Joint and marginal specification tests for conditional mean and variance models

This article proposes a class of joint and marginal spectral diagnostic tests for parametric conditional means and variances of linear and nonlinear time series models. The use of joint and marginal tests is motivated from the fact that marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified. The new tests are based on a generalized spectral approach and do not need to choose a lag order depending on the sample size or to smooth the data. Moreover, the proposed tests are robust to higher order dependence of unknown form, in particular to conditional skewness and kurtosis. It turns out that the asymptotic null distributions of the new tests depend on the data generating process. Hence, we implement the tests with the assistance of a wild bootstrap procedure. A simulation study compares the finite sample performance of the proposed and competing tests, and shows that our tests can play a valuable role in time series modeling. Finally, an application to the S&P 500 highlights the merits of our approach.     

On the Kuiper test for normality with mean and variance unknown

Summary A modified form of the Kuiper statistic V _n is developed for testing the composite hypothesis that a sample of size n comes from a normal population with unspecified mean and variance. Its distribution is derived using Monte Carlo methods. Power comparison with the adjusted Kuiper test proposed by L outer and K oerts [6] indicates that our test is superior with respect to certain alternatives.     

On the Kuiper test for normality with mean and variance unknown

Summary If one wants to test the hypothesis as to whether a set of observations comes from a completely specified continuous distribution or not, one can use the Kuiper test. But if one or more parameters have to be estimated, the standard tables for the Kuiper test are no longer valid. This paper presents a table to use with the Kuiper statistic for testing whether a sample comes from a normal distribution when the mean and variance are to be estimated from the sample. The critical points are obtained by means of Monte-Carlo calculation; the power of the test is estimated by simulation; and the results of the powers for several alternative distributions are compared with the estimated powers of the Kolmogorov-Smirnov test.     

Robust feature screening for varying coefficient models via quantile partial correlation

This article is concerned with feature screening for varying coefficient models with ultrahigh-dimensional predictors. We propose a new sure independence screening method based on quantile partial correlation (QPC-SIS), which is quite robust against outliers and heavy-tailed distributions. Then we establish the sure screening property for the QPC-SIS, and conduct simulations to examine its finite sample performance. The results of simulation study indicate that the QPC-SIS performs better than other methods like sure independent screening (SIS), sure independent ranking and screening, distance correlation-sure independent screening, conditional correlation sure independence screening and nonparametric independent screening, which shows the validity and rationality of QPC-SIS.     

Cointegrating rank selection in models with time-varying variance

Reduced rank regression (RRR) models with time varying heterogeneity are considered. Standard information criteria for selecting cointegrating rank are shown to be weakly consistent in semiparametric RRR models in which the errors have general nonparametric short memory components and shifting volatility provided the penalty coefficient _Cn→∞$C_n\to \infty$ and _Cn/n→0$C_n/n\to 0$ as n→∞$n\to \infty$. The AIC criterion is inconsistent and its limit distribution is given. The results extend those in Cheng and Phillips (2009a) and are useful in empirical work where structural breaks or time evolution in the error variances is present. An empirical application to exchange rate data is provided.     

On new variance approximations for linear models with inequality constraints

In this paper, we examine the estimation of linear models subject to inequality constraints with a special focus on new variance approximations for the estimated parameters. For models with one inequality restriction, the proposed variance formulas are exact. The variance approximations proposed in this paper can be used in regression analysis, Kalman filtering, and balancing national accounts, when inequality constraints are to be incorporated in the estimation procedure.     

Fitting nonlinear time-series models with applications to stochastic variance models

New strategies for the implementation of maximum likelihood estimation of nonlinear time series models are suggested. They make use of recent work on the EM algorithm and iterative simulation techniques. The estimation procedures are applied to the problem of fitting stochastic variance models to exchange rate data.     

Conditional empirical likelihood for quantile regression models

In this paper, we propose a new Bayesian quantile regression estimator using conditional empirical likelihood as the working likelihood function. We show that the proposed estimator is asymptotically efficient and the confidence interval constructed is asymptotically valid. Our estimator has low computation cost since the posterior distribution function has explicit form. The finite sample performance of the proposed estimator is evaluated through Monte Carlo studies.     

Conditional implicit mean and the law of iterated integrals

This paper presents a new framework which generalizes the concept of conditional expectation to mean values which are implicitly defined as unique solutions to some functional equation. We call such a mean value an implicit mean. The implicit mean and its very special example, the quasi-linear mean, have been extensively applied to economics and decision theory. This paper provides a procedure of defining the conditional implicit mean and then analyzes its properties. In particular, we show that the conditional implicit mean is in general “biased” in the sense that an analogue of the law of iterated expectations does not hold and we characterize the quasi-linear mean as the only implicit mean which is “unbiased”.     

Conditional empirical likelihood estimation and inference for quantile regression models

This paper considers two empirical likelihood-based estimation, inference, and specification testing methods for quantile regression models. First, we apply the method of conditional empirical likelihood (CEL) by Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667–1714] and Zhang and Gijbels [2003. Sieve empirical likelihood and extensions of the generalized least squares. Scandinavian Journal of Statistics 30, 1–24] to quantile regression models. Second, to avoid practical problems of the CEL method induced by the discontinuity in parameters of CEL, we propose a smoothed counterpart of CEL, called smoothed conditional empirical likelihood (SCEL). We derive asymptotic properties of the CEL and SCEL estimators, parameter hypothesis tests, and model specification tests. Important features are (i) the CEL and SCEL estimators are asymptotically efficient and do not require preliminary weight estimation; (ii) by inverting the CEL and SCEL ratio parameter hypothesis tests, asymptotically valid confidence intervals can be obtained without estimating the asymptotic variances of the estimators; and (iii) in contrast to CEL, the SCEL method can be implemented by some standard Newton-type optimization. Simulation results demonstrate that the SCEL method in particular compares favorably with existing alternatives.     

Conditional independence and log linear models for multi-dimensional contingency tables

Interference about conditional independence in relation to log linear models are discussed for contingency tables. The parameters and likelihood ratios for a log linear model with a dependent variable are shown to be identical to those for a multivariate model. An approximaate method of calculating log likelihood ratios, even when all dimensions of the table have more than two levels (no binary variables) is derived. The implications for sociological “causal” models are discussed.     

The mean and variance of the likelihood ratio criterion for the multinomial distribution

Summary The mean and variance of — 2 In A for the multinomial distribution are derived in closed form. A comparison is made betweenSchaffer's [1957] approximations to the moments and the exact moments for varying sample size.     

Monitoring the mean and the variance of a stationary process

We deal with the problem of how deviations in the mean or the variance of a time series can be detected. Several simultaneous control charts are introduced which are based on EWMA (exponentially weighted moving average) statistics for the mean and the empirical variance. The combined X − S² EWMA chart is extended to time series. Further simultaneous charts are considered. The comparision of these schemes shows that the residual attempt must be favored if a variance change is present.     

Sequential approach to simultaneous estimation of the mean and variance

The problem of simultaneous estimation of the mean and variance of a normal distribution has been studied. We propose a semi-circular region _n ={(a,b)':b>0} of radiusd, which has approximately a preassigned coverage probability. Asymptotic efficiency and asymptotic consistency (asd0) of our proposed sequential procedures have been proved.Research partially supported by U.S. Army Research Grant No. DAAG29-76-G-0038.     

Adjusted Pearson Chi-Square feature screening for multi-classification with ultrahigh dimensional data

Huang et al. (J Bus Econ Stat 32:237–244, 2014) first proposed a Pearson Chi-Square based feature screening procedure tailored to multi-classification problem with ultrahigh dimensional categorical covariates, which is a common problem in practice but has seldom been discussed in the literature. However, their work establishes the sure screening property only in a limited setting. Moreover, the p value based adjustments when the number of categories involved by each covariate is different do not work well in several practical situations. In this paper, we propose an adjusted Pearson Chi-Square feature screening procedure and a modified method for tuning parameter selection. Theoretically, we establish the sure screening property of the proposed method in general settings. Empirically, the proposed method is more successful than Pearson Chi-Square feature screening in handling non-equal numbers of covariate categories in finite samples. Results of three simulation studies and one real data analysis are presented. Our work together with Huang et al. (J Bus Econ Stat 32:237–244, 2014) establishes a solid theoretical foundation and empirical evidence for the family of Pearson Chi-Square based feature screening methods.     

The analysis of unbalanced linear models with variance components

Statistical inference for fixed effects, random effects and components of variance in an unbalanced linear model with variance components will be discussed. Variance components will be estimated by Restricted Maximum Likelihood. Iterative procedures for computing the estimates, such as Fisher scoring and the EM-algorithm, are described.     

Efficiencies for optimum designs when transforming the response in nonlinear models with nonconstant variance

Often the responses from mechanistic models have to be transformed to achieve error distributions that are symmetric and have constant variance. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples given of optimum designs for particular values of λ, together with the efficiency of these designs as λ varies. Approaches to finding designs robust to variations in λ are indicated and exemplified.     

Nonparametric independence feature screening for ultrahigh-dimensional survival data

With the explosion of digital information, high-dimensional data is frequently collected in prevalent domains, in which the dimension of covariates can be much larger than the sample size. Many effective methods have been developed to reduce the dimension of such data recently, however, few methods might perform well for survival data with censoring. In this article, we develop a novel nonparametric feature screening procedure based on ultrahigh-dimensional survival data by incorporating the inverse probability weighting scheme to tackle the issue of censoring. The proposed method is model-free and hence can be implemented for extensive survival models. Moreover, it is robust to heterogeneity and invariant to monotone increasing transformations of the response. The sure screening property and ranking consistency property are also established under mild conditions. The competence and robustness of our method is further confirmed through comprehensive simulation studies and an analysis of a real data example.     

Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling

This paper considers the problem of procuring reliable information on sensitive quantitative characteristics without exposing respondents’ identities. A generalized optional randomized response procedure is proposed, which enables us to estimate some unknown population parameters unbiasedly. In particular, conditions for the assurance of unbiased estimations of mean, variance and sensitivity level are studied. Efficiency comparisons are also carried out to study the performance of the proposed procedure.