Limit theorems for identically distributed martingale difference

Abstract

In this paper, we establish some general results for the strong law of large numbers and the complete convergence of martingale difference which include the well-known Marcinkiewicz–Zygmund strong law and Spitzer complete convergence.     

Testing the Martingale Difference Hypothesis

Abstract

In this paper we consider testing that an economic time series follows a martingale difference process. The martingale difference hypothesis has typically been tested using information contained in the second moments of a process, that is, using test statistics based on the sample autocovariances or periodograms. Tests based on these statistics are inconsistent since they cannot detect nonlinear alternatives. In this paper we consider tests that detect linear and nonlinear alternatives. Given that the asymptotic distributions of the considered tests statistics depend on the data generating process, we propose to implement the tests using a modified wild bootstrap procedure. The paper theoretically justifies the proposed tests and examines their finite sample behavior by means of Monte Carlo experiments.     

Optimal investment-consumption and life insurance with capital constraints

Abstract

The aim of this paper is to solve an optimal investment, consumption and life insurance problem when the investor is restricted to capital guarantee. We consider an incomplete market described by a jump-diffusion model with stochastic volatility. Using the martingale approach, we prove the existence of the optimal strategy and the optimal martingale measure and we obtain the explicit solutions for the power utility functions.     

Strong consistency of conditional least squares estimators in multiple regime threshold autoregressive models

In this paper we analyse the conditional least squares estimators of the parameters of a multiple regime threshold AR(1) model and prove that under certain conditions these are strongly consistent. We assume that the error process in each regime is amartingale difference sequence. Then we deal with strong consistency of the natural estimator of the error variance in each regime.     

Consistency of information criteria for model selection with missing data

ABSTRACT

In this paper, we investigate the consistency of the Expectation Maximization (EM) algorithm-based information criteria for model selection with missing data. The criteria correspond to a penalization of the conditional expectation of the complete data log-likelihood given the observed data and with respect to the missing data conditional density. We present asymptotic properties related to maximum likelihood estimation in the presence of incomplete data and we provide sufficient conditions for the consistency of model selection by minimizing the information criteria. Their finite sample performance is illustrated through simulation and real data studies.     

High dimensional asymptotics for the naive Hotelling T2 statistic in pattern recognition

Abstract

This paper examines the high dimensional asymptotics of the naive Hotelling T² statistic. Naive Bayes has been utilized in high dimensional pattern recognition as a method to avoid singularities in the estimated covariance matrix. The naive Hotelling T² statistic, which is equivalent to the estimator of the naive canonical correlation, is a statistically important quantity in naive Bayes and its high dimensional behavior has been studied under several conditions. In this paper, asymptotic normality of the naive Hotelling T² statistic under a high dimension low sample size setting is developed using the central limit theorem of a martingale difference sequence.     

Optimal investment,consumption, and life insurance in an incomplete market

ABSTRACT

In this article, we solve an optimal insurance-consumption-investment problem for a wage earner in an incomplete market, where the stock price has a mean-reverting drift. By using the martingale method, we analyze this problem and derive the optimal strategies. Explicit solutions are found for both power and logarithmic utilities.     

Complete convergence for weighted sums of mixingale sequences and statistical applications

In this paper, the complete convergence of weighted sums of L^r-mixingale is established, from which the complete convergence of martingale differences is also derived. As statistical applications, non parametric regression model and simpler linear errors-in-variables model with mixingale errors are discussed.     

A note on nonlinear regression for the autoregressive moving average with non-hd errors

Estimation by nonlinear regression of the parameters for the stationary and invertible autoregressive moving average (ARMA) model with mixing or martingale difference errors is considered. Simple proofs of consistency and asymptotic normality for the nonlinear least squares estimator are given by exploiting results from nonlinear estimation theory and mixing and mixingale theory.     

A goodness-of-fit test for the stratified proportional hazards model for survival data

Abstract

Goodness-of-fit testing is addressed in the stratified proportional hazards model for survival data. A test statistic based on within-strata cumulative sums of martingale residuals over covariates is proposed and its asymptotic distribution is derived under the null hypothesis of model adequacy. A Monte Carlo procedure is proposed to approximate the critical value of the test. Simulation studies are conducted to examine finite-sample performance of the proposed statistic.     

On the complete convergence for martingale difference sequence

In this paper, we establish several generalized results on complete convergence for martingale difference sequence, which include some well-known results.     

ASYMPTOTIC PROPERTIES OF AN OPTIMAL ESTIMATING FUNCTION APPROACH TO THE ANALYSIS OF MARK RECAPTURE DATA

ABSTRACT

A drawback of non parametric estimators of the size of a closed population in the presence of heterogeneous capture probabilities has been their lack of analytic tractability. Here we show that the martingale estimating function/sample coverage approach to estimating the size of a closed population with heterogeneous capture probabilities is mathematically tractable and develop its large sample properties.     

Least squares estimation of the parameters of a stochastic difference equation with polynomial regression component

In this paper, strong consistency of least squares estimates of the coefficients of stochastic difference equations with polynomial regression components, has been established, using martingale arguments, under autoregressive, partially explosive and purely explosive situations. The asymptotic normality of these estimates have also been discussed in this paper.     

Semiparametric regression estimation for longitudinal data in models with martingale difference error's structure

An inherent characteristic of longitudinal data is the dependence among the observations within the same subject. For exhibiting dependencies among the observations within the same subject, this paper considers a semiparametric partially linear regression model for longitudinal data based on martingale difference error's structure. We establish a strong consistency for the least squares estimator of a parametric component and the estimator of a non-parametric function under some mild conditions. A simulation study shows the performance of the proposed estimator in finite samples.     

Semiparametric Efficient Estimation of the Mean of a Time Series in the Presence of Conditional Heterogeneity of Unknown Form

Abstract

We obtain semiparametric efficiency bounds for estimation of a location parameter in a time series model where the innovations are stationary and ergodic conditionally symmetric martingale differences but otherwise possess general dependence and distributions of unknown form. We then describe an iterative estimator that achieves this bound when the conditional density functions of the sample are known. Finally, we develop a “semi-adaptive” estimator that achieves the bound when these densities are unknown by the investigator. This estimator employs nonparametric kernel estimates of the densities. Monte Carlo results are reported.     

Recursive kernel estimate of the conditional quantile for functional ergodic data

ABSTRACT

In this article, we study the recursive kernel estimator of the conditional quantile of a scalar response variable Y given a random variable (rv) X taking values in a semi-metric space. Two estimators are considered. While the first one is given by inverting the double-kernel estimate of the conditional distribution function, the second estimator is obtained by using the robust approach. We establish the almost complete consistency of these estimates when the observations are sampled from a functional ergodic process. Finally, a simulation study is carried out to illustrate the finite sample performance of these estimators.     

High-dimensional posterior consistency of the Bayesian lasso

ABSTRACT

This paper considers posterior consistency in the context of high-dimensional variable selection using the Bayesian lasso algorithm. In a frequentist setting, consistency is perhaps the most basic property that we expect any reasonable estimator to achieve. However, in a Bayesian setting, consistency is often ignored or taken for granted, especially in more complex hierarchical Bayesian models. In this paper, we have derived sufficient conditions for posterior consistency in the Bayesian lasso model with the orthogonal design, where the number of parameters grows with the sample size.     

A Modified Empirical Martingale Simulation for Financial Derivative Pricing

This article extends the empirical martingale simulation (EMS) method from using a risk-neutral measure to using a dynamic measure for financial derivative pricing. Although the EMS is shown to be capable of obtaining consistent estimate of financial derivative prices in a more efficient way than the standard Monte Carlo simulation procedure, it can proceed only under a risk-neutral framework. In practice, however, it is cumbersome to obtain the explicit expression of a risk-neutral model when dealing with a complex model. To alleviate this difficulty, we compute the financial derivative prices under the dynamic model and impose the martingale property on the simulated sample paths of both the change of measure process and the underlying asset prices under the dynamic P measure. Hence, we call this modification the empirical P-martingale simulation (EPMS). The strong consistency of the EPMS is established and its efficiency is performed by simulation in the GARCH framework. Simulation results shows that EPMS has the similar variance reduction as the EMS method in option pricing if the risk-neutral model can be obtained, and is more efficient than the standard Monte Carlo simulation in most cases.     

A pivot function and its limiting distribution: applications in goodness of fit and testing hypothesis

ABSTRACT

In this paper, a pivot function which is in terms of the sample and the underlying population distribution is introduced. It is assumed that the population distribution is continuous and strictly increasing on its support. Then, the martingale central limit theorem is applied to prove that limiting distribution of the pivot function is the standard normal. Interestingly, this result provides a unified procedure that can be applied for the goodness of fit, and for the purpose of parametric and nonparametric inferences, for the populations having distribution functions that are continuous and strictly increasing on their supports. The method is fairly simple and can be easily applied.     

Upper bounds for ruin probabilities under model uncertainty

Abstract

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.