Median Analysis of Repeated Measures Associated with Recurrent Events in Presence of Terminal Event

Recurrent events are often encountered in medical follow up studies. In addition, such recurrences have other quantities associated with them that are of considerable interest, for instance medical costs of the repeated hospitalizations and tumor size in cancer recurrences. These processes can be viewed as point processes, i.e. processes with arbitrary positive jump at each recurrence. An analysis of the mean function for such point processes have been proposed in the literature. However, such point processes are often skewed, leading to median as a more appropriate measure than the mean. Furthermore, the analysis of recurrent event data is often complicated by the presence of death. We propose a semiparametric model for assessing the effect of covariates on the quantiles of the point processes. We investigate both the finite sample as well as the large sample properties of the proposed estimators. We conclude with a real data analysis of the medical cost associated with the treatment of ovarian cancer.

CONDITIONAL JUMP DYNAMICS IN STOCK RETURNS: EVIDENCE FROM MIST STOCK EXCHANGES

This paper applies a conditional jump model that was proposed by Chan and Maheu (2002) to examine the stock market dynamics of Mexico, Indonesia, South Korea, and Turkey (MIST). We find that the conditional jump intensity parameter estimates are statistically significant and change dramatically between two sample periods. We show that a high probability of jumps today predicts a high probability of jumps in the next period. The impact of a previous shock to the next period's jump intensity is found to be higher in Turkey compared to other MIST countries. Contrary to the previous literature, we discover that after a stock market crash, it is more likely to see a negative jump (drop) again in the stock exchanges of Mexico and Indonesia. Only in Turkey, it is more likely to see a positive jump after market crashes.

Extinction and explosion of nonlinear Markov branching processes

This paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

The behavior near the origin of the supremum functional in a process with stationary independent increments

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z 1(t) = X(t), Z 2(t) = sup0≦s≦t X(s) and Z 3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: lim t↓0 t —1 P[Zi (t) > x] = M +(x) at all points of continuity of M +, the Lévy measure of X.

The behavior near the origin of the supremum functional in a process with stationary independent increments

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z1(t) = X(t), Z2(t) = sup0≦s≦tX(s) and Z3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: limt↓0t—1P[Zi(t) > x] = M+(x) at all points of continuity of M+, the Lévy measure of X.

The structure of extremal processes

An extremal-Fprocess {Y(t);t≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed likeFin the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration ofF.Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps ofY(t) in (t1,t2] follows a Poisson distribution with parameter logt2/t1. The time transformationt→etgives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the inter-jump times. WhenFis an extreme value distribution theY-process has special properties. The most important is that ifF(x) = exp {—e–x} thenY(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour ofY(t) and its jump times.

The structure of extremal processes

An extremal-F process {Y (t); t ≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed like F in the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration of F. Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps of Y (t) in (t1, t2] follows a Poisson distribution with parameter log t2/t1. The time transformation t→ et gives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the inter-jump times. When F is an extreme value distribution the Y-process has special properties. The most important is that if F(x) = exp {—e–x} then Y(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour of Y(t) and its jump times.