An omnibus two-sample test for ranked-set sampling data

We develop an omnibus two-sample test for ranked-set sampling (RSS) data. The test statistic is the conditional probability of seeing the observed sequence of ranks in the combined sample, given the observed sequences within the separate samples. We compare the test to existing tests under perfect rankings, finding that it can outperform existing tests in terms of power, particularly when the set size is large. The test does not maintain its level under imperfect rankings. However, one can create a permutation version of the test that is comparable in power to the basic test under perfect rankings and also maintains its level under imperfect rankings. Both tests extend naturally to judgment post-stratification, unbalanced RSS, and even RSS with multiple set sizes. Interestingly, the tests have no simple random sampling analog.     

The extended two-sample problem: nonparametric case

The classical two-sample problem is extended here to the case where the distribution functions of the observable random variables are specified functions of unknown distribution functions and the null hypotheses to be tested or the parameters to be estimated relate to these unknown distributions. Various properties of the proposed rank tests and derived estimates are studied.     

Wilcoxon–Mann–Whitney test for stratified samples and Efron's paradox dice

Two-treatment multi-center clinical trials are the most common type of clinical trials in practice. The aim of this paper is to discuss a curious property of certain standard nonparametric procedures used in the analysis of such clinical trials. Different analyses of a simulated data example are presented, which lead to contrasting and surprising results. The source of the potentially misleading outcome is then explored while relating the simulated data with the concept of Efron's paradox dice and the notion of nontransitivity. With the root of the problem established, an alternate nonparametric method from the literature is shown to address the problem. Finally, pointing out an interpretational concern of using the alternate procedure, a modification to this procedure is also suggested and corresponding theoretical results are presented.     

A class of count models and a new consistent test for the Poisson distribution

We define a class of count distributions which includes the Poisson as well as many alternative count models. Then the empirical probability generating function is utilized to construct a test for the Poisson distribution, which is consistent against this class of alternatives. The limit distribution of the test statistic is derived in case of a general underlying distribution, and efficiency considerations are addressed. A simulation study indicates that the new test is comparable in performance to more complicated omnibus tests.     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

A nonparametric inverse probability weighted estimation for functional data with missing response data at random

This paper considers the nonparametric inverse probability weighted estimation for functional data with missing response data at random. Under mild conditions, the asymptotic properties of the proposed estimation method are established. Based on the resampling method, the estimation of the asymptotic variance of the proposed estimator is obtained. Finally, the finite sample properties of the proposed estimation method are investigated via Monte Carlo simulation studies. A real data analysis is given to illustrate the use of the proposed method.     

Asymptotic normality and mean consistency for the weighted estimator in nonparametric regression models

In this paper, we mainly study the asymptotic properties of weighted estimator for the nonparametric regression model based on linearly negative quadrant dependent (LNQD, for short) errors. We obtain the rate of uniformly asymptotic normality of the weighted estimator which is nearly $O\left(n^{?1∕4}\right)$ when the moment condition is appropriate. The results generalize the corresponding ones of Yang (2003) from NA samples to LNQD samples and improve or extend the corresponding one of Li et al. (2012) for LNQD samples. Moreover, we obtain some results on mean consistency, uniformly mean consistency, and the rate of mean consistency for the weighted estimator. Finally we carry out some simulations to verify the validity of our results.     

Asymptotic properties of the MAMSE adaptive likelihood weights

The weighted likelihood is a generalization of the likelihood designed to borrow strength from similar populations while making minimal assumptions. If the weights are properly chosen, the maximum weighted likelihood estimate may perform better than the maximum likelihood estimate (MLE). In a previous article, the minimum averaged mean squared error (MAMSE) weights are proposed and simulations show that they allow to outperform the MLE in many cases. In this paper, we study the asymptotic properties of the MAMSE weights. In particular, we prove that the MAMSE-weighted mixture of empirical distribution functions converges uniformly to the target distribution and that the maximum weighted likelihood estimate is strongly consistent. A short simulation illustrates the use of bootstrap in this context.     

Near-exact distributions for the sphericity likelihood ratio test statistic

In this paper three near-exact distributions are developed for the sphericity test statistic. The exact probability density function of this statistic is usually represented through the use of the Meijer G function, which renders the computation of quantiles impossible even for a moderately large number of variables. The main purpose of this paper is to obtain near-exact distributions that lie closer to the exact distribution than the asymptotic distributions while, at the same time, correspond to density and cumulative distribution functions practical to use, allowing for an easy determination of quantiles. In addition to this, two asymptotic distributions that lie closer to the exact distribution than the existing ones were developed. Two measures are considered to evaluate the proximity between the exact and the asymptotic and near-exact distributions developed. As a reference we use the saddlepoint approximations developed by Butler et al. [1993. Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc., Ser. B 55, 171–183] as well as the asymptotic distribution proposed by Box.     

Goodness-of-fit tests for a heavy tailed distribution

We study the Kolmogorov–Smirnov test, Berk–Jones test, score test and their integrated versions in the context of testing the goodness-of-fit of a heavy tailed distribution function. A comparison of these tests is conducted via Bahadur efficiency and simulations.     

The consistency of a matching test

Let $\text{X}{}^1{}_1\text{?X}{}^1{}_2\text{???X}{}^1{}_{\mathrm{n}}$ be the order statistics of a random sample from a distribution on [0, 1]. Let A_k, the kth match, be the event that $\text{X}{}^1{}_{\mathrm{k}}\text{?(}\text{(k?1)}\text{n}\text{k}\text{n}$], and let S_n be the total number of matches. The consistency of S_n for testing uniform df, U, against df G≠U is investigated, and it is shown that S_n is consistent if the intersection of G with U has Lebesgue measure zero. It is also consistent against a sequence of alternatives approaching U at a rate less faster than $\text{n}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}$.     

Bias bound for the minimax estimator

The bias bound function of an estimator is an important quantity in order to perform globally robust inference. We show how to evaluate the exact bias bound for the minimax estimator of the location parameter for a wide class of unimodal symmetric location and scale family. We show, by an example, how to obtain an upper bound of the bias bound for a unimodal asymmetric location and scale family. We provide the exact bias bound of the minimum distance/disparity estimators under a contamination neighborhood generated from the same distance.     

Nonparametric estimation of conditional expectation

Denote the integer lattice points in the N  -dimensional Euclidean space by Z^N${\mathbb{Z}}^N$ and assume that (_Xi,_Yi)$(X_{\mathbf{i}},Y_{\mathbf{i}})$, i∈Z^N$\mathbf{i}\in {\mathbb{Z}}^N$ is a mixing random field. Estimators of the conditional expectation r(x)=E[_Yi|_Xi=x]$r(\mathbf{x})=\mathrm{E}[Y_{\mathbf{i}}|X_{\mathbf{i}}=\mathbf{x}]$ by nearest neighbor methods are established and investigated. The main analytical result of this study is that, under general mixing assumptions, the estimators considered are asymptotically normal. Many difficulties arise since points in higher dimensional space N?2$N?2$ cannot be linearly ordered. Our result applies to many situations where parametric methods cannot be adopted with confidence.     

Testing for changes in the covariance structure of linear processes

We consider several procedures to detect changes in the mean or the covariance structure of a linear process. The tests are based on the weighted CUSUM process. The limit distributions of the test statistics are derived under the no change null hypothesis. We develop new strong and weak approximations for the sample mean as well as the sample correlations of linear processes. A small Monte Carlo simulation illustrates the applicability of our results.     

Multi-arm response-adaptive designs for circular responses

A multi-arm response-adaptive allocation design is developed for circular treatment outcomes. Several exact and asymptotic properties of the design are studied. Stage-wise treatment selection procedures based on the proposed response-adaptive design are also suggested to exclude the worse performing treatment(s) at earlier stages. Detailed simulation study is carried out to evaluate the proposed selection procedures. The applicability of the proposed methodologies is illustrated through a real clinical trial data on cataract surgery.     

Distribution-free tests for parallelism and concurrence in two-sample regression problem

This paper suggests some distribution-free methods for testing hypothesis of parallelism and concurrence of two linear regressions. We assume that the independent variable x is equally spaced. The proposed procedures are compared with nonparametric competitors and the normal theory t-test.     

Laws of iterated logarithm for quasi-maximum likelihood estimator in generalized linear model

For the case that the expectation of the response variable Y   is correctly specified in the generalized linear model (GLM), under some regular assumptions, we obtain and prove the law of the iterated logarithm and Chung type law of the iterated logarithm for the quasi-maximum likelihood estimator (QMLE) _βn${\beta }_n$ in this model.     

The convolution theorem for estimating linear functionals in indirect nonparametric regression models

Nonparametric regression—directly or indirectly observed—is one of the important statistical models. On one hand it contains two infinite dimensional parameters (the regression function and the error density), and on the other it is of rather simple structure. Therefore, it may serve as an interesting paradigm for illustrating or developing abstract statistical theory for non-Euclidean parameters. In this paper estimation of a linear functional of the indirectly observed regression function is considered, when a deterministic design is used. It should be noted that any Fourier coefficient of an expansion of the regression function in an orthonormal basis is such a functional. Because the design is deterministic the observables are independent but not identically distributed. Local asymptotic normality is established and applied to prove Hájek's convolution theorem for this functional. Pertinent references are Beran [1977. Robust location estimates. Ann. Statist. 5, 431–444] and McNeney and Wellner [2000. Application of convolution theorems in semiparametric models with non-i.i.d. data. J. Statist. Plann. Inference 91, 441–480]. For purposes explained above, however, the paper is kept self-contained and full proofs are provided.     

Variance function estimation of a one-dimensional nonstationary process

We propose a flexible nonparametric estimation of a variance function from a one-dimensional process where the process errors are nonstationary and correlated. Due to nonstationarity a local variogram is defined, and its asymptotic properties are derived. We include a bandwidth selection method for smoothing taking into account the correlations in the errors. We compare the proposed difference-based nonparametric approach with Anderes and Stein(2011)’s local-likelihood approach. Our method has a smaller integrated MSE, easily fixes the boundary bias, and requires far less computing time than the likelihood-based method.