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 Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

Bayes estimation and prediction of the two-parameter gamma distribution

In this article, the Bayes estimates of two-parameter gamma distribution are considered. It is well known that the Bayes estimators of the two-parameter gamma distribution do not have compact form. In this paper, it is assumed that the scale parameter has a gamma prior and the shape parameter has any log-concave prior, and they are independently distributed. Under the above priors, we use Gibbs sampling technique to generate samples from the posterior density function. Based on the generated samples, we can compute the Bayes estimates of the unknown parameters and can also construct HPD credible intervals. We also compute the approximate Bayes estimates using Lindley's approximation under the assumption of gamma priors of the shape parameter. Monte Carlo simulations are performed to compare the performances of the Bayes estimators with the classical estimators. One data analysis is performed for illustrative purposes. We further discuss the Bayesian prediction of future observation based on the observed sample and it is seen that the Gibbs sampling technique can be used quite effectively for estimating the posterior predictive density and also for constructing predictive intervals of the order statistics from the future sample.     

Profile Hellinger distance estimation

The successful application of the Hellinger distance approach to fully parametric models is well known. The corresponding optimal estimators, known as minimum Hellinger distance (MHD) estimators, are efficient and have excellent robustness properties [Beran R. Minimum Hellinger distance estimators for parametric models. Ann Statist. 1977;5:445–463]. This combination of efficiency and robustness makes MHD estimators appealing in practice. However, their application to semiparametric statistical models, which have a nuisance parameter (typically of infinite dimension), has not been fully studied. In this paper, we investigate a methodology to extend the MHD approach to general semiparametric models. We introduce the profile Hellinger distance and use it to construct a minimum profile Hellinger distance estimator of the finite-dimensional parameter of interest. This approach is analogous in some sense to the profile likelihood approach. We investigate the asymptotic properties such as the asymptotic normality, efficiency, and adaptivity of the proposed estimator. We also investigate its robustness properties. We present its small-sample properties using a Monte Carlo study.     

Alternatives to post‐processing posterior predictive p values

The posterior predictive p value (ppp) was invented as a Bayesian counterpart to classical p values. The methodology can be applied to discrepancy measures involving both data and parameters and can, hence, be targeted to check for various modeling assumptions. The interpretation can, however, be difficult since the distribution of the ppp value under modeling assumptions varies substantially between cases. A calibration procedure has been suggested, treating the ppp value as a test statistic in a prior predictive test. In this paper, we suggest that a prior predictive test may instead be based on the expected posterior discrepancy, which is somewhat simpler, both conceptually and computationally. Since both these methods require the simulation of a large posterior parameter sample for each of an equally large prior predictive data sample, we furthermore suggest to look for ways to match the given discrepancy by a computation‐saving conflict measure. This approach is also based on simulations but only requires sampling from two different distributions representing two contrasting information sources about a model parameter. The conflict measure methodology is also more flexible in that it handles non‐informative priors without difficulty. We compare the different approaches theoretically in some simple models and in a more complex applied example.     

Combined Exponentially Weighted Moving Average Charts for the Mean and Variance Based on the Predictive Distribution

This article develops combined exponentially weighted moving average (EWMA) charts for the mean and variance of a normal distribution. A Bayesian approach is used to incorporate parameter uncertainty. We first use a Bayesian predictive distribution to construct the control chart, and we then use a sampling theory approach to evaluate it under various hypothetical specifications for the data generation model. Simulations are used to compare the proposed charts for different values of both the weighing constant for the exponentially weighted moving averages and for the size of the calibration sample that is used to estimate the in-statistical-control process parameters. We also examine the separate performance of the EWMA chart for the variance.     

Improved estimation of common location of two exponential populations with order restricted scale parameters using censored samples

ABSTRACT

Estimation of common location parameter of two exponential populations is considered when the scale parameters are ordered using type-II censored samples. A general inadmissibility result is proved which helps in deriving improved estimators. Further, a class of estimators dominating the MLE has been derived by an application of integrated expression of risk difference (IERD) approach of Kubokawa. A discussion regarding extending the results to a general k( ? 2) populations has been done. Finally, all the proposed estimators are compared through simulation.     

Asymptotic distribution of regression correlation coefficient for Poisson regression model

This study treats an asymptotic distribution for measures of predictive power for generalized linear models (GLMs). We focus on the regression correlation coefficient (RCC) that is one of the measures of predictive power. The RCC, proposed by Zheng and Agresti is a population value and a generalization of the population value for the coefficient of determination. Therefore, the RCC is easy to interpret and familiar. Recently, Takahashi and Kurosawa provided an explicit form of the RCC and proposed a new RCC estimator for a Poisson regression model. They also showed the validity of the new estimator compared with other estimators. This study discusses the new statistical properties of the RCC for the Poisson regression model. Furthermore, we show an asymptotic normality of the RCC estimator.     

PMSE dominance of the positive-part shrinkage estimator in a regression model with proxy variables

Consider a linear regression model with some relevant regressors are unobservable. In such a situation, we estimate the model by using the proxy variables as regressors or by simply omitting the relevant regressors. In this paper, we derive the explicit formula of predictive mean squared error (PMSE) of a general family of shrinkage estimators of regression coefficients. It is shown analytically that the positive-part shrinkage estimator dominates the ordinary shrinkage estimator even when proxy variables are used in place of the unobserved variables. Also, as an example, our result is applied to the double k-class estimator proposed by Ullah and Ullah (Double k-class estimators of coefficients in linear regression. Econometrica. 1978;46:705–722). Our numerical results show that the positive-part double k-class estimator with proxy variables has preferable PMSE performance.     

Causal Reasoning from Longitudinal Data*

Abstract. This paper reviews some of the key statistical ideas that are encountered when trying to find empirical support to causal interpretations and conclusions, by applying statistical methods on experimental or observational longitudinal data. In such data, typically a collection of individuals are followed over time, then each one has registered a sequence of covariate measurements along with values of control variables that in the analysis are to be interpreted as causes, and finally the individual outcomes or responses are reported. Particular attention is given to the potentially important problem of confounding. We provide conditions under which, at least in principle, unconfounded estimation of the causal effects can be accomplished. Our approach for dealing with causal problems is entirely probabilistic, and we apply Bayesian ideas and techniques to deal with the corresponding statistical inference. In particular, we use the general framework of marked point processes for setting up the probability models, and consider posterior predictive distributions as providing the natural summary measures for assessing the causal effects. We also draw connections to relevant recent work in this area, notably to Judea Pearl's formulations based on graphical models and his calculus of so‐called do‐probabilities. Two examples illustrating different aspects of causal reasoning are discussed in detail.     

Estimation from Burr type XII distribution using progressive first-failure censored data

In this paper, a new life test plan called a progressively first-failure-censoring scheme introduced by Wu and Ku? [On estimation based on progressive first-failure-censored sampling, Comput. Statist. Data Anal. 53(10) (2009), pp. 3659–3670] is considered. Based on this type of censoring, the maximum likelihood (ML) and Bayes estimates for some survival time parameters namely reliability and hazard functions, as well as the parameters of the Burr-XII distribution are obtained. The Bayes estimators relative to both the symmetric and asymmetric loss functions are discussed. We use the conjugate prior for the one-shape parameter and discrete prior for the other parameter. Exact and approximate confidence intervals with the exact confidence region for the two-shape parameters are derived. A numerical example using the real data set is provided to illustrate the proposed estimation methods developed here. The ML and the different Bayes estimates are compared via a Monte Carlo simulation study.     

An asymptotic equivalence of the cross-data and predictive estimators

Abstract

Estimators using multiplicative tuning parameters for maximum likelihood estimators in cross-validation are called cross-data estimators in this paper. Single-sample versions of the cross-data estimators have been called predictive estimators in literatures, which are given by maximizing the expected log-likelihood, where the two-fold expectations are taken over the distributions of future and current data using maximum likelihood estimators based on current data. An asymptotic equivalence of the cross-data and predictive estimators is shown, which guarantees an optimality of the predictive estimator when an unknown population parameter vector is replaced by the sample counterpart. Examples using typical statistical distributions are shown.     

Pitman closest equivariant estimators and predictors under location–scale models

For location, scale and location–scale models, which are common in practical applications, we derive optimum equivariant estimators and predictors using the Pitman closeness criterion. This approach is very robust with respect to the choice of the loss function as it only requires the loss function to be strictly monotone. We also prove that, in general, the Pitman closeness comparison of any two equivariant predictors depends on the unknown parameter only through a maximal invariant, and hence it is independent of the parameter when the parameter space is transitive. We present several examples illustrating applications of our theoretical results.     

On predictive distributions and Bayesian networks

In this paper we are interested in discrete prediction problems for a decision-theoretic setting, where the task is to compute the predictive distribution for a finite set of possible alternatives. This question is first addressed in a general Bayesian framework, where we consider a set of probability distributions defined by some parametric model class. Given a prior distribution on the model parameters and a set of sample data, one possible approach for determining a predictive distribution is to fix the parameters to the instantiation with the maximum a posteriori probability. A more accurate predictive distribution can be obtained by computing the evidence (marginal likelihood), i.e., the integral over all the individual parameter instantiations. As an alternative to these two approaches, we demonstrate how to use Rissanen's new definition of stochastic complexity for determining predictive distributions, and show how the evidence predictive distribution with Jeffrey's prior approaches the new stochastic complexity predictive distribution in the limit with increasing amount of sample data. To compare the alternative approaches in practice, each of the predictive distributions discussed is instantiated in the Bayesian network model family case. In particular, to determine Jeffrey's prior for this model family, we show how to compute the (expected) Fisher information matrix for a fixed but arbitrary Bayesian network structure. In the empirical part of the paper the predictive distributions are compared by using the simple tree-structured Naive Bayes model, which is used in the experiments for computational reasons. The experimentation with several public domain classification datasets suggest that the evidence approach produces the most accurate predictions in the log-score sense. The evidence-based methods are also quite robust in the sense that they predict surprisingly well even when only a small fraction of the full training set is used.     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

A Combined Nonlinear Programming Model and Kibria Method for Choosing Ridge Parameter Regression

Ridge regression solves multicollinearity problems by introducing a biasing parameter that is called ridge parameter; it shrinks the estimates as well as their standard errors in order to reach acceptable results. Many methods are available for estimating a ridge parameter. This article has considered some of these methods and also proposed a combined nonlinear programming model and Kibria method. A simulation study has been made to evaluate the performance of the proposed estimators based on the minimum mean squared error criterion. The simulation study indicates that under certain conditions the proposed estimators outperform the least squares (LS) estimators and other popular existing estimators. Moreover, the new proposed model is applied on dataset that suffers also from the presence of heteroscedastic errors.     

Shrinkage averaging estimation

We discuss the impact of tuning parameter selection uncertainty in the context of shrinkage estimation and propose a methodology to account for problems arising from this issue: Transferring established concepts from model averaging to shrinkage estimation yields the concept of shrinkage averaging estimation (SAE) which reflects the idea of using weighted combinations of shrinkage estimators with different tuning parameters to improve overall stability, predictive performance and standard errors of shrinkage estimators. Two distinct approaches for an appropriate weight choice, both of which are inspired by concepts from the recent literature of model averaging, are presented: The first approach relates to an optimal weight choice with regard to the predictive performance of the final weighted estimator and its implementation can be realized via quadratic programming. The second approach has a fairly different motivation and considers the construction of weights via a resampling experiment. Focusing on Ridge, Lasso and Random Lasso estimators, the properties of the proposed shrinkage averaging estimators resulting from these strategies are explored by means of Monte-Carlo studies and are compared to traditional approaches where the tuning parameter is simply selected via cross validation criteria. The results show that the proposed SAE methodology can improve an estimators’ overall performance and reveal and incorporate tuning parameter uncertainty. As an illustration, selected methods are applied to some recent data from a study on leadership behavior in life science companies.     

The adaptive calibration of testing p-values

Abstract

In statistical hypothesis testing, a p-value is expected to be distributed as the uniform distribution on the interval (0, 1) under the null hypothesis. However, some p-values, such as the generalized p-value and the posterior predictive p-value, cannot be assured of this property. In this paper, we propose an adaptive p-value calibration approach, and show that the calibrated p-value is asymptotically distributed as the uniform distribution. For Behrens–Fisher problem and goodness-of-fit test under a normal model, the calibrated p-values are constructed and their behavior is evaluated numerically. Simulations show that the calibrated p-values are superior than original ones.     

The p-values for one-sided hypothesis testing in univariate linear calibration

In this article, we focus on the one-sided hypothesis testing for the univariate linear calibration, where a normally distributed response variable and an explanatory variable are involved. The observations of the response variable corresponding to known values of the explanatory variable are used to make inferences on a single unknown value of the explanatory variable. We apply the generalized inference to the calibration problem, and take the generalized p-value as the test statistic to develop a new p-value for one-sided hypothesis testing, which we refer to as the one-sided posterior predictive p-value. The behavior of the one-sided posterior predictive p-value is numerically compared with that of the generalized p-value, and simulations show that the proposed p-value is quite satisfactory in the frequentist performance.     

Objective Bayesian Analysis for Log-logistic Distribution

In this article, Bayesian approach is applied to estimate the parameters of Log-logistic distribution under reference prior and Jeffreys’ prior. The reference prior is derived and it is found that the reference prior is also a second-order matching priors as for the case of any parameter of interest. The Bayesian estimators cannot be obtained in explicit forms. Metropolis within Gibbs sampling algorithm is used to obtain the Bayesian estimators. The Bayesian estimates are compared with the maximum likelihood estimates via simulation study. A real dataset is considered for illustrative purposes.