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The log-logistic distribution is one of the popular distributions in life-testing applications. This article develops an acceptance sampling procedure for the log-logistic lifetime distribution based on grouped data when the shape parameter is given. Both producer and consumer risks are considered to develop the ordinary, approximate and simulated sampling plans. Some of the proposed sampling plans are tabulated; moreover, those three types of sampling plans are compared with each other under the same censoring rates. The use of these tables is illustrated by an example. [ABSTRACT FROM AUTHOR]

In this paper, the estimation of parameters based on a progressively type-I interval censored sample from a two-parameter Weibull distribution is studied. Different methods of estimation are discussed. They include the maximum likelihood estimation, method of moments, estimation based on Weibull probability plot, mid-point approximation method and one-step approximation method. The estimation procedures are discussed in details and compared via Monte Carlo simulations in terms of their biases and mean square errors. Some recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of estimation developed here. [ABSTRACT FROM AUTHOR]

Motivated by certain models arising in the field of statistical process control and acceptance sampling, we study the waiting-time distribution for the first occurrence of a pattern belonging to a specific family of patterns in a bivariate sequence of trinomial trials. Such a setup can be applied in cases where the final quality of the output of an industrial process is determined by two correlated variables that are used to classify the two measured characteristics as "quite close to target quality," "of satisfactory quality," and "far from target quality." A typical case where this happens is whenever registering the exact value of the characteristic of interest is expensive or difficult (e.g., when the observations come in the form of grouped data, due to the use of m-step gauges). The exact distribution of the waiting time is derived by using a Markov chain-embedding technique, which is then used in order to determine the most appropriate design for the quality-control problems discussed. [ABSTRACT FROM AUTHOR]

This paper considers a life test under progressive type I group censoring with a Weibull failure time distribution. The maximum likelihood method is used to derive the estimators of the parameters of the failure time distribution. In practice, several variables, such as the number of test units, the number of inspections, and the length of inspection interval are related to the precision of estimation and the cost of experiment. An inappropriate setting of these decision variables not only wastes the resources of the experiment but also reduces the precision of estimation. One problem arising from designing a life test is the restricted budget of experiment. Therefore, under the constraint that the total cost of experiment does not exceed a pre-determined budget, this paper provides an algorithm to solve the optimal decision variables by considering three different criteria. An example is discussed to illustrate the proposed method. The sensitivity analysis is also studied. [ABSTRACT FROM AUTHOR]

We present estimators for semiparametric regression models where the dependent variable is grouped, that is, known to fall in a specified group with observable thresholds while its true value remains latent. Income, weeks unemployed, and treatment length are examples of such variables. Because the model is not amenable to direct estimation, estimators are derived from a transform in which the index emerges as the partially linear component in a vector of identities. √n asymptotic normality of the proposed estimator is derived. The analytical results are applied to study physicians' provision of charity care, using data in which charity care is grouped. [ABSTRACT FROM AUTHOR]

Abstract: The term grouped data refers to the case where the exact value of the characteristic of interest is either unknown or difficult to register. In the present article we study a model that can be used for the simultaneous control of two (possibly correlated) variables whose values have been registered in the form of grouped data. The exact distribution of the waiting time for an out of control signal through the suggested scheme and its ARL are investigated by using a Markov Chain embedding methodology and by establishing a recurrence scheme for the respective tail probabilities. A detailed study of the performance of the scheme is also carried out when the characteristics of interest follow the Marsall–Olkin’s bivariate Exponential Distribution. [Copyright &y& Elsevier]

Estimation concerning the parameters of the lognormal distribution are discussed for progressively Type I interval censored samples. Maximum likelihood estimators as well as approximate Bayes estimators of the parameters are developed. The approximate asymptotic variance covariance matrix of the maximum likelihood estimates is given and the posterior risks for the Bayes estimates are derived. Illustrative examples involving real and simulated data sets show that the Bayes estimates provide more consistent results than the maximum likelihood estimates for the data discussed. [ABSTRACT FROM AUTHOR]

For reducing the experimental time and considering the purpose of administrative convenience, the article establishes an acceptance sampling procedure for the Rayleigh lifetime distribution under a step-stress test with an equally-spaced interval censoring scheme. Both producer and consumer risks are considered and an algorithm is provided to develop the ordinary acceptance sampling plans. A numerical study is conducted for evaluating the performance of the proposed method. Moreover, a numerical example is used for illustration. [ABSTRACT FROM AUTHOR]

Abstract: Some traditional life tests result in no or very few failures by the end of test. In such cases, one approach is to do life testing at higher-than-usual stress conditions in order to obtain failures quickly. This paper discusses a k-level step-stress accelerated life test under type I progressive group-censoring with random removals. An exponential failure time distribution with mean life that is a log-linear function of stress and a cumulative exposure model are considered. We derive the maximum likelihood estimators of the model parameters and establish the asymptotic properties of the estimators. We investigate four selection criteria which enable us to obtain the optimum test plans. One is to minimize the asymptotic variance of the maximum likelihood estimator of the logarithm of the mean lifetime at use-condition, and the other three criteria are to maximize the determinant, trace and the smallest eigenvalue of Fisher''s information matrix. Some numerical studies are discussed to illustrate the proposed criteria. [Copyright &y& Elsevier]

The basic approach to estimating the Gini and extended Gini indices is to approximate the Lorenz curve by a number of linear segments, and then estimate the Gini coefficients as the areas (or weighted areas) between the linear segments and the 45-degree line. We show that the estimator for the extended Gini, obtained from the Lorenz curve (Chotikapanich and Griffiths, 2001) is algebraically identical to a covariance-based estimator. The advantages of the covariance-based estimators are twofold; first, they are easy to compute, using any standard statistical software, and second, the covariance-based estimators allow for the decomposition of the Gini index of a sum of variables (or populations). [ABSTRACT FROM AUTHOR]

Modern databases enable the estimation of conditional distributions for Y given X in a specified interval. Confidentiality requirements typically mean that individual observations cannot be released and output may be restricted to Y values for X within specified intervals. The width of such an interval could have a major impact upon the quality of the estimates. In this article, we develop an improved estimation procedure to reduce the impact of interval width, using a new variable whose distribution does not vary over the X interval in question. The method is illustrated using data on housing prices and income tax payment. [ABSTRACT FROM AUTHOR]

In the context of the univariate Gaussian mixture with grouped data, it is shown that the global maximum of the likelihood may correspond to a situation where a Dirac lies in any non-empty interval. Existence of a domain of attraction near such a maximizer is discussed and we establish that the expectation-maximization (EM) iterates move extremely slowly inside this domain. These theoretical results are illustrated both by some Monte-Carlo experiments and by a real data set. To help practitioners identify and discard these potentially dangerous degenerate maximizers, a specific stopping rule for EM is proposed. [ABSTRACT FROM AUTHOR]

This article considers the problem of testing the validity of the assumption that the underlying distribution of life is Pareto. For complete and censored samples, the relationship between the Pareto and the exponential distributions could be of vital importance to test for the validity of this assumption. For grouped uncensored data the classical Pearson χ2 test based on the multinomial model can be used. Attention is confined in this article to handle grouped data with withdrawals within intervals. Graphical as well as analytical procedures will be presented. Maximum likelihood estimators for the parameters of the Pareto distribution based on grouped data will be derived. [ABSTRACT FROM AUTHOR]

In this paper, we propose the conditional optimal spacing defined as the optimal spacing after specifying a predetermined order statistic. If we specify a censoring time, then the optimal inspection times for grouped inspection can be determined from this conditional optimal spacing. We take an example of exponential distribution, and provide a simple method of finding the conditional optimal spacing. [ABSTRACT FROM AUTHOR]

Abstract: We consider goodness of fit tests for the Rayleigh distribution with grouped data. New Kolmogrov–Smirnov type tests are suggested and compared with the traditional chi-square and likelihood ratio tests. The results show that some of the suggested tests have a good power performance as compared with the traditional ones. [Copyright &y& Elsevier]

The minimum Hellinger distance estimator is known to have desirable properties in terms of robustness and efficiency. We propose an approximate minimum Hellinger distance estimator by adapting the approach to grouped data from a continuous distribution. It is easier to compute the approximate version for either the continuous data or the grouped data. Given certain conditions on the model distribution and reasonable grouping rules, the approximate minimum Hellinger distance estimator is shown to be consistent and asymptotically normal. Furthermore, it is robust and can be asymptotically as efficient as the maximum likelihood estimator. The merit of the estimator is demonstrated through simulation studies and real data examples. [ABSTRACT FROM AUTHOR]

Mixed-effects models have become a popular approach for the analysis of grouped data that arise in many areas as diverse as clinical trials, epidemiology, and sociology. Examples of grouped data include longitudinal data, repeated measures, and multilevel data. In the case of linear mixed-effects (LME) models, the likelihood function can be expressed in closed form, with efficient computational algorithms having been proposed for maximum likelihood and restricted maximum likelihood estimation. For nonlinear mixed-effects (NLME) models and generalized linear mixed models (GLMM5), however, the likelihood function does not have a closed form. Different likelihood approximations, with varying degrees of accuracy and computational complexity, have been proposed for these models. This article describes algorithms for one such approximation, the adaptive Gaussian quadrature (AGQ), for GLMMs which scale up efficiently to multilevel models with arbitrary number of levels. The proposed algorithms greatly reduce the computational complexity and the memory usage for approximating the multilevel GLMM likelihood, when compared to a direct application of a single-level AGQ approximation algorithm to the multilevel case. The accuracy of the associated estimates is evaluated and compared to that of estimates obtained from other approximations via simulation studies. [ABSTRACT FROM AUTHOR]

Accelerated life testing of products is used to get information quickly on their lifetime distributions. This paper discusses a k-stage step-stress accelerated life test under progressive type I censoring with grouped data. An exponential lifetime distribution with mean life that is a log-linear function of stress is considered. A cumulative exposure model is also assumed. We use the maximum likelihood method to obtain the estimators of the model parameters. The methods for obtaining the optimum test plan are investigated using the variance-optimality and D-optimality criteria. Some numerical studies are discussed to illustrate the proposed criteria. [ABSTRACT FROM AUTHOR]

Various inequality and social welfare measures often depend heavily on the choice of a distribution of income. Picking a distribution that best fits the data involves throwing away information and does not allow for the fact that a wrong choice can be made. Instead, Bayesian model averaging utilizes a weighted average of the results from a number of income distributions, with each weight given by the probability that a distribution is ‘correct’. In this study, prior densities are placed on mean income, the mode of income and the Gini coefficient for Australian income units with one parent (1997–8). Then, using grouped sample data on incomes, posterior densities for the mean and mode of income and the Gini coefficient are derived for a variety of income distributions. The model-averaged results from these income distributions are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Grouped data are common but often improperly treated in welfare and child welfare research. Conventional regression models are not appropriate for analysis of this type of data, because the presence of intra-class correlation among study subjects from the same group violates the assumption that observations are independent of one another. This study demonstrates the advantages of using hierarchical linear modeling (HLM) to analyze grouped data found in the 1997 Child Development Supplement to the Panel Study of Income Dynamics. Specifically, this article presents an HLM example investigating intergenerational dependence on welfare and its relation to child academic achievement. Results show that HLM is a robust and flexible tool that can effectively test various types of research hypotheses, particularly those concerning multilevel influences and macro-to-micro relations. The study shows that early educational intervention is essential in improving child academic achievement for children receiving welfare, particularly for those who used welfare for most of their own childhood and whose caregivers never used welfare. [Copyright &y& Elsevier]

A new incomplete data model, the trunsored model, in lifetime analysis is introduced. This model can be regarded as a unified model of the censored and truncated models. Using the model, we can not only estimate the ratio of the fragile population to the mixed fragile and durable populations, but also test a hypothesis that the ratio is equal to a prescribed value. A central point of the paper is that such a test can easily be realized through the newly introduced trunsored model, because it has been difficult to do such a hypothesis test under only the framework of censored and truncated models. Therefore, the relationship of the trunsored model to the censored and truncated models is clarified because the trunsored model unifies the censored and truncated models. The paper also shows how to obtain the estimates of the parameters in lifetime estimation, and corresponding confidence intervals for the fragile population. Typical examples applied to electronic board failures. and to breast cancer data, for lifetime estimation are demonstrated, and successfully worked using the trunsored model. [ABSTRACT FROM AUTHOR]

In several applications data are grouped and there are within-group correlations. With continuous data, there are several available models that are often used; with counting data, the Poisson distribution is the natural choice. In this paper a mixed log-linear model based on a Poisson–Poisson conditional distribution is presented. The initial model is a conditional model for the mean of the response variable, and the marginal model is formed thereafter. Random effects with Poisson distribution are introduced and a variance-covariance matrix for the response vector is formed embodying the covariance structure induced by the grouping of the data.

Abstract: Length and weight data are often analyzed in fisheries science to derive a parametric weight–length relationship for estimating biomass and to develop indices of condition for comparing the ‘wellness’ of different populations of fish. However, analysis of such data often ignores the inherent spatial and temporal grouping of the observations, and hence, the data hierarchy. This paper proposes the use of linear mixed-effects models as an effective means of analyzing and comparing weight–length relationships and indices of condition when there are many groups. The use of simple linear regression (where grouping is ignored), ANCOVA (where group effects are incorporated as fixed-effects), and linear mixed-effects models (where group effects are random-effects) are compared using data for Atlantic sea scallops (Placopecten magellanicus). The group means of residuals is proposed as a measure of relative weight or index of population condition. Linear mixed-effects models should be used to analyze grouped data because the variability among groups is ignored in simple linear regression and ANCOVA. Also, it is important that explanatory variables be incorporated in analyses of grouped data because their influence may mask the true differences among groups. [Copyright &y& Elsevier]

In this paper we propose inference methods based on the EM algorithm for estimating the parameters of a weakly parameterised competing risks model with masked causes of failure and second‐stage data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the cause‐specific hazard functions and of the masking probabilities is performed via an EM algorithm. Both the E‐ and M‐steps can be solved in closed form under the full model and under some restricted models of interest. We illustrate the flexibility of the method by showing how grouped data and tests of common hypotheses in the literature on missing cause of death can be handled. The method is applied to a real dataset and the asymptotic and robustness properties of the estimators are investigated through simulation. [ABSTRACT FROM AUTHOR]

A new method along with Bayesian approach for estimating the parameter in the distribution function F(x;θ) by using grouped data is developed in this paper. The support of F(x;θ) is divided into disjointed intervals as -∞ = x0 < x1 < · · · < xk-1 < xk = +∞. Grouped data are the numbers of observations falling in the intervals. This method can be applied to estimate not only the parameter in one population model but also the parameters in multi-population model which is subject to the order restrictions. For it is not easy to present the prior distribution of the parameter θ in F(x; θ) by grouped data directly, it is considered the prior distribution of the probabilities of observations falling in the intervals denoted by Pj(θ) = F(xj;θ) - F(xj-1;θ),j = 1, · · ·, k. The probabilities follow the multivariate distribution and can be regarded as the function of the parameter θ. Pearson divergence D(p; q) is introduced to scale the distance between the probabilities Pj(θ),j = 1 · · ·, k and the samples from the posterior distribution (Dirichlet distribution) of the probabilities. Then by minimizing the Pearson divergence D(p;q), the 'posterior' samples of the parameter θ can be obtained, through which statistical inference including Bayesian-Pearson Divergence Estimator of the parameter can be processed. Simulations and a numerical example employing this method are presented. [ABSTRACT FROM AUTHOR]

In the present paper, an estimator of the shape parameter of the Pareto failure model is presented using grouped data. This estimator is based on obtaining the parameter in terms of the hazard rate, then replacing the unknown hazard rate by a grouped data estimator available in the literature. Death records are given as a numerical illustration in the medical context. The relation between the hazard rate and the income elasticity is derived. This relation allows the presentation of the same estimator in terms of the income elasticity so that it could be used in an economic context. Two illustrations are presented using income data. Simulated data are generated to compare the estimator with the maximum likelihood estimator. [ABSTRACT FROM AUTHOR]

A general frequency modulated (GFM) signal characterizes the vibrations produced by compressors, turbines, propellers, gears, and other rotating machines in a dynamic environment. A GFM signal is defined as the composition of a real or complex, periodic, or almost-periodic carrier function with a real, differentiable modulation function. A GFM signal therefore contains sinusoids whose frequencies are (possibly nonintegral) multiples of a fundamental; to distinguish a GFM signal from a set of unrelated sinusoids, it is necessary to track them as a group. This paper develops the general frequency modulation tracker (GFMT) for one or more GFM signals in noise using the expectation/conditional maximization (ECM) algorithm that is an extension of the expectation-maximization (EM) algorithm. Three advantages of this approach are that the ratios (harmonic numbers) of the carrier functions do not need to be known a priori, that the parameters of multiple signals are estimated simultaneously, and that the GFMT algorithm exploits knowledge of the noise spectrum so that a separate normalization procedure is not required. Several simulated examples are presented to illustrate the algorithm's performance. [ABSTRACT FROM AUTHOR]

We introduce in this paper an iterative estimation procedure based on conditional medians valid to fit linear models when, on the one hand, the distribution of errors, assumed to be known, may be general and, on the other, the dependent data stem from different sources and, consequently, may be either non-grouped or grouped with different classification criteria. The procedure requires us at each step to interpolate the grouped data and is similar to the EM algorithm with normal errors. The expectation step has been replaced by a median-based step which avoids doing awkward integration with general errors and, also, we have substituted for the maximisation step, a natural one which only coincides with it when the errors are normally distributed. With these modifications, we have proved that the iterative estimating algorithm converges to a point which is unique and non-dependent on the starting values. Finally, our final estimate, being a Huber type M-estimator, may enjoy good stochastic asymptotic properties which have also been investigated in detail. [Copyright &y& Elsevier]

In this paper we introduce an iterative estimation procedure based on conditional modes suitable to fit linear models when errors are known to be unimodal and, moreover, the dependent data stem from different sources and, consequently, may be either non-grouped or grouped with different classification criteria. The procedure requires, at each step, the imputation of the exact values of the grouped data and runs by means of a process that is similar to the EM algorithm with normal errors. The expectation step has been substituted with a mode step that avoids awkward integration with general errors and, in addition, we have substituted the maximisation step with a natural one which only coincides with it when the error distribution is normal. Notwithstanding the former modifications, we have proved that, on the one hand, the iterative estimating algorithm converges to a point which is unique and non-dependent on the starting values and, on the other hand, our final estimate, being an M-estimator, may enjoy good stochastic asymptotic properties such as consistency, boundness in L 2, and limit normality. [ABSTRACT FROM AUTHOR]