ST227 Survival Models – Part III Estimation in the Markov Model
Department of Statistics, London School of Economics
January 31, 2019
Copyright By PowCoder代写 加微信 powcoder
1 Markov Models: Data and Estimation 1.1 Introduction
We now have mathematical models for various situations that arise in insurance, e.g. a model of mortality with states fAlive; Deadg, a model for PHI with states fW ell; Ill; Deadg.
These models depend on transition intensities gh, g 6= h between x
states g and h.
The values of these transition intensities are unknown – they are para- meters in the models.
Thus, we need some data which we can use to estimate the values of the transition intensities gh.
Wewillstartwiththe2-statemodelofmortalitywithstatesfAlive;Deadg, and work through this example in detail.
It turns out that the answers for the 2-state model give a very good guide to what happens in the multi-state case.
1.2 Data and notation for the 2-state model
Suppose we have data on n independent lives all aged between x and
x + 1. We suppose for life i:
x+i = x+bi =
x+ti = i =
age at which observation begins, 0 i < 1
age at which observation ends, if life i does not die, 0 bi < 1 age at which observation stops, by death or censoring
It will often be useful to work with the time spent under observation, so we deÖne
ti i; the observed waiting time Also, we deÖne
1, if life i is observed to die
0; if life i is not observed to die
Censoring is a form of missing data problem in which time to event is not observed for reasons such as termination of study before all recruited subjects have shown the event of interest or the subject has left the study prior to experiencing an event.
Censoring is the key feature of survival data (indeed survival analysis might be deÖned as the analysis of censored data) and the mechanisms which give rise to censoring play an important part in statistical inference.
Censoring is present when we do not observe the exact length of a lifetime, but observe only that its length falls within some interval. This can happen in several ways which we are going to discuss in detail in the following lectures.
The following diagrams may be helpful.
Case 1. The ith life does not die, i.e. reaches the end of the observation period at x + bi.
Inthiscase,di=0,ti=bi, i=bi i.
Case 2. The ith life dies at x + ti before the end of the observation period.
Inthiscase,di=1, i