Martingale Structure Research Articles

Solution Representations for Poisson’s Equation, Martingale Structure, and the Markov Chain Central Limit Theorem

The solution of Poisson’s equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study three different representations for the solution for countable state space irreducible Markov chains, two based on entry time expectations, and the other based on a potential kernel. Our consideration of null recurrent chains allows us to extend our theory to positive recurrent nonexplosive Markov jump processes. We also develop the martingale structure induced by these solutions to Poisson’s equation, under minimal conditions, and establish verifiable Lyapunov conditions to support our theory. Finally, we provide a central limit theorem for Markov dependent sums, under conditions weaker than have previously appeared in the literature.

Two-sample inference procedures under nonproportional hazards.

We introduce a new two-sample inference procedure to assess the relative performance of two groups over time. Our model-free method does not assume proportional hazards, making it suitable for scenarios where nonproportional hazards may exist. Our procedure includes a diagnostic tau plot to identify changes in hazard timing and a formal inference procedure. The tau-based measures we develop are clinically meaningful and provide interpretable estimands to summarize the treatment effect over time. Our proposed statistic is a U-statistic and exhibits a martingale structure, allowing us to construct confidence intervals and perform hypothesis testing. Our approach is robust with respect to the censoring distribution. We also demonstrate how our method can be applied for sensitivity analysis in scenarios with missing tail information due to insufficient follow-up. Without censoring, Kendall's tau estimator we propose reduces to the Wilcoxon-Mann-Whitney statistic. We evaluate our method using simulations to compare its performance with the restricted mean survival time and log-rank statistics. We also apply our approach to data from several published oncology clinical trials where nonproportional hazards may exist.

Almost sure stability criterion for continuous-time linear systems with uniformly distributed uncertainty

Assuming the uncertain parameter of a linear polytopic system to be uniformly distributed, we can prove that the system is almost sure exponentially stable if and only if its averaging system is stable. Computing the mathematical expectation of the state-transition matrix and characterizing the martingale structure embedded in its variance is the key for deriving this stability criterion.

Quantitative central limit theorems for the parabolic Anderson model driven by colored noises

In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use It\^o's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincar\'e inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (2020 \emph{Electron. J. Probab.}) and Nualart-Song-Zheng (2021 \emph{ALEA, Lat. Am. J. Probab. Math. Stat.}).

Martingale Structure for General Thermodynamic Functionals of Diffusion Processes Under Second-Order Averaging

Novel hidden thermodynamic structures have recently been uncovered during the investigation of nonequilibrium thermodynamics for multiscale stochastic processes. Here we reveal the martingale structure for a general thermodynamic functional of inhomogeneous singularly perturbed diffusion processes under second-order averaging, where a general thermodynamic functional is defined as the logarithmic Radon–Nykodim derivative between the laws of the original process and a comparable process (forward case) or its time reversal (backward case). In the forward case, we prove that the regular and anomalous parts of a thermodynamic functional are orthogonal martingales. In the backward case, while the regular part may not be a martingale, we prove that the anomalous part is still a martingale. With the aid of the martingale structure, we prove the integral fluctuation theorem satisfied by the regular and anomalous parts of a general thermodynamic functional. Further extensions and applications to stochastic thermodynamics are also discussed, including the martingale structure and fluctuation theorems for the regular and anomalous parts of entropy production and housekeeping heat in the absence or presence of odd variables.

Dynamic inference in general nested case-control designs.

Nested case-control designs are attractive in studies with a time-to-event endpoint if the outcome is rare or if interest lies in evaluating expensive covariates. The appeal is that these designs restrict to small subsets of all patients at risk just prior to the observed event times. Only these small subsets need to be evaluated. Typically, the controls are selected at random and methods for time-simultaneous inference have been proposed in the literature. However, the martingale structure behind nested case-control designs allows for more powerful and flexible non-standard sampling designs. We exploit that structure to find simultaneous confidence bands based on wild bootstrap resampling procedures within this general class of designs. We show in a simulation study that the intended coverage probability is obtained for confidence bands for cumulative baseline hazard functions. We apply our methods to observational data about hospital-acquiredinfections.

NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS

Nonparametric identification of the Mixed Hazard model is shown. The setup allows for covariates that are random, time-varying, satisfy a rich path structure and are censored by events. For each set of model parameters, an observed process is constructed. The process corresponding to the true model parameters is a martingale, the ones corresponding to incorrect model parameters are not. The unique martingale structure yields a family of moment conditions that only the true parameters can satisfy. These moments identify the model and suggest a GMM estimation approach. The moments do not require use of the hazard function.

GEL estimation and tests of spatial autoregressive models

This paper considers the generalized empirical likelihood (GEL) estimation and tests of high order spatial autoregressive (SAR) models by exploring an inherent martingale structure. The GEL estimator has the same asymptotic distribution as the generalized method of moments estimator explored with same moment conditions for estimation, but circumvents a first step estimation of the optimal weighting matrix with a preliminary estimator, and thus can be robust to unknown heteroskedasticity and non-normality. While the GEL removes the asymptotic bias from the preliminary estimator and partially removes the bias due to the correlation between the moment conditions and their Jacobian, the empirical likelihood as a special member of GELs further partially removes the bias from estimating the second moment matrix. We also formulate the GEL overidentification test, Moran’s I test, and GEL ratio tests for parameter restrictions and non-nested hypotheses. While some of the conventional tests might not be robust to non-normality and/or unknown heteroskedasticity, the corresponding GEL tests can be robust.

Stationary solutions to the compressible Navier–Stokes system driven by stochastic forces

We study the long-time behavior of solutions to a stochastically driven Navier–Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue–Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. An essential tool in order to obtain the global-in-time estimate is the stationarity of solutions on each approximation level, which provides a certain regularizing effect. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory, due to the underlying martingale structure of the noise.

WEAK CONVERGENCE TO STOCHASTIC INTEGRALS UNDER PRIMITIVE CONDITIONS IN NONLINEAR ECONOMETRIC MODELS

Limit theory with stochastic integrals plays a major role in time series econometrics. In earlier contributions on weak convergence to stochastic integrals, the literature commonly uses martingale and semi-martingale structures. Liang, Phillips, Wang, and Wang (2016) (see also Wang (2015), Chap. 4.5) currently extended weak convergence to stochastic integrals by allowing for a linear process or a α-mixing sequence in innovations. While these martingale, linear process and α-mixing structures have wide relevance, they are not sufficiently general to cover many econometric applications that have endogeneity and nonlinearity. This paper provides new conditions for weak convergence to stochastic integrals. Our frameworks allow for long memory processes, causal processes, and near-epoch dependence in innovations, which have applications in a wide range of econometric areas such as TAR, bilinear, and other nonlinear models.

Two-color balanced affine urn models with multiple drawings

We study a class of balanced urn schemes on balls of two colors (say white and black). At each drawing, a sample of size m≥1 is taken out from the urn, and ball addition rules are applied. We consider these multiple drawings under sampling with or without replacement. We further classify ball addition matrices according to the structure of the expected value into affine and nonaffine classes. We give a necessary and sufficient condition for a scheme to be in the affine subclass, for which we get explicit results for the expected value and second moment of the number of white balls after n steps and an asymptotic expansion of the variance. Moreover, we uncover a martingale structure. This unifies several earlier works focused on special cases of urn models with multiple drawings [5,6,17,19,20,22] as well as the special case of sample size m=1. The class is parametrized by Λ, specified by the ratio of the two eigenvalues of a “reduced” ball replacement matrix and the sample size. We categorize the class into small-index urns (Λ<12), critical-index urns (Λ=12), and large-index urns (Λ>12), and triangular urns. We obtain central limit theorems for small- and critical-index urns and prove almost-sure convergence for triangular and large-index urns. Moreover, we discuss the moment structure of large-index urns and triangular urns.

Robust estimation in the additive hazards model

ABSTRACTIn the additive hazards model the hazard function of a survival variable T is modeled additively as , where λ0(t) is a common non parametric baseline hazard function and is a vector of independent variables. For this model, the pioneering work of Lin and Ying (1994) develops a closed-form estimator for the regression parameter from a new estimating equation. That equation has a similar structure to the corresponding partial likelihood score function for the multiplicative model (Cox 1972) in that it exploits a martingale structure and it allows estimation of separate from the baseline hazard function. Their estimator is asymptotically normal and highly efficient. However, a potential drawback is that it is very sensitive to outliers. In this paper we propose a family of robust alternatives for estimation of the parameter in the additive hazards model which is robust to outliers and still highly efficient and asymptotically normal. We prove Fisher-consistency, obtain the influence function, and illustrate the estimation with simulated and real data. The latter corresponds to the time-honored Welsh Nickels Refiners dataset first introduced by Doll et al. (1970) and subsequently analyzed by Breslow and Day (1987) and Lin and Ying (1994), among others.

Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes

In this paper we propose a general derivative pricing framework that employs decoupled time-changed (DTC) Lévy processes to model the underlying assets of contingent claims. A DTC Lévy process is a generalized time-changed Lévy process whose continuous and pure jump parts are allowed to follow separate random time scalings; we devise the martingale structure for a DTC Lévy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying Lévy decomposition allows the introduction of asset price models consistent with the assumption of a correlated pair of continuous and jump market activity rates; we study one illustrative DTC model of this kind based on the so-called Wishart process. The theory we develop is applied to the problem of pricing not only claims that depend on the price or the volatility of an underlying asset, but also more sophisticated derivatives whose payoffs rely on the joint performance of these two financial variables, such as the target volatility option. We solve the pricing problem through a Fourier-inversion method. Numerical analyses validating our techniques are provided. In particular, we present some evidence that correlating the activity rates could be beneficial for modeling the volatility skew dynamics.

The compulsive gambler process

In the compulsive gambler process there is a finite set of agents who meet pairwise at random times ($i$ and $j$ meet at times of a rate-$\nu_{ij}$ Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other's money. We introduce this process and describe some of its basic properties. Some properties are rather obvious (martingale structure; comparison with Kingman coalescent) while others are more subtle (an "exchangeable over the money elements" property, and a construction reminiscent of the Donnelly-Kurtz look-down construction). Several directions for possible future research are described. One - where agents meet neighbors in a sparse graph - is studied here, and another - a continuous-space extension called the metric coalescent - is studied in Lanoue (2014).

Adaptive likelihood ratio approaches for the detection of space–time disease clusters

A methodology based on adaptive likelihood ratios (ALRs) for the detection of emerging disease clusters is presented. The martingale structure of the regular likelihood ratio is preserved by the ALR. The upper limit for the false alarm rate of the proposed method depends only on the quantity of evaluated cluster candidates. Thus Monte Carlo simulations are not required to validate the procedures’ statistical significance, allowing the construction of a fast computational algorithm to detect clusters. The number of evaluated clusters is also significantly reduced, through the use of an adaptive approach to prune many unpromising clusters. This further increases the computational speed. Performance is evaluated through simulations to measure the average detection delay and the probability of correct cluster detection. We present applications for thyroid cancer in New Mexico and hanseniasis in children in the Brazilian Amazon.

Partial marginal likelihood estimation for general transformation models

We consider a large class of transformation models introduced by Gu et al. (2005) [14]. They proposed an estimation procedure for calculating the maximum partial marginal likelihood estimator (MPMLE) of regression parameters. A big advantage of MPMLE is that it avoids estimating two infinitely dimensional nuisance parameters: baseline and censoring survival functions. And they showed the validity of MPMLE through extensive simulations. In this paper, we establish the asymptotic properties of MPMLE in the general transformation models for either right or left censored data. The difficulty in establishing these asymptotic results comes from the fact that the score function derived from the partial marginal likelihood does not have ordinary independence or martingale structure. We develop a novel discretization method to resolve the difficulty. The estimation procedure is further examined using simulation studies and the analysis of the ACTG019 data.

Asymptotic distribution of the nonparametric distribution estimator based on a martingale approach in doubly censored data

For analysis of time-to-event data with incomplete information beyond right-censoring, many generalizations of the inference of the distribution and regression model have been proposed. However, the development of martingale approaches in this area has not progressed greatly, while for right-censored data such an approach has spread widely to study the asymptotic properties of estimators and to derive regression diagnosis methods. In this paper, focusing on doubly censored data, we discuss a martingale approach for inference of the nonparametric maximum likelihood estimator (NPMLE). We formulate a martingale structure of the NPMLE using a score function of the semiparametric profile likelihood. Finally, an expression of the asymptotic distribution of the NPMLE is derived more conveniently without depending on an infinite matrix expression as in previous research. A further useful point is that a variance-covariance formula of the NPMLE computable in a larger sample is obtained as an empirical version of the limit form presented here.

Quantile Regression for Doubly Censored Data

Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we propose a new analysis strategy for such doubly censored data by adopting a quantile regression model. We develop computationally simple estimation and inference procedures by appropriately using the embedded martingale structure. Asymptotic properties, including the uniform consistency and weak convergence, are established for the resulting estimators. Moreover, we propose conditional inference to address the special identifiability issues attached to the double censoring setting. We further show that the proposed method can be readily adapted to handle left truncation. Simulation studies demonstrate good finite-sample performance of the new inferential procedures. The practical utility of our method is illustrated by an analysis of the onset of the most commonly investigated respiratory infection, Pseudomonas aeruginosa, in children with cystic fibrosis through the use of the U.S. Cystic Fibrosis Registry.

‘SPREADABLE ARRAYS AND MARTINGALE STRUCTURES’

Abstract

On the concentration and the convergence rate with a moment condition in first passage percolation

We consider the first passage percolation model on the Z d lattice. In this model, we assign independently to each edge e a non-negative passage time t ( e ) with a common distribution F . Let a 0 , n be the passage time from the origin to ( n , 0 , … , 0 ) . Under the exponential tail assumption, Kesten (1993) [9] and Talagrand (1995) [12] investigated the concentration of a 0 , n from its mean using different methods. With this concentration and the exponential tail assumption, Alexander (1993) [1] gave an estimate for the convergence rate for E a 0 , n . In this paper, focusing on a moment condition, we reinvestigate the concentration and the convergence rate for a 0 , n using a special martingale structure.