Consistency in Concave Regression

Hanson, D. L.; Pledger, Gordon

doi:10.1214/aos/1176343640

Cited by 143 publications

(78 citation statements)

References 3 publications

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“…The estimation of concave regression functions (same context as above except that m(·) is known to be concave) has also been extensively considered (see e.g., Hildreth, 1954 andPledger, 1976) and its distribution is known in the least squares case (see Wang, 1993). Finally, algorithms that extend Hildreth's to estimate a regression curve under inequality restrictions have been proposed by Dykstra (1983) and Ruud (1997), again in the constrained least squares context.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric option pricing under shape restrictions

Aı̈t-Sahalia

Duarte

2003

Journal of Econometrics

289

171

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Frequently, economic theory places shape restrictions on functional relationships between economic variables. This paper develops a method to constrain the values of the ÿrst and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P 500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose.

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Section: Introductionmentioning

confidence: 99%

Nonparametric option pricing under shape restrictions

Aı̈t-Sahalia

Duarte

2003

Journal of Econometrics

289

171

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“…Indeed, it is easy to verify that the univariate CNLS formulation (5) by Hanson and Pledger (1976) is obtained as a special case of (6) when m = 1. We would conjecture that the known statistical properties of the univariate CNLS estimator (consistency, rate of convergence) carry over to the multivariate setting, but this remains to be formally shown.…”

Section: Stage 1: Cnls Estimationmentioning

confidence: 96%

“…In this respect, the recent studies Kuosmanen (2008) and Kuosmanen and Johnson (2010) have shown that DEA can be understood as a constrained special case of nonparametric least squares subject to shape constraints. More specifically, Kuosmanen and Johnson (2010) prove formally that the classic outputoriented DEA estimator can be computed in the singleoutput case by solving the convex nonparametric least squares (CNLS) problem (Hildreth 1954;Hanson and Pledger 1976;Groeneboom et al 2001a,b;Kuosmanen 2008) subject to monotonicity and concavity constraints that characterize the frontier, and a sign constraint on the regression residuals. Thus, DEA can be naturally viewed as a nonparametric counterpart to the parametric programming approach of Aigner and Chu (1968).…”

Section: Introductionmentioning

confidence: 99%

Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints

2010

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The field of productive efficiency analysis is currently divided between two main paradigms: the deterministic, nonparametric Data Envelopment Analysis (DEA) and the parametric Stochastic Frontier Analysis (SFA). This paper examines an encompassing semiparametric frontier model that combines the DEA-type nonparametric frontier, which satisfies monotonicity and concavity, with the SFA-style stochastic homoskedastic composite error term. To estimate this model, a new twostage method is proposed, referred to as Stochastic Nonsmooth Envelopment of Data (StoNED). The first stage of the StoNED method applies convex nonparametric least squares (CNLS) to estimate the shape of the frontier without any assumptions about its functional form or smoothness. In the second stage, the conditional expectations of inefficiency are estimated based on the CNLS residuals, using the method of moments or pseudolikelihood techniques. Although in a cross-sectional setting distinguishing inefficiency from noise in general requires distributional assumptions, we also show how these can be relaxed in our approach if panel data are available. Performance of the StoNED method is examined using Monte Carlo simulations.

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“…For more details about theoretical aspects, see Dümbgen et al (2004), where it is clarified how to apply also an isotonic constraints onf (see also Hanson and Pledger (1976); Aguilera et al (2011)). …”

Section: Endmentioning

confidence: 99%

Estimation procedures for exchangeable Marshall copulas with hydrological application

Durante

Okhrin

2014

Stoch Environ Res Risk Assess

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Complex phenomena in environmental sciences can be conveniently represented by several inter-dependent random variables. In order to describe such situations, copula-based models have been studied during the last year. In this paper, we consider a novel family of bivariate copulas, called exchangeable Marshall copulas. Such copulas describe both positive and (upper) tail association between random variables. Specifically, inference procedures for the family of exchangeable Marshall copulas are introduced, based on the estimation of their (univariate) generator. Moreover, the performance of the proposed methodologies is shown in a simulation study. Finally, an illustration describes how the proposed procedures can be useful in a hydrological application.

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Consistency in Concave Regression

Cited by 143 publications

References 3 publications

Nonparametric option pricing under shape restrictions

Nonparametric option pricing under shape restrictions

Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints

Estimation procedures for exchangeable Marshall copulas with hydrological application

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