Second order asymptotic optimality of estimators for a density with finite cusps

We consider i.i.d. samples from a continuous density with finite cusps. Then we obtain the bound for the second order asymptotic distribution of all asymptotically median unbiased estimators. Further we get the second order asymptotic distribution of a bias-adjusted maximum likelihood estimator, and we see that it is not generally second order asymptotically efficient.     

The 3/2th and 2nd order asymptotic efficiency of maximum probability estimators in non-regular cases

In this paper we consider the estimation problem on independent and identically distributed observations from a location parameter family generated by a density which is positive and symmetric on a finite interval, with a jump and a nonnegative right differential coefficient at the left endpoit. It is shown that the maximum probability estimator (MPE) is 3/2th order two-sided asymptotically efficient at a point in the sense that it has the most concentration probability around the true parameter at the point in the class of 3/2th order asymptotically median unbiased (AMU) estimators only when the right differential coefficient vanishes at the left endpoint. The second order upper bound for the concentration probability of second order AMU estimators is also given. Further, it is shown that the MPE is second order two-sided asymptotically efficient at a point in the above case only.Research supported by University of Tsukuba Project Research.     

Higher order asymptotics in estimation for two-sided Weibull type distributions

We consider the estimation problem of a location parameter on a sample of size n from a two-sided Weibull type density % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOzaiaacIcacaWG4bGaeyOeI0IaeqiUdeNaaiykaiabg2da9iaa% doeacaGGOaGaeqySdeMaaiykaiGacwgacaGG4bGaaiiCaiaacIcacq% GHsislcaGG8bGaamiEaiabgkHiTiabeI7aXjaacYhadaahaaWcbeqa% aiabeg7aHbaakiaacMcaaaa!52AD!\[f(x - \theta ) = C(\alpha )\exp ( - |x - \theta |^\alpha )\] for –<x<, –<< and 1<a<3/2, where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Gaam4qaiaacIcacqaHXoqycaGGPaGaeyypa0JaeqySdeMaai4laiaa% cUhacaaIYaGaeu4KdCKaaiikaiaaigdacaGGVaGaeqySdeMaaiykai% aac2haaaa!4B0E!\[C(\alpha ) = \alpha /\{ 2\Gamma (1/\alpha )\} \]. Then the bound for the distribution of asymptotically median unbiased estimators is obtained up to the 2a-th order, i.e., the order % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOBamaaCaaaleqabaGaeyOeI0IaaiikaiaaikdacqaHXoqycqGH% sislcaaIXaGaaiykaiaac+cacaaIYaaaaaaa!4444!\[n^{ - (2\alpha - 1)/2} \]. The asymptotic distribution of a maximum likelihood estimator (MLE) is also calculated up to the 2a-th order. It is shown that the MLE is not 2a-th order asymptotically efficient. The amount of the loss of asymptotic information of the MLE is given.     

A characterization of second order efficiency in a curved exponential family

Summary The optimality of estimation method is investigated in a curved exponential family. A risk function, which is an extension of a residual sum of squares in regression analysis, is introduced. It is shown that second order efficiency of an estimation method is equivalent to attain the minimum among limiting risks of all estimation methods.     

Second order asymptotic comparison of estimators of a common parameter in the double exponential case

Summary The problem to estimate a common parameter for the pooled sample from the double exponential distributions is discussed in the presence of nuisance parameters. The maximum likelihood estimator, a weighted median, a weighted mean and others are asymptotically compared up to the second order, i.e. the ordern ^−1/2 with the asymptotic expansions of their distributions. University of Electro-communications     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

Inverse probability weighted estimation of local average treatment effects: A higher order MSE expansion

We consider a new inverse probability weighted estimator for the local average treatment effect parameter where the instrument propensity score is estimated by local polynomial regression. We derive its asymptotics and provide a higher order expansion of its asymptotic MSE.     

A nonlinear fourth order diffusion problem: Convergence to the steady state and non-negativity of solutions

The paper deals with nonlinear diffusion, both time-dependent and time-independent. The spatial terms in the partial differential equation (p.d.e.) contain a second order nonlinear part (where the non-negative diffusivity depends on the dependent variable) and a fourth order linear part. In the context of non-null, time-independent boundary conditions, convergence of the unsteady to the steady state is established. The second part of the paper discusses criteria on data ensuring non-negativity of the solutions. This is done for the steady state irrespective of the spatial dimension; and it is done for the unsteady state for the one-dimensional rectilinear case only, using a result from the first part of the paper.     

A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

We consider the numerical solution of a fourth‐order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth‐order parabolic equation, we perform an implicit discretization in time and a C⁰ Interior Penalty Discontinuous Galerkin (C⁰IPDG) discretization in space. The C⁰IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C⁰IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C⁰IPDG method will be shown in the second part of this paper.