Standard Asymptotic Results Research Articles

Regression analysis of longitudinal data with random change point.

A great deal of literature has been established for regression analysis of longitudinal data and in particular, many methods have been proposed for the situation where there exist some change points. However, most of these methods only apply to continuous response and focus on the situations where the change point only occurs on the response or the trend of the individual trajectory. In this article, we propose a new joint modeling approach that allows not only the change point to vary for different subjects or be subject-specific but also the effect heterogeneity of the covariates before and after the change point. The method combines a generalized linear mixed effect model with a random change point for the longitudinal response and a log-linear regression model for the random change point. For inference, a maximum likelihood estimation procedure is developed and the asymptotic properties of the resulting estimators, which differ from the standard asymptotic results, are established. A simulation study is conducted and suggests that the proposed method works well for practical situations. An application to a set of real data on COVID-19 is provided.

Doubly-Robust Inference in R using drtmle

Inverse probability of treatment weighted estimators and doubly robust estimators (including augmented inverse probability of treatment weight and targeted minimum loss estimators) are widely used in causal inference to estimate and draw inference about the average effect of a treatment. As an intermediate step, these estimators require estimation of key nuisance parameters, which are often regression functions. Typically, regressions are estimated using maximum likelihood and parametric models. Confidence intervals and p-values may be computed based on standard asymptotic results, such as the central limit theorem, the delta method, and the nonparametric bootstrap. However, in high-dimensional settings, maximum likelihood estimation often breaks down and standard procedures no longer yield correct inference. Instead, we may rely on adaptive estimators of nuisance parameters to construct flexible regression estimators. However, use of adaptive estimators poses a challenge for performing statistical inference about an estimated treatment effect. While doubly robust estimators facilitate inference when all relevant regression functions are consistently estimated, the same cannot be said when at least one nuisance estimator is inconsistent. drtmle implements doubly robust confidence intervals and hypothesis tests for targeted minimum loss estimates of the average treatment effect, in addition to several other recently proposed estimators of the average treatment effect.

Small Sample Corrections for Wald Tests in Latent Variable Models

Summary Latent variable models are commonly used in psychology and increasingly used for analysing brain imaging data. Such studies typically involve a small number of participants (n<100), where standard asymptotic results often fail to control the type 1 error appropriately. The paper presents two corrections improving the control of the type 1 error of Wald tests in latent variable models estimated by using maximum likelihood. First, we derive a correction for the bias of the maximum likelihood estimator of the variance parameters. This enables us to estimate corrected standard errors for model parameters and corrected Wald statistics. Second, we use a Student t-distribution instead of a Gaussian distribution to account for the variability of the variance estimator. The degrees of freedom of the Student t-distributions are estimated by using a Satterthwaite approximation. A simulation study based on data from two published brain imaging studies demonstrates that combining these two corrections provides superior control of the type 1 error rate compared with the uncorrected Wald test, despite being conservative for some parameters. The methods proposed are implemented in the R package lavaSearch2, which is available from https://cran.r-project.org/web/packages/lavaSearch2.

Fast permutation tests and related methods, for association between rare variants and binary outcomes.

In large-scale genetic studies, a primary aim is to test for an association between genetic variants and a disease outcome. The variants of interest are often rare and appear with low frequency among subjects. In this situation, statistical tests based on standard asymptotic results do not adequately control the type I error rate, especially if the case : control ratio is unbalanced. In this article, we propose the use of permutation and approximate unconditional tests for testing association with rare variants. We use novel analytical calculations to efficiently approximate the true type I error rate under common study designs, and in numerical studies show that the proposed classes of tests significantly improve upon standard testing methods. We also illustrate our methods in data from a recent case-control study for genetic causes of a severe side effect of a common drug treatment.

Long Memory Factor Model: A Semi-Parametric Approach

This paper considers the estimation of memory parameters of latent factors in a high-dimensional factor model. We first show that factors can be consistently estimated by principal component analysis (PCA). Then, we treat the estimated factors as observable and apply the local Whittle estimator to find the factor memory. By imposing additional restrictions on bandwidth and sample size, we are able to achieve the standard asymptotic results (consistency and asymptotic normality). Numerical evidence suggests that models with stationary and nonstationary factors have to be treated separately. A model-selection approach is thus proposed for practical applications. Simulations show that our procedure has good finite sample properties. Finally, we provide an empirical application to the Fama-French three-factor model.

Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization

In a recent paper, we introduced a trust-region method with variable norms for unconstrained minimization, we proved standard asymptotic convergence results, and we discussed the impact of this method in global optimization. Here we will show that, with a simple modification with respect to the sufficient descent condition and replacing the trust-region approach with a suitable cubic regularization, the complexity of this method for finding approximate first-order stationary points is $$O(\varepsilon ^{-3/2})$$O(ź-3/2). We also prove a complexity result with respect to second-order stationarity. Some numerical experiments are also presented to illustrate the effect of the modification on practical performance.

Likelihood ratio tests for a dose-response effect using multiple nonlinear regression models.

We consider the problem of testing for a dose-related effect based on a candidate set of (typically nonlinear) dose-response models using likelihood-ratio tests. For the considered models this reduces to assessing whether the slope parameter in these nonlinear regression models is zero or not. A technical problem is that the null distribution (when the slope is zero) depends on non-identifiable parameters, so that standard asymptotic results on the distribution of the likelihood-ratio test no longer apply. Asymptotic solutions for this problem have been extensively discussed in the literature. The resulting approximations however are not of simple form and require simulation to calculate the asymptotic distribution. In addition, their appropriateness might be doubtful for the case of a small sample size. Direct simulation to approximate the null distribution is numerically unstable due to the non identifiability of some parameters. In this article, we derive a numerical algorithm to approximate the exact distribution of the likelihood-ratio test under multiple models for normally distributed data. The algorithm uses methods from differential geometry and can be used to evaluate the distribution under the null hypothesis, but also allows for power and sample size calculations. We compare the proposed testing approach to the MCP-Mod methodology and alternative methods for testing for a dose-related trend in a dose-finding example data set and simulations.

Estimation and testing for multiple regulation of multivariate mixed outcomes.

Considerable interest has recently been focused on studying multiple phenotypes simultaneously in both epidemiological and genomic studies, either to capture the multidimensionality of complex disorders or to understand shared etiology of related disorders. We seek to identify multiple regulators or predictors that are associated with multiple outcomes when these outcomes may be measured on very different scales or composed of a mixture of continuous, binary, and not-fully observed elements. We first propose an estimation technique to put all effects on similar scales, and we induce sparsity on the estimated effects. We provide standard asymptotic results for this estimator and show that resampling can be used to quantify uncertainty in finite samples. We finally provide a multiple testing procedure which can be geared specifically to the types of multiple regulators of interest, and we establish that, under standard regularity conditions, the familywise error rate will approach 0 as sample size diverges. Simulation results indicate that our approach can improve over unregularized methods both in reducing bias in estimation and improving power for testing.

On likelihood ratio testing for penalized splines

Penalized spline regression using a mixed effects representation is one of the most popular nonparametric regression tools to estimate an unknown regression function $$f(\cdot )$$ . In this context testing for polynomial regression against a general alternative is equivalent to testing for a zero variance component. In this paper, we fill the gap between different published null distributions of the corresponding restricted likelihood ratio test under different assumptions. We show that: (1) the asymptotic scenario is determined by the choice of the penalty and not by the choice of the spline basis or number of knots; (2) non-standard asymptotic results correspond to common penalized spline penalties on derivatives of $$f(\cdot )$$ , which ensure good power properties; and (3) standard asymptotic results correspond to penalized spline penalties on $$f(\cdot )$$ itself, which lead to sizeable power losses under smooth alternatives. We provide simple and easy to use guidelines for the restricted likelihood ratio test in this context.

Testing for Common Conditionally Heteroskedastic Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that Hansen's (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a fifty-fifty mixture of ... and ..., where H is the number of moment conditions, as opposed to a .... With more than two assets, this distribution lies between the ... and ... (p denotes the number of parameters). These results show that ignoring the lack of first-order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets. A Monte Carlo study illustrates these findings. (ProQuest: ... denotes formulae/symbols omitted.)

Testing for Common GARCH Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that the Hansen’s (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain empirical process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a half-half mixture of x_(H-1)^2and x_H^2, where H is the number of moment conditions, as opposed to a x_(H-1)^2. With more than two assets, this distribution lies between the x_(H-p)^2 and x_H^2 (p, the number of parameters). These results show that ignoring the lack of first order identification of the moment condition model leads to oversized tests with possibly increasing over-rejection rate with the number of assets. A Monte Carlo study illustrates these findings.

Estimation in Discretely Observed Diffusions Killed at a Threshold

Abstract. Parameter estimation in diffusion processes from discrete observations up to a first‐passage time is clearly of practical relevance, but does not seem to have been studied so far. In neuroscience, many models for the membrane potential evolution involve the presence of an upper threshold. Data are modelled as discretely observed diffusions which are killed when the threshold is reached. Statistical inference is often based on a misspecified likelihood ignoring the presence of the threshold causing severe bias, e.g. the bias incurred in the drift parameters of the Ornstein–Uhlenbeck model for biological relevant parameters can be up to 25–100 per cent. We compute or approximate the likelihood function of the killed process. When estimating from a single trajectory, considerable bias may still be present, and the distribution of the estimates can be heavily skewed and with a huge variance. Parametric bootstrap is effective in correcting the bias. Standard asymptotic results do not apply, but consistency and asymptotic normality may be recovered when multiple trajectories are observed, if the mean first‐passage time through the threshold is finite. Numerical examples illustrate the results and an experimental data set of intracellular recordings of the membrane potential of a motoneuron is analysed.

Correcting MM estimates for “fat” data sets

Regression MM estimates require the estimation of the error scale, and the determination of a constant that controls the efficiency. These two steps are based on the asymptotic results that are derived assuming that the number of predictors p remains fixed while the number of observations n tends to infinity, which means assuming that the ratio p / n is “small”. However, many high-dimensional data sets have a “large” value of p / n (say, ≥0.2). It is shown that the standard asymptotic results do not hold if p / n is large; namely that (a) the estimated scale underestimates the true error scale, and (b) that even if the scale is correctly estimated, the actual efficiency can be much lower than the nominal one. To overcome these drawbacks simple corrections for the scale and for the efficiency controlling constant are proposed, and it is demonstrated that these corrections improve on the estimate’s performance under both normal and contaminated data.

Empirical Likelihood Method for Censored Median Regression Models

Based on the semiparametric median regression analysis for the right-censored data developed by Ying et al. (1995), an empirical likelihood based inferential procedure for the regression coefficients is proposed. The limiting distribution of the proposed log-empirical likelihood ratio test statistic follows a chi-squared distribution, which corresponds to the standard asymptotic results of the empirical likelihood method. The inference about the subsets of the entire regression coefficients vector is discussed. The proposed method is illustrated by some simulation studies.

Haplotype‐Based Regression Analysis and Inference of Case–Control Studies with Unphased Genotypes and Measurement Errors in Environmental Exposures

It is widely believed that risks of many complex diseases are determined by genetic susceptibilities, environmental exposures, and their interaction. Chatterjee and Carroll (2005, Biometrika 92, 399-418) developed an efficient retrospective maximum-likelihood method for analysis of case-control studies that exploits an assumption of gene-environment independence and leaves the distribution of the environmental covariates to be completely nonparametric. Spinka, Carroll, and Chatterjee (2005, Genetic Epidemiology 29, 108-127) extended this approach to studies where certain types of genetic information, such as haplotype phases, may be missing on some subjects. We further extend this approach to situations when some of the environmental exposures are measured with error. Using a polychotomous logistic regression model, we allow disease status to have K+ 1 levels. We propose use of a pseudolikelihood and a related EM algorithm for parameter estimation. We prove consistency and derive the resulting asymptotic covariance matrix of parameter estimates when the variance of the measurement error is known and when it is estimated using replications. Inferences with measurement error corrections are complicated by the fact that the Wald test often behaves poorly in the presence of large amounts of measurement error. The likelihood-ratio (LR) techniques are known to be a good alternative. However, the LR tests are not technically correct in this setting because the likelihood function is based on an incorrect model, i.e., a prospective model in a retrospective sampling scheme. We corrected standard asymptotic results to account for the fact that the LR test is based on a likelihood-type function. The performance of the proposed method is illustrated using simulation studies emphasizing the case when genetic information is in the form of haplotypes and missing data arises from haplotype-phase ambiguity. An application of our method is illustrated using a population-based case-control study of the association between calcium intake and the risk of colorectal adenoma.

Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

We consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley–Hotelling and Bartlett–Nanda–Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed. Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels. For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable. As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results.

Adjusted Nelson–Aalen Estimates With Retrospective Matching

In certain situations, randomized trials are not possible, but formal comparison of survival in two or more groups is still desirable. Although no substitute for randomization, application of the matched case-control design to survival studies may minimize the potential bias. Comparable estimates for each treatment group can be obtained by matching or stratifying according to prognostic variables, and weighting each individual to make stratum distributions the same over the treatment groups. Here we develop a weighted Nelson–Aalen estimator and a corresponding survival estimator. When data are closely matched, the stratum effects should be considered as random, leading to additional variance components, and standard asymptotic results do not apply. We derive a variance estimator and show that, asymptotically, the estimator converges to a Gaussian process. Simulations show that the asymptotic results are a good approximation in moderate-sized samples. It should be noted that ignoring the heterogeneity will lead to underestimation of the variance. An application is presented in a clinical study where data on patients receiving standard treatments are obtained for comparison with patients receiving a novel treatment, with retrospective matching to obtain comparable estimates.

Near unit root and the relationship between inflation and interest rates: A reexamination of the Fisher effect

Tests of the Fisher effect are plagued by high persistence in interest rates. Instead of standard regression analysis and asymptotic results, methods relying on local-to-unity asymptotics are employed in testing for the Fisher effect with monthly U.S. data covering the period 1953:1–1990:12. These procedures are extensions of a recently presented method (Cavanagh, Elliott and Stock (1995)) based on simultaneous confidence intervals, and they have the advantage of being asymptotically valid whether interest rates are integrated of order one or zero, or near unit root processes. Taking appropriately account of the near unit root problem the findings in most of the previous literature are reconfirmed. There is support for the Fisher effect in the interest rate targeting period (1953:1–1979:10) of the Federal Reserve but not in the 1979:11–1990:12 period.

Asymptotic Inference in Censored Regression Models Revisited

This paper establishes that regressors in the models with censored dependent variables need not be bounded for the standard asymptotic results to apply. Thus regressors which grow monotonically with the observation index may be acceptable. It also purports to provide an upper bound on the rate at which regressors may grow.

Fixed-Domain Asymptotics for Spatial Periodograms

Abstract The periodogram for a spatial process observed on a lattice is often used to estimate the spectral density. The bases for such estimators are two asymptotic properties that periodograms commonly possess: (1) the periodogram at a particular frequency is approximately unbiased for the spectral density, and (2) the correlation of the periodogram at distinct frequencies is approximately zero. For spatial data, it is often appropriate to use fixed-domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases. Using fixed-domain asymptotics, this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results. But by appropriately filtering the data before computing the periodogram, it is possible to obtain results similar to the standard asymptotic results for spatial periodograms.