Estimation and Prediction for Exponentiated Family of Distributions Based on Records

In this article, a family of distributions, namely the exponentiated family of distributions, is defined and for the unknown parameters, different point estimates are derived based on record statistics. Prediction for future record values is presented from a Bayesian view point. Two numerical examples and a Monte Carlo simulation study are presented to illustrate the results.     

Bayesian prediction of order statistics based on k-record values from exponential distribution

Bayesian prediction of order statistics as well as the mean of a future sample based on observed record values from an exponential distribution are discussed. Several Bayesian prediction intervals and point predictors are derived. Finally, some numerical computations are presented for illustrating all the proposed inferential procedures.     

Bayesian and frequentist prediction limits for the Poisson distribution

Prediction limits for Poisson distribution are useful in real life when predicting the occurrences of some phenomena, for example, the number of infections from a disease per year among school children, or the number of hospitalizations per year among patients with cardiovascular disease. In order to allocate the right resources and to estimate the associated cost, one would want to know the worst (i.e., an upper limit) and the best (i.e., the lower limit) scenarios. Under the Poisson distribution, we construct the optimal frequentist and Bayesian prediction limits, and assess frequentist properties of the Bayesian prediction limits. We show that Bayesian upper prediction limit derived from uniform prior distribution and Bayesian lower prediction limit derived from modified Jeffreys non informative prior coincide with their respective frequentist limits. This is not the case for the Bayesian lower prediction limit derived from a uniform prior and the Bayesian upper prediction limit derived from a modified Jeffreys prior distribution. Furthermore, it is shown that not all Bayesian prediction limits derived from a proper prior can be interpreted in a frequentist context. Using a counterexample, we state a sufficient condition and show that Bayesian prediction limits derived from proper priors satisfying our condition cannot be interpreted in a frequentist context. Analysis of simulated data and data on Atlantic tropical storm occurrences are presented.     

Some extensions of zero-inflated models and Bayesian tests for them

Zero-inflated power series distribution is commonly used for modelling count data with extra zeros. Inflation at point zero has been investigated and several tests for zero inflation have been examined. However sometimes, inflation occurs at a point apart from zero. In this case, we say inflation occurs at an arbitrary point j. The j-inflation has been discussed less than zero inflation. In this paper, inflation at an arbitrary point j is studied with more details and a Bayesian test for detecting inflation at point j is presented. The Bayesian method is extended to inflation at arbitrary points i and j. The relationship between the distribution for inflation at point j, inflation at points i and j and missing value imputation is studied. It is shown how to obtain a proper estimate of the population variance if a mean-imputed missing at random data set is used. Some simulation studies are conducted and the proposed Bayesian test is applied on two real data sets.     

Statistical analysis for Kumaraswamy’s distribution based on record data

In this paper we review some results that have been derived on record values for some well known probability density functions and based on m records from Kumaraswamy’s distribution we obtain estimators for the two parameters and the future sth record value. These estimates are derived using the maximum likelihood and Bayesian approaches. In the Bayesian approach, the two parameters are assumed to be random variables and estimators for the parameters and for the future sth record value are obtained, when we have observed m past record values, using the well known squared error loss (SEL) function and a linear exponential (LINEX) loss function. The findings are illustrated with actual and computer generated data.     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

Inference and prediction for the burr type x model based on records

Based on record values, the maximum likelihood, minimum variance unbiased and Bayes estimators of the one parameter of the Burr type X distribution are computed and compared. The Bayesian and non-Bayesian confidence intervals for this parameter are also presented. A Bayesian prediction interval for the sth future record is obtained in a closed form. Based on simulated record values, numerical computations and comparisons between the different estimators are given     

A Bayesian method of sample size determination with practical applications

Summary.  The problem motivating the paper is the determination of sample size in clinical trials under normal likelihoods and at the substantive testing stage of a financial audit where normality is not an appropriate assumption. A combination of analytical and simulation-based techniques within the Bayesian framework is proposed. The framework accommodates two different prior distributions: one is the general purpose fitting prior distribution that is used in Bayesian analysis and the other is the expert subjective prior distribution, the sampling prior which is believed to generate the parameter values which in turn generate the data. We obtain many theoretical results and one key result is that typical non-informative prior distributions lead to very small sample sizes. In contrast, a very informative prior distribution may either lead to a very small or a very large sample size depending on the location of the centre of the prior distribution and the hypothesized value of the parameter. The methods that are developed are quite general and can be applied to other sample size determination problems. Some numerical illustrations which bring out many other aspects of the optimum sample size are given.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times

ABSTRACT

The maximum likelihood and Bayesian approaches for estimating the parameters and the prediction of future record values for the Kumaraswamy distribution has been considered when the lower record values along with the number of observations following the record values (inter-record-times) have been observed. The Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. The Bayes and the maximum likelihood estimates are compared by using the estimated risk through Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record values arising from the Kumaraswamy distribution based on record values with their corresponding inter-record times and only record values. The comparison of the derived predictors are carried out by using Monte Carlo simulations. Real data are analysed for an illustration of the findings.     

On diagnostic devices for proposing half-logistic and inverse half-logistic models using generalized (k) record values

In this work we consider the generalized upper (k) record values (GURV’s) and generalized lower (k) record values (GLRV’s) arising from half-logistic distribution (HLD) and inverse half-logistic distribution (IHLD). We derive some characterization results of HLD based on some moment relations of generalized upper (k) record values and those of generalized lower (k) record values and accordingly devised some diagnostic tools to identify HLD as a model to the distribution of a population. Similar characterization theorems and diagnostic tools are developed for IHLD as well. Simulation studies are conducted to validate the diagnostic tools devised for both HLD and IHLD.     

Confidence sets as programming problems

Optimizing criteria for choosing a confidence set for a parameter are formulated as mathematical programming problems. The two optimizing criteria, probability of coverage and size of set, give rise to a pair of inverse programming problems. Several examples are worked out. The programming problems are then formulated to allow the incorporation of partial information about the parameter. By varying the family of prior distributions, a continuum of problems from the frequency approach to a Bayesian approach is obtained. Some examples are considered in which the family of priors contains more than one but not all prior distributions.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Recurrence relations for single and product moments of record values from generalized pareto distribution

In this paper we establish some recurrence relations satisfied by single and product moments of upper record values from the generalized Pareto distribution. It is shown that these relations may be used to obtain all the single and product moments of all record values in a simple recursive manner. We also show that similar results established recently by Balakrishnan and Ahsanullah (1993) for the upper record values from the exponential distribution may be deduced by letting the shape parameter p tend to 0.     

Two-sided Bayesian and frequentist tolerance intervals: general asymptotic results with applications

It is well known that that the construction of two-sided tolerance intervals is far more challenging than that of their one-sided counterparts. In a general framework of parametric models, we derive asymptotic results leading to explicit formulae for two-sided Bayesian and frequentist tolerance intervals. In the process, probability matching priors for such intervals are characterized and their role in finding frequentist tolerance intervals via a Bayesian route is indicated. Furthermore, in situations where matching priors are hard to obtain, we develop purely frequentist tolerance intervals as well. The findings are applied to real data. Simulation studies are seen to lend support to the asymptotic results in finite samples.     

Bayesian Estimation for the Exponentiated Weibull Model via Markov Chain Monte Carlo Simulation

Bayesian estimation for the two unknown parameters and the reliability function of the exponentiated Weibull model are obtained based on generalized order statistics. Markov chain Monte Carlo (MCMC) methods are considered to compute the Bayes estimates of the target parameters. Our computations are based on the balanced loss function which contains the symmetric and asymmetric loss functions as special cases. The results have been specialized to the progressively Type-II censored data and upper record values. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation.     

A weakly informative prior for Bayesian dynamic model selection with applications in fMRI

In recent years, Bayesian statistics methods in neuroscience have been showing important advances. In particular, detection of brain signals for studying the complexity of the brain is an active area of research. Functional magnetic resonance imagining (fMRI) is an important tool to determine which parts of the brain are activated by different types of physical behavior. According to recent results, there is evidence that the values of the connectivity brain signal parameters are close to zero and due to the nature of time series fMRI data with high-frequency behavior, Bayesian dynamic models for identifying sparsity are indeed far-reaching. We propose a multivariate Bayesian dynamic approach for model selection and shrinkage estimation of the connectivity parameters. We describe the coupling or lead-lag between any pair of regions by using mixture priors for the connectivity parameters and propose a new weakly informative default prior for the state variances. This framework produces one-step-ahead proper posterior predictive results and induces shrinkage and robustness suitable for fMRI data in the presence of sparsity. To explore the performance of the proposed methodology, we present simulation studies and an application to functional magnetic resonance imaging data.     

Flexible objective Bayesian linear regression with applications in survival analysis

We study objective Bayesian inference for linear regression models with residual errors distributed according to the class of two-piece scale mixtures of normal distributions. These models allow for capturing departures from the usual assumption of normality of the errors in terms of heavy tails, asymmetry, and certain types of heteroscedasticity. We propose a general non-informative, scale-invariant, prior structure and provide sufficient conditions for the propriety of the posterior distribution of the model parameters, which cover cases when the response variables are censored. These results allow us to apply the proposed models in the context of survival analysis. This paper represents an extension to the Bayesian framework of the models proposed in [16]. We present a simulation study that shows good frequentist properties of the posterior credible intervals as well as point estimators associated to the proposed priors. We illustrate the performance of these models with real data in the context of survival analysis of cancer patients.     

A Bayesian analysis for the multivariate point null testing problem

A Bayesian test for the point null testing problem in the multivariate case is developed. A procedure to get the mixed distribution using the prior density is suggested. For comparisons between the Bayesian and classical approaches, lower bounds on posterior probabilities of the null hypothesis, over some reasonable classes of prior distributions, are computed and compared with the p-value of the classical test. With our procedure, a better approximation is obtained because the p-value is in the range of the Bayesian measures of evidence.     

Classical and Bayesian estimation of P(X<Y) using upper record values from Kumaraswamy’s distribution

In this paper, maximum likelihood and Bayesian approaches have been used to obtain the estimation of \(P(X<Y)\) based on a set of upper record values from Kumaraswamy distribution. The existence and uniqueness of the maximum likelihood estimates of the Kumaraswamy distribution parameters are obtained. Confidence intervals, exact and approximate, as well as Bayesian credible intervals are constructed. Bayes estimators have been developed under symmetric (squared error) and asymmetric (LINEX) loss functions using the conjugate and non informative prior distributions. The approximation forms of Lindley (Trabajos de Estadistica 3:281–288, 1980) and Tierney and Kadane (J Am Stat Assoc 81:82–86, 1986) are used for the Bayesian cases. Monte Carlo simulations are performed to compare the different proposed methods.