In this article, we introduce a conditional marginal model for longitudinal data, in which the residuals form a martingale difference sequence. This model allows us to consider a rich class of estimating equations, which contains several estimating equations proposed in the literature. A particular sequence of estimating equations in this class contains a random matrix $\mathcal{R}_{i-1}^*(β)$, as a replacement for the ``true'' conditional correlation matrix of the $i$-th individual. Using the approach of [12], we identify some sufficient conditions under which this particular sequence of equations is asymptotically optimal (in our class). In the second part of the article, we identify a second set of conditions, under which we prove the existence and strong consistency of a sequence of estimators of $β$, defined as roots of estimation equations which are martingale transforms (in particular, roots of the sequence of asymptotically optimal equations).
We consider the marginal models of Liang and Zeger [Biometrika 73 (1986) 13-22] for the analysis of longitudinal data and we develop a theory of statistical inference for such models. We prove the existence, weak consistency and asymptotic normality of a sequence of estimators defined as roots of pseudo-likelihood equations.
In this article we study the existence and strong consistency of GEE estimators, when the generalized estimating functions are martingales with random coefficients. Furthermore, we characterize estimating functions which are asymptotically optimal.
We study the existence, strong consistency and asymptotic normality of estimators obtained from estimating functions, that are p-dimensional martingale transforms. The problem is motivated by the analysis of evolutionary clustered data, with distributions belonging to the exponential family, and which may also vary in terms of other component series. Within a quasi-likelihood approach, we construct estimating equations, which accommodate different forms of dependency among the components of the response vector and establish multivariate extensions of results on linear and generalized linear mo…
Consider the design based situation where an $r$-regular set is sampled on a random lattice. A fast algorithm for estimating the integrated mean curvature based on this observation is to use a weighted sum of $2\times \dotsm \times 2$ configuration counts. We show that for a randomly translated lattice, no asymptotically unbiased estimator of this type exists in dimension greater than or equal to three, while for stationary isotropic lattices, asymptotically unbiased estimators are plenty. Both results follow from a general formula that we state and prove, describing the asymptotic behavior of…
Within a high-frequency framework, we propose a non-parametric approach to estimate a family of copulas associated to a time-changed Brownian motion. We show that our estimator is consistent and asymptotically mixed-Gaussian. Furthermore, we test its finite-sample accuracy via Monte Carlo.
Construction of just-in-time adaptive interventions, such as prompts delivered by mobile apps to promote and maintain behavioral change, requires knowledge about time-varying moderated effects to inform when and how we deliver intervention options. Micro-randomized trials (MRT) have emerged as a sequentially randomized design to gather requisite data for effect estimation. The existing literature (Qian et al., 2020; Boruvka et al., 2018; Dempsey et al., 2020) has defined a general class of causal estimands, referred to as "causal excursion effects", to assess the time-varying moderated effect.…
In this article we propose a new variable selection method for analyzing data collected from longitudinal sample surveys. The procedure is based on the survey-weighted quadratic inference function, which was recently introduced as an alternative to the survey-weighted generalized estimating function. Under the joint model-design framework, we introduce the penalized survey-weighted quadratic inference estimator and obtain sufficient conditions for the existence, weak consistency, sparsity and asymptotic normality. To illustrate the finite sample performance of the model selection procedure, we…
We study the least squares estimator in the residual variance estimation context. We show that the mean squared differences of paired observations are asymptotically normally distributed. We further establish that, by regressing the mean squared differences of these paired observations on the squared distances between paired covariates via a simple least squares procedure, the resulting variance estimator is not only asymptotically normal and root-$n$ consistent, but also reaches the optimal bound in terms of estimation variance. We also demonstrate the advantage of the least squares estimator…
Given a random sample from a distribution with density function that depends on an unknown parameter $θ$, we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback--Leibler loss function $D(f_θ||{\hat{f}})=\int{f_θ \log{(f_θ/ hat{f})}}$ is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk…
A new single-index model that reflects the time-dynamic effects of the single index is proposed for longitudinal and functional response data, possibly measured with errors, for both longitudinal and time-invariant covariates. With appropriate initial estimates of the parametric index, the proposed estimator is shown to be $\sqrt{n}$-consistent and asymptotically normally distributed. We also address the nonparametric estimation of regression functions and provide estimates with optimal convergence rates. One advantage of the new approach is that the same bandwidth is used to estimate both the…
In this article, we propose a new method for analyzing longitudinal data which contain responses that are missing at random. This method consists in solving the generalized estimating equation (GEE) of Liang and Zeger (1986) in which the incomplete responses are replaced by values adjusted using the inverse probability weights proposed in Yi, Ma and Carroll (2012). We show that the root estimator is consistent and asymptotically normal, essentially under the some conditions on the marginal distribution and the surrogate correlation matrix as those presented in Xie and Yang (2003) in the case o…
We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size increases to infinity. Furthermore, we show that the limiting estimator is consistent and asymptotically efficient, as expected. The method applies to semiparametric regression models with unspecified covariances among the observations. In the special case of linear models, the procedure reduces to iterative reweighted least squares. Finite sample performance of the…
In many families of distributions, maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is not easily available. The score matching estimator of Hyvärinen (2005) provides an alternative where this normalization constant is not required. The corresponding estimating equations become linear for an exponential family. The score matching estimator is shown to be consistent and asymptotically normally distributed for such models, although not necessarily efficient. Gaussian linear concentration models are examples …
In this paper, we propose an estimator of the second-order parameter of randomly right-truncated Pareto-type distributions data and establish its consistency and asymptotic normality. Moreover, we derive an asymptotically unbiased estimator of the tail index and study its asymptotic behaviour. Our considerations are based on a useful Gaussian approximation of the tail product-limit process recently given by Benchaira et al. [Tail product-limit process for truncated data with application to extreme value index estimation. Extremes, 2016; 19: 219-251] and the results of Gomes et al. [Semi-parame…
S&P 500 index data sampled at one-minute intervals over the course of 11.5 years (January 1989- May 2000) is analyzed, and in particular the Hurst parameter over segments of stationarity (the time period over which the Hurst parameter is almost constant) is estimated. An asymptotically unbiased and efficient estimator using the log-scale spectrum is employed. The estimator is asymptotically Gaussian and the variance of the estimate that is obtained from a data segment of $N$ points is of order $\frac{1}{N}$. Wavelet analysis is tailor made for the high frequency data set, since it has low comp…
We consider three problems in high-dimensional Gaussian linear mixed models. Without any assumptions on the design for the fixed effects, we construct an asymptotic $F$-statistic for testing whether a collection of random effects is zero, derive an asymptotic confidence interval for a single random effect at the parametric rate $\sqrt{n}$, and propose an empirical Bayes estimator for a part of the mean vector in ANOVA type models that performs asymptotically as well as the oracle Bayes estimator. We support our results with numerical simulations and provide comparisons with oracle estimators. …
We study the problem of estimating parameters of the limiting equation of a multiscale diffusion in the case of averaging and homogenization, given data from the corresponding multiscale system. First, we review some recent results that make use of the maximum likelihood of the limiting equation. In particular, it has been shown that in the averaging case, the MLE will be asymptotically consistent in the limit while in the homogenization case, the MLE will be asymptotically consistent only if we subsample the data. Then, we focus on the problem of estimating the diffusion coefficient. We sugge…
We show that the full-sample bootstrap is asymptotically valid for constructing confidence intervals for high-quantiles, tail probabilities, and other tail parameters of a univariate distribution. This resolves the doubts that have been raised about the validity of such bootstrap methods. In our extensive simulation study, the overall performance of the bootstrap method was better than that of the standard asymptotic method, indicating that the bootstrap method is at least as good, if not better than, the asymptotic method for inference. This paper also lays the foundation for developing boots…
A parameter estimation problem is considered, in which dispersed sensors transmit to the statistician partial information regarding their observations. The sensors observe the paths of continuous semimartingales, whose drifts are linear with respect to a common parameter. A novel estimating scheme is suggested, according to which each sensor transmits only one-bit messages at stopping times of its local filtration. The proposed estimator is shown to be consistent and, for a large class of processes, asymptotically optimal, in the sense that its asymptotic distribution is the same as the exact …
Measurement errors usually arise during the longitudinal data collection process. Ignoring the effects of measurement errors will lead to invalid estimates. The Lifestyle Education for Activity and Nutrition (LEAN) study was designed to assess the effectiveness of intervention for enhancing weight loss over nine months. The covariates systolic blood pressure (SBP) and diastolic blood pressure (DBP) were measured at baseline, month 4, and month 9. At each assessment time, there were two replicate measurements for SBP and DBP. The replicate measurement errors of SBP follow different distribution…
The problem of change-point estimation is considered under a general framework where the data are generated by unknown stationary ergodic process distributions. In this context, the consistent estimation of the number of change-points is provably impossible. However, it is shown that a consistent clustering method may be used to estimate the number of change points, under the additional constraint that the correct number of process distributions that generate the data is provided. This additional parameter has a natural interpretation in many real-world applications. An algorithm is proposed t…
