Monotone Estimator Research Articles

A class of grouped Brunk estimators and penalized spline estimators for monotone regression

We study a class of monotone univariate regression estimators. We use B-splines to approximate an underlying regression function and estimate spline coefficients based on grouped data. We investigate asymptotic properties of two monotone estimators: a grouped Brunk estimator and a penalized monotone estimator. These estimators are consistent at the boundary and their mean square errors achieve optimal convergence rates under suitable assumptions of the true regression function. Asymptotic distributions are developed and are shown to be independent of spline degrees and the number of knots. Simulation results and car data illustrate performance of the proposed estimators.

Method for estimating network structure of short-term traffic flow forecasting model

Artificial neural network forecasting model is an efficient method to forecast the short-term traffic flow,but it is hard to choose the proper input variables and the number of hidden neurons. A data-driven algorithm was proposed to estimate the network structure of neural network forecasting model. According to the chaotic characteristics of the short-term traffic flow,phase space reconstruction was introduced to choose input variables reasonably. Then the number of hidden neurons can be estimated by the fast monotonic value estimation method. The experiment at result demonstrates the efficiency of the proposed algorithm on estimating the network structure of forecasting model.

Smoothing under Diffeomorphic Constraints with Homeomorphic Splines

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one-dimensional setting, incorporating diffeomorphic constraints is equivalent to imposing monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to make monotone any unconstrained estimator of a regression function and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching.

Empirical Bayes estimation for additive hazards regression models

We develop a novel empirical Bayesian framework for the semiparametric additive hazards regression model. The integrated likelihood, obtained by integration over the unknown prior of the nonparametric baseline cumulative hazard, can be maximized using standard statistical software. Unlike the corresponding full Bayes method, our empirical Bayes estimators of regression parameters, survival curves and their corresponding standard errors have easily computed closed-form expressions and require no elicitation of hyperparameters of the prior. The method guarantees a monotone estimator of the survival function and accommodates time-varying regression coefficients and covariates. To facilitate frequentist-type inference based on large-sample approximation, we present the asymptotic properties of the semiparametric empirical Bayes estimates. We illustrate the implementation and advantages of our methodology with a reanalysis of a survival dataset and a simulation study.

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

In a recent paper Dette, Neumeyer and Pilz (H. Dette, N. Neumeyer, and K.F. Pilz, A simple nonparametric estimator of a strictly monotone regression function, Bernoulli 12 (2006), pp. 469–490) proposed a new nonparametric estimate of a smooth monotone regression function. This method is based on a non-decreasing rearrangement of an arbitrary unconstrained nonparametric estimator. Under the assumption of a twice continuously differentiable regression function, the estimate is first-order asymptotic equivalent to the unconstrained estimate and other types of smooth monotone estimates. In this note, we provide a more refined asymptotic analysis of the monotone regression estimate. In the case where the regression function is increasing but only once continuously differentiable, we prove the asymptotic normality of an appropriately standardised version of the estimate, where the asymptotic variance is of order n −2/3−ϵ, the bias is of order n −1/3+ϵ and ϵ > 0 is small. Therefore, the rate of convergence of the new estimate is worse than the rate n −1/3 of the (unsmoothed) monotone least squares estimate. On the other hand, if the derivative of the regression function is Hölder continuous, the rate of convergence of the new estimate is faster than n −1/3.

DISCOUNT CURVE ESTIMATION BY MONOTONIZING MCCULLOCH SPLINES

In this paper a new method for monotone estimation of discount curves is proposed. The main idea of this approach is a simple modification of the commonly used (unconstrained) McCulloch Spline. We construct an integrated density estimate from the predicted values of the discount curve. It can be shown that this statistic is an estimate of the inverse of the discount function and the final estimate can be obtained by a numerical inversion. The resulting procedure is simple and we have implemented it in Excel and VBA, respectively. The performance is illustrated by several examples, in which the curve was previously estimated with an unconstrained McCulloch Spline.

Estimation of smooth regression functions in monotone response models

We consider the estimation of smooth regression functions in a class of conditionally parametric co-variate-response models. Independent and identically distributed observations are available from the distribution of ( Z , X ) , where Z is a real-valued co-variate with some unknown distribution, and the response X conditional on Z is distributed according to the density p ( · , ψ ( Z ) ) , where p ( · , θ ) is a one-parameter exponential family. The function ψ is a smooth monotone function. Under this formulation, the regression function E ( X | Z ) is monotone in the co-variate Z (and can be expressed as a one–one function of ψ ); hence the term “monotone response model”. Using a penalized least squares approach that incorporates both monotonicity and smoothness, we develop a scheme for producing smooth monotone estimates of the regression function and also the function ψ across this entire class of models. Point-wise asymptotic normality of this estimator is established, with the rate of convergence depending on the smoothing parameter. This enables construction of Wald-type (point-wise) as well as pivotal confidence sets for ψ and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed.

A second Marshall inequality in convex estimation

We prove a second Marshall inequality for adaptive convex density estimation via least squares. The result completes the first inequality proved recently by Dümbgen et al. [2007. Marshall's lemma for convex density estimation. IMS Lecture Notes—Monograph Series, submitted for publication. Preprint available at 〈 http://arxiv.org/abs/math.ST/0609277 〉 ], and is very similar to the original Marshall inequality in monotone estimation.

Consistent estimation of a convex density at the origin

Motivated by Hampel’s birds migration problem, Groeneboom, Jongbloed, and Wellner [7] established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density, g 0, at a fixed point t 0 > 0. However, estimation of the distribution function of the birds’ resting times involves estimation of g′0 at 0, a boundary point at which the estimators are not consistent.In this paper, we focus on the Least Squares estimator, \(\tilde g_n \). Our goal is to show that consistent estimators of both g 0(0) and g′0(0) can be based solely on \(\tilde g_n \). Following the idea of Kulikov and Lopuhaä [14] in monotone estimation, we show that it suffices to take \(\tilde g_n \)(n −α) and \(\tilde g'_n \)(n −α), with α ∈ (0, 1/3). We establish their joint asymptotic distributions and show that α = 1/5 should be taken as it yields the fastest rates of convergence.

On the estimation of a monotone conditional variance in nonparametric regression

A monotone estimate of the conditional variance function in a heteroscedastic, nonparametric regression model is proposed. The method is based on the application of a kernel density estimate to an unconstrained estimate of the variance function and yields an estimate of the inverse variance function. The final monotone estimate of the variance function is obtained by an inversion of this function. The method is applicable to a broad class of nonparametric estimates of the conditional variance and particularly attractive to users of conventional kernel methods, because it does not require constrained optimization techniques. The approach is also illustrated by means of a simulation study.

A note on uniform consistency of monotone function estimators

Recently, Dette et al. [A simple nonparametric estimator of a strictly increasing regression function. Bernoulli 12, 469–490] proposed a new monotone estimator for strictly increasing nonparametric regression functions and proved asymptotic normality. We explain two modifications of their method that can be used to obtain monotone versions of any nonparametric function estimators, for instance estimators of densities, variance functions or hazard rates. The method is appealing to practitioners because they can use their favorite method of function estimation (kernel smoothing, wavelets, orthogonal series, etc.) and obtain a monotone estimator that inherits desirable properties of the original estimator. In particular, we show that both monotone estimators share the same rates of uniform convergence (almost sure or in probability) as the original estimator.

Monotone Smoothing with Application to Dose-Response Curve

Motivated by problems that arise in dose-response curve estimation, we developed a new method to estimate a monotone curve. The resulting monotone estimator is obtained by combining techniques from smoothing splines with nonnegativity properties of cubic B-splines. Numerical experiments are given to exemplify the method.

A simple nonparametric estimator of a strictly monotone regression function

A new method for monotone estimation of a regression function is proposed, which is potentially attractive to users of conventional smoothing methods. The main idea of the new approach is to construct a density estimate from the estimated values m̂ (i/N) ( i =1,…,N ) of the regression function and to use these `data' for the calculation of an estimate of the inverse of the regression function. The final estimate is then obtained by a numerical inversion. Compared to the currently available techniques for monotone estimation the new method does not require constrained optimization. We prove asymptotic normality of the new estimate and compare the asymptotic properties with the unconstrained estimate. In particular, it is shown that for kernel estimates or local polynomials the bandwidths in the procedure can be chosen such that the monotone estimate is first-order asymptotically equivalent to the unconstrained estimate. We also illustrate the performance of the new procedure by means of a simulation study.

On the Distance Between Cumulative Sum Diagram and Its Greatest Convex Minorant for Unequally Spaced Design Points

Abstract. The supremum difference between the cumulative sum diagram, and its greatest convex minorant (GCM), in case of non‐parametric isotonic regression is considered. When the regression function is strictly increasing, and the design points are unequally spaced, but approximate a positive density in even a slow rate (n−1/3), then the difference is shown to shrink in a very rapid (close to n−2/3) rate. The result is analogous to the corresponding result in case of a monotone density estimation established by Kiefer and Wolfowitz, but uses entirely different representation. The limit distribution of the GCM as a process on the unit interval is obtained when the design variables are i.i.d. with a positive density. Finally, a pointwise asymptotic normality result is proved for the smooth monotone estimator, obtained by the convolution of a kernel with the classical monotone estimator.

A Note on Nonparametric Estimation of the Effective Dose in Quantal Bioassay

For the common binary response model, we propose a direct method for the nonparametric estimation of the effective dose level. The estimator is obtained by the composition of a nonparametric estimate of the quantile response curve and a classical density estimate. The new method yields a simple and reliable monotone estimate of the effective dose-level curve α → EDα and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series estimators, or smoothing splines. Moreover, it is computationally very efficient, because it does not require a numerical inversion of a monotonized estimate of the quantile dose-response curve. We prove asymptotic normality of the new estimate and compare it with an available alternative estimate (based on a monotonized nonparametric estimate of the dose-response curve and calculation of the inverse function) by means of a simulation study.

Inverse boosting for monotone regression functions

A new method for the estimation of smooth monotone regression functions is proposed. It is assumed that the monotonicity may come from some physical or economic reason. A monotone estimator of an integral (of a positive function) using a gradient boosting method is derived. The proposed method generates a sequence of fits without monotone constraints and combines them to form a monotone estimate. An advantage of the proposed algorithm is that one can use a popular smoothing technique without the monotone constraint as the base learner. The performance of the proposed procedure is demonstrated on both simulated and real data sets.

Compound estimation of a monotone sequence

In this paper we consider simultaneous estimation of a monotone sequence of parameters in the context of compound decision theory and obtain a class of monotone estimators with the strong asymptotic property that their compound risk converges uniformly to zero, for a large class of loss functions including the square error loss function and the absolute deviation loss function, as the number of parameters increases. We also show that when the number of parameters is fixed our estimators converge uniformly in probability to the parameters they are estimating, as the number of observations increases. As examples we consider estimation of forces of mortality with censored data in single and double decrement environments.

Tail Behavior and Breakdown Properties of Equivariant Estimators of Location

For translation and scale equivariant estimators of location, inequalities connecting tail behavior and the finite-sample breakdown point are proved, analogous to those established by He et al. (1990, Econometrika, 58, 1195–1214) for monotone and translation equivariant estimators. Some other inequalities are given as well, enabling to establish refined bounds and in some cases exact values for the tail behavior under heavy- and light-tailed distributions. The inequalities cover translation and scale equivariant estimators in great generality, and they involve new breakdown-related quantities, whose relations to the breakdown point are discussed. The worth of tail-behavior considerations in robustness theory is demonstrated on examples, showing the impact of the basic two techniques in robust estimation: trimming and averaging. The mathematical language employs notions from regular variation theory.

Pseudo-likelihood estimates of the cumulative risk of an autosomal dominant disease from a kin-cohort study.

Wacholder et al. [1998: Am J Epidemiol 148:623-629] and Struewing et al. [1997: N Engl J Med 336:1401-1408] have recently proposed a design called the kin-cohort design to estimate the probability of developing disease (penetrance) associated with an autosomal dominant gene. In this design, volunteers (probands) agree to be genotyped and one also determines the disease history (phenotype) of first-degree relatives of the proband. They used this design to estimate that the chance of developing breast cancer by age 70 in Ashkenazi Jewish women who carried mutations of the genes BRCA1 or BRCA2 was 0.56, a figure that was lower than previously estimated from highly affected families. The method that they used to estimate the cumulative risk of breast cancer, while asymptotically correct, does not necessarily produce monotone estimates in small samples. To obtain monotone, weakly parametric estimates, we consider separate piecewise exponential models for carriers and non-carriers. As the number of intervals on which constant hazards are assumed increases, however, the maximum likelihood score equations become unstable and difficult to solve. We, therefore, developed alternative pseudo-likelihood procedures that are readily solvable for piecewise exponential models with many intervals. We study these techniques through simulations and a re-analysis of a portion of the data used by Struewing et al. [1997] and discuss possible extensions.

Maximum likelihood estimation of smooth monotone and unimodal densities

We study the nonparametric estimation of univariate monotone and unimodal densities usingthe maximum smoothed likelihood approach. The monotone estimator is the derivative of the least concave majorant of the distribution correspondingto a kernel estimator.We prove that the mapping on distributions $\Phi$ with density $\varphi$, $$\varphi \mapsto \text{the derivative of the least concave majorant of $\Phi},$$ is a contraction in all $L^P$ norms $(1 \leq p \leq \infty)$, and some other “distances” such as the Hellinger and Kullback–Leibler distances. The contractivity implies error bounds for monotone density estimation. Almost the same error bounds hold for unimodal estimation.