Nonparametric Estimation for FBSDEs Models with Applications in Finance

As a continuous-time model, forward-backward stochastic differential equations (in short FBSDEs) have been successfully applied in mathematical finance, e.g., European option pricing for either a small or a large investor in a Markovian market. However, the correct FBSDEs model for a specific topic can neither be provided automatically by financial market nor derived from theory of mathematical finance. In this article, a nonparametric FBSDEs model is adopted for its flexibility and robustness, and the estimators of the functional coefficients of the FBSDEs model are obtained. The asymptotic properties of the estimators are also discussed. A simulation is performed to test the feasibility of our method.     

Empirical bayes approach to parameter identification in linear stochastic difference equations

Empirical Bayesian parameter estimators and predictors for linear stochastic difference equations are constructed and discussed. Some properties as consistency and asymptotic optimality are investigated. The given methods are illustrated by the example of a univariate first order autoregressive process.     

Robust statistical inference: weighted likelihoods or usual m-estimation?

We review the weighted likelihood estimating equations methodology introduced by Markatou, Basu and Lindsay (1995). and Basu, Markatou and Lindsay (1995) and compare it, in the case of symmetric and asymmetric contamination, with Huber's M-estimators of location. The simulation study shows that the weighted likelihood estimating equations estimator is at least as competitive as Huber's M-estimators in the case of symmetric contamination. In the case of asymmetric contamination it may be superior than Huber's M-estimators     

Statistical Properties of Generalized Method-of-Moments Estimators of Structural Parameters Obtained From Financial Market Data

The article examines the properties of generalized method of moments GMM estimators of utility function parameters. The research strategy is to apply the GMM procedure to generated data on asset returns from stochastic exchange economies; discrete methods and Markov chain models are used to approximate the solutions to the integral equations for the asset prices. The findings are as follows: (a) There is variance/bias trade-off regarding the number of lags used to form instruments; with short lags, the estimates of utility function parameters are nearly asymptotically optimal, but with longer lags the estimates concentrate around biased values and confidence intervals become misleading, (b) The test of the overidentifying restrictions performs well in small samples; if anything, the test is biased toward acceptance of the null hypothesis.     

Rensis Likert 1903–1981

The equivalence of several methods of solving linear simultaneous equations is demonstrated for use in teaching applied animal breeding. Matrix notation represents equations concisely and allows ease of algebraic treatment.     

We Must Preach what is Practiced

Much of what we teach in statistics is seldom used in practice. Much of what students need we do not teach.     

Goodness-of-Fit Statistics for General Linear Regression Equations in the Presence of Replicated Responses

Goodness-of-fit statistics for general multiple-linear-regression equations are reviewed for the case of replicated responses. A modification of the coefficient of determination is recommended. This statistic has 1.0 as its achievable upper bound and has the coefficient of determination as a special case. It indicates more effectively how close a general-linear-regression equation is relative to the best possible one and is particularly useful when the purpose is to ascertain whether higher-order terms of a given set of explanatory variables are required. Other goodness-of-fit statistics that take into account the variation within replicated responses are reviewed. An illustration example is presented.     

Generalized Gauss–Hermite filtering

We consider a generalization of the Gauss–Hermite filter (GHF), where the filter density is represented by a Hermite expansion with leading Gaussian term (GGHF). Thus, the usual GHF is included as a special case. The moment equations for the time update are solved stepwise by Gauss–Hermite integration, and the measurement update is computed by the Bayes formula, again using numerical integration. The performance of the filter is compared numerically with the GHF, the UKF (unscented Kalman filter) and the EKF (extended Kalman filter) and leads to a lower mean squared filter error.     

Efficient Computation of Reduced Regression Models

We consider settings where it is of interest to fit and assess regression submodels that arise as various explanatory variables are excluded from a larger regression model. The larger model is referred to as the full model; the submodels are the reduced models. We show that a computationally efficient approximation to the regression estimates under any reduced model can be obtained from a simple weighted least squares (WLS) approach based on the estimated regression parameters and covariance matrix from the full model. This WLS approach can be considered an extension to unbiased estimating equations of a first-order Taylor series approach proposed by Lawless and Singhal. Using data from the 2010 Nationwide Inpatient Sample (NIS), a 20% weighted, stratified, cluster sample of approximately 8 million hospital stays from approximately 1000 hospitals, we illustrate the WLS approach when fitting interval censored regression models to estimate the effect of type of surgery (robotic versus nonrobotic surgery) on hospital length-of-stay while adjusting for three sets of covariates: patient-level characteristics, hospital characteristics, and zip-code level characteristics. Ordinarily, standard fitting of the reduced models to the NIS data takes approximately 10 hours; using the proposed WLS approach, the reduced models take seconds to fit.     

Efficiency of an adaptive estimator of a log-zero-poisson parameter

An estimator, λ is proposed for the parameter λ of the log-zero-Poisson distribution. While it is not a consistent estimator of λ in the usual statistical sense, it is shown to be quite close to the maximum likelihood estimates for many of the 35 sets of data on which it is tried. Since obtaining maximum likelihood estimates is extremely difficult for this and other contagious distributions, this estimate will act at least as an initial estimate in solving the likelihood equations iteratively. A lesson learned from this experience is that in the area of contagious distributions, variability is so large that attention should be focused directly on the mean squared error and not on consistency or unbiasedness, whether for small samples or for the asymptotic case. Sample sizes for some of the data considered in the paper are in hundreds. The fact that the estimator which is not a consistent estimator of λ is closer to the maximum likeli-hood estimator than the consistent moment estimator shows that the variability is large enough to not permit consistency to materialize even for such large sample sizes usually available in actual practice.