Exact likelihood inference for two populations from two-parameter exponential distributions under joint Type-II censoring

In this article, we develop exact inference for two populations that have a two-parameter exponential distribution with the same location parameter and different scale parameters when Type-II censoring is implemented on the two samples in a combined manner. We obtain the conditional maximum likelihood estimators (MLEs) of the three parameters. We then derive the exact distributions of these MLEs along with their moment generating functions. Based on general entropy loss function, Bayesian study about the parameters is presented. Finally, some simulation results and an illustrative example are presented to illustrate the methods of inference developed here.     

CONDITIONAL INFERENCE PROCEDURES FOR THE PARETO AND POWER FUNCTION DISTRIBUTIONS BASED ON TYPE-II RIGHT CENSORED SAMPLES

In this paper we consider conditional inference procedures for the Pareto and power function distributions. We develop procedures for obtaining confidence intervals for the location and scale parameters as well as upper and lower n probability tolerance intervals for a proportion g, given a Type-II right censored sample from the corresponding distribution. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since, for each distribution, the procedures assume that a shape parameter x is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in x.     

APPLICATIONS AND EXTENSIONS OF BASU'S RESULTS ON MAXIMAL ANCILLARITY

Maximality of ancillaries is important in both conditional inference without nuisance parameters and marginal inference with nuisance parameters. We extend results of Basu (1959) to the more classical former case and discuss the different nature of ancillaries in these two contexts. We apply the results to a general ancillary independent of a sufficient statistic. Finally, we discuss difficulties in finding necessary conditions for maximality.     

Likelihood Based Inference in Non-linear Regression Models Using the p* and R* Approach

We develop second order asymptotic results for likelihood-based inference in Gaussian non-linear regression models. We provide an approximation to the conditional density of the maximum likelihood estimator given an approximate ancillary statistic (the affine ancillary). From this approximation, we derive a statistic to test an hypothesis on one component of the parameter. This test statistic is an adjustment of the signed log-likelihood ratio statistic. The distributional approximations (for the maximum likelihood estimator and for the test statistic) are of second order in large deviation regions.     

Approximate conditional inference and the linear functional model

Approximate conditional inference is developed for the slope parameter of the linear functional model with two variables. It is shown that the model can be transformed so that the slope parameter becomes an angle and nuisance parameters are radial distances. If the nuisance parameters are known an exact confidence interval based on a location-type conditional distribution is available for the angle. More gen¬erally, confidence distributions are used to average the conditional distribution over the nuisance parameters yielding an approximate conditional confidence interval that reflects the precision indicated by the data. An example is analyzed.     

Multivariate Tests of Mean-Variance Efficiency and Spanning With a Large Number of Assets and Time-Varying Covariances

We develop a finite-sample procedure to test the mean-variance efficiency and spanning hypotheses, without imposing any parametric assumptions on the distribution of model disturbances. In so doing, we provide an exact distribution-free method to test uniform linear restrictions in multivariate linear regression models. The framework allows for unknown forms of nonnormalities as well as time-varying conditional variances and covariances among the model disturbances. We derive exact bounds on the null distribution of joint F statistics to deal with the presence of nuisance parameters, and we show how to implement the resulting generalized nonparametric bounds tests with Monte Carlo resampling techniques. In sharp contrast to the usual tests that are not even computable when the number of test assets is too large, the power of the proposed test procedure potentially increases along both the time and cross-sectional dimensions.     

Tests for Equality of Parameters of Two Exponential Distributions with Common Known Coefficient of Variation

ABSTRACT

This article considers the problem of testing equality of parameters of two exponential distributions having common known coefficient of variation, both under unconditional and conditional setup. Unconditional tests based on BLUE'S and LRT are considered. Using the Conditionality Principle of Fisher, an UMP conditional test for one-sided alternative is derived by conditioning on an ancillary. This test is seen to be uniformly more powerful than unconditional tests in certain given ranges of ancillary. Simulation studies on the power functions of the tests are done for this purpose.     

Linear calibra and conditional inference

Approximate conditional inference is developed for the linear calibration problem. It is shown that this problem can be transformed so that the primary parameter is an angle, the nuisance parameter is a radial distance, and the density is rotationally symmetric. Were the nuisance parameter known, exact location confidence intervals would be available by location of structural arguments. A confidence distribution is used to average out the nuisance parameter yielding an approximate confidence interval that involves a precision indicator derived from the radial distance. Some difficulties with the ordinary solution are avoided by the conditional procedure.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

Approximate Studentization with marginal and conditional inference

Many inference problems lead naturally to a marginal or conditional measure of departure that depends on a nuisance parameter. As a device for first-order elimination of the nuisance parameter, we suggest averaging with respect to an exact or approximate confidence distribution function. It is shown that for many standard problems where an exact answer is available by other methods, the averaging method reproduces the exact answer. Moreover, for the gamma-mean problem, where the exact answer is not explicitly available, the averaging method gives results that agree closely with those obtained from higher-order asymptotic methods. Examples are discussed; detailed asymptotic calculations will be examined elsewhere.     

Exact likelihood inference for two exponential populations based on a joint generalized Type-I hybrid censored sample

Following the work of Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring. Comm Statist Theory Methods. 1988;17:1857–1870], several results have been developed regarding the exact likelihood inference of exponential parameters based on different forms of censored samples. In this paper, the conditional maximum likelihood estimators (MLEs) of two exponential mean parameters are derived under joint generalized Type-I hybrid censoring on the two samples. The moment generating functions (MGFs) and the exact densities of the conditional MLEs are obtained, using which exact confidence intervals are then developed for the model parameters. We also derive the means, variances, and mean squared errors of these estimates. An efficient computational method is developed based on the joint MGF. Finally, an example is presented to illustrate the methods of inference developed here.     

Exact Likelihood Inference for k Exponential Populations Under Joint Type-II Censoring

In this paper, when a jointly Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. The moment generating functions and the exact densities of these MLEs are obtained using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are also discussed. An empirical comparison of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

A Simple Analysis of the Exact Probability Matching Prior in the Location-Scale Model

It has long been asserted that in univariate location-scale models, when concerned with inference for either the location or scale parameter, the use of the inverse of the scale parameter as a Bayesian prior yields posterior credible sets that have exactly the correct frequentist confidence set interpretation. This claim dates to at least Peers, and has subsequently been noted by various authors, with varying degrees of justification. We present a simple, direct demonstration of the exact matching property of the posterior credible sets derived under use of this prior in the univariate location-scale model. This is done by establishing an equivalence between the conditional frequentist and posterior densities of the pivotal quantities on which conditional frequentist inferences are based.     

Covariate Classification Using Dependent Samples

The two-population classification problem using dependent samples is extended when covariates are available for classification. Analysis is done using a conditional model, under a multivariate normal set-up, given the covariates. The conditional model considered here includes the parameter structure relevant to growth models. Likelihood ratio or plug-in likelihood ratio classification rules are derived depending on the knowledge of the parameters in the model. For exact distribution of the classification statistics, they are reduced to forms suitable for application of standard results.     

A New Inference Framework for Dependency Networks

Dependency networks (DNs) have been receiving more attention recently because their structures and parameters can be easily learned from data. The full conditional distributions (FCDs) are known conditions of DNs. Gibbs sampling is currently the most popular inference method on DNs. However, sampling methods converge slowly and it can be hard to diagnose their convergence. In this article, we introduce a set of linear equations to describe the relations between joint probability distributions (JPDs) and FCDs. These equations provide a novel perspective to understand reasoning on DNs. Based on these linear equations, we develop both exact and approximate algorithms for inference on DNs. Experiments show that the proposed approximate algorithms can produce effective results by maintaining low computational complexity.     

Conditions for Non-confounding and Collapsibility without Knowledge of Completely Constructed Causal Diagrams

In this paper, we discuss several concepts in causal inference in terms of causal diagrams proposed by Pearl (1993 , 1995a , b ), and we give conditions for non-confounding, homogeneity and collapsibility for causal effects without knowledge of a completely constructed causal diagram. We first introduce the concepts of non-confounding, conditional non-confounding, uniform non-confounding, homogeneity, collapsibility and strong collapsibility for causal effects, then we present necessary and sufficient conditions for uniform non-confounding, homegeneity and collapsibilities, and finally we show sufficient conditions for non-confounding, conditional non-confounding and uniform non-confounding.     

Bayesian inference over ICA models: application to multibiometric score fusion with quality estimates

Bayesian networks are not well-formulated for continuous variables. The majority of recent works dealing with Bayesian inference are restricted only to special types of continuous variables such as the conditional linear Gaussian model for Gaussian variables. In this context, an exact Bayesian inference algorithm for clusters of continuous variables which may be approximated by independent component analysis models is proposed. The complexity in memory space is linear and the overfitting problem is attenuated, while the inference time is still exponential. Experiments for multibiometric score fusion with quality estimates are conducted, and it is observed that the performances are satisfactory compared to some known fusion techniques.     

Exact Likelihood Inference for k Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of k competing products with regard to their reliability. In this paper, when a joint progressively Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. Their conditional moment generating functions and exact densities are obtained, using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are discussed. An empirical evaluation of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities and average widths. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.