Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function

Baraud, Yannick; Huet, Sylvie; Laurent, Béatrice

doi:10.1214/009053604000000896

Cited by 43 publications

(80 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Hall and Heckman (2000) (from now on, HH) developed a test based on the slopes of local linear estimates of f . The list of other papers includes Schlee (1982), Bowman, Jones, and Gijbels (1998), Dumbgen and Spokoiny (2001), Durot (2003), Beraud, Huet, and Laurent (2005), and Wang and Meyer (2011). Lee, Linton, and Whang (2009) and Delgado and Escanciano (2010) derived tests of stochastic monotonicity, which means that the conditional cdf of Y given X, F Y |X (y, x), is (weakly) decreasing in x for any fixed y.…”

Section: Introductionmentioning

confidence: 99%

Testing regression monotonicity in econometric models

Chetverikov

2012

View full text Add to dashboard Cite

Abstract. Monotonicity is a key qualitative prediction of a wide array of economic models derived via robust comparative statics. It is therefore important to design effective and practical econometric methods for testing this prediction in empirical analysis. This paper develops a general nonparametric framework for testing monotonicity of a regression function. Using this framework, a broad class of new tests is introduced, which gives an empirical researcher a lot of flexibility to incorporate ex ante information she might have. The paper also develops new methods for simulating critical values, which are based on the combination of a bootstrap procedure and new selection algorithms. These methods yield tests that have correct asymptotic size and are asymptotically nonconservative. It is also shown how to obtain an adaptive rate optimal test that has the best attainable rate of uniform consistency against models whose regression function has Lipschitz-continuous first-order derivatives and that automatically adapts to the unknown smoothness of the regression function. Simulations show that the power of the new tests in many cases significantly exceeds that of some prior tests, e.g. that of Ghosal, Sen, and Van der Vaart (2000). An application of the developed procedures to the dataset of Ellison and Ellison (2011) shows that there is some evidence of strategic entry deterrence in pharmaceutical industry where incumbents may use strategic investment to prevent generic entries when their patents expire.

show abstract

Section: Introductionmentioning

confidence: 99%

Testing regression monotonicity in econometric models

Chetverikov

2012

View full text Add to dashboard Cite

show abstract

“…Various types of hypothesis tests are proposed with different test statistics. However, most work in the literature has focused on observing the behavior of the test statistic as n → ∞ with a fixed value of r (Yatchew, 1992;Hall & Jeckman, 2000;Baraud et al, 2005) or imposed a condition that requires the normality of the ϵ i j 's (Bartholomew, 1959;Shapiro, 1988;Baraud et al, 2005). For example, the MSE has been studied as a test statistics in Shapiro (1988), but the behavior of the test statistic is studied only for the case where n → ∞ with r fixed.…”

Section: Test Of Positivitymentioning

confidence: 99%

Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

Lim¹,

Tavarez²

2017

IJSP

View full text Add to dashboard Cite

We propose a new method of testing for a function's convexity, monotonicity, or positivity, based on some noisy observations of the function made over a finite set T of points in the domain, where the observations can be made multiple times at each point in T . One of the traditional approaches to the test of a function's shape characteristic is to fit a convex, a monotone, or a positive function, depending on the shape characteristic we wish to test for, to the data set minimizing the sum of squared errors, and to compute the sum of squared differences (SSD) between the fit and the data set. While the traditional approach proceeds by observing the SSD as the number of points in T increases to infinity, we propose observing the SSD as r, the number of observations taken at each point in T , increases to infinity. This new way of observing the asymptotic behavior of the SSD leads to a test procedure that does not require the estimation of any additional parameters, and hence, is easy to implement. The proposed test procedure is proven to achieve a prescribed power as r → ∞. Numerical examples illustrate that the proposed test successfully detects the convexity/monotonicity/positivity of a function, as well as the non-convexity/non-monotonicity/non-positivity of a function.

show abstract

“…Thus, if the scale parameter β doesn't depend on X, again the case [2] is satisfied. If instead β is some convex or concave function of X, we are in cases [3] or [4] depending on the sign of γ − ln(ln(1/α)).…”

Section: Corollary 2 Letmentioning

confidence: 99%

“…For example, the case of Y having a chi-squared distribution with k = k(X) degrees of freedom is coherent with case [2], since the mode, for k ≥ 2, is k−2 = E (Y | X)−2. More generally, if the distribution of Y conditional to X = x is Gamma(k(x), θ(x)), then the difference between the conditional expectation and the mode is just θ(x), so that cases [2], [3] or [4] are encountered for θ being respectively at most linear, convex or concave in x.…”

Section: Corollary 2 Letmentioning

confidence: 99%

A Refined Jensen's Inequality in Hilbert Spaces and Empirical Approximations

Leorato

2008

SSRN Journal

View full text Add to dashboard Cite

Let f : X → R be a convex mapping and X a Hilbert space. In this paper we prove the following refinement of Jensen's inequality:for every A, B such that E(X |X ∈ A ) = E(X |X ∈ B ) and B ⊂ A. Expectations of Hilbert space valued random elements are defined by means of the Pettis integrals. Our result generalizes a result of Karlin and Novikov (1963), who derived it for X = R. The inverse implication is also true if P is an absolutely continuous probability measure. A convexity criterion based on the Jensen-type inequalities follows and we study its asymptotic accuracy when the empirical distribution function based on a n−dimensional sample approximates the unknown distribution function. Some statistical applications are addressed, such as nonparametric estimation and testing for convex regression functions or other functionals.

show abstract

Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function

Cited by 43 publications

References 12 publications

Testing regression monotonicity in econometric models

Testing regression monotonicity in econometric models

Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

A Refined Jensen's Inequality in Hilbert Spaces and Empirical Approximations

Contact Info

Product

Resources

About