Shape constrained kernel density estimation

In this paper, a method for estimating monotone, convex and log-concave densities is proposed. The estimation procedure consists of an unconstrained kernel estimator which is modi?ed in a second step with respect to the desired shape constraint by using monotone rearrangements. It is shown that the resulting estimate is a density itself and shares the asymptotic properties of the unconstrained estimate. A short simulation study shows the ?nite sample behavior. --Convexity,log-concavity,monotone rearrangements,monotonicity,nonparametric density estimation

Central limit theorems for the integrated squared error of derivative estimators

A central limit theorem for the weighted integrated squared error of kernel type estimators of the first two derivatives of a nonparametric regression function is proved by using results for martingale differences and U-statistics. The results focus on the setting of the Nadaraya-Watson estimator but can also be transfered to local polynomial estimates. --central limit theorem,integrated squared error,kernel estimates,local polynomial estimate,Nadaraya-Watson estimate,nonparametric regression

Testing strict monotonicity in nonparametric regression

A new test for strict monotonicity of the regression function is proposed which is based on a composition of an estimate of the inverse of the regression function with a common regression estimate. This composition is equal to the identity if and only if the ?true? regression function is strictly monotone, and a test based on an L2-distance is investigated. The asymptotic normality of the corresponding test statistic is established under the null hypothesis of strict monotonicity. --nonparametric regression,strictly monotone regression,goodness-of-fit test

Estimating a convex function in nonparametric regression

A new nonparametric estimate of a convex regression function is proposed and its stochastic properties are studied. The method starts with an unconstrained estimate of the derivative of the regression function, which is firstly isotonized and then integrated. We prove asymptotic normality of the new estimate and show that it is first order asymptotically equivalent to the initial unconstrained estimate if the regression function is in fact convex. If convexity is not present the method estimates a convex function whose derivative has the same Lp-norm as the derivative of the (non-convex) underlying regression function. The finite sample properties of the new estimate are investigated by means of a simulation study and the application of the new method is demonstrated in two data examples. --nonparametric regression,order restricted inference,convexity,Nadaraya-Watson estimate,nondecreasing rearrangement

Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators

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Shape constrained estimators in inverse regression models with convolution-type operator

In this paper we are concerned with shape restricted estimation in inverse regression problems with convolution-type operator. We use increasing rearrangements to compute increasingand convex estimates from an (in principle arbitrary) unconstrained estimate of the unknown regression function. An advantage of our approach is that it is not necessary that prior shape information is known to be valid on the complete domain of the regression function. Instead, it is sufficient if it holds on some compact interval. A simulation study shows that the shape restricted estimate on the respective interval is significantly less sensitive to moderate undersmoothing than the unconstrained estimate, which substantially improves applicability of estimates based on data-driven bandwidth estimators. Finally, we demonstrate the application of the increasing estimator by the estimation of the luminosity profile of an elliptical galaxy. Here, a major interest is in reconstructing the central peak of the profile, which, due to its small size, requires to select the bandwidth as small as possible. --convexity,increasing rearrangements,image reconstruction,inverse problems,monotonicity,order restricted inference,regression estimation,shape restrictions

A note on testing the covariance matrix for large dimension

We consider the problem of testing hypotheses regarding the covariance matrix of multivariate normal data, if the sample size s and dimension n satisfy lim [n,s→∞] n/s = y. Recently, several tests have been proposed in the case, where the sample size and dimension are of the same order, that is y ∈ (0,∞). In this paper we consider the cases y = 0 and y = ∞. It is demonstrated that standard techniques are not applicable to deal with these cases. A new technique is introduced, which is of its own interest, and is used to derive the asymptotic distribution of the test statistics in the extreme cases y = 0 and y = ∞. --sphericity test,random matrices,Wishart distribution

A note on estimating a monotone regression by combining kernel and density estimates

n.a. --order restricted inference,monotone estimation,greatest convex minorant,Nadaraya-Watson estimate