Parameter Estimations With Mean-uncertainty And Applications Under Nonlinear Expectation Framework

In this paper we focus on the optimal unbiased estimators for maximal distribution under nonlinear expectation framework.By means of simulation and real data analysis,we obtain outcomes from the application of "the worst case risk measure" compared with those under classic circumstances.It has been proven that for sample from maximal distribution,the optimal unbiased estimators are the first and last of the order statistics.For their simulations,we actually use data generated by uniform distribution,asymptotically approximating the desired estimators,by means of data with more than one distributions and thereafter some unpredictable true random numbers generated from nature,which intuitively turns out that the deviation between two approximations maintain a steady state in their deviation as the sample size grows large.As for the mean uncertainty,we mainly take into consideration residual errors of upper expectation regression,i.e.IE(Y|X)= g(β,X)Suppose the upper expectation of residuals is μ,with the assistant of mini-max-risk regression,quadratic valuation techniques,the consistent estimation is achieved when a penalty is applied to ensure the convexity of the objective function.Moreover,instead of being constrained in linear form,such upper-expectation-regression approaches are extended to partial linear models.In statistical sense,the partially linear model not only takes into account the obvious factors but also the latent,unobservable ones,which influence the covariates and the response variables in one way or another.Unlike the prevent methods such as the backfitting algorithm and the profile likelihood approach,we here discuss an old way going along with penalized smoothing spline parameter and owning some approximation in probability with general cross validation estimator.Moreover,under mean-uncertainty,one can also estimate by upper expectation procedure via some penalized approach,in that way,when the class of distributions has some desirable properties,the estimation is quite consistent to classical estimators with certain probability measure.Lastly,considering the relationship between the price and the volume of transactions of financial assets,our real data analysis is carried out when the mean uncertainty is attached to the regression models,aiming at some plausible adjustments in investment strategies.