INVERSE PROBLEMS IN MARKOV MODELS AND MEDICAL INSURANCE Spivak - PowerPoint PPT Presentation

INVERSE PROBLEMS IN MARKOV MODELS AND MEDICAL INSURANCE Spivak Semyen, Abdyusheva Svetlana Bashkir State University Frunze str. 32, Ufa, 450074, Russia Phone: +7 3472-236162 Fax: +7 3472-236680 Email: s.spivak@ bashnet.ru

DIRECT AND INVERSE PROBLEM Tw o mutually opposite problems arise w hen using Markov processes for modeling. DIRECT PROBLEM is to calculate probabilities of corresponding states and other characteristics of process. It is assumed that parameters of model are available. INVERSE PROBLEM is to evaluate parameters of model by using experimental output data. In Markov model input parameters are forces of transition from state to state. When dealing w ith queuing problems or insurance models these values are unknow n in advance. Meanw hile, statistical information about output results for some models can be found in special literature.

ESTIMATING THE PROBABILITIES As stated above, many traditional problems of actuarial analysis can be considered in terms of multi-state processes. It is assumed that at any time the individual is in one of a number of states. The current state of the individual or movement (transition) from one state to another may have some financial impact. The task is to quantify this impact. That is, w e need to estimate the probabilities of being in a particular state. Markov process is a very convenient tool for calculation the occurrence probabilities of any events. Figure 1 represents the scheme including three possible states: "active", "disabled" and "dead" commonly used in modeling disability insurance. In this case premiums are payable w hile the insured is in state 1, and benefits are payable w hile he is in state 2. 1.alive 2. disabled 3. dead Figure 1. Model w ith three states.

CHAPMAN - KOLMOGOROV EQUATIONS We define the transition probability function + ≡ + = = ∈ p ( , s s t ) Pr{ X s ( t ) j | X s ( ) i }, , i j {1,2,... }, n ij n ∑ + = ( , ) 1 p s s t and assume, that f o r a l l t > 0 . ij = j 1 We also assume the existence of the limits + p ( , t t h ) i, j ∈ {1,2 … n}, i ≠ j. = ij µ ( ) t lim ij → h h 0 At i ≠ j µ ij is an intensity of transition from state i in state j. It can be easily ≥ 0 seen that at s, t, u ? n ∑ i, j ∈ { 1 , 2 , … n } . + + = + + + + p ( , s s t u ) p ( , s s t p ) ( s t s , t u ), (1) ij il lj = l 1 (1) is known as Chapman-Kolmogorov equations. T r a n s i t i o n p r o b a b i l i t y f u n c t i o n s a r e u s e d i n c a l c u l a t i o n s o f a c t u a r i a l v a l u e s . T h e f o r c e s o f t r a n s i t i o n a n d t h e t r a n s i t i o n p r o b a b i l i t y f u n c t i o n s a r e r e l a t e d b y t h e C h a p m a n - K o l m o g o r o v f o r w a r d a n d b a c k w a r d e q u a t i o n s ∂ ∂ n n ∑ ∑ + = − µ + + = + µ + p ( , s s t ) ( ) s p ( , s s t ) (3) p ( , s s t ) p ( , s s t ) ( s t ) (2) ∂ ij li lj ∂ ij il lj t t = = l 1 l 1 respectively, with boundary conditions p ( , ) s s d = , ij ij =  1, ???? i j , =  d where ij 0, ????? . 

HOMOGENEOUS MARKOV PROCESSES The explicit form of probability transition functions can be ?obtained w hen µ ij (t)= µ ij for all t. Such Markovian process is uniform (homogeneous) in time, or stationary. An assumption that transition forces are constant implies that the time spent in each state has exponential distribution and the functions p ij (s,s+t) are identical for all s, so that they can be w ritten simply as p ij . In case of step functions for transition forces, it is possible to divide the time interval of our research into intervals on w hich parameters are constant . In order to solutions (the probabilities functions) to be continuous, it is necessary to join the solutions from different intervals. When the transition forces are know n, the problem reduces to the straightforw ard task: solving Chapman-Kolmogorov equations. For models w ith constant intensities it is not so complicated irrespective of number of possible system states. How ever, if the transition forces are unknow n, there is an inverse task, that is to estimate the transition forces from some statistical data.

THE PARAMETERS ESTIMATION One more assumption, w hich w ould make a solution much simpler, is that w e have the complete information about each observable individual’s history. In [ Industrial Accident Insurance: Actuarial Foundations. Ed. by V.N. Baskakov, Moscow , Academia, 2001 ] the model of many states is applied to modeling the process of disability insurance. The estimates of transition forces are carried out by the maximum likelihood method w ith the complete data – all individual’s transition states and time spent in them are assumed know n. How ever, it is very rare w hen w e have the complete data. In the problems of life insurance experimental information is likely to be represented in so-called mortality tables only. From them one can obtain the information on values of probability of death in a particula r age for individuals of a particular group. One of the most common methods of statistical estimation is the least squares method. The idea being is that the unknow n parameters are estimated by minimizing the standard error vector.

EXAMPLE 1 Let us consider an example of applying this approach to estimation of mortality of smokers and non-smokers. The statistical data for calculations w as taken from [Benjamin B., Michaelson R., Mortality differences betw een smokers and non-smokers, Journal of the Institute of Actuaries, Vol. 115, Part III, ? 461, 1988] . The BAJ provides w ith mortality tables for four groups: males/females and smokers/non-smokers. Here is an example: Mortality table for non-smokers (males). Age x l x d x q x 25 96570 65 0.00067 26 96505 63 0.00066 27 96442 62 0.00065 28 96380 62 0.00065 29 96318 64 0.00066 30 96254 65 0.00067 31 96189 67 0.00070 32 96122 70 0.00072 33 96052 73 0.00076 34 95979 77 0.00080 35 95902 82 0.00086 The data for male smokers and non-smokers, age 25-35 was used for this research.

The model is the three-states process shown in Figure 2: s x 1.non-smoker 2.smoker g x µ x N S µ x S 3.dead Figure 2. Model for Smokers / Non-Smokers This model includes four unknown parameters: s x , g x , µ x NS , µ x S – transition intensities. Mortality tables give us the values of death probabilities for age x. In Figure 2 these are the probabilities of transition from state 1 into state 3, and from state 2 into state 3. Chapman-Kolmogorov equations for this scheme are: ì d p ï ï ∑ = p µ + p µ = ï p 1 11 12 21 11 11 Norm condition: dt ï ij ï ï i , j d p ï ï p µ p µ = + ï 12 1 2 22 11 12 The initial conditions can be derived as follows: dt ï ï ï In the initial moment (t=0) the individual does not smoke, d p ï ï p µ p µ = + i.e. is in a state 1: ï 13 1 1 13 12 23 ï dt ï ï í = = = ï d p p (0) 1, p (0) 0, p (0) 0, ï p µ p µ = + 1 1 12 1 3 ï 21 2 1 11 22 21 ï dt = = = ï p (0) 1, p (0) 0, p (0) 0. ï 21 22 2 3 ï d p ï p µ p µ = + ï 22 21 12 22 22 ï dt ï ï ï d p ï p µ p µ = + ï ï 23 21 13 22 23 dt î ï ï

The least squares method gives the follow ing numerical estimates of the transition µ 12 = 0,0164, µ 21 = 0,0592, µ 13 =0,00107, µ 23 = 0,00921. intensities: In the case considered parameters of the model w ill depend on age x. In the more general case they w ill depend on the duration of stay in the particular state (i.e. they w ill differ for individuals of the same age but w ith the different past information). This case is obviously more complicated and requires more data about observed individuals. So, let us assume that the intensities of transition betw een the states differ for different ages, i.e. µ ij ≅µ ij (?). Individual’s age x is a discrete number (1, 2, … years). It is assumed that during one life year transition intensities stay constant, i.e. µ ij (?) functions are ???????-??????????. If w e consider an individual’s age as a continuos parameter and not discrete , we can try to build “smooth” transition intensities. It is possible to calculate the values of the transition functions in the discrete points from a given data and then interpolate these points by a smooth enough curve . We can consider the [0, 1] interval as a subclass on w hich w e are looking for an approximate solution (this is justified by the prior considerations). The follow ing numerical estimates of the transition intensities [2] w ere obtained by the trial and error method. They w ere approximated, and the follow ing results w ere o btained: µ 12 ∈ [0,000171;0,00036], µ 21 ∈ [0,000121;0,00051], µ 13 ∈ [0,000671;0,00086], µ 23 ∈ [0,001180;0,00193]. As the above solution is not unique, w e have to estimate the interval of values of µ ij, in w hich w e obtain a good approximation of the statistical data. The analysis of the corresponding Markov scheme and the set of equations show s that the substantial changes in µ 13, µ 23 w ill cause the large changes in the values of p 13, p 23 . The changes in µ 12, µ 21 w ill influence these variables in the much smaller extend.