A generalized model of logistic regression for clustered data

This paper proposes a generalized logistic regression model that can account for the correlation among responses on subunits. The subunits may arise as data on multiple observations within an individual. This method generalizes earlier work by Rosner (1984 a,b) and others. Methodological generalizations include: (1) the use of the more general Polya-Eggenberger distribution instead of the beta-binomial distribution to model the correlation structure, so that cases with negative, positive, or zero intraclass correlation can be handled; (2) a stepwise approach; (3) linear and non-linear regression; and, (4) the inclusion of the case of a truncated distribution. The model can accommodate missing data and covariates on the unit and subunit level. The derivative-free simplex algorithm is used to estimate the parameters.

The model is applied to data describing the progression of obstruction in coronary disease where multiple arterial segments are studied for each patient. The correlation in response that may exist for these multiple segments is accounted for in the analyses while attempting to examine associations with individual-specific (e.g., history of diabetes) and segment-specific (e.g., initial percent stenosis) covariates. Analyses were performed on a data set describing 382 patients with unoperated coronary artery disease and two coronary angiograms separated by at least one month and on a data set describing 284 patients undergoing percutaneous transluminal coronary angioplasty and studied by coronary angiograms.     

An Empirical Bayes Method for Multivariate Meta-analysis with an Application in Clinical Trials

We propose an empirical Bayes method for evaluating overall and study-specific treatment effects in multivariate meta-analysis with binary outcome. Instead of modeling transformed proportions or risks via commonly used multivariate general or generalized linear models, we directly model the risks without any transformation. The exact posterior distribution of the study-specific relative risk is derived. The hyperparameters in the posterior distribution can be inferred through an empirical Bayes procedure. As our method does not rely on the choice of transformation, it provides a flexible alternative to the existing methods and in addition, the correlation parameter can be intuitively interpreted as the correlation coefficient between risks.     

On the beta-correlated binomial (bcb) distribution - a three parameter generalization of the binomial distribution

A modification of the beta-correlated binomial (BCB) distribution of Paul (1985) is proposed. This modification over- comes some theoretical difficulties encountered in the original BCB distribution.     

Bayesian analysis of the difference of two proportions

In this article we give the expression of the prior distribution of p₁-P₂, where P₁ and P₂ and independent proportions with a beta prior each. The expression derived for the posterior distribution of P₁-P₂ then shows the closure of the beta-difference family for independent dual Bernoulli samples. Other bayesian results are also presented.     

Generalisation of Mood's Test to Factorial Designs to Detect Heterogeneity of Variances

Mood's test, which is a relatively old test (and the oldest non‐parametric test among those tests in its class) for determining heterogeneity of variance, is still being widely used in different areas such as biometry, biostatistics and medicine. Although it is a popular test, it is not suitable for use on a two‐way factorial design. In this paper, Mood's test is generalised to the 2 × 2 factorial design setting and its performance is compared with that of Klotz's test. The power and robustness of these tests are examined in detail by means of a simulation study with 10,000 replications. Based on the simulation results, the generalised Mood's and Klotz's tests can especially be recommended in settings in which the parent distribution is symmetric. As an example application we analyse data from a multi‐factor agricultural system that involves chilli peppers, nematodes and yellow nutsedge. This example dataset suggests that the performance of the generalised Mood test is in agreement with that of the generalised Klotz's test.     

Extended generalised Pareto models for tail estimation

The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.     

A generalised NGINAR(1) process with inflated‐parameter geometric counting series

In this paper we propose a new stationary first‐order non‐negative integer valued autoregressive process with geometric marginals based on a generalised version of the negative binomial thinning operator. In this manner we obtain another process that we refer to as a generalised stationary integer‐valued autoregressive process of the first order with geometric marginals. This new process will enable one to tackle the problem of overdispersion inherent in the analysis of integer‐valued time series data, and contains the new geometric process as a particular case. In addition various properties of the new process, such as conditional distribution, autocorrelation structure and innovation structure, are derived. We discuss conditional maximum likelihood estimation of the model parameters. We evaluate the performance of the conditional maximum likelihood estimators by a Monte Carlo study. The proposed process is fitted to time series of number of weekly sales (economics) and weekly number of syphilis cases (medicine) illustrating its capabilities in challenging cases of highly overdispersed count data.     

Identifiability of a generalized mabkov-polya damage model

In the present paper, certain random damage models are examined, such as the generalized MARKOV-POLY A (GMP), the Quasi-Binomial, and the Quasi-Hypergeo-metric, in which an integer random variable N is reduced to B. Following JANAEDAN (1973 b) who has characterized the Multivariate Hypergeometric distribution in terms of the Multinomial, we have shown that under the GMP damage model, the distributions of N and B both belong to the family of the generalised POLYA-EGGENBERGER (GPE) distributions. We have also shown that the damage model can be uniquely identified as the GMPD given that B and N belong to the same GPE family. A physical interpretation of the result is given     

Simulating Media Exposure Distributions

We examine three media exposure distribution (e.d.) simulation methods. The first is based on the maximum likelihood estimate of an individual's exposure, the second on ‘personal probability’ (Greene 1970) and the third on a dependent Bernoulli trials model (Klotz 1973). The last method uses population exposure probabilities rather than individual exposure probabilities, thereby markedly reducing computation time. Magazine exposure data are used to compare the accuracy and computation times of the simulation methods with a log–linear e.d. model (Danaher 1988b) and the popular Metheringham (1964) model based on the beta–binomial distribution (BBD). The results show that the simulation methods are not as accurate as the log– linear model but are more accurate than Metheringham's model, However, all the simulation methods take less computation time than the log–linear model for schedules with more than six magazines, making them viable competitors for large schedule sizes     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

An Easily Implemented Generalised Inverse Gaussian Generator

Using the ‘ratio’ method an easily implemented algorithm is derived for the generalised inverse Gaussian distribution. Computer timings and efficiency calculations show that the procedure is fast over a wide range of distribution parameter values.     

Deviance residuals in generalised log-gamma regression models with censored observations

In this article, we compare three residuals based on the deviance component in generalised log-gamma regression models with censored observations. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and the empirical distribution of each residual is displayed and compared with the standard normal distribution. For all cases studied, the empirical distributions of the proposed residuals are in general symmetric around zero, but only a martingale-type residual presented negligible kurtosis for the majority of the cases studied. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for the martingale-type residual in generalised log-gamma regression models with censored data. A lifetime data set is analysed under log-gamma regression models and a model checking based on the martingale-type residual is performed.     

Loss-based approach to two-piece location-scale distributions with applications to dependent data

Two-piece location-scale models are used for modeling data presenting departures from symmetry. In this paper, we propose an objective Bayesian methodology for the tail parameter of two particular distributions of the above family: the skewed exponential power distribution and the skewed generalised logistic distribution. We apply the proposed objective approach to time series models and linear regression models where the error terms follow the distributions object of study. The performance of the proposed approach is illustrated through simulation experiments and real data analysis. The methodology yields improvements in density forecasts, as shown by the analysis we carry out on the electricity prices in Nordpool markets.

     

Comparison of methods of estimation of parameters of the weibull distribution

The generalised least squares, maximum likelihood, Bain-Antle 1 and 2, and two mixed methods of estimating the parameters of the two-parameter Weibull distribution are compared. The comparison is made using (a) the observed relative efficiency of parameter estimates and (b) themean squared relative error in estimated quantiles, to summarize the results of 1000 simulated samples of sizes 10 and 25. The results are that: generalised least squares is the best method of estimating the shape parameter ß the best method of estimating the scale parameter a depends onthe size of ß for quantile estimation maximum likelihood is best Bain-Antle 2 is uniformly the worst of the methods.     

On the first time of ruin in two-dimensional discrete time risk model with dependent claim occurrences

This article is concerned with a two-dimensional discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain a recursive expression for the finite time non ruin probability under such a dependence among claim occurrences. For an illustration, we define a bivariate compound beta-binomial risk model and present numerical results on this model by comparing the corresponding results of the bivariate compound binomial risk model.     

Mortality,longevity and experiments with the Lee–Carter model

The paper reviews the Lee-Carter modelling framework, illustrated with an application, and then extends the framework through the development of a wider class of generalised, parametric, non-linear models. The choice of error distribution is also generalised. These extensions permit the modelling and extrapolation of age-specific cohort effects as well as the more familiar age-specific period effects: the age-period-cohort version of the model is discussed with a worked example. The paper also provides a comparative study of simulation strategies for assessing risk in mortality rate predictions and the associated forecast estimates of life expectancy and annuity values in both period and cohort perspectives.     

Empirical bayes testing of a distribution function with d1richlet process priors

This paper considers the generalised empirical Bayes two-action (testing) and multiple action problems concerning a distribution function The Dirichiet process priors p of Ferguson have been used as the prior distributions on the space of distribution functions on the real line. The two-action component problem is considered in detail and when p is unknown partially empirical Bayes procedures {6 } which are asymptotically optimal with rates 0{jT1/2) and OCCmCn+l))     

Generalised Smooth Tests of Goodness of Fit Utilising L‐moments

In this paper we present a semiparametric test of goodness of fit which is based on the method of L‐moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L‐moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L‐moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.     

A new continuous distribution on the unit interval applied to modelling the points ratio of football teams

We introduce a new flexible distribution to deal with variables on the unit interval based on a transformation of the sinh–arcsinh distribution, which accommodates different degrees of skewness and kurtosis and becomes an interesting alternative to model this type of data. We also include this new distribution into the generalised additive models for location, scale and shape (GAMLSS) framework in order to develop and fit its regression model. For different parameter settings, some simulations are performed to investigate the behaviour of the estimators. The potentiality of the new regression model is illustrated by means of a real dataset related to the points rate of football teams at the end of a championship from the four most important leagues in the world: Barclays Premier League (England), Bundesliga (Germany), Serie A (Italy) and BBVA league (Spain) during three seasons (2011–2012, 2012–2013 and 2013–2014).     

Sequential change detection in the presence of unknown parameters

It is commonly required to detect change points in sequences of random variables. In the most difficult setting of this problem, change detection must be performed sequentially with new observations being constantly received over time. Further, the parameters of both the pre- and post- change distributions may be unknown. In Hawkins and Zamba (Technometrics 47(2):164–173, 2005), the sequential generalised likelihood ratio test was introduced for detecting changes in this context, under the assumption that the observations follow a Gaussian distribution. However, we show that the asymptotic approximation used in their test statistic leads to it being conservative even when a large numbers of observations is available. We propose an improved procedure which is more efficient, in the sense of detecting changes faster, in all situations. We also show that similar issues arise in other parametric change detection contexts, which we illustrate by introducing a novel monitoring procedure for sequences of Exponentially distributed random variable, which is an important topic in time-to-failure modelling.