + cov(Z,Y)
cov(a*X, b*Y)
a*b*cov(X, Y)
Mean residual live
Time series: Exponential smoothing
Individual risk model -> Formula and assumptions
Maximum likelihood estimator of markov chain transition probabilities including 95% confidence interval
Define compound poisson process
