PARAMETRIC ESTIMATION OF HAZARD FUNCTIONS WITH STOCHASTIC COVARIATE PROCESSES

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1 Sankhyā : The Indian Journal of Statitic 1999, Volume 61, Serie A, Pt. 2, pp PARAMETRIC ESTIMATION OF HAZARD FUNCTIONS WITH STOCHASTIC COVARIATE PROCESSES By SIMEON M. BERMAN Courant Intitute of Mathematical Science, New York and HALINA FRYDMAN Stern School of Buine, New York SUMMARY. Let X(t, t, be a real or vector valued tochatic proce and T a random killing-time of the proce which generally depend on the ample function. In the context of urvival analyi, T repreent the time to a precribed event (e.g. ytem failure, time of dieae ymptom, etc. and X(t i a tochatic covariate proce, oberved up to time T. The conditional ditribution of T, given X(t, t, i aumed to be of a known functional form with an unknown vector parameter θ; however, the ditribution of X( are not pecified. For an arbitrary fixed α > the obervable data from a ingle realization of T and X( i min(t, α, X(t, t min(t, α. For n 1 the maximum likelihood etimator of θ i baed on n independent copie of the obervable data. It i hown that olution of the likelihood equation are conitent and aymptotically normal and efficient under pecified regularity condition on the hazard function aociated with the conditional ditribution of T. The Fiher information matrix i repreented in term of the hazard function. The form of the hazard function i very general, and i not retricted to the commonly conidered cae where it depend on X( only through the preent point X(t. Furthermore, the proce X( i a general, not necearily Markovian proce. 1. Introduction Let (Ω, P be a probability pace, and let the following family of random variable be defined on it: a real or vector valued tochatic proce X(t, t, and a nonnegative real random variable T with the property P (T > t X(, = P (T > t X(, t... (1.1 for all t >. Such a random variable i called a killing-time of the proce. The baic condition aumed here on the probability meaure P i: Paper received. March 1998; revied October AMS (1991 ubject claification. 62F12, 62M9, 62P1. Key word and phrae. Aymptotic ditribution of etimator, hazard function, killing-time, likelihood function, tochatic marker proce.

2 parametric etimation of hazard function 175 For ome integer m 1, there i an unknown parameter θ R m uch that the conditional probability meaure in (1.1 depend on θ: P θ (T > t X(, ; and the retriction of P to the igma-field generated by X(,, doe not depend on θ. The theme of thi work i the etimation of the unknown parameter θ on the bai of obervation of T and the value of X(, for T. The latter retriction on the value of X( i a conequence of the role of T a a time at which the proce i killed, o that it i not obervable after that time. More preciely, we precribe a contant α, < α, uch that a ingle obervation of the proce i defined a min(t, α, X(, min(t, α.... (1.2 Let (T i, X i (,, i = 1,..., n, be n independent copie of (T, X(, ; then, the et of independent copie of the ubproce (1.2 i the obervation et. The introduction of the contant α allow the poibility of a bounded ampling interval. The object of thi invetigation i to determine the condition on the family of meaure P uch that the maximum likelihood etimator ˆθ of θ i conitent and aymptotically normal and efficient. The reult are guided by the fact that the likelihood function of a ingle obervation (1.2 factor into a product of the conditional likelihood of T, given the X-value, and the marginal likelihood of the X-value, where the latter doe not depend on θ. Thu the likelihood function i eentially baed on the conditional denity obtained from (1.1. We will conider the cae where the conditional ditribution of T ha a mooth denity function. We define the conditional hazard function of T a q(t, X(, θ = d dt P θ(t > t X(, P θ (T > t X(,.... (1.3 Under the aumption (1.1, q(t, X(, θ depend on X( only for X(, t. It follow from the definition of q that the conditional denity of T, which i the eential factor of the ingle-obervation likelihood function, i repreentable a ( q(t, X(, θ exp q(, X(, θ d.... (1.4 Therefore, the condition for the propertie of the maximum likelihood etimator are tated in term of condition on q. Thi work wa motivated by the etimation iue ariing in the area of urvival analyi concerned with the modelling of the hazard of failure a a function of a marker, a time-varying covariate repreented by a tochatic proce. Here the killing-time i identified a the urvival-time. Knowledge of the hazard function q and the current and pat value of the marker proce can, in principle, yield urvival probabilitie for the future.

3 176 imeon m. berman and halina frydman Marker model have been applied in medicine to tudy the relationhip of urvival to the clinically obervable variable. Some recent paper on the etimation and other apect of marker model in the context of tudying the progreion of AIDS are Fuaro et al (1993, Jewell and Kalbfleich (1992, Self and Pawitan (1992, and Tiati et al (1995. Fuaro et al (1993 conider nonparametric etimation of marker-dependent hazard function. Jewell and Kalbfleich (1992 dicu a number of methodological iue aociated with marker model in medicine. Self and Pawitan (1992, Tiati et al (1995, and Yahin and Manton (1997, conider etimation in marker model when the marker proce i oberved periodically. Marker-dependent hazard function have alo emerged recently a an important ingredient of a new cla of model in finance for pricing financial intrument ubject to the rik of default (ee Lando (1996 and the reference therein. One of the major iue in thi cla of model i the pecification and etimation of the hazard-rate of default a a function of time-varying tate variable, repreented by tochatic procee. Here the killing-time i identified a the time of default and the tochatic proce X(t, t > decribe the evolution of a tate variable known to influence the likelihood of default. For example in Jarrow et al (1997, the hazard-rate of default of a defaultable bond i modelled a a function of it credit rating which i repreented by a continuou time Markov chain. The main reult, Theorem 4.1, pecifie the condition on the hazard function that are ufficient for the aymptotic propertie of ˆθ, the maximum likelihood etimator of θ. We how that if the hazard function q(t, x(, θ, which i, of coure, a tochatic proce, atifie pecified regularity condition, then the condition in the hypothei of the claical theorem of Cramer (1946, p. 5 are valid, and o, by the latter theorem, ˆθ i conitent and aymptotically normal and efficient. The limiting covariance matrix i alo expreed a a functional of q. Our reult i related to the general reult in Anderen, Borgan, Gill and Keiding (1992, p on maximum likelihood etimation of parametrically pecified intenity procee. While it may be poible to etablih the aymptotic propertie of ˆθ in our cae by howing that our et of ufficient condition implie the et of condition given in Anderen et al (1992, Condition VI.1.1, p. 42, we choe intead the impler and more natural route of directly verifying Cramer condition. Our reult employ the claical central limit theorem, the law of large number, and a imple tochatic integral with repect to a proce with conditionally orthogonal increment. Section 2 contain an analyi of the likelihood function of a ingle realization (1.2. The logarithmic derivative with repect to the parameter (θ j are repreented a tochatic integral with repect to a proce with conditionally orthogonal increment. Section 3 retate Cramer theorem in a form that i more directly applicable to our framework. Section 4 contain the main reult, pecifying the condition

4 parametric etimation of hazard function 177 on the hazard function that are ufficient for the aymptotic propertie of ˆθ. Section 5 include everal new example illutrating the application of Theorem 4.1. In the example it i aumed that the hazard function depend on X(, t only through X(t. Under additional aumption that X(t i a Markov proce the limiting covariance matrix i explicitly calculated in everal example uing the reult on the joint ditribution of (X(T, T obtained in Berman and Frydman (1996. Statitical inference for ˆθ when a ingle obervation of the proce conit of a ubet of (1.2, including the cae when a marker proce i oberved only periodically, will be conidered in a forthcoming paper. 2. Likelihood Function Let T be a killing-time for a proce X uch that the conditional ditribution of T given X( depend on an unknown parameter θ. Suppoe that the conditional denity, p(t, X(, θ = d dt P θ(t > t X(,... (2.1 exit for each t >, almot urely. If the finite-dimenional ditribution of X( have denitie, then, for any finite et t, t 1,... t k of poitive number, the joint denity of (T, X(t j, j =, 1,..., k at a point (t, x j, j =, 1,..., k i of the form f(t, x, x 1,..., x k ; θ = E [ p(t, X(, θ X(tj = x j, j =, 1,..., k ] h(x,..., x k,... (2.2 where h(x,..., x k i the joint denity of X(t j at x j, j =, 1,... k, and doe not depend on θ. The function f(t, X(t,..., X(t k ; θ i the likelihood function of a ingle oberved realization (T, X(t,..., X(t k. The likelihood function for a ample of n independent copie i a product of the correponding inglerealization likelihood function (2.2. In the method of maximum likelihood, the function h in (2.2 may be ignored becaue it doe not depend on θ. Therefore, it uffice to repreent the ingle-realization likelihood a E [ p(t, X(, θ X(t j, j =, 1,..., k ].... (2.3 at t = T. Under the aumption (1.1 and the definition (2.1, it follow that p(t, X(, θ i meaurable with repect to the igma-field generated by X(, t; therefore, E(p(t, X(, θ X(, t = p(t, X(, θ.... (2.4

5 178 imeon m. berman and halina frydman For arbitrary t >, the conditional hazard function (1.3 i then, by (1.4, and If q(t, X(, θ = d dt log p(, X(, θ d;... (2.5 log p(t, X(, θ = log q(t, X(, θ Lemma 2.1. Define the random function t q(, X(, θ d.... (2.6 N(t = 1(T t, Y (t = 1(T > t,... (2.7 M(t = N(t q(, X(, θ Y ( d.... (2.8 E [q(, X(, θ] 2 d <, for all t >,... (2.9 then M(t, t, ha, under the conditional probability meaure P ( X(,, orthogonal increment, and, with probability 1, for < t. [ E (M(t M( 2 ] X(u, u = Proof. By the definition (2.7 of N and Y : p(u, X(, θ du... (2.1 E [(N(t N( 2 ] X( = E [ N(t N( ] X( = P ( < T t t X( = p(u, X(, θ du; and { E N(t { = E = q(u, X(, θ Y (u du X( q(u, X(, θ 1(u < T t du X( q(u, X(, θ u p(v, X(, θ dv du;

6 parametric etimation of hazard function 179 and E [ ( { ( = 2E = 2 2 ] q(u, X(, θ Y (u du X( u q(u, X(, θ q(v, X(, θ Y (v dv u p(v, X(, θ dv du. q(u, X(, θ du X( The expectation are defined by virtue of (2.9. Thi confirm (2.1. The reult E M(t and the orthogonality of the increment follow by imilar calculation. Suppoe, for almot all ample function X(, that the lat term in (2.6 ha partial derivative with repect to the component θ j of θ, and that the differentiation may be done under the integral ign; then log p(t, X(, θ = log q(t, X(, θ log q(, X(, θ q(, X(, θ d. It follow that for almot all X(, the core function (/ log p(t, X(, θ at t = T i repreentable a log q(, X(, θ d M(,... (2.11 where the tochatic integral with repect to M( i well defined, by virtue of Lemma 2.1, under the aumption ( 2 log q(, X(, θ p(, X(, θ d <.... ( Cramer Theorem on the Solution of the Likelihood Equation Let Y be a random vector or tochatic proce, and let f(y, θ be the likelihood function of a ingle realization of Y. Here f i a known functional of (Y, θ, and θ = (θ 1,..., θ m i an unknown real-vector parameter. A i well-known in the theory of maximum likelihood, f i unique except for factor that do not depend on θ. Let θ be the true value of θ, and let B be an arbitrary neighborhood of θ. Let Y i, i = 1, 2,... be independent copie of

7 18 imeon m. berman and halina frydman Y and put L(θ = n f(y i, θ, the likelihood function for the ample of n independent realization. The following reult i equivalent to the vector verion of the claical theorem of Cramer (1946, page 5, on the conitency and aymptotic normality of maximum likelihood etimator. Lemma 3.1. Suppoe that the following condition hold: L.3.1. The partial derivative of log f, D (1 log f = (/ log f, D (2 log f = ( 2 / θ h log f, and D (3 log f = ( 3 / θ h θ l log f exit and are continuou for θ B, almot urely for j, h, l = 1,... m. L.3.2. We have ( E ( log f =, E θ=θ θ j the matrix (σ jh =, where 2 log f <, for j = 1,..., m ; θ=θ... (3.1 [ σ jh = E log f ] log f, j, h = 1,..., m,... (3.2 θ h θ=θ i trictly poitive definite; and σ jh i alo equal to which i aumed to be finite. ( 2 log f E,... (3.3 θ h θ=θ L.3.3. There exit a real-valued functional H(Y uch that up θ B D (3 log f H(Y, for all third-order partial derivative and almot all Y, and E H(Y <. Then the ytem of likelihood equation (/ log L(θ =, 1 j m, ha a olution ˆθ, depending on n, uch that ˆθ θ in probability, for n, and ( n (ˆθ θ d N, 1, where i defined in L.3.2. The ubject of thi paper i the maximum likelihood etimation of the unknown parameter θ on the bai of the et of independent realization of the ubproce (1.2. A hown in Section 2, the unknown parameter θ enter the likelihood function only through the conditional ditribution of T, given the obervation on X(. Hence, by the uniquene of the likelihood function up to factor that do not depend on θ, the ingle-realization likelihood function f in the tatement of Lemma 3.1 may be taken to be the function (2.3. The logarithmic derivative are repreentable in term of the aociated hazard function q by mean of the tochatic integral (2.11. The condition L.3.1 L.3.3 in the hypothei of Lemma 3.1 will be hown to be fulfilled under correponding condition on the hazard function.

8 parametric etimation of hazard function 181 A in mot application of maximum likelihood, the matrix = (θ i generally unknown and ha to be etimated from the ame data ued to etimate θ. If the underlying proce X(t i pecified o that (θ can be determined by mathematical analyi, then the natural etimator of i (ˆθ, where ˆθ i the etimator of θ. (ˆθ i called the parametric etimator. If X(t i itelf not pecified or if cannot be ufficiently identified a a function of θ, then it can often be etimated by applying the law of large number a will be hown at the end of Section 4. Such an etimator i called nonparametric. 4. Main Reult The ample data conit of a et of n independent copie of (1.2 for ome α >. The likelihood function of a ingle realization (1.2 i obtained by a modification of (1.4 to account for the length α of the obervation interval. In view of the two poibilitie, T α and T > α, we obtain the likelihood function of one realization, f(t, X(, θ hence = 1(T α q(t, X(, θ exp ( T q(, X(, θ d ( + 1(T > α exp q(, X(, θ d ( = (q(t, X(, θ 1(T α exp 1(T > q(, X(, θ d ; log f = 1(T α log q(t, X(, θ 1(T > q(, X(, θ d.... (4.1 If the derivative of 1(T > q(, X(, θ d with repect to θ j can be taken inide the integral, then log f = 1(T α log q(t, X(, θ 1(T > q(, X(, θ log q(, X(, θ d.... (4.2 The following theorem furnihe condition on q which are ufficient for the condition on the function f aumed in Lemma 3.1. Theorem 4.1. Aume the following three et of condition: L.4.1. Let D (i, i = 1, 2, 3 be the differential operator defined in L.3.1. Then for almot all realization (T, X( the function D (i log q(t, X(, θ

9 182 imeon m. berman and halina frydman and D (i α 1(T > q(, X(, θ d, i = 1, 2, 3, exit and are continuou for θ B, and furthermore, D (i 1(T > q(, X(, θ d = 1(T > D (i q(, X(, θ d. L.4.2. For every j, h = 1,..., m, and θ B. { α 2 E log q(, X(, θ p(, X(, θ d <, and { 2 2 E log q(, X(, θ θ h p(, X(, θ d < ; and the following (nonnegative definite matrix i trictly poitive definite at θ = θ : { α E log q(, X(, θ θ h log q(, X(, θ p(, X(, θ d, j, h = 1,..., m.... (4.3 L.4.3. There are nonnegative Borel function H 1 (t, X( and H 2 (t, X( uch that for all third-order derivative D (3, and all θ B, and all t α, and almot all ample function X(, we have the inequalitie D (3 log q(t, X(, θ H 1 (t, X(, D (3 q(t, X(, θ H 2 (t, X(, E { H 1 (t, X( p(t, X(, θ dt <, { E H 2 (t, X( p(, X(, θ dt <. t Then the condition L.3.1, L.3.2, and L.3.3 of Lemma 3.1 hold for f = f(t, X(, θ, and with a the matrix with entrie (4.3. Proof. It i obviou that L.4.1 implie L.3.1 and that L.4.3 implie L.3.3 in the cae where log f i given by (4.2. log f in (4.2 i repre- There remain only to prove L.3.2. Under L.4.1, entable a log q(t, X(, θ dm(t,... (4.4

10 parametric etimation of hazard function 183 where M(t i given by (2.8, which i analogou to (2.11 except that the upper limit α i inerted in the place of. By the conditional orthogonality of the increment of M and formula (2.1, the expected value of (4.4 i equal to, and the variance i equal to the (finite integral (4.3 with j = h. Thi confirm (3.1. For the confirmation of the poitive definitene of, oberve that, by the repreentation (4.4 of (/ log f, the covariance σ jh in (3.2 i equal to { E log q(t, X(, θ dm(t θ h log q(t, X(, θ dm(t at θ = θ, which i equal to the integral (4.3. The reulting matrix i, by hypothei, trictly poitive definite. Finally, we verify (3.3. Recall the general reult from elementary calculu for an arbitrary mooth function f of everal variable: 2 { f 2 log f = f + log f log f... (4.5 θ h θ h θ h From (4.2 it follow under condition L.4.1 that 2 log f = 1(T α 2 log q(t, X(, θ θ h θ h which, by (4.5 with q in the place of f, i equal to 1(T α 2 log q(t, X(, θ θ h { 2 log q(, X(, θ θ h + 1(T > 2 q(, X(, θ θ h d, 1(T > q(, X(, θ log q(, X(, θ By the definition (2.8 of M(t, the previou um i repreentable a log q(t, X(, θ 1(T α + log q(t, X(, θ θ h { 2 log q(, X(, θ log q(, X(, θ + θ h log q(, X(, θ d. θ h log q(, X(, θ dm(. θ h By Lemma 2.1, it follow that { 2 { log f E θ h X( log q(t, X(, θ log q(t, X(, θ = E 1(T α θ h X(. By taking expectation over X(, we ee that the preceding expreion become the entry (j, h of the matrix defined by (4.3. Thi operation i jutified under L.4.2.

11 184 imeon m. berman and halina frydman The element (4.3 of i repreentable a { σ jh = E 1(T α log q(t, X(, θ log q(t, X(, θ.... (4.6 θ h If (T i, X i (, i = 1,..., n are the independent copie of (T, X( ued in the etimation procedure, then, by the Law of Large Number, 1 n n 1(T i α log q(t i, X i (, θ θ h log q(t i, X i (, θ i a conitent nonparametric etimator of σ jh. If the ditribution of X( are known and σ jh can be explicitly calculated a a function σ jh (θ, then, by Theorem 4.1, σ jh (ˆθ i a conitent parametric etimator of σ jh (θ if the latter function i continuou. 5. Example We conider the cla of hazard function q where the dependence on X(, t, i limited to X(t alone: There i a real-valued function q(t, x, θ uch that q(t, X(, θ = q(t, X(t, θ. Other model, where the dependence of q(t, X(, θ on X( i through ome functional of X(, t, uch a X(t a (a > or X( d, are certainly plauible. However, in mot application the dependence of q on X(, t, i limited to X(t alone. In particular we firt take q to be of the form q(t, x, θ = m θ j q j (t, x 1(x J j,... (5.1 j=1 where J j, j = 1,..., m, are dijoint Borel et forming a decompoition of the real line, (q j are poitive function, and 1( i the indicator function. From the relation q = 1(x J j q j (t, x, log q = 1(x J j /θ j, and from (4.2 it follow that the likelihood equation for θ j ha the olution ˆθ j = n 1(X i(t i J j, T i α n 1(T i >, X i ( J j q j (, X i ( d,... (5.2 where (T i, X i (, i = 1,..., n are the independent oberved realization. From the elementary form of log q/ it i imple to formulate general condition under which Theorem 4.1 can be applied.

12 parametric etimation of hazard function 185 The information matrix (σ jh in (4.6 take the form σ jh = 1 P (T α, X(T J θj 2 j, for j = h, =, for j h. According to the dicuion following (4.6, the nonparametric etimator of σ jj i 1 n n ˆθ 1 (T i α, X i (T i J j, j 2 where ˆθ j i given by (5.2. The parametric etimator i obtained, under the aumption that P (T α, X(T J j = P θ (T α, X(T J j can be calculated a a function of θ, by uing the etimator ˆθ in the place of θ : ˆθ 2 j Pˆθ (T α, X(T J j.... (5.4 If m = 1 in (5.1, then q(t, x, θ = θq(t, x, and (5.2 and (5.3 aume the impler form: n ˆθ = 1(T i α 1 (T i > q(, X i ( d,... (5.5 and n σ 11 = 1 P (T α.... (5.6 θ2 Next we dicu the calculation of (5.3 and (5.6 in pecific cae. In each cae it i eay to verify the condition of Theorem 4.1. In Example 5.1 and 5.2 we ue the reult on the joint ditribution of (X(T, T obtained in Berman and Frydman (1996. Example 5.3 employ the reult from Yahin (1985 and Yahin and Manton (1997 on the formula for the marginal urvival function of T. In the final example q ha a form different from (5.1. Example 5.1. Let X(t, t, be a Poion proce with parameter λ. Take q(t, x, θ = θx; then, a hown by Berman and Frydman (1996, P ( T α X( = i [ = 1 exp λα θαi + λ ( 1 e θα ] θ and the parametric etimator of i obtained from thi by (5.6. Example 5.2. Suppoe that X(t, t, i a continuou-time Markov chain with tate pace S = (1,..., m, matrix generator A, and the killing-rate function q(t, x, θ = m j=1 θ j 1(x = j. Then, by (5.2, ˆθ j = n 1 (T i α, X i (T i = j n 1 (T i >, X i ( = j d,

13 186 imeon m. berman and halina frydman and σ jj (θ θ i given by (5.3. Put D = diag (θ 1,..., θ m ; then, by the reult of Berman and Frydman (1996, for any h, j S, P ( T < α, X(T = j X( = h = θj [( I e α(a D (D A 1] hj o that, for any initial ditribution (π h on S, P (T < α, X(T = j = θ j m h=1 π h [( I e α(a D (D A 1] hj Thi model i related to that of Jarrow, Lando and Turnbull (1997. Example 5.3. Let X(t, t, be a Gauian diffuion defined by dx(t = [a (t + a 1 (t X(t] dt + σ(t dw (t,... (5.7 where X( i a contant, and a (t, a 1 (t, b(t are known function of time, and define q(t, x, θ = θx (5.8 Yahin (1985 howed that in thi cae the ditribution of X(t conditioned on T > t i alo Gauian with the mean and variance function atifying a et of ordinary differential equation. Thi reult combined with the general formula in Yahin and Manton (1997 for the marginal urvival function: ( P (T > t = exp E[q(u, X(u, θ T > u] du make the calculation of P (T > t tractable in the model given by (5.7 and (5.8. In particular, uppoe that X(t i a Brownian motion with variance parameter σ 2 (i.e. a (t = a 1 (t, σ(t = σ. Then the calculation give o that (5.6 i of the form ( P (T > t = coh 2 1/2 θ σα ( θ [1 2 coh 2 ] 1/2 θ σα.. Thi model i related to that of Yahin and Manton (1997. Example 5.4. Conider the hazard function q(t, x, θ = a + (x θ 2,

14 parametric etimation of hazard function 187 where a > i known and θ i a real-valued unknown parameter. (Thi i not in the pecial cla (5.1. The likelihood equation baed on the independent realization (T i, X i (, i = 1,..., n, i, by (4.2, equivalent to n Ti α X i ( d n θ = n (T i α n { Xi(T i 1(T i α (X i(t i θ 2 +a 1(T i α (X i(t i θ 2 +a where T α = min(t, α. The latter i olvable by ucceive approximation. By (4.6, [ ] 2 (X(T θ 1(T α σ 11 = 4E (X(T θ 2. + a Thi model i related to that of Yahin and Manton (1997. Remark 5.5. The hazard function h (t + βx(t and h (t(1 + βx(t were conidered by Jewell and Kalbfleich (1992 and Self and Pawitan (1992, repectively. In their model β i an unknown parameter and h (t an unknown function; thu, thee are emi-parametric model. By contrat, our model i purely parametric ince the functional form of q i aumed to be given. Acknowledgement. The work of Halina Frydman wa partially upported by ummer reearch grant from the NYU Salomon Center. She alo thank Biotatitic Department at the Univerity of Copenhagen for providing hopitable environment during her work on thi paper., Reference Anderen P.K., Borgan O., Gill R. and Keiding N. (1993. Statitical Model Baed on Counting Procee. Springer, New York. Berman, S. and Frydman H. (1996. Ditribution aociated with Markov procee with killing. Communication in Statitic: Stochatic Model, Cramer H. (1946. Mathematical Method of Statitic. Princeton Univerity Pre, Princeton, New Jerey. Fuaro E.F., Nielen J.P, and Sheike T.H. (1993. Marker-dependent hazard etimation: An application to Aid, Statitic in Medicine, 12, Jarrow R., Lando D. and Turnbull S. (1997. A Markov model for the term tructure of credit rik pread. The Review of Financial Studie, Jewell N.P. and Kalbfleich J.D. (1992. Marker model in urvival analyi and application to iue aociated with Aid. in Aid Epidemiology: Methodological Iue, N. Jewell, K. Dietz, V. Farewell, Editor, Birkhäuer, Boton. Lando D. (1996. Modeling bond and derivative with default rik. Working paper, Intitute of Mathematical Statitic, Univerity of Copenhagen. Tiati A.A., De Gruttola V. and Wulfohn M.S. (1995. Modelling the relationhip of urvival to longitudinal data meaured with error. Application to urvival and CD4 count in patient with Aid. Jour. Amer. Statit. Aoc., 9, Self S. and Pawitan Y. (1992. Modelling a marker of dieae progreion and onet of dieae. In AIDS Epidemiology: Methodological Iue, N. Jewell, K. Dietz, V. Farewell, Editor , Birkhäuer, Boton.

15 188 imeon m. berman and halina frydman Yahin A.I. (1985. Dynamic in urvival analyi: conditional Gauian property veru Cameron-Martin formula. In Statitic and Control of Stochatic Procee, N. V. Krylov, R. S. Lipter and A. A. Novikov, Editor, , Springer. Yahin A.I., and Manton K.G (1997. Effect of unoberved and partially oberved covariate procee on ytem failure: a review of model and etimation trategie. Statitical Science, 12, Simeon M. Berman Courant Intitute of Mathematical Science New York Univerity 251 Mercer Street New York, NY berman@cim.nyu.edu Halina Frydman Stern School of Buine New York Univerity 44 Wet 4th Street New York, NY hfrydman@tern.nyu.edu