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Nowadays, the confidentiality of data and information is of great importance for many companies and organizations. For this reason, they may prefer not to release exact data, but instead to grant researchers access to approximate data. For example, rather than providing the exact measurements of their clients, they may only provide researchers with grouped data, that is, the number of clients falling in each of a set of non-overlapping measurement intervals. The challenge is to estimate the mean and variance structure of the hidden ungrouped data based on the observed grouped data. To tackle this problem, this work considers the exact observed data likelihood and applies the Expectation-Maximization (EM) and Monte Carlo EM (MCEM) algorithms for cases where the hidden data follow a univariate, bivariate, or multivariate normal distribution. Simulation studies are conducted to evaluate the performance of the proposed EM and MCEM algorithms. The well-known Galton data set is considered as an application example. [ABSTRACT FROM AUTHOR]

We introduce a new command, summclust, that summarizes the cluster structure of the dataset for linear regression models with clustered disturbances. The key unit of observation for such a model is the cluster. We therefore propose cluster-level measures of leverage, partial leverage, and influence and show how to compute them quickly in most cases. The measures of leverage and partial leverage can be used as diagnostic tools to identify datasets and regression designs in which cluster–robust inference is likely to be challenging. The measures of influence can provide valuable information about how the results depend on the data in the various clusters. We also show how to calculate two jackknife variance matrix estimators efficiently as a by-product of our other computations. These estimators, which are already available in Stata, are generally more conservative than conventional variance matrix estimators. The summclust command computes all the quantities that we discuss. [ABSTRACT FROM AUTHOR]

Introduction: Surveys are common research tools, and questionnaires revisions are a common occurrence in longitudinal studies. Revisions can, at times, introduce systematic shifts in measures of interest. We formulate that questionnaire revision are a stochastic process with transition matrices. Thus, revision shifts can be reduced by first estimating these transition matrices, which can be utilized in estimation of interested measures. Materials and method: An ideal survey response model is defined by mapping between the true value of a participant's response to an interval in the grouped data type scale. A population completed surveys multiple times, as modeled with multiple stochastic process. This included stochastic processes related to true values and intervals. While multiple factors contribute to changes in survey responses, here, we explored the method that can mitigate the effects of questionnaire revision. We proposed the Version Alignment Method (VAM), a data preprocessing tool, which can separate the transitions according to revisions from all transitions via solving an optimization problem and using the revision-related transitions to remove the revision effect. To verify VAM, we used simulation data to study the estimation error and a real life MJ dataset containing large amounts of long-term questionnaire responses with several questionnaire revisions to study its feasibility. Result: We compared the difference of the annual average between consecutive years. Without adjustment, the difference is 0.593 when the revision occurred, while VAM brought it down to 0.115, where difference between years without revision was in the 0.005, 0.125 range. Furthermore, our method rendered the responses to the same set of intervals, thus comparing the relative frequency of items before and after revisions became possible. The average estimation error in L infinity was 0.0044 which occupied the 95% CI which was constructed by bootstrap analysis. Conclusion: Questionnaire revisions can induce different response bias and information loss, thus causing inconsistencies in the estimated measures. Conventional methods can only partly remedy this issue. Our proposal, VAM, can estimate the aggregate difference of all revision-related systematic errors and can reduce the differences, thus reducing inconsistencies in the final estimations of longitudinal studies. [ABSTRACT FROM AUTHOR]

A wide range of machine-learning-based approaches have been developed in the past decade, increasing our ability to accurately model nonlinear and nonadditive response surfaces. This has improved performance for inferential tasks such as estimating average treatment effects in situations where standard parametric models may not fit the data well. These methods have also shown promise for the related task of identifying heterogeneous treatment effects. However, the estimation of both overall and heterogeneous treatment effects can be hampered when data are structured within groups if we fail to correctly model the dependence between observations. Most machine learning methods do not readily accommodate such structure. This paper introduces a new algorithm, stan4bart, that combines the flexibility of Bayesian Additive Regression Trees (BART) for fitting nonlinear response surfaces with the computational and statistical efficiencies of using Stan for the parametric components of the model. We demonstrate how stan4bart can be used to estimate average, subgroup, and individual-level treatment effects with stronger performance than other flexible approaches that ignore the multilevel structure of the data as well as multilevel approaches that have strict parametric forms. [ABSTRACT FROM AUTHOR]

Objective: The aim of this study is to examine the effectiveness of linear mixed-effects (LME) model, one of the traditional models used in the classification of clustered data, and mixed-effects machine learning models, which are the latest approaches. Material and Methods: For the simulation, various data sets were created with different number of groups (250, 500, 1000) and different sample sizes (5000, 10000, 15000). Within the scope of the simulation, LME model, mixed-effects random forest (MERF) and Gaussian process boosting (GPBoost) models were compared in terms of root mean square error (RMSE) on two functions. Results: When the error variance (EV) is 4 for the linear function, sample size is small and the number of groups is high, RMSE of MERF model is smaller. In all other scenarios, RMSE of the linear model was smaller and. In cases where EV for the nonlinear function is 1, the sample size is small and the number of groups is high, RMSE of MERF is smaller. In all other scenarios (while EV was 1), RMSE of GPBoost model was small, p<0.05 for the difference with LME, and p>0.05 for MERF. In cases where EV is 4 for the nonlinear function and the sample size and groups is high, RMSE of MERF is smaller. In all other scenarios (while EV was 4), RMSE of GPBoost model was smaller. Conclusion: As a conclusion, for a nonlinear function, GPBoost performed better than MERF and LME methods in terms of RMSE and time. However, when a linear function is considered, LME gives a better result. [ABSTRACT FROM AUTHOR]

The paper provides a Bayesian alternative to transform the abridged age-specific fertility data into an unabridged one. A model suitable for a binary type of fertility data is considered for the study. The success probabilities of the considered model are assumed to vary in accordance with some established parametric age-specific fertility models. Bayes analysis is done using uniform priors for the unknown parameters of the considered models. Since the posterior analysis involves analytically intractable form of integrals, the paper employs the Gibbs sampler algorithm with intermediate Metropolis-Hastings steps to draw the corresponding posterior samples. Numerical illustration is provided using grouped age-specific fertility rate data. Finally, a few standard Bayesian model comparison tools are used in order to recommend an appropriate model for the data in hand, keeping in view the intended objectives. [ABSTRACT FROM AUTHOR]

Researchers often need to work with categorical income data. The typical nonparametric (including midpoint) and parametric estimation methods used to estimate summary statistics both have advantages, but they carry assumptions that cause them to deviate in important ways from real-world income distributions. The method introduced here, random empirical distribution imputation (REDI), imputes discrete observations using binned income data, while also calculating summary statistics. REDI achieves this through random cold-deck imputation from a real-world reference data set (demonstrated here using the Current Population Survey Annual Social and Economic Supplement). This method can be used to reconcile bins between data sets or across years and handle top incomes. REDI has other advantages for computing values of an income distribution that is nonparametric, bin consistent, area and variance preserving, continuous, and computationally fast. The author provides proof of concept using two years of the American Community Survey. The method is available as the redi command for Stata. [ABSTRACT FROM AUTHOR]

To understand how income inequality affects individuals and communities, researchers must have accurate measures of income inequality at lower geographic levels, such as counties, school districts, and census tracts. Studies of income inequality, however, are constrained by the tabular format in which censuses publish income data. In this article, the author proposes a new method, Lorenz interpolation, for estimating income inequality from binned income data. Using public microsample data from the American Community Survey (ACS), the author shows that Lorenz interpolation produces more accurate and reliable income inequality estimates than do alternative estimation methods. Then, using restricted ACS income data obtained through a Federal Statistical Research Data Center, the author evaluates the accuracy of Lorenz interpolation at the census tract and school district levels. Lorenz interpolation produces reliable school district–level estimates, but the method produces less reliable estimates for some income inequality measures at the tract level. These findings indicate that researchers should refrain from estimating tract-level income inequality measures from tabular data. They also show that aggregating tract income distributions to higher geographic levels can produce valid estimates of income inequality. [ABSTRACT FROM AUTHOR]

Background: Time to age at first marriage of women is the duration of time until the age at which they started living with their first partner. Time to age at first marriage is widely considered a proxy indicator for the age at which women begin to be exposed to the risks inherent in sexual activity. The purpose is to model the determinant of time to age at first marriage among women in Ethiopia using Cox models with mixed effects. Methods: The 2016 Ethiopian Demography and Health survey sample was selected using a two-stage cluster design. The data set in this study were obtained from the Demography and Health survey conducted in Ethiopia in 2016. In this study, we used Cox models with mixed effects. Results: Of all 15,683 women aged 15–49 years, 11,405 (72.72%) were married with the median and mean age at first marriage 17 years and 18 years, respectively. Cox frailty survival model showed that residence, educational level, occupation, work status of women& head education level of households were the most significant factors whereas religion, access to media and wealth index of a household of women were not significant factors at 5% level of significance. The significant clustering effect showed that heterogeneity among the regions on age at first marriage was present. Conclusions: The present study determined the duration of time until the age at first marriage and indicated relevant solutions for marriage-related problems of women aged 15–49 years in Ethiopia. Women residing in rural area of Ethiopia and had lower education level were married earlier. Therefore, programs to reduce the high rate of early marriage in Ethiopia should give attention to women education and women residing in rural area. Plain language summary: Time to age at first marriage of women is the duration of time until the age at which they started living with their first partner. African women are more likely to marry earlier than other continent women, which causes high fertility due to their long period of exposure to the risk of pregnancy. Even though Sub-Sahara Africa accounts for the highest rate of age at first marriage among countries in the Africa continent, comparably the case is very worse in Ethiopia. Furthermore, there is no study about the determinant of time to age at first marriage in Ethiopia using advanced models like Cox Model with Mixed effects. This model allows for the analysis of data with complex patterns of variability, with a focus on nested sources of variability. Very often it makes sense to use such a model to represent the variability within and between groups. For instance, in this study, our interest is not only knowing the significant effect of determinant factors on time to age at first marriage among women in Ethiopia but also the variability of time to age at first marriage within the region of Ethiopia and between regions of Ethiopia. We have used a dataset for this study from Ethiopia Demography and Health Survey which was conducted in 2016. The study helps to indicate relevant solutions for women's marriage-related problems (predominantly, women's reproductive health problems) in Ethiopia and it provides input for further studies in Ethiopia. [ABSTRACT FROM AUTHOR]

In medical and social science research, reliability of testing methods measured through inter‐ and intraobserver agreement is critical in disease diagnosis. Often comparison of agreement across multiple testing methods is sought in situations where testing is carried out on the same experimental units rendering the outcomes to be correlated. In this article, we first developed a Bayesian method for comparing dependent agreement measures under a grouped data setting. Simulation studies showed that the proposed methodology outperforms the competing methods in terms of power, while maintaining a decent type I error rate. We further developed a Bayesian joint model for comparing dependent agreement measures adjusting for subject and rater‐level heterogeneity. Simulation studies indicate that our model outperforms a competing method that is used in this context. The developed methodology was implemented on a key measure on a dichotomous rating scale from a study with six raters evaluating three classification methods for chest radiographs for pneumoconiosis developed by the International Labor Office. [ABSTRACT FROM AUTHOR]

Grouped data are frequently used in several fields of study. In this work, we use the expectation-maximization (EM) algorithm for fitting the skew-normal (SN) mixture model to the grouped data. Implementing the EM algorithm requires computing the one-dimensional integrals for each group or class. Our simulation study and real data analyses reveal that the EM algorithm not only always converges but also can be implemented in just a few seconds even when the number of components is large, contrary to the Bayesian paradigm that is computationally expensive. The accuracy of the EM algorithm and superiority of the SN mixture model over the traditional normal mixture model in modelling grouped data are demonstrated through the simulation and three real data illustrations. For implementing the EM algorithm, we use the package called ForestFit developed for R environment available at . [ABSTRACT FROM AUTHOR]

This paper investigates the statistical analysis of grouped accelerated temperature cycling test data when the product lifetime follows a Weibull distribution. A log-linear acceleration equation is derived from the Coffin-Manson model. The problem is transformed to a constant-stress accelerated life test with grouped data and multiple acceleration variables. The Jeffreys prior and reference priors are derived. Maximum likelihood estimation and Bayesian estimation with objective priors are obtained by applying the technique of data augmentation. A simulation study shows that both of these two methods perform well when sample size is large, and the Bayesian method gives better performance under small sample sizes. [ABSTRACT FROM AUTHOR]

Virtual power plants (VPPs) of secondary distribution network have come into existence due to the installation of generators in the premises of domestic consumers and the recent legislative emphasis on net‐zero energy operation. A predetermined operation of VPP as required by a system operator is possible only when its power output is assessed beforehand. A simple approach for assessment based on load profiling using descriptive statistical parameters such as z‐scores of the load data is proposed in this study. The proposed method collates historical customer data according to seasons, days of the week and specified time‐intervals of the day. Then, z‐scores are computed for each of the historical load and generator data. The calculated z‐scores are subsequently grouped into specified clusters. The grouped data in each of the clusters are then employed to predict the VPP output. Demonstration of the assessment of VPP output using available Pecan street data is presented to validate the proposed approach. [ABSTRACT FROM AUTHOR]

The observed increase in economic inequality, where the major concern is relative to the huge growth of the highest incomes, motivates to revisit classical measures of inequality and to offer new ways to synthesize the variability of the entire income distribution. The idea is to provide policy makers a way to contrast the economic position of the group of the poorer p percent of the population and to compare their mean income to the one owned by the p percent of the richest. The new measure is still a Lorenz-based one, but the significant focus is based here in equally sized and opposite parts of the population whose difference is so remarkable nowadays. We then highlight the specific information given by the new inequality measure and curve, by comparing it to the widely employed Lorenz curve and Gini index and the more recent Zenga approach, and provide an application to Italian data on household income, wealth, and consumption along the years 1980–2012. The effects of estimating inequality indices and curves from grouped data are also discussed. [ABSTRACT FROM AUTHOR]

In studies of clinical phenotypes, such as dementia, disability, and frailty, participants are typically assessed at in-person clinic visits. Thus, the precise time of onset for the phenotype is unknown. The discreteness of the clinic visits yields grouped event time data. We investigate how to perform a risk factor analysis in the case of grouped data. Since visits can be months to years apart, numbers of ties can be large, causing the exact tie-handling method of the Cox model to be computationally infeasible. We propose two, new, computationally efficient approximations to the exact method: Laplace approximation and an analytic approximation. Through extensive simulation studies, we compare these new methods to the Prentice-Gloeckler model and the Cox model using Efron's and Breslow's tie-handling methods. In addition, we compare the methods in an application to a large cohort study (N = 3,605) on the development of clinical frailty in older adults. In our simulations, the Laplace approximation has low bias in all settings, and the analytic approximation has low bias in settings where the regression coefficient is not large in magnitude. Their corresponding confidence intervals also have approximately the nominal coverage probability. In the data application, the results from the approximations are nearly identical to that of the Prentice-Gloeckler model. [ABSTRACT FROM AUTHOR]

Rate regulation plays an important role in the financial service of the auto insurance industry. Modelling the Size of Loss distributions, particularly the large loss distribution at the aggregate level, is a key component of rate regulation to produce a benchmark for the industry. However, traditionally actuarial practice is constructing the group-based Size of Loss with irregular intervals. The frequency values associated with each group are counted to form a frequency distribution. However, this approach is limited by lacking a parametric distribution that can capture the underlying randomness of the Size of Loss. In this work, we propose a de-grouping method based on the simulation of the uniform random variate, with and without constraint on knowing the average incurred loss per claim count. Using the de-grouping method, we simulate the individual observation based on the limited information available from the grouped Size of Loss. We have successfully identified the Exponentiated Weibull distribution as a good candidate as it outperforms other heavy-tailed distributions being considered. Our findings provide critical guidance and direction for the practical use of the proposed de-grouping method in auto insurance rate regulation practice and other fields in business and economics. [ABSTRACT FROM AUTHOR]

Multivariable models for prediction or estimating associations with an outcome are rarely built in isolation. Instead, they are based upon a mixture of covariates that have been evaluated in earlier studies (eg, age, sex, or common biomarkers) and covariates that were collected specifically for the current study (eg, a panel of novel biomarkers or other hypothesized risk factors). For that context, we present the multistep elastic net (MSN), which considers penalized regression with variables that can be qualitatively grouped based upon their degree of prior research support: established predictors vs unestablished predictors. The MSN chooses between uniform penalization of all predictors (the standard elastic net) and weaker penalization of the established predictors in a cross-validated framework and includes the option to impose zero penalty on the established predictors. In simulation studies that reflect the motivating context, we show the comparability or superiority of the MSN over the standard elastic net, the Integrative LASSO with Penalty Factors, the sparse group lasso, and the group lasso, and we investigate the importance of not penalizing the established predictors at all. We demonstrate the MSN to update a prediction model for pediatric ECMO patient mortality. [ABSTRACT FROM AUTHOR]

Weibull mixture models are widely used in a variety of fields for modeling phenomena caused by heterogeneous sources. We focus on circumstances in which original observations are not available, and instead the data comes in the form of a grouping of the original observations. We illustrate EM algorithm for fitting Weibull mixture models for grouped data and propose a bootstrap likelihood ratio test (LRT) for determining the number of subpopulations in a mixture model. The effectiveness of the LRT methods are investigated via simulation. We illustrate the utility of these methods by applying them to two grouped data applications. [ABSTRACT FROM AUTHOR]

Background: Time- and dose-to-event phenotypes used in basic science and translational studies are commonly measured imprecisely or incompletely due to limitations of the experimental design or data collection schema. For example, drug-induced toxicities are not reported by the actual time or dose triggering the event, but rather are inferred from the cycle or dose to which the event is attributed. This exemplifies a prevalent type of imprecise measurement called grouped failure time, where times or doses are restricted to discrete increments. Failure to appropriately account for the grouped nature of the data, when present, may lead to biased analyses. Results: We present groupedSurv, an R package which implements a statistically rigorous and computationally efficient approach for conducting genome-wide analyses based on grouped failure time phenotypes. Our approach accommodates adjustments for baseline covariates, and analysis at the variant or gene level. We illustrate the statistical properties of the approach and computational performance of the package by simulation. We present the results of a reanalysis of a published genome-wide study to identify common germline variants associated with the risk of taxane-induced peripheral neuropathy in breast cancer patients. Conclusions: groupedSurv enables fast and rigorous genome-wide analysis on the basis of grouped failure time phenotypes at the variant, gene or pathway level. The package is freely available under a public license through the Comprehensive R Archive Network. [ABSTRACT FROM AUTHOR]

Sampling within a given interval with a constraint has not been previously considered. Standard parametric simulation engines require knowledge of the parameters of the distribution from which a sample is drawn. These methods are limited if additional constrains are required for the simulated data. We propose a method that generates the targeted number of individual observations within a given interval with a constraint that the average value of observations is known. This method is further extended to a grouped data setting, as a way of data de-grouping, when the frequency and average value of observations are provided for each group. Several simulation studies are employed to evaluate the performance of the proposed method, in case of both a single interval and grouped data, for different simulation settings. Furthermore, the proposed method is evaluated in the parameter recovery when different distributions are fitted to the de-grouped data. This method is found to be superior to the uniform method previously used in data de-grouping. The results of the simulation study are promising and they show that this method can be used successfully in the applications where data de-grouping requires that the average value of observations is maintained in each group. The application of the proposed method is illustrated on a real data of insurance losses for bodily injury claims. [ABSTRACT FROM AUTHOR]

This article distinguishes three measures of intergenerational economic mobility that emerge when the population is divided into groups: overall individual mobility, within-group mobility, and between-group mobility. We clarify their properties and the relationship between them. We then evaluate Clark’s use of surname between-group persistence as a preferred measure of intergenerational mobility in the book The Son Also Rises: Surnames and the History of Social Mobility. We show that aggregate surname-level intergenerational persistence cannot be compared with individual persistence because group-level income averages captures diverse individual-level and group-level factors impossible to disentangle without additional identifying information. Furthermore, measures of group persistence do not address the problem of measurement error leading to attenuation bias, which is Clark’s rationale to study surname mobility. An empirical example partitioning the population into groups based on racial/ethnic origins and a simulation clarify the relationship between these different measures of mobility. [ABSTRACT FROM AUTHOR]

Researchers studying income inequality, economic segregation, and other subjects must often rely on grouped data—that is, data in which thousands or millions of observations have been reduced to counts of units by specified income brackets. The distribution of households within the brackets is unknown, and highest incomes are often included in an open-ended top bracket, such as “$200,000 and above.” Common approaches to this estimation problem include calculating midpoint estimators with an assumed Pareto distribution in the top bracket and fitting a flexible multiple-parameter distribution to the data. The authors describe a new method, mean-constrained integration over brackets (MCIB), that is far more accurate than those methods using only the bracket counts and the overall mean of the data. On the basis of an analysis of 297 metropolitan areas, MCIB produces estimates of the standard deviation, Gini coefficient, and Theil index that are correlated at 0.997, 0.998, and 0.991, respectively, with the parameters calculated from the underlying individual record data. Similar levels of accuracy are obtained for percentiles of the distribution and the shares of income by quintiles of the distribution. The technique can easily be extended to other distributional parameters and inequality statistics. [ABSTRACT FROM AUTHOR]

Abstract: Control charts for variables are a reference tool for the statistical monitoring of a quantitative variable. However, there are processes in which the exact measurement of the variable is highly complex or costly. In these cases, one option is to base the monitoring on the number of units that are classified by a gauge as being above, below, or between a pair of reference limits. With this approach, the process control achieves the economy and agility of control by attributes. In the literature, one can find different alternatives to such schemes. In this paper, a more general formulation is introduced that subsumes many of the previously proposed schemes into a new control scheme by means of an additional parameter. Through a specially designed optimization procedure, it is possible to obtain the parameters that maximize the scheme's performance against a specified process shift. [ABSTRACT FROM AUTHOR]

Based on the competing risks grouped data, Bayesian estimation approach is considered for the parameters of the Weibull distribution and the related specific hazard and survival functions. The estimation procedures are carried out under the square error loss (SELF) and linear exponential loss (LINEX) functions. High posterior (HPD) credible intervals for the specified param-eters are also obtained. The derived estimators are in explicit closed forms. Their properties and performance are illustrated through an application to real lifetime's data and an extended simulation study. Overall results indi-cate that, the Bayesian estimators are dominated other estimators obtained by other methods and are recommended when continuous life testing is not available. [ABSTRACT FROM AUTHOR]

The Zenga index of inequality is a new proposal of measuring this phenomenon. The properties and empirical applications of Zenga index (and the underlying Zenga curve) have been recently widely investigated. Its advantages (and differences, as compared to other existing measures) have been pointed out. However, one of the possible problems associated with the use of the Zenga index has not yet been appropriately addressed. Namely, the Zenga index assumes different values depending on whether it is applied to grouped or ungrouped data. As it may seem that due to contemporary computers power it is not necessary to group data, the problem still exists as a conceptual one. Moreover, in some situations - such as applying equivalence scales - avoiding grouping of the data is not possible even in principle. The problem is stated, illustrated by simple numerical examples and briefly discussed in this paper. [ABSTRACT FROM AUTHOR]

Grouped data are commonly encountered in applications. All data from a continuous population are grouped due to rounding of the individual observations. The Bernstein polynomial model is proposed as an approximate model in this paper for estimating a univariate density function based on grouped data. The coefficients of the Bernstein polynomial, as the mixture proportions of beta distributions, can be estimated using an EM algorithm. The optimal degree of the Bernstein polynomial can be determined using a change-point estimation method. The rate of convergence of the proposed density estimate to the true density is proved to be almost parametric by an acceptance–rejection argument used for generating random numbers. The proposed method is compared with some existing methods in a simulation study and is applied to the Chicken Embryo Data. [ABSTRACT FROM PUBLISHER]

It is widely believed that the median is “usually” between the mean and the mode for skewed unimodal distributions. However, this inequality is not always true, especially with grouped data. Unavailability of complete raw data further necessitates the importance of evaluating this characteristic in grouped data. There is a gap in the current statistical literature on assessing mean–median–mode inequality for grouped data. The study aims to evaluate the relationship between the mean, median, and mode with unimodal grouped data; derive conditions for their inequalities; and present their application. [ABSTRACT FROM PUBLISHER]

In this article I develop a wild bootstrap procedure for cluster-robust inference in linear quantile regression models. I show that the bootstrap leads to asymptotically valid inference on the entire quantile regression process in a setting with a large number of small, heterogeneous clusters and provides consistent estimates of the asymptotic covariance function of that process. The proposed bootstrap procedure is easy to implement and performs well even when the number of clusters is much smaller than the sample size. An application to Project STAR data is provided. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Finite Mixture of Regressions (FMR) are among the most widely used models for dealing with heterogeneity in regression problems. FMR is a model-based clustering approach that models the data by assigning individual observations to one of the K latent regression clusters. In some applications, it is desired to cluster groups of observations together rather than individually. We present an extension to the regular FMR, that we call the Grouped Mixture of Regressions (GMR), which allows for a known group structure among observations, in addition to their possibly unknown heterogeneity. The research is motivated by a large financial longitudinal dataset from a brand of automotive dealerships across the United States. The task was to cluster the dealers based on their financial performance and also improve performance prediction for individual dealers. We derive a fast and salable algorithm for estimating the model parameters using Expectation–Maximization (EM). We also show how the group structure can improve prediction by sharing information among members of each group, as reflected in the posterior predictive density under GMR. The performance of the approach is assessed using both synthetic data as well as the dealership data. Among our findings are the superior predictive performance of GMR and the ability to correctly detect the number of clusters using simple cross-validation. • Performance evaluation of automotive dealerships and retail stores in general. • Clustering automotive dealerships and retail stores in general. • Finite mixture of regressions with group structure. • Posterior predictive density for FMR with group structure. [ABSTRACT FROM AUTHOR]

This paper introduces a phase-type software reliability model (PH-SRM) and develops parameter estimation algorithms with grouped data. The PH-SRM is one of the most flexible models, which contains the existing non-homogeneous Poisson process (NHPP) models, and can approximate any type of NHPP-based models with high accuracy. Hence PH-SRM is promising to reduce the effort to select the best models in software reliability assessment. However, PH-SRM may involve many parameters compared to typical NHPP models. Thus the efficient parameter estimation algorithm is required. This paper enhances the parameter estimation algorithms for PH-SRM, so that they can handle grouped data. The grouped data is commonly applied to collect the data such as the number of bugs per day in practice. Thus the presented algorithms are helpful for the reliability assessment in practical software development project. Concretely, we consider the EM (expectation-maximization) algorithm for PH-SRM with both fault-detection time and grouped data. Finally, we examine performance of PH-SRM from the viewpoints of fitting ability. [ABSTRACT FROM AUTHOR]

Interval-censored data are very common in the reliability and lifetime data analysis. This paper investigates the performance of different estimation procedures for a special type of interval-censored data, i.e. grouped data, from three widely used lifetime distributions. The approaches considered here include the maximum likelihood estimation, the minimum distance estimation based on chi-square criterion, the moment estimation based on imputation (IM) method and an ad hoc estimation procedure. Although IM-based techniques are extensively used recently, we show that this method is not always effective. It is found that the ad hoc estimation procedure is equivalent to the minimum distance estimation with another distance metric and more effective in the simulation. The procedures of different approaches are presented and their performances are investigated by Monte Carlo simulation for various combinations of sample sizes and parameter settings. The numerical results provide guidelines to analyse grouped data for practitioners when they need to choose a good estimation approach. [ABSTRACT FROM AUTHOR]

Researchers often estimate income inequality by using data that give only the number of cases (e.g., families or households) whose incomes fall in “bins” such as $ 0 to $9,999, $10,000 to $14,999, . . . , ≥$200,000. We find that popular methods for estimating inequality from binned incomes are not robust in small samples, where popular methods can produce infinite, undefined, or arbitrarily large estimates. To solve these and other problems, we develop two improved estimators: a robust Pareto midpoint estimator (RPME) and a multimodel generalized beta estimator (MGBE). In a broad evaluation using U.S. national, state, and county data from 1970 to 2009, we find that both estimators produce very good estimates of the mean and Gini coefficient but less accurate estimates of the Theil index and mean log deviation. Neither estimator is uniformly more accurate, but the RPME is much faster, which may be a consideration when many estimates must be obtained from many data sets. We have made the methods available as the rpme and mgbe command for Stata and the binequality package for R. [ABSTRACT FROM AUTHOR]

Ungrouping binned data can be desirable for many reasons: Bins can be too coarse to allow for accurate analysis; comparisons can be hindered when different grouping approaches are used in different histograms; and the last interval is often wide and open-ended and, thus, covers a lot of information in the tail area. Age group-specific disease incidence rates and abridged life tables are examples of binned data. We propose a versatile method for ungrouping histograms that assumes that only the underlying distribution is smooth. Because of this modest assumption, the approach is suitable for most applications. The method is based on the composite link model, with a penalty added to ensure the smoothness of the target distribution. Estimates are obtained by maximizing a penalized likelihood. This maximization is performed efficiently by a version of the iteratively reweighted least-squares algorithm. Optimal values of the smoothing parameter are chosen by minimizing Akaike’s Information Criterion.We demonstrate the performance of this method in a simulation study and provide several examples that illustrate the approach. Wide, open-ended intervals can be handled properly. The method can be extended to the estimation of rates when both the event counts and the exposures to risk are grouped. [ABSTRACT FROM AUTHOR]

In this article, the two-step method for prediction, which was proposed by Li et al. (2012), is extended for modelling grouped data, which besides having near-collinear explanatory variables, also having different mean structure, i.e. the mean structure of some part of the data is more complex than other parts. In the first step, inspired by partial least squares regression (PLS), the information for explanatory variables is summarized by a multilinear model with Krylov structured design matrices, which for different groups have different size. The multilinear model is similar to the classical growth curve model except that the design matrices are unknown and are functions of the dispersion matrix. Under such a multilinear model, natural estimators for mean and dispersion matrices are proposed. In the second step, the response is predicted through a conditional predictor where the estimators obtained in the first step are utilized. [ABSTRACT FROM AUTHOR]

This article outlines an alternative maximum likelihood gamma parameter estimator for grouped and truncated data. Such data might be produced by traditional disdrometer instruments for recording rain drop sizes. The method is an adaptation of a log-linear generalized linear model ( GLM), forming an iterative process which quickly converges to very near maximum likelihood estimates. Simulations with a Joss-Waldvogel bin regime show that the estimator has little bias. Application to observed data from two different types of disdrometers yields results practically equal to published maximum likelihood estimates. [ABSTRACT FROM AUTHOR]

In this paper, the optimal design of a progressively group-censored life test with the restriction of experimental budget is developed for the Chen distribution is considered. The maximum likelihood estimates, approximate confidence intervals for the parameters based on progressively group-censored sample are obtained. Wu et al.'s (2008a) approach is used to determine the number of test units, the number of inspections and the length of inspection interval of a life test under a pre-determined budget of experiment such that the determinant of the asymptotic variances-covariance matrix of estimators of parameters is minimum. A numerical example is presented and the sensitivity analysis is also studied. [ABSTRACT FROM AUTHOR]

62H12, 62H12, 60E05, 60E05, quartile estimation, generalised lambda distribution, grouped data, quartile-based inequality measures We show that interval estimators for quantile-based inequality measures are suited to this type of data. Keywords and phrases: quartile estimation, generalised lambda distribution, grouped data, quartile-based inequalitymeasures. [Extracted from the article]

Abstract: In this paper, we consider the multiple testing problems for grouped hypotheses. Two procedures are proposed based on weighted p-values, where the weights for p-values are obtained by maximizing a power-related objective function. We find that the proposed procedures can control the false discovery rate asymptotically, and are more powerful than existing methods asymptotically. We further examine their performances with extensive simulations. For illustration, we apply the proposed methods to the adequate yearly progress data. [Copyright &y& Elsevier]

We consider asymmetric kernel estimates based on grouped data. We propose an iterated scheme for constructing such an estimator and apply an iterated smoothed bootstrap approach for bandwidth selection. We compare our approach with competing methods in estimating actuarial loss models using both simulations and data studies. The simulation results show that with this new method, the estimated density from grouped data matches the true density more closely than with competing approaches. [ABSTRACT FROM AUTHOR]

In this article, optimal design under the restriction of pre-determined budget of experiment is developed for the Pareto distribution when the life test is progressively group censored. We use the maximum-likelihood method to obtain the point estimator of the Pareto parameter. We propose two approaches to decide the number of test units, the number of inspections, and the length of inspection interval under limited budget such that the asymptotic variance of estimator of Pareto parameter is minimum. A numerical example is given to illustrate the proposed method. Some sensitivity analysis is also studied. [ABSTRACT FROM AUTHOR]

We consider the estimation problem under the Lehmann model with interval-censored data, but focus on the computational issues. There are two methods for computing the semi-parametric maximum likelihood estimator (SMLE) under the Lehmann model (or called Cox model): the Newton-Raphson (NR) method and the profile likelihood (PL) method. We show that they often do not get close to the SMLE. We propose several approach to overcome the computational difficulty and apply our method to a breast cancer research data set. [ABSTRACT FROM AUTHOR]

Coarse data is a general type of incomplete data that includes grouped data, censored data, and missing data. The likelihood-based estimation approach with coarse data is challenging because the likelihood function is in integral form. The Monte Carlo EM algorithm of Wei & Tanner [Wei & Tanner (1990). Journal of the American Statistical Association, 85, 699-704] is adapted to compute the maximum likelihood estimator in the presence of coarse data. Stochastic coarse data is also covered and the computation can be implemented using the parametric fractional imputation method proposed by Kim [Kim (2011). Biometrika, 98, 119-132]. Results from a limited simulation study are presented. The proposed method is also applied to the Korean Longitudinal Study of Aging (KLoSA). The Canadian Journal of Statistics 40: 604-618; 2012 © 2012 Statistical Society of Canada [ABSTRACT FROM AUTHOR]

In recent advances of Weibull generalization, the odd Weibull family has been shown to be useful not only for lifetime data modeling, but also discriminating between Weibull and inverse Weibull distributions. This three-parameter distribution accommodates seven different hazard shapes and a wide variety of shapes of the density function including bimodality. In addition, the odd Weibull parameters can be estimated in two different ways since the inverse transformation of the family does not change its density function. In this article, we adapted this two-way estimation method for analyzing grouped, censored, and truncated data that frequently encountered in survival analysis. [ABSTRACT FROM AUTHOR]

We consider the task of Bayesian inference of the mean of normal observations when the available data have been discretized and when no prior knowledge about the mean and the variance exists. An application is presented which illustrates that the discretization of the data should not be ignored when their variability is of the order of the discretization step. We show that the standard (noninformative) prior for location-scale family distributions is no longer appropriate. We work out the reference prior of Berger and Bernardo, which leads to different and more reasonable results. However, for this prior the posterior also shows some non-desirable properties. We argue that this is due to the inherent difficulty of the considered problem, which also affects other methods of inference. We therefore complement our analysis by an empirical Bayes approach. While such proceeding overcomes the disadvantages of the standard and reference priors and appears to provide a reasonable inference, it may raise conceptual concerns. We conclude that it is difficult to provide a widely accepted prior for the considered problem. [ABSTRACT FROM AUTHOR]

In this paper, a progressive type I group-censoring life test for Burr XII distribution is considered. We use the maximum likelihood method to obtain the point estimators of the Burr XII parameters. The approximate confidence intervals for the parameters of Burr XII distribution are also obtained. We use modified algorithm proposed by Kus et al. [20] to decide the number of test units, number of inspections, and length of inspection interval under a restricted budget of experiment such that the determinant of the asymptotic variances-covariance of estimators of parameters is minimum. A numerical example is presented to illustrate the proposed approach. The sensitivity analysis is also investigated. [ABSTRACT FROM AUTHOR]