Pivotal Quantities Research Articles

Note on the family of proportional reversed hazard distributions

For many years, authors have been interested in developing methods for generating distributions that provide a flexible family to model lifetime variables. This paper proposes an exact inference approach to the family of proportional reversed hazard distributions based on the pivotal quantity, which yields exact confidence intervals with the shortest-length as well as reasonable estimators for the family of proportional reversed hazard distributions. In addition, the approach is extended to functions related to the unknown parameters using a generalized pivotal quantity such as the the goodness of fit test and entropy issues without computational complexity. The proposed method is illustrated through Monte Carlo simulations and real data analysis.

ESTIMASI SELANG KEPERCAYAAN NILAI UJIAN NASIONAL BERBASIS KOMPETENSI BERDASARKAN MODEL REGRESI SEMIPARAMETRIK MULTIRESPON TRUNCATED SPLINE

Confidence interval estimation is important in statistical inference for the parameters of the regression model, but the theory of confidence interval estimation for multi-response semiparametric regression model parameters based on the truncated spline estimator has not been examined. In this study, we estimate the confidence interval of the multi-response semiparametric regression model based on the truncated spline estimator by using pivotal quantity method with the central limit theorem approach. This confidence interval theory is applied to data of competency-based national exam (UNBK) scores in West Nusa Tenggara Province where its UNBK in the lowest position among other provinces in Indonesia. The method used for estimating parameters is weighted least square. The best model is determined based on the Generalized Cross Validation (GCV) minimum value. Based on the estimated 95% confidence interval of parameters of the multi-response truncated spline semiparametric regression model, the results showed that the insignificant factors affecting the UNBK scores were gender and parental education duration while the report card of scores and USBK scores had a positive effect on the UNBK scores but only the UNBK scores of mathematics that report card of scores factor has a negative effect on it.

Prediction Methods for Future Failure Times Based on Type-II Right-Censored Samples from New Pareto-Type Distribution

In this paper, we consider the prediction problem of the future failure times based on observed data from two-parameter Pareto-type distribution under Type-II right-censored samples. Different point predictors including best unbiased, maximum likelihood and conditional median predictors of the future failure times are obtained. The corresponding prediction intervals using pivotal quantity, conditional argument and shortest-length based method are also developed. The so-obtained point predictors and prediction intervals are compared via experimental numerical simulation. The optimality criteria considered for comparison purposes are prediction bias and mean square prediction error for point predictors and average length and coverage probability for prediction intervals. Real-life data representing the duration of the failure times of mechanical components as reported by Murthy (in: Wiley series in probability and statistics, Wiley, Hoboken, 2004) are analyzed for illustrative purposes.

Reliability analysis for accelerated degradation data based on the Wiener process with random effects

AbstractOn the basis of the principle of degradation mechanism invariance, a Wiener degradation process with random drift parameter is used to model the data collected from the constant stress accelerated degradation test. Small‐sample statistical inference method for this model is proposed. On the basis of Fisher's method, a test statistic is proposed to test if there is unit‐to‐unit variability in the population. For reliability inference, the quantities of interest are the quantile function, the reliability function, and the mean time to failure at the designed stress level. Because it is challenging to obtain exact confidence intervals (CIs) for these quantities, a regression type of model is used to construct pivotal quantities, and we develop generalized confidence intervals (GCIs) procedure for those quantities of interest. Generalized prediction interval for future degradation value at designed stress level is also discussed. A Monte Carlo simulation study is used to demonstrate the benefits of our procedures. Through simulation comparison, it is found that the coverage proportions of the proposed GCIs are better than that of the Wald CIs and GCIs have good properties even when there are only a small number of test samples available. Finally, a real example is used to illustrate the developed procedures.

Inference for confidence sets of the generalized inverted exponential distribution under [formula omitted]-record values

Based on the k-record values, confidence sets are explored for the parameters of the generalized inverted exponential distribution. Series of exact balanced confidence intervals and exact confidence regions are constructed using pivotal quantities. In order to obtain minimum-size confidence sets, constrained optimization problems are also discussed, and the associated nonlinear programming procedures are established by minimizing Lagrangian functions. Shortest-length confidence intervals and smallest-area confidence regions for the unknown parameters can be obtained by simultaneously solving nonlinear system. Finally, two real data examples and a simulation study are provided for illustrative purposes.

Pivotal Inference for the Inverted Exponentiated Rayleigh Distribution Based on Progressive Type-II Censored Data

In this article, the pivotal inference is proposed to estimate the two unknown parameters of the inverse exponentiated Rayleigh distribution based on progressive censored data. We derive the point estimator and construct an interval estimator using the pivotal quantity method. To compare the performance of this proposed method and the traditional maximum likelihood estimation method, a simulation study is conducted. The simulation results show that the proposed method performs better in terms of bias and mean squared error. Finally, a real dataset is used to illustrate the proposed approaches.

Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring

Two estimation problems are studied based on the general progressively censored samples, and the distributions from the inverted scale family (ISF) are considered as prospective life distributions. One is the exact interval estimation for the unknown parameter θ , which is achieved by constructing the pivotal quantity. Through Monte Carlo simulations, the average 90 % and 95 % confidence intervals are obtained, and the validity of the above interval estimation is illustrated with a numerical example. The other is the estimation of R = P ( Y < X ) in the case of ISF. The maximum likelihood estimator (MLE) as well as approximate maximum likelihood estimator (AMLE) is obtained, together with the corresponding R-symmetric asymptotic confidence intervals. With Bootstrap methods, we also propose two R-asymmetric confidence intervals, which have a good performance for small samples. Furthermore, assuming the scale parameters follow independent gamma priors, the Bayesian estimator as well as the HPD credible interval of R is thus acquired. Finally, we make an evaluation on the effectiveness of the proposed estimations through Monte Carlo simulations and provide an illustrative example of two real datasets.

A New Technique of Invariant Statistical Embedding and Averaging Via Pivotal Quantities for Intelligent Constructing Efficient Statistical Decisions under Parametric Uncertainty

In the present paper, a new technique of invariant embedding of sample statistics in a decision criterion (performance index) and averaging this criterion via pivotal quantities is proposed for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, the technique of invariant statistical embedding and averaging via pivotal quantities (ISE&APQ) is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. The aim of the present paper is to show how the technique of ISE&APQ may be employed in the particular case of optimization, estimation, or improvement of statistical decisions under parametric uncertainty. To illustrate the proposed technique of ISE&APQ, application examples are given.

Minimum Volume Confidence Regions for Parameters of Exponential Distributions from Different Samples

Independent samples from different exponential distributions are available in many statistical applications, for example, in queuing or reliability models, where the random variables describe waiting times and lifetimes, respectively. When two samples are observed, inference is then carried out for two location and two scale parameters. In multivariate setups including common location and common scale parameter assumptions, we provide confidence regions for the parameters of interest, which have minimum Lebesgue measure among all those based on the usual pivotal quantities and with the same or higher confidence level; in particular, they improve in terms of area (volume) upon the standard ‘trapezoidal’ confidence regions being constructed by combining independent univariate pivot statistics. The proposed confidence regions do not require any factorization of the overall confidence level, and their calculations need simple Monte Carlo simulations, only. Although focusing on two complete samples, generalizations of the results to more than two and doubly type-II censored samples are possible.

Can Bayesian, confidence distribution and frequentist inference agree?

We discuss and characterise connections between frequentist, confidence distribution and objective Bayesian inference, when considering higher-order asymptotics, matching priors, and confidence distributions based on pivotal quantities. The focus is on testing precise or sharp null hypotheses on a scalar parameter of interest. Moreover, we illustrate that the application of these procedures requires little additional effort compared to the application of standard first-order theory. In this respect, using the R software, we indicate how to perform in practice the computation with three examples in the context of data from inter-laboratory studies, of the stress–strength reliability, and of a growth curve from dose–response data.

Statistical properties on generalized half-logistic distribution

This paper addresses statistical properties of the generalized half-logistic distribution. We first establish the relations between the generalized half-logistic distribution and some well-known probability distributions, and then derive the moments of a random variable from the generalized half-logistic distribution. In addition, the maximum likelihood method is developed to provide the maximum likelihood estimators and approximate confidence intervals for unknown parameters. Note that the shape parameter can be the parameter of interest because the generalized half-logistic distribution has different shapes according to the value of the shape parameter. Therefore, we provide an unbiased estimator and an exact confidence interval based on a pivotal quantity for the shape parameter of the generalized half-logistic distribution, which is more efficient than the maximum likelihood estimation. Finally, we show that the generalized half-logistic distribution is available as a failure time model through real data analysis.

Comparison of two treatments on a covariate variable

ABSTRACT In clinical trials, the efficacy of treatment might be dependent on the value of a covariate variable. Therefore, it might be possible to detect the region over the covariate variable where the two treatments under investigation do not have significantly different efficacy or the region of superiority of one treatment. The non-significant region can be verified to be a confidence interval for the abscissa of the intersection point of two regression lines, and each of the complementary regions of the confidence interval corresponds to a region of superiority. In this study, we develop a method of constructing the confidence interval based on the concept of a generalized pivotal quantity, so as to perform the task of detecting the possible three regions for a clinical trial. Two real-world examples are given to illustrate the application of our proposed method, and a simulation study is conducted to evaluate its performance.

Entropy-based pivotal statistics for multi-sample problems in planar shape

Morphometric data come from several natural and man-made phenomena; e.g., biological processes and medical image processing. The analysis of these data—known as statistical shape analysis (SSA)—requires tailored methods because the majority of multivariate techniques are for the Euclidean space. An important branch at the SSA consists in using landmark data in two dimensions, called planar shape. Hypothesis tests in the last context have been proposed to check difference of mean shapes in multiple samples. A common assumption in these tests is that involved samples have the same covariance matrix, but few works are devoted to assess the former hypothesis. This paper addresses the proposal of new entropy-based multi-samples tests for variability. To formulate these tests, we assume that pre-shape data are well described by the complex Bingham ( $${\mathbb {C}} B$$ ) distribution. We derive expressions for the Renyi and Shannon entropies for the $${\mathbb {C}} B$$ and complex Watson models. Moreover, these quantities are understood as entropy-based tests to assess if multiple spherical samples have the same degree of disorder, a kind of variability. To quantify the performance of proposed tests, we perform a Monte Carlo study, adopting empirical test sizes and powers as comparison criteria. An application to real data using the second thoracic vertebra T2 of mouses is done to assess possible effects of body weight on the shape of mouse vertebra. Numerical results indicate that the test based on the optimized Renyi entropy may consist in a good tool to recognize variability similarity on planar-shape data.

A Novel Intelligent Technique of Invariant Statistical Embedding and Averaging via Pivotal Quantities for Optimization or Improvement of Statistical Decision Rules under Parametric Uncertainty

In the present paper, for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty, a new technique of invariant embedding of sample statistics in a decision criterion and averaging this criterion over pivots’ probability distributions is proposed. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, the technique of invariant statistical embedding and averaging via pivotal quantities (ISE&APQ) is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. The aim of the present paper is to show how the technique of ISE&APQ may be employed in the particular case of optimization, estimation, or improvement of statistical decisions under parametric uncertainty. To illustrate the proposed technique of ISE&APQ, illustrative examples of intelligent constructing exact statistical tolerance limits for prediction of future outcomes coming from log-location-scale distributions under parametric uncertainty are given

Reliability inference for a multicomponent stress–strength model based on Kumaraswamy distribution

In this paper, inference for a multicomponent stress–strength (MSS) model is studied under censored data. When both latent strength and stress random variables follow Kumaraswamy distributions with common shape parameters, the maximum likelihood estimate of MSS reliability is established and associated approximate confidence interval is constructed using the asymptotic distribution theory and delta method. Moreover, pivotal quantities based generalized point and confidence interval estimates are presented for the MSS reliability. Furthermore, likelihood and generalized pivotal based estimates are also presented when the strength and stress variables have unequal shape parameters. For complementary and comparison, bootstrap confidence intervals are provided as well under common and unequal parameter cases. In addition, to compare the equivalence between strength and stress shape parameters, the likelihood ratio test for hypothesis of interest is also discussed. Finally, simulation study and a real data example are provided to investigate the performance of proposed procedures.

Confidence limits for compliance testing using mixed acceptance criteria

AbstractFor manufactured items sold by weight or volume, this article considers mixed acceptance criteria that put limits on the sample mean or on an upper confidence limit based on the sample mean and on the number of individual sample units that are nonconforming. For a normally distributed quality characteristic of interest, this article develops lower confidence limits for the mixed acceptance criteria applying the concept of a generalized pivotal quantity and applying a bias‐corrected and accelerated parametric bootstrap. The accuracy of the confidence limits is assessed using estimated coverage probabilities, and the results are illustrated with an example.

Reliability analysis for stress-strength model from a general family of truncated distributions under censored data

Under progressive Type-II censoring, inference of stress-strength reliability (SSR) is studied for a general family of lower truncated distributions. When the lifetime models of the strength and stress variables have arbitrary and common parameters, maximum likelihood and pivotal quantities based generalized estimators of SSR are established, respectively. Confidence intervals are constructed based on generalized pivotal quantities and bootstrap technique under different parameter cases as well. In addition, to compare the equivalence of the strength and stress parameters, likelihood ratio testing of interested parameters is provided as a complementary. Simulation studies and two real-life data examples are provided to investigate the performance of proposed methods.

Statistical inference for constant‐stress accelerated life tests with dependent competing risks from Marshall‐Olkin bivariate exponential distribution

AbstractThis paper considers a constant‐stress accelerated dependent competing risks model under Type‐II censoring. The dependent structure between competing risks is modeled by a Marshall‐Olkin bivariate exponential distribution, and the accelerated model is described by the power rule model. The point and interval estimation of the model parameters and the reliability function under the normal usage condition at mission time are obtained by using the maximum likelihood estimation method and the bootstrap sampling technique. Moreover, the pivotal quantities based estimation are adopted to estimate the model parameters and the generalized confidence intervals. As a comparison, we also consider the Bayes estimation and the highest posterior density credible intervals for the model parameters based on conjugate priors and importance sampling method, respectively. To illustrate the proposed methodology, a Monte Carlo simulation is used to study the performances of different estimation methods. Finally, a dataset is analyzed for illustrative purpose and a comparison with the original results is also given.

Nonconformance probability of an air quality index

Air pollution is one of the most important global environmental issues. Taiwan Environmental Protection Agency (EPA) is currently using an Air Quality Index (AQI) to measure and monitor its national air quality. The main objective of this study is to assess the air quality per hour every month in Taichung City of Taiwan from 2014 to 2016 based on the nonconformance probability of the AQI index. The nonconformance probability is defined as the probability that a characteristic of interest falls outside of an acceptance region. A lower confidence bound for the nonconformance probability is applied to test whether the AQI index value exceeds a warning threshold, and then the government could issue warnings according to the decision made by such statistical inference. An unbalanced two-way random effects model is presented for fitting the AQI index values. We evaluate three different lower confidence bound construction methods, including a t-based, an adjusted t-based and a generalized pivotal quantity (GPQ) based methods, through a detailed simulation study. Finally, a hybrid method of the t-based and the adjusted t-based estimators is recommended for practical use.

Inference for the two-parameter exponential distribution with generalized order statistics

ABSTRACT Inferences about estimation and prediction of the two-parameter exponential distribution are based on generalized order statistics. Point and interval estimates are used for scale and location parameters. Unbiased point predictors and reconstructors are based upon pivotal quantities. The mean square error and Pitman’s measure help assess the closeness of estimators and predictors. Point estimators of scale and location parameters and point predictors of future observations are computed in the application of durations until remission of 20 leukemia patients and in the application until failure of air-conditioning systems.