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33
Nonparametric Kernel Regression Subject To Monotonicity Constraints
 Annals of Statistics
, 1999
"... . We suggest a biasedbootstrap method for monotonising general linear, kerneltype estimators, for example local linear estimators and NadarayaWatson estimators. Attributes of our approach include the fact that it produces smooth estimates, that is applicable to a particularly wide range of estimat ..."
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Cited by 41 (2 self)
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. We suggest a biasedbootstrap method for monotonising general linear, kerneltype estimators, for example local linear estimators and NadarayaWatson estimators. Attributes of our approach include the fact that it produces smooth estimates, that is applicable to a particularly wide range of estimator types, and that it can be employed after the smoothing step has been implemented. Therefore, an experimenter may use his or her favourite kernel estimator, and their favourite bandwidth selector, to construct the basic nonparametric smoother, and then use our technique to render it monotone in a smooth way. Since our method is based on maximising fidelity to the conventional empirical approach, subject to monotonicity, then if the original kernel smoother were monotone we would not modify it. More generally, we would adjust it by adjoining weights to data values so as to make least possible change, in the sense of a distance measure, subject to imposing the constraint of monotonicity. KEY...
Testing Monotonicity Of Regression
, 1998
"... this article, we study this problem and construct asymptotically valid tests. Our test statistics are suitable functionals of a stochastic process which may be viewed as a local version of Kendall's tau statistic and have simple natural interpretations. The process involved is a degreetwo Upro ..."
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Cited by 11 (0 self)
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this article, we study this problem and construct asymptotically valid tests. Our test statistics are suitable functionals of a stochastic process which may be viewed as a local version of Kendall's tau statistic and have simple natural interpretations. The process involved is a degreetwo Uprocess, as in Nolan and Pollard (1987). The asymptotic behaviour of the test statistics are studied in three major steps: Approximation of the Uprocess by the empirical process defined by the H'ajek projection, strong approximation of the empirical process by a Gaussian process and finally the extreme value theory for stationary Gaussian processes. The paper is organized as follows. In Section 2, we introduce two different types of test statistics. We also formally describe the model and the hypothesis and explain the notation and regularity conditions in this section. In Section 3, we investigate the asymptotic behaviour of the Uprocess and establish the Gaussian process approximation. Section 4 is devoted to the study of the limiting distribution of the first test statistics using the extreme value theory for stationary Gaussian processes and the results of Section 3. In Section 5, we show that this test is consistent against all alternatives and also determine the minimal rate so that alternatives further apart than this rate can be effectively tested. The second test statistic is studied in Section 6. Technical proofs are presented in Section 7 and the appendix. 2. The Test Statistics
Nonparametric state price density estimation using constrained least squares and the bootstrap, Journal of Econometrics, in print. 36 649 Discussion Paper Series For a complete list of Discussion Papers published by the SFB 649, please visit http://sfb649
, 2005
"... The economic theory of option pricing imposes constraints on the structure of call functions and state price densities. Except in a few polar cases, it does not prescribe functional forms. This paper proposes a nonparametric estimator of option pricing models which incorporates various restrictions ..."
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Cited by 10 (2 self)
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The economic theory of option pricing imposes constraints on the structure of call functions and state price densities. Except in a few polar cases, it does not prescribe functional forms. This paper proposes a nonparametric estimator of option pricing models which incorporates various restrictions within a single least squares procedure thus permitting investigation of a wide variety of model specifications and constraints. Among these we consider monotonicity and convexity of the call function and integration to one of the state price density. The procedure easily accommodates heteroskedasticity of the residuals. The bootstrap is used to produce confidence intervals for the call function and its first two derivatives. We apply the techniques to option pricing data on the DAX. Keywords: option pricing, state price density estimation, nonparametric least squares, bootstrap inference, monotonicity, convexity
Comparing the Shapes of Regression Functions
, 2000
"... Introduction The shape of a function is often of key interest in regression analysis. For instance, local maxima, or `bumps', are scientifically important features in the shape of a re1 gression function. In evolutionary biology, an individual's chance of survival typically depends on the ..."
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Cited by 10 (1 self)
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Introduction The shape of a function is often of key interest in regression analysis. For instance, local maxima, or `bumps', are scientifically important features in the shape of a re1 gression function. In evolutionary biology, an individual's chance of survival typically depends on the value of a physical trait. If the chance of survival attains a global maximum, then the trait will evolve toward the optimal value. If there are two local maxima, then the trait will evolve differently, with two commonly occurring values emerging. Thus the shape of the curve determines the type of evolution. The presence of bumps can raise scientific questions. Growth curves of U.S. children clearly show two local maxima in the rate of growth. In Swiss children, however, these growth spurts seem to be either missing or less prominent; see Ramsay, Bock & Gasser (1995). Are the Swiss and U.S. growth rates similar in shape? If not, how and why do their shapes differ? Can we group countri
Testing for Monotone Increasing Hazard Rate
 Annals of Statistics
, 2005
"... A test of the null hypothesis that a hazard rate is monotone nondecreasing, versus the alternative that it is not, is proposed. Both the test statistic and the means of calibrating it are new. Unlike previous approaches, neither is based on the assumption that the null distribution is exponential. I ..."
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Cited by 5 (0 self)
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A test of the null hypothesis that a hazard rate is monotone nondecreasing, versus the alternative that it is not, is proposed. Both the test statistic and the means of calibrating it are new. Unlike previous approaches, neither is based on the assumption that the null distribution is exponential. Instead, empirical information is used to effectively identify and eliminate from further consideration parts of the line where the hazard rate is clearly increasing; and to confine subsequent attention only to those parts that remain. This produces a test with greater apparent power, without the excessive conservatism of exponentialbased tests. Our approach to calibration borrows from ideas used in certain tests for unimodality of a density, in that a bandwidth is increased until a distribution with the desired properties is obtained. However, the test statistic does not involve any smoothing, and is, in fact, based directly on an assessment of convexity of the distribution function, using the conventional empirical distribution. The test is shown to have optimal power properties in difficult cases, where it is called upon to detect a small departure, in the form of a bump, from monotonicity. More general theoretical properties of the test and its numerical performance are explored. 1. Introduction. Estimation
Testing For Monotonicity Of A Regression Mean Without Selecting A Bandwidth
, 1998
"... . A new approach to testing for monotonicity of a regression mean, not requiring computation of a curve estimator or a bandwidth, is suggested. It is based on the notion of `running gradients' over short intervals, although from some viewpoints it may be regarded as an analogue for monotonicity ..."
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Cited by 4 (3 self)
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. A new approach to testing for monotonicity of a regression mean, not requiring computation of a curve estimator or a bandwidth, is suggested. It is based on the notion of `running gradients' over short intervals, although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip/excess mass approach for testing modality hypotheses about densities. Like the latter methods, the new technique does not suffer difficulties caused by almostflat parts of the target function. In fact, it is calibrated so as to work well for flat response curves, and as a result it has relatively good power properties in boundary cases where the curve exhibits shoulders. In this respect, as well as in its construction, the `running gradients' approach differs from alternative techniques based on the notion of a critical bandwidth. KEYWORDS. Bootstrap, calibration, curve estimation, Monte Carlo, response curve, running gradient. SHORT TITLE. Testing for monotonicity. 1 The man...
Testing of Monotonicity in Regression Models
 Mimeograph Series, Operations Research, Statistics
, 1990
"... In data anaysis concerning the investigation of the relationship between a dependent variable Y and an independent variable X, one may wish to determine whether this relationship is monotone or not. This determination may be of interest in itself, or it may form part of a (nonparametric) regression ..."
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Cited by 2 (0 self)
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In data anaysis concerning the investigation of the relationship between a dependent variable Y and an independent variable X, one may wish to determine whether this relationship is monotone or not. This determination may be of interest in itself, or it may form part of a (nonparametric) regression analysis which relies on monotonicity of the true regression function. In this paper we generalize the test of positive correlation by proposing a test statistic for monotonicity based on fitting a parametric model, say a higher order polynomial, to the data with and without the monotonicity constraint. The statistic has an asymptotic chibarsquared distribution under the null hypothesis that the true regression function is on the boundary of the space of monotone functions. Based on the theoretical results, an algorithm is developed for testing the significance of the statistic, and it is shown to perform well in several null and nonnull settings. Extensions to fitting regression splines ...
A heuristic approach for discovering reference models by mining process model variants
, 2009
"... Abstract. Recently, a new generation of adaptive ProcessAware Information Systems (PAISs) has emerged, which enables structural process changes during runtime while preserving PAIS robustness and consistency. Such flexibility, in turn, leads to a large number of process variants derived from the ..."
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Cited by 2 (1 self)
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Abstract. Recently, a new generation of adaptive ProcessAware Information Systems (PAISs) has emerged, which enables structural process changes during runtime while preserving PAIS robustness and consistency. Such flexibility, in turn, leads to a large number of process variants derived from the same model, but differing in structure. Generally, such variants are expensive to configure and maintain. This paper provides a heuristic search algorithm which fosters learning from past process changes by mining process variants. The algorithm discovers a reference model based on which the need for future process configuration and adaptation can be reduced. It additionally provides the flexibility to control the process evolution procedure, i.e., we can control to what degree the discovered reference model differs from the original one. As benefit, we can not only control the effort for updating the reference model, but also gain the flexibility to perform only the most important adaptations of the current reference model. Our mining algorithm is implemented and evaluated by a simulation using more than 7000 process models. Simulation results indicate strong performance and scalability of our algorithm even when facing largesized process models. 1
A Test for Multimodality of Regression Derivatives with Application to Nonparametric Growth Regressions
 Journal of Applied Econometrics
"... This paper presents a method to test for multimodality of an estimated kernel density of parameter estimates from a locallinear leastsquares regression derivative. The procedure is laid out in seven simple steps and a suggestion for implementation is proposed. A Monte Carlo exercise is used to ex ..."
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Cited by 2 (2 self)
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This paper presents a method to test for multimodality of an estimated kernel density of parameter estimates from a locallinear leastsquares regression derivative. The procedure is laid out in seven simple steps and a suggestion for implementation is proposed. A Monte Carlo exercise is used to examine the finite sample properties of the test along with those from a calibrated version of it which corrects for the conservative nature of Silvermantype tests. The test is included in a study on nonparametric growth regressions. The results show that in the estimation of unconditional βconvergence, the distribution of the parameter estimates is multimodal with one mode in the negative region (primarily OECD economies) and possibly two modes in the positive region (primarily nonOECD economies) of the parameter estimates. The results for conditional βconvergence show that the density is predominantly negative and unimodal. Finally, the application attempts to determine why particular observations posess positive marginal effects on initial income in both the unconditional and conditional frameworks.