Via generalized interval arithmetic, we propose a Generalized Interval Arithmetic Center and Range (GIA-CR) model for random intervals, where parameters in the model satisfy linear inequality constraints. We construct a constrained estimator of the parameter vector and develop asymptotically uniformly valid tests for linear equality constraints on the parameters in the model. We conduct a simulation study to examine the finite sample performance of our estimator and tests. Furthermore, we propose a coefficient of determination for the GIA-CR model. As a separate contribution, we establish the asymptotic distribution of the constrained estimator in Blanco-Fernández (2015, Multiple Set Arithmetic-Based Linear Regression Models for Interval-Valued Variables) in which the parameters satisfy an increasing number of random inequality constraints.

This paper develops a wavelet (spectral) approach to testing the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations.

In this paper, we propose nonparametric estimators ofsharp bounds on the distribution of treatmenteffects of a binary treatment and establish theirasymptotic distributions. We note the possiblefailure of the standard bootstrap with the samesample size and apply thefewer-than-n bootstrap to makinginferences on these bounds. The finite sampleperformances of the confidence intervals for thebounds based on normal critical values, the standardbootstrap, and the fewer-than-nbootstrap are investigated via a simulation study.Finally we establish sharp bounds on the treatmenteffect distribution when covariates areavailable.

Let F denote a distribution functiondefined on the probability space (Ω,,P), which is absolutelycontinuous with respect to the Lebesgue measure inRd with probabilitydensity function f. Letf0(·,β) be aparametric density function that depends on anunknown p × 1 vector β. In thispaper, we consider tests of the goodness-of-fit off0(·,β) forf(·) for some β based on (i) theintegrated squared difference between a kernelestimate of f(·) and thequasimaximum likelihood estimate off0(·,β) denoted byIn and (ii) theintegrated squared difference between a kernelestimate of f(·) and thecorresponding kernel smoothed estimate off0(·, β) denoted byJn. It is shown inthis paper that the amount of smoothing applied tothe data in constructing the kernel estimate off(·) determines the form of thetest statistic based onIn. For each testdeveloped, we also examine its asymptotic propertiesincluding consistency and the local power property.In particular, we show that tests developed in thispaper, except the first one, are more powerful thanthe Kolmogorov-Smirnov test under the sequence oflocal alternatives introduced in Rosenblatt [12],although they are less powerful than theKolmogorov-Smirnov test under the sequence of Pitmanalternatives. A small simulation study is carriedout to examine the finite sample performance of oneof these tests.

In this paper, we address two important issues in semiparametric survival model selection for censored data generated by the Archimedean copula family: method of estimating the parametric copulas and data reuse. We demonstrate that for selection among candidate copula models that might all be misspecified, estimators of the parametric copulas based on minimizing the selection criterion function may be preferred to other estimators. To handle the issue of data reuse, we put model selection in the context of hypothesis testing and propose a simple test for model selection from a finite number of parametric copulas. Results from a simulation study and two empirical illustrations confirm our theoretical findings.We thank the editor Peter Phillips, three anonymous referees, and Hal White for their comments, which greatly improved the paper. An earlier version of this paper was presented at the 2005 World Congress Meetings of the Econometric Society, the 2005 Joint Statistical Meetings, the University of Waterloo, and the University of Western Ontario. Chen acknowledges support from the National Science Foundation and the C.V. Starr Center at NYU. Fan acknowledges support from the National Science Foundation. We thank Demian Pouzo for excellent research assistance on the numerical work in this paper and Weijing Wang for providing us with the Fortran code for computing the bivariate Kaplan–Meier estimator of Dabrowska (1988).

This paper proposes a bootstrap test for the correct specification of parametric conditional distributions. It extends Zheng's test (Zheng, 2000, Econometric Theory 16, 667–691) to allow for discrete dependent variables and for mixed discrete and continuous conditional variables. We establish the asymptotic null distribution of the test statistic with data-driven stochastic smoothing parameters. By smoothing both the discrete and continuous variables via the method of cross-validation, our test has the advantage of automatically removing irrelevant variables from the estimate of the conditional density function and, as a consequence, enjoys substantial power gains in finite samples, as confirmed by our simulation results. The simulation results also reveal that the bootstrap test successfully overcomes the size distortion problem associated with Zheng's test.We are grateful for the insightful comments from three referees and a co-editor that greatly improved the paper. Li's research is partially supported by the Private Enterprise Research Center, Texas A&M University. Fan is grateful to the National Science Foundation for research support.

We point out the close relationship between the integrated conditional moment tests in Bierens (1982, Journal of Econometrics 20, 105–134) and Bierens and Ploberger (1997, Econometrica 65, 1129–1152) with the complex-valued exponential weight function and the kernel-based tests in Härdle and Mammen (1993, Annals of Statistics 21, 1926–1947), Li and Wang (1998, Journal of Econometrics 87, 145–165), and Zheng (1996, Journal of Econometrics 75, 263–289). It is well established that the integrated conditional moment tests of Bierens (1982) and Bierens and Ploberger (1997) are more powerful than kernel-based nonparametric tests against Pitman local alternatives. In this paper we analyze the power properties of the kernel-based tests and the integrated conditional moment tests for a sequence of “singular” local alternatives, and show that the kernel-based tests can be more powerful than the integrated conditional moment tests for these “singular” local alternatives. These results suggest that integrated conditional moment tests and kernel-based tests should be viewed as complements to each other. Results from a simulation study are in agreement with the theoretical results.

In this paper, we study the bias-corrected test developed in Fan(1994). It is based on the integrated squared difference between akernel estimator of the unknown density function of a random vectorand a kernel smoothed estimator of the parametric density functionto be tested under the null hypothesis. We provide an alternativeasymptotic approximation of the finite-sample distribution of thistest by fixing the smoothing parameter. In contrast to the normalapproximation obtained in Fan (1994) in which the smoothingparameter shrinks to zero as the sample size grows to infinity, weobtain a non-normal asymptotic distribution for the bias-correctedtest. A parametric bootstrap procedure is proposed to approximatethe critical values of this test. We show both analytically and bysimulation that the proposed bootstrap procedure works. Consistencyand local power properties of the bias-corrected test with a fixedsmoothing parameter are also discussed.