Linear Mixed Effects (LME) models have been widely applied in clustered data analysis in many areas including marketing research, clinical trials, and biomedical studies. Inference can be conducted using maximum likelihood approach if assuming Normal distributions on the random effects. However, in many applications of economy, business and medicine, it is often essential to impose constraints on the regression parameters after taking their real-world interpretations into account. Therefore, in this paper we extend the classical (unconstrained) LME models to allow for sign constraints on its overall coefficients. We propose to assume a symmetric doubly truncated Normal (SDTN) distribution on the random effects instead of the unconstrained Normal distribution which is often found in classical literature. With the aforementioned change, difficulty has dramatically increased as the exact distribution of the dependent variable becomes analytically intractable. We then develop likelihood-based approaches to estimate the unknown model parameters utilizing the approximation of its exact distribution. Simulation studies have shown that the proposed constrained model not only improves real-world interpretations of results, but also achieves satisfactory performance on model fits as compared to the existing model.
Doubly truncated data are found in astronomy, econometrics and survival analysis literature. They arise when each observation is confined to an interval, i.e., only those which fall within their respective intervals are observed along with the intervals. Unlike the more widely studied one-sided truncation that can be handled effectively by the counting process-based approach, doubly truncated data are much more difficult to handle. In their analysis of an astronomical data set, Efron and Petrosian (1999) proposed some nonparametric methods, including a generalization of Kendall's tau test, for…
Due to measurement noise, a common problem in in various fields is how to estimate the ratio of two functions. We consider this problem of estimating the ratio of two functions in a nonparametric regression model. Assuming the noise is normally distributed, this is equivalent to estimating the ratio of the means of two normally distributed random variables. We identified a consistent estimator that gives the mean squared loss of order $O(1/n)$ ($n$ is the sample size) when conditioned on a highly probable event. We also present our result applied to both the real data from EAPS and on simulate…
We propose a new approach for estimating the parameters of a probability distribution. It consists on combining two new methods of estimation. The first is based on the definition of a new distance measuring the difference between variations of two distributions on a finite number of points from their support and on using this measure for estimation purposes by the method of minimum distance. For the second method, given an empirical discrete distribution, we build up an auxiliary discrete theoretical distribution having the same support of the first and depending on the same parameters of the…
Best practices for estimating and reporting epidemiological delay distributions of infectious diseases using public health surveillance and healthcare data
It is proved that the sum of n independent but non-identically distributed doubly truncated Normal distributions converges in distribution to a Normal distribution. It is also shown how the result can be applied in estimating a constrained mixed effects model.
Compositional data, which is data consisting of fractions or probabilities, is common in many fields including ecology, economics, physical science and political science. If these data would otherwise be normally distributed, their spread can be conveniently represented by a multivariate normal distribution truncated to the non-negative space under a unit simplex. Here this distribution is called the simplex-truncated multivariate normal distribution. For calculations on truncated distributions, it is often useful to obtain rapid estimates of their integral, mean and covariance; these quantiti…
A new and an enriched JPEG algorithm is provided for identifying redundancies in a sequence of irregular noisy data points which also accommodates a reference-free criterion function. Our main contribution is by formulating analytically (instead of approximating) the inverse of the transpose of JPEGwavelet transform without involving matrices which are computationally cumbersome. The algorithm is suitable for the widely-spread situations where the original data distribution is unobservable such as in cases where there is deficient representation of the entire population in the training data (i…
A conventional linear model for functional data involves expressing a response variable $Y$ in terms of the explanatory function $X(t)$, via the model: $Y=a+\int_I b(t)X(t)dt+\hbox{error}$, where $a$ is a scalar, $b$ is an unknown function and $I=[0, α]$ is a compact interval. However, in some problems the support of $b$ or $X$, $I_1$ say, is a proper and unknown subset of $I$, and is a quantity of particular practical interest. In this paper, motivated by a real-data example involving particulate emissions, we develop methods for estimating $I_1$. We give particular emphasis to the case $I_1=…
Louisiana State University（路易斯安那州立大学）；Southern University and A&M College（南方大学及农工学院）
In this paper, we have established a new framework of truncated inverse sampling for estimating mean values of non-negative random variables such as binomial, Poisson, hyper-geometrical, and bounded variables. We have derived explicit formulas and computational methods for designing sampling schemes to ensure prescribed levels of precision and confidence for point estimators. Moreover, we have developed interval estimation methods.
We present `latentcor`, an R package for correlation estimation from data with mixed variable types. Mixed variables types, including continuous, binary, ordinal, zero-inflated, or truncated data are routinely collected in many areas of science. Accurate estimation of correlations among such variables is often the first critical step in statistical analysis workflows. Pearson correlation as the default choice is not well suited for mixed data types as the underlying normality assumption is violated. The concept of semi-parametric latent Gaussian copula models, on the other hand, provides a uni…
Truncated linear regression is a classical challenge in Statistics, wherein a label, $y = w^T x + \varepsilon$, and its corresponding feature vector, $x \in \mathbb{R}^k$, are only observed if the label falls in some subset $S \subseteq \mathbb{R}$; otherwise the existence of the pair $(x, y)$ is hidden from observation. Linear regression with truncated observations has remained a challenge, in its general form, since the early works of~\citet{tobin1958estimation,amemiya1973regression}. When the distribution of the error is normal with known variance, recent work of~\citet{daskalakis2019trunca…
The purpose of this work is to improve the efficiency in estimating the average causal effect (ACE) on the survival scale where right-censoring exists and high-dimensional covariate information is available. We propose new estimators using regularized survival regression and survival random forests (SRF) to make the adjustment for the high dimensional covariates to improve efficiency. We study the behavior of the adjusted estimator under mild assumptions and show theoretical guarantees that the proposed estimators are more efficient than the unadjusted ones asymptotically when using SRF for ad…
We consider an empirical likelihood framework for inference for a statistical model based on an informative sampling design and population-level information. The population-level information is summarized in the form of estimating equations and incorporated into the inference through additional constraints. Covariate information is incorporated both through the weights and the estimating equations. The estimator is based on conditional weights. We show that under usual conditions, with population size increasing unbounded, the estimates are strongly consistent, asymptotically unbiased, and nor…
Bimodal truncated count distributions are frequently observed in aggregate survey data and in user ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an additional mode. Modeling bimodal behavior in discrete data is useful for various purposes, from comparing shapes of different samples (or survey questions) to predicting future ratings by new raters. The Poisson distribution is the most common distribution for fitting count data and can be modified to achieve mixtures of truncated Poisson distributions. However,…
Statistical methods with empirical likelihood (EL) are appealing and effective especially in conjunction with estimating equations through which useful data information can be adaptively and flexibly incorporated. It is also known in the literature that EL approaches encounter difficulties when dealing with problems having high-dimensional model parameters and estimating equations. To overcome the challenges, we begin our study with a careful investigation on high-dimensional EL from a new scope targeting at estimating a high-dimensional sparse model parameters. We show that the new scope prov…
Bayesian Inference for Left-Truncated Log-Logistic Distributions for Time-to-event Data Analysis
NCIAS, Arizona State University（国家癌症研究所亚利桑那州立大学分校）；Center for Biostatistics, Department of Biomedical Informatics, The Ohio State University（生物统计中心，生物医学信息学系，俄亥俄州立大学）；Department of Electrical and Computer Engineering, The University of Texas at Austin（电气与计算机工程系，德克萨斯大学奥斯汀分校）；Department of Mathematics, The University of Hawaii at Mānoa（数学系，夏威夷大学毛伊分校）
Time-smoothed inverse probability weighted estimation of effects of generalized time-varying treatment strategies on repeated outcomes truncated by death
Quantitative portfolio allocation requires the accurate and tractable estimation of covariances between a large number of assets, whose histories can greatly vary in length. Such data are said to follow a monotone missingness pattern, under which the likelihood has a convenient factorization. Upon further assuming that asset returns are multivariate normally distributed, with histories at least as long as the total asset count, maximum likelihood (ML) estimates are easily obtained by performing repeated ordinary least squares (OLS) regressions, one for each asset. Things get more interesting w…
University of Colorado, Boulder（科罗拉多大学博尔德分校）；ETH Zürich（苏黎世联邦理工学院）
Quantities with right-skewed distributions are ubiquitous in complex social systems, including political conflict, economics and social networks, and these systems sometimes produce extremely large events. For instance, the 9/11 terrorist events produced nearly 3000 fatalities, nearly six times more than the next largest event. But, was this enormous loss of life statistically unlikely given modern terrorism's historical record? Accurately estimating the probability of such an event is complicated by the large fluctuations in the empirical distribution's upper tail. We present a generic statis…
The Aalen-Johansen estimator generalizes the Kaplan-Meier estimator for independently left-truncated and right-censored survival data to estimating the transition probability matrix of a time-inhomogeneous Markov model with finite state space. Such multi-state models have a wide range of applications for modelling complex courses of a disease over the course of time, but the Markov assumption may often be in doubt. If censoring is entirely unrelated to the multi-state data, it has been noted that the Aalen-Johansen estimator, standardized by the initial empirical distribution of the multi-stat…
Wavelet thresholding generally assumes independent, identically distributed normal errors when estimating functions in a nonparametric regression setting. VisuShrink and SureShrink are just two of the many common thresholding methods based on this assumption. When the errors are not normally distributed, however, few methods have been proposed. A distribution-free method for thresholding wavelet coefficients in nonparametric regression is described, which unlike some other non-normal error thresholding methods, does not assume the form of the non-normal distribution is known. Improvements are …
Statistical inference on the explained variation of an outcome by a set of covariates is of particular interest in practice. When the covariates are of moderate to high-dimension and the effects are not sparse, several approaches have been proposed for estimation and inference. One major problem with the existing approaches is that the inference procedures are not robust to the normality assumption on the covariates and the residual errors. In this paper, we propose an estimating equation approach to the estimation and inference on the explained variation in the high-dimensional linear model. …
We present a Bayesian mixture model for estimating the joint distribution of mixed ordinal, nominal, and continuous data conditional on a set of fixed variables. The model uses multivariate normal and categorical mixture kernels for the random variables. It induces dependence between the random and fixed variables through the means of the multivariate normal mixture kernels and via a truncated local Dirichlet process. The latter encourages observations with similar values of the fixed variables to share mixture components. Using a simulation of data fusion, we illustrate that the model can est…
