EMPIRICAL LIKELIHOOD‐BASED INFERENCES FOR PARTIALLY LINEAR MODELS WITH MISSING COVARIATES

This paper considers statistical inference for partially linear models Y = X ^? β +ν(Z) +? when the linear covariate X is missing with missing probability π depending upon (Y, Z). We propose empirical likelihood‐based statistics to construct confidence regions for β and ν(z). The resulting empirical likelihood ratio statistics are shown to be asymptotically chi‐squared‐distributed. The finite‐sample performance of the proposed statistics is assessed by simulation experiments. The proposed methods are applied to a dataset from an AIDS clinical trial.     

Data Dependent Cells Chi‐Square Test With Recurrent Events

We consider a recurrent event wherein the inter‐event times are independent and identically distributed with a common absolutely continuous distribution function F. In this article, interest is in the problem of testing the null hypothesis that F belongs to some parametric family where the q‐dimensional parameter is unknown. We propose a general Chi‐squared test in which cell boundaries are data dependent. An estimator of the parameter obtained by minimizing a quadratic form resulting from a properly scaled vector of differences between Observed and Expected frequencies is used to construct the test. This estimator is known as the minimum chi‐square estimator. Large sample properties of the proposed test statistic are established using empirical processes tools. A simulation study is conducted to assess the performance of the test under parameter misspecification, and our procedures are applied to a fleet of Boeing 720 jet planes' air conditioning system failures.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Iso‐chi‐squared testing of 2 × k: ordered tables

The Pearson chi‐squared statistic for testing the equality of two multinomial populations when the categories are nominal is much less appropriate for ordinal categories. Test statistics typically used in this context are based on scorings of the ordinal levels, but the results of these tests are highly dependent on the choice of scores. The authors propose a test which naturally modifies the Pearson chi‐squared statistic to incorporate the ordinal information. The proposed test statistic does not depend on the scores and under the null hypothesis of equality of populations, it is asymptotically equivalent to the likelihood ratio test against the alternative of two‐sided likelihood ratio ordering.     

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

In this article, we point out some interesting relations between the exact test and the score test for a binomial proportion p. Based on the properties of the tests, we propose some approximate as well as exact methods of computing sample sizes required for the tests to attain a specified power. Sample sizes required for the tests are tabulated for various values of p to attain a power of 0.80 at level 0.05. We also propose approximate and exact methods of computing sample sizes needed to construct confidence intervals with a given precision. Using the proposed exact methods, sample sizes required to construct 95% confidence intervals with various precisions are tabulated for p = .05(.05).5. The approximate methods for computing sample sizes for score confidence intervals are very satisfactory and the results coincide with those of the exact methods for many cases.     

Two-step calibration of design weights under two auxiliary variables in sample survey

Calibration on the available auxiliary variables is widely used to increase the precision of the estimates of parameters. Singh and Sedory [Two-step calibration of design weights in survey sampling. Commun Stat Theory Methods. 2016;45(12):3510–3523.] considered the problem of calibration of design weights under two-step for single auxiliary variable. For a given sample, design weights and calibrated weights are set proportional to each other, in the first step. While, in the second step, the value of proportionality constant is determined on the basis of objectives of individual investigator/user for, for example, to get minimum mean squared error or reduction of bias. In this paper, we have suggested to use two auxiliary variables for two-step calibration of the design weights and compared the results with single auxiliary variable for different sample sizes based on simulated and real-life data set. The simulated and real-life application results show that two-auxiliary variables based two-step calibration estimator outperforms the estimator under single auxiliary variable in terms of minimum mean squared error.     

A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.     

A Family of Goodness‐of‐Fit Tests for Copulas Based on Characteristic Functions

A general class of rank statistics based on the characteristic function is introduced for testing goodness‐of‐fit hypotheses about the copula of a continuous random vector. These statistics are defined as L ₂ weighted functional distances between a nonparametric estimator and a semi‐parametric estimator of the characteristic function associated with a copula. It is shown that these statistics behave asymptotically as degenerate V ‐statistics of order four and that the limit distributions have representations in terms of weighted sums of independent chi‐square variables. The consistency of the tests against general alternatives is established and an asymptotically valid parametric bootstrap is suggested for the computation of the critical values of the tests. The behaviour of the new tests in small and moderate sample sizes is investigated with the help of simulations and compared with a competing test based on the empirical copula. Finally, the methodology is illustrated on a five‐dimensional data set.     

Goodness-of-fit test based on empirical characterisitc function

This paper discusses a goodness-of-fit test that uses the integral of the squared modulus of the difference between the empirical characteristic function of the sample data and the characteristic function of the hypothesized distribution. Monte Carlo procedures are employed to obtain the empirical percentage points for testing the fit of normal, logistic and exponential distributions with unknown location and scale parameters. Results of Monte Carlo power comparisons with other well-developed goodness-of-fit tests are summarized. Tne proposed test is shown to have superior power for testing the fit of the logistic distibotion (for moderate sample sizes) against a wide range of alternative distributions.     

Comparison of Goodness-of-Fit Measures in Probit Regression Model

This article examines several goodness-of-fit measures in the binary probit regression model. Existing pseudo-R ² measures are reviewed, two modified and one new pseudo-R ² measure are proposed. For the probit regression model, empirical comparisons are made for different goodness-of-fit measures with the squared sample correlation coefficient of the observed response and the predicted probabilities. As an illustration, the goodness-of-fit measures are applied to a “paid labor force” data set.     

A Method for Measuring Distance From a Training Data Set

A new method is proposed for measuring the distance between a training data set and a single, new observation. The novel distance measure reflects the expected squared prediction error when a quantitative response variable is predicted on the basis of the training data set using the distance weighted k-nearest-neighbor method. The simulation presented here shows that the distance measure correlates well with the true expected squared prediction error in practice. The distance measure can be applied, for example, in assessing the uncertainty of prediction.     

Likelihood inference for small variance components

The authors explore likelihood‐based methods for making inferences about the components of variance in a general normal mixed linear model. In particular, they use local asymptotic approximations to construct confidence intervals for the components of variance when the components are close to the boundary of the parameter space. In the process, they explore the question of how to profile the restricted likelihood (REML). Also, they show that general REML estimates are less likely to fall on the boundary of the parameter space than maximum‐likelihood estimates and that the likelihood‐ratio test based on the local asymptotic approximation has higher power than the likelihood‐ratio test based on the usual chi‐squared approximation. They examine the finite‐sample properties of the proposed intervals by means of a simulation study.     

Testing for Bivariate Extreme Dependence Using Kendall's Process

Abstract. New tests for the hypothesis of bivariate extreme‐value dependence are proposed. All test statistics that are investigated are continuous functionals of either Kendall's process or its version with estimated parameters. The procedures considered are based on linear combinations of moments and on Cramér–von Mises distances. A suitably adapted version of the multiplier central limit theorem for Kendall's process enables the computation of asymptotically valid p‐values. The power of the tests is evaluated for small, moderate and large sample sizes, as well as asymptotically, under local alternatives. An illustration with a real data set is presented.     

A powerful affine invariant test for multivariate normality based on interpoint distances of principal components

In this paper, a probability plots class of tests for multivariate normality is introduced. Based on independent standardized principal components of a d-variate normal data set, we obtained the sum of squared differences between corresponding observations of an ordered set of each principal component observations and the set of the population pth quantiles of the standard normal distribution. We proposed the sum of these d-sums of squared differences as an appropriate statistic for testing multivariate normality. We evaluated empirical critical values of the statistic and compared its power with those of some highly regarded techniques with a wonderful result.     

Homogeneity test based on ranked set samples

Abstract

The homogeneity hypothesis is investigated in a location family of distributions. A moment-based test is introduced based on data collected from a ranked set sampling scheme. The asymptotic distribution of the proposed test statistic is determined and the performance of the test is studied via simulation. Furthermore, for small sample sizes, the bootstrap procedure is used to distinguish the homogeneity of data. An illustrative example is also presented to explain the proposed procedures in this paper.     

Testing and Merging Information for Effect Size Estimation

A large-sample test for testing the equality of two effect sizes is presented. The null and non-null distributions of the proposed test statistic are derived. Further, the problem of estimating the effect size is considered when it is a priori suspected that two effect sizes may be close to each other. The combined data from all the samples leads to more efficient estimator of the effect size. We propose a basis for optimally combining estimation problems when there is uncertainty concerning the appropriate statistical model-estimator to use in representing the sampling process. The objective here is to produce natural adaptive estimators with some good statistical properties. In the context of two bivariate statistical models, the expressions for the asymptotic mean squared error of the proposed estimators are derived and compared with the parallel expressions for the benchmark estimators. We demonstrate that the suggested preliminary test estimator has superior asymptotic mean squared error performance relative to the benchmark and pooled estimators. A simulation study and application of the methodology to real data are presented.     

Estimation of odds ratios under order restrictions

A new method for estimating a set of odds ratios under an order restriction based on estimating equations is proposed. The method is applied to those of the conditional maximum likelihood estimators and the Mantel-Haenszel estimators. The estimators derived from the conditional likelihood estimating equations are shown to maximize the conditional likelihoods. It is also seen that the restricted estimators converge almost surely to the respective odds ratios when the respective sample sizes become large regularly. The restricted estimators are compared with the unrestricted maximum likelihood estimators by a Monte Carlo simulation. The simulation studies show that the restricted estimates improve the mean squared errors remarkably, while the Mantel-Haenszel type estimates are competitive with the conditional maximum likelihood estimates, being slightly worse.     

Variance Components Testing in ANOVA‐Type Mixed Models

The purpose of this article is threefold. First, variance components testing for ANOVA ‐type mixed models is considered, in which response may not be divided into independent sub‐vectors, whereas most of existing methods are for models where response can be divided into independent sub‐vectors. Second, testing that a certain subset of variance components is zero. Third, as normality is often violated in practice, it is desirable to construct tests under very mild assumptions. To achieve these goals, an adaptive difference‐based test and an adaptive trace‐based test are constructed. The test statistics are asymptotically normal under the null hypothesis, are consistent against all global alternatives and can detect local alternatives distinct from the null at a rate as close to n ^{? 1 ∕ 2} as possible with n being the sample size. Moreover, when the dimensions of variance components in different sets are bounded, we develop a test with chi‐square as its limiting null distribution. The finite sample performance of the tests is examined via simulations, and a real data set is analysed for illustration.     

Bayesian inference and prediction of the Pareto distribution based on ordered ranked set sampling

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- and m-cycle. A simulation study and real data are applied to show the proposed results.