Set Of Regularity Conditions Research Articles

The martingale comparison method for Markov processes

AbstractComparison results for Markov processes with respect to function-class-induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.

The wisdom of the crowd and prediction markets

Thanks to digital innovation, the wisdom of the crowd, which aims at gathering information (e.g. Wikipedia) and making a prediction (e.g. using prediction markets) from a group’s aggregated inputs, has been widely appreciated. An innovative survey design, based on a Bayesian learning framework, called the Bayesian truth serum (BTS), was proposed previously to reduce the bias in the simple majority rule by asking additional survey questions. A natural question is whether we can extend the BTS framework to prediction markets (not just polls). To do so, this paper proposes two estimators, one based on a prediction market alone and the other based on both the market and a poll question. We show that both estimators are consistent within the BTS framework, under different sets of regularity conditions. Simulations are conducted to examine the convergence of different estimators. A real data set of sports betting is used to demonstrate the effectiveness of one estimator.

A class of asymptotically efficient estimators based on sample spacings

In this paper, we consider general classes of estimators based on higher-order sample spacings, called the Generalized Spacings Estimators. Such classes of estimators are obtained by minimizing the Csiszár divergence between the empirical and true distributions for various convex functions, include the “maximum spacing estimators” as well as the maximum likelihood estimators (MLEs) as special cases, and are especially useful when the latter do not exist. These results generalize several earlier studies on spacings-based estimation, by utilizing non-overlapping spacings that are of an order which increases with the sample size. These estimators are shown to be consistent as well as asymptotically normal under a fairly general set of regularity conditions. When the step size and the number of spacings grow with the sample size, an asymptotically efficient class of estimators, called the “Minimum Power Divergence Estimators,” are shown to exist. Simulation studies give further support to the performance of these asymptotically efficient estimators in finite samples and compare well relative to the MLEs. Unlike the MLEs, some of these estimators are also shown to be quite robust under heavy contamination.

Pricing Interest Rate-Sensitive Credit Portfolio Derivatives

In this paper we present a modelling framework for portfolio credit risk which incorporates the dependence between risk-free interest-rates and the default loss process. The contribution in this approach is that - besides the traditional diffusion based covariation between loss intensities and interest-rates - a direct dependence between interest-rates and the loss process is allowed, in particular default-free interest-rates can also depend on the loss history of the credit portfolio. Amongst other things this enables us to capture the effect that economy-wide default events are likely to have on government bond markets and/or central banks' interest-rate policies. Similar to Schonbucher (2005), the model is set up using a set of loss contingent forward interest-rates fn(t; T) and loss-contingent forward credit protection rates Fn(t; T) to parameterize the market prices of default-free bonds and credit-sensitive assets such as CDOs. We show that (up to weak regularity conditions), existence of such a parametrization is necessary and sufficient for the absence of static arbitrage opportunities in the underlying assets. We also give necessary conditions and sufficient conditions on the dynamics of the parametrization which ensure absence of dynamic arbitrage opportunities in the model. Similar to the HJM drift restrictions for default-free interest-rates, these conditions take the form of restrictions on the drifts of fn(t; T) and Fn(t; T), together with a set of regularity conditions.

Convergence rate of Gibbs sampler and its application

Based on the convergence rate defined by the Pearson– χ 2 distance, this paper discusses properties of different Gibbs sampling schemes. Under a set of regularity conditions, it is proved in this paper that the rate of convergence on systematic scan Gibbs samplers is the norm of a forward operator. We also discuss that the collapsed Gibbs sampler has a faster convergence rate than the systematic scan Gibbs sampler as proposed by Liu et al. Based on the definition of convergence rate of the Pearson– χ 2 distance, this paper proved this result quantitatively. According to Theorem 2, we also proved that the convergence rate defined with the spectral radius of matrix by Robert and Shau is equivalent to the corresponding radius of the forward operator.

A steady-state model for the spread of HIV among drug users

This paper proposes a new approach to model the spread of HIV/AIDS among intravenous drug users (IVDUs). The focus is on a group of n IVDUs within which infective contacts occur, and which evolves in discrete time, subject to group splitting, immigration, and emigration. We are interested in finding the probability distribution of the ultimate number Y n of HIV infectives produced by the group as time tends to infinity, and obtain a stochastic recursive equation for it. Although, on the surface, the process resembles a branching process, our results cannot be obtained using techniques from the theory of branching processes. We use the probability metrics approach to obtain limit theorems for the normalized sequence L n = (Y n-EY n)n −1 2 . Finally, we consider the behavior of L n under different sets of regularity conditions, when for example L n = (Y n-EY n)n −1 α tends to an α-stable distribution.

ON COURNOT-BERTRAND MIXED DUOPOLIES*

This paper deals with a situation where a quantity-setting Coumot firm and a price-setting Bertrand firm coexist in the same industry. Under a set of regularity conditions on demand and cost, we compute equilibrium prices on Cournot-Bertrand “mixed” duopolies, and compare them with those of “pure” Cournot and Bertrand duopolies.

On the Asymptotics of Constrained $M$-Estimation

Limit theorems for an $M$-estimate constrained to lie in a closed subset of $\mathbb{R}^d$ are given under two different sets of regularity conditions. A consistent sequence of global optimizers converges under Chernoff regularity of the parameter set. A $\sqrt n$-consistent sequence of local optimizers converges under Clarke regularity of the parameter set. In either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value. Limit theorems for the optimal value are also obtained, agreeing with Chernoff's result in the case of maximum likelihood with global optimizers.

A unified method to calculate the moments of the least squares estimators in nonlinear regression

A unified method and a set of regularity conditions are presented in this paper to calculate the moments of the least squares estimators, residuals, and fitting errors in nonlinear regression. The results given by Box, Clarke and other authors have been greatly improved and developed. The key point of the method is that we find a series of stochastic expansions related to the estimators. These expansions consist of curvature measures and independent standard normal random variables; therefore they are easy to understand and to deal with from statistical and geometrical point of view.