Goodness-of-Fit Test for Monotone Functions

Abstract

ABSTRACT. In this article, we develop a test for the null hypothesis that a real-valued function belongs to a given parametric set against the non-parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right-censoring model with monotone hazard rate. The criterion for testing is an L L L L p -distance between a Grenander-type non-parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.

Limit Theory in Monotone Function Estimation

“…Another application is connected to the problem of testing monotonicity of a function of interest [1,24,27,49,12]. Other tests can be built based on the limiting global behavior of isotonic estimators, to test goodness of fit [32,31], or equality of several monotone functions [28], whereas some other tests are more connected to the local limit behavior of isotonic estimators, such as likelihood ratio tests [9,10,82]. Another application of local limiting distribution of isotonic estimator is construction of confidence intervals for monotone functions [48].…”

A Density-Based Empirical Likelihood Ratio Approach for Goodness-of-fit Tests in Decreasing Densities

“…Another popular approach for the underlying problem is to reject the null hypothesis if an appropriate non-parametric estimator is far enough from the parametric estimator computed under the null hypothesis. For more information about these two approaches and the related references, Durot and Reboul 67 [7] provided a comprehensive review. Durot and Tocquet [8] carried out an study on a goodness-of-fit test for a decreasing regression model: the test suggests rejecting the null hypothesis that the monotone regression model belongs to a parametric family against the alternative if the L 1 -distance between the null hypothesis and the Brunk estimator is large enough.…”

“…Ducharme and Fontez [6] adapted the smooth method of Neyman [17] to test a regression with a positive and increasing mean. Durot and Reboul [7] developed a test for the null hypothesis indicating a real-valued function of interest is a member of a parametric set against the non-parametric alternative within the monotone constraint, say decreasing. They introduced a general model for the test which covers the monotone density, regression and hazard rate with right-censoring models.…”

The nonparametric bootstrap for the current status model

“…Although it is argued in [8] that the naive bootstrap will not work for their goodness-of-fit test for monotone functions, based on the Grenander estimator, no theoretical justification for this conjecture is given. Other examples where a smooth bootstrap procedure is used, are the likelihood ratio type two-sample test for current status data proposed by [11] and the test for equality of functions under monotonicity constraints proposed by [7].…”