Method Of Variance Estimates Recovery Research Articles

Testing the ratio of two Poisson means based on an inferential model

The ratio of two Poisson means is commonly used in biological, epidemiological, and medical. In this article, we consider the problem of testing the ratio of two Poisson means and propose a valid and efficient test based on the inference model (IM). The simulation studies show that the type I error and power of the IM-based test are competitive or even better than the six recommended tests: The likelihood ratio, mid-p, logarithmic transformation, and three tests based on the method of variance estimates recovery (MOVER). For R < 1 , the Rao-MOVER and fiducial-MOVER tests are slightly liberal, the mid-p test is somewhat conservative. For R > 1 , the Rao-MOVER and fiducial-MOVER tests are slightly conservative. The likelihood ratio and logarithmic transformation tests cannot control the type I error well. Therefore, the IM test could be recommended for practical applications. A real numerical example is presented to illustrate the flexibility of the proposed test.

Confidence intervals for functions of signal-to-noise ratio with application to economics and finance

PurposeConfidence intervals play a crucial role in economics and finance, providing a credible range of values for an unknown parameter along with a corresponding level of certainty. Their applications encompass economic forecasting, market research, financial forecasting, econometric analysis, policy analysis, financial reporting, investment decision-making, credit risk assessment and consumer confidence surveys. Signal-to-noise ratio (SNR) finds applications in economics and finance across various domains such as economic forecasting, financial modeling, market analysis and risk assessment. A high SNR indicates a robust and dependable signal, simplifying the process of making well-informed decisions. On the other hand, a low SNR indicates a weak signal that could be obscured by noise, so decision-making procedures need to take this into serious consideration. This research focuses on the development of confidence intervals for functions derived from the SNR and explores their application in the fields of economics and finance.Design/methodology/approachThe construction of the confidence intervals involved the application of various methodologies. For the SNR, confidence intervals were formed using the generalized confidence interval (GCI), large sample and Bayesian approaches. The difference between SNRs was estimated through the GCI, large sample, method of variance estimates recovery (MOVER), parametric bootstrap and Bayesian approaches. Additionally, confidence intervals for the common SNR were constructed using the GCI, adjusted MOVER, computational and Bayesian approaches. The performance of these confidence intervals was assessed using coverage probability and average length, evaluated through Monte Carlo simulation.FindingsThe GCI approach demonstrated superior performance over other approaches in terms of both coverage probability and average length for the SNR and the difference between SNRs. Hence, employing the GCI approach is advised for constructing confidence intervals for these parameters. As for the common SNR, the Bayesian approach exhibited the shortest average length. Consequently, the Bayesian approach is recommended for constructing confidence intervals for the common SNR.Originality/valueThis research presents confidence intervals for functions of the SNR to assess SNR estimation in the fields of economics and finance.

The Simultaneous Confidence Interval for the Ratios of the Coefficients of Variation of Multiple Inverse Gaussian Distributions and Its Application to PM2.5 Data

Due to slash/burn agricultural activity and frequent forest fires, PM2.5 has become a significant air pollution problem in Thailand, especially in the north and north east regions. Since its dispersion differs both spatially and temporally, estimating PM2.5 concentrations discretely by area, for which the inverse Gaussian distribution is suitable, can provide valuable information. Herein, we provide derivations of the simultaneous confidence interval for the ratios of the coefficients of variation of multiple inverse Gaussian distributions using the generalized confidence interval, the Bayesian interval based on the Jeffreys’ rule prior, the fiducial interval, and the method of variance estimates recovery. The efficacies of these methods were compared by considering the coverage probability and average length obtained from simulation results of daily PM2.5 datasets. The findings indicate that in most instances, the fiducial method with the highest posterior density demonstrated a superior performance. However, in certain scenarios, the Bayesian approach using the Jeffreys’ rule prior for the highest posterior density yielded favorable results.

Interval estimation of relative risks for combined unilateral and bilateral correlated data

ABSTRACT Measurements are generally collected as unilateral or bilateral data in clinical trials, epidemiology, or observational studies. For example, in ophthalmology studies, the primary outcome is often obtained from one eye or both eyes of an individual. In medical studies, the relative risk is usually the parameter of interest and is commonly used. In this article, we develop three confidence intervals for the relative risk for combined unilateral and bilateral correlated data under the equal dependence assumption. The proposed confidence intervals are based on maximum likelihood estimates of parameters derived using the Fisher scoring method. Simulation studies are conducted to evaluate the performance of proposed confidence intervals with respect to the empirical coverage probability, the mean interval width, and the ratio of mesial non-coverage probability to the distal non-coverage probability. We also compare the proposed methods with the confidence interval based on the method of variance estimates recovery and the confidence interval obtained from the modified Poisson regression model with correlated binary data. We recommend the score confidence interval for general applications because it best controls converge probabilities at the 95% level with reasonable mean interval width. We illustrate the methods with a real-world example.

Simultaneous Confidence Intervals for All Pairwise Differences between Means of Weibull Distributions

The Weibull distribution is a continuous probability distribution that finds wide application in various fields for analyzing real-world data. Specifically, wind speed data often adhere to the Weibull distribution. In our study, our aim is to compare the mean wind speed datasets from different areas in Thailand. To achieve this, we proposed simultaneous confidence intervals for all pairwise differences between the means of Weibull distributions. The generalized confidence interval (GCI), method of variance estimates recovery (MOVER), and a Bayesian approach, utilizing both gamma and uniform prior distributions, are proposed to construct simultaneous confidence intervals. Through simulations, we find that the Bayesian highest posterior density (HPD) interval using a gamma prior distribution demonstrates satisfactory performance, while the GCI proves to be a viable alternative. Finally, we applied these proposed approaches to real wind speed data in northeastern and southern Thailand to illustrate their effectiveness and practicality.

Estimating average wind speed in Thailand using confidence intervals for common mean of several Weibull distributions.

The Weibull distribution has been used to analyze data from many fields, including engineering, survival and lifetime analysis, and weather forecasting, particularly wind speed data. It is useful to measure the central tendency of wind speed data in specific locations using statistical parameters for instance the mean to accurately forecast the severity of future catastrophic events. In particular, the common mean of several independent wind speed samples collected from different locations is a useful statistic. To explore wind speed data from several areas in Surat Thani province, a large province in southern Thailand, we constructed estimates of the confidence interval for the common mean of several Weibull distributions using the Bayesian equitailed confidence interval and the highest posterior density interval using the gamma prior. Their performances are compared with those of the generalized confidence interval and the adjusted method of variance estimates recovery based on their coverage probabilities and expected lengths. The results demonstrate that when the common mean is small and the sample size is large, the Bayesian highest posterior density interval performed the best since its coverage probabilities were higher than the nominal confidence level and it provided the shortest expected lengths. Moreover, the generalized confidence interval performed well in some scenarios whereas adjusted method of variance estimates recovery did not. The approaches were used to estimate the common mean of real wind speed datasets from several areas in Surat Thani province, Thailand, fitted to Weibull distributions. These results support the simulation results in that the Bayesian methods performed the best. Hence, the Bayesian highest posterior density interval is the most appropriate method for establishing the confidence interval for the common mean of several Weibull distributions.

Estimation of the Confidence Interval for the Ratio of the Coefficients of Variation of Two Weibull Distributions and Its Application to Wind Speed Data

The Weibull distribution, one of the most significant distributions with applications in numerous fields, is associated with numerous distributions such as generalized gamma distribution, exponential distribution, and Rayleigh distribution, which are asymmetric. Nevertheless, it shares a close relationship with a normal distribution where a process of transformation allows them to become symmetric. The Weibull distribution is commonly used to study the failure of components and phenomena. It has been applied to a variety of scenarios, including failure time, claims amount, unemployment duration, survival time, and especially wind speed data. A suitable area for installing a wind turbine requires a wind speed that is both sufficiently high and consistent, and so comparing the variation in wind speed in two areas is eminently desirable. In this paper, methods to estimate the confidence interval for the ratio of the coefficients of variation of two Weibull distributions are proposed and applied to compare the variation in wind speed in two areas. The methods are the generalized confidence interval (GCI), the method of variance estimates recovery (MOVER), and Bayesian methods based on the gamma and uniform priors. The Bayesian methods comprise the equal-tailed confidence interval and the highest posterior density (HPD) interval. The effectiveness of the methods was evaluated in terms of their coverage probabilities and expected lengths and also empirically applied to wind speed datasets from two different areas in Thailand. The results indicate that the HPD interval based on the uniform prior outperformed the others in most of the scenarios tested and so it is suggested for estimating the confidence interval for the ratio of the coefficients of variation of two Weibull distributions.

Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions

In situations where several positive random variables cannot be described using symmetrical distributions, a positively asymmetric distribution which has garnered much attention for studying them is the Birnbaum-Saunders (BS) distribution. This distribution was originally proposed to study fatigue over time in materials and has become widely employed for reliability and fatigue studies. In statistics, the coefficient of variation (CV) is employed to measure relative variation. Furthermore, comparing the CVs of several samples from BS distributions is an important approach to assess the variation among them. Herein, we propose estimation methods for the simultaneous confidence intervals (SCIs) for all pairwise differences between the CVs of multiple BS distributions based on the percentile bootstrap, the generalized confidence interval (GCI), the method of variance estimates recovery (MOVER) based on the asymptotic confidence interval (ACI) and GCI, Bayesian credible interval, and the highest posterior density (HPD) interval. The coverage probabilities and average lengths of the proposed methods were examined via a simulation study to determine their performance. The results demonstrate that GCI and the MOVER based on the GCI method provided satisfactory performances in almost every case studied. Particulate matter ≤ 2.5 μm (PM2.5) concentration datasets from three areas in northern Thailand were used to illustrate the effectiveness of the proposed methods.

Simultaneous Confidence Intervals for the Ratios of the Means of Zero-Inflated Gamma Distributions and Its Application

Heavy rain in September (the middle of the rainy season in Thailand) can cause unexpected events and natural disasters such as flooding in many areas of the country. Rainfall series that contain both zero and positive values belong to the zero-inflated gamma distribution, which combines the binomial and gamma distributions. Precipitation in various areas of a country can be estimated by using simultaneous confidence intervals (CIs) for the ratios of the means of multiple zero-inflated gamma populations. Herein, we propose six simultaneous CIs constructed using the fiducial generalized CI method, Bayesian and highest posterior density (HPD) interval methods based on the Jeffreys’rule or uniform prior, and method of variance estimates recovery (MOVER). The performances of the proposed simultaneous CI methods were evaluated using a Monte Carlo simulation in terms of the coverage probabilities and expected lengths. The results from a comparative simulation study show that the HPD interval based on the Jeffreys’rule prior performed the best in most cases, while in some situations, the fiducial generalized CI performed well. All of the methods were applied to estimate the simultaneous CIs for the ratios of the means of natural rainfall data from six regions in Thailand.

Estimation of common percentile of rainfall datasets in Thailand using delta-lognormal distributions.

Weighted percentiles in many areas can be used to investigate the overall trend in a particular context. In this article, the confidence intervals for the common percentile are constructed to estimate rainfall in Thailand. The confidence interval for the common percentile help to indicate intensity of rainfall. Herein, four new approaches for estimating confidence intervals for the common percentile of several delta-lognormal distributions are presented: the fiducial generalized confidence interval, the adjusted method of variance estimates recovery, and two Bayesian approaches using fiducial quantity and approximate fiducial distribution. The Monte Carlo simulation was used to evaluate the coverage probabilities and average lengths via the R statistical program. The proposed confidence intervals are compared in terms of their coverage probabilities and average lengths, and the results of a comparative study based on these metrics indicate that one of the Bayesian confidence intervals is better than the others. The efficacies of the approaches are also illustrated by applying them to daily rainfall datasets from various regions in Thailand.

Confidence Intervals for Common Coefficient of Variation of Several Birnbaum–Saunders Distributions

The Birnbaum–Saunders (BS) distribution, also known as the fatigue life distribution, is right-skewed and used to model the failure times of industrial components. It has received much attention due to its attractive properties and its relationship to the normal distribution (which is symmetric). Furthermore, the coefficient of variation (CV) is commonly used to analyze variation within a dataset. In some situations, the independent samples are collected from different instruments or laboratories. Consequently, it is of importance to make inference for the common CV. To this end, confidence intervals based on the generalized confidence interval (GCI), method of variance estimates recovery (MOVER), large-sample (LS), Bayesian credible interval (BayCrI), and highest posterior density interval (HPDI) methods are proposed herein to estimate the common CV of several BS distributions. Their performances in terms of their coverage probabilities and average lengths were investigated by using Monte Carlo simulation. The simulation results indicate that the HPDI-based confidence interval outperformed the others in all of the investigated scenarios. Finally, the efficacies of the proposed confidence intervals are illustrated by applying them to real datasets of PM10 (particulate matter ≤ 10 μm) concentrations from three pollution monitoring stations in Chiang Mai, Thailand.

Confidence Intervals Based on the Difference of Medians for Independent Log-Normal Distributions

In this paper, we study the inferences of the difference of medians for two independent log-normal distributions. These methods include traditional methods such as the parametric bootstrap approach, the normal approximation approach, the method of variance estimates recovery approach, and the generalized confidence interval approach. The simultaneous confidence intervals for the difference in the median for more than two independent log-normal distributions are also discussed. Our simulation studies evaluate the performances of the proposed confidence intervals in terms of coverage probabilities and average lengths. We find that the parametric bootstrap approach would be a suitable choice for smaller sample sizes for the two independent distributions and multiple independent distributions. However, the method of variance estimates recovery and normal approximation approaches are alternative competitors for constructing simultaneous confidence intervals, especially when the populations have large variance. We also include two practical applications demonstrating the use of the techniques on observed data, where one data set works for the PM2.5 mass concentrations in Bangkapi and Dindaeng in Thailand and the other data came from the study of nitrogen-bound bovine serum albumin produced by three groups of diabetic mice. Both applications show that the confidence intervals from the parametric bootstrap approach have the smallest length.

Confidence interval estimation of the common mean of several gamma populations.

Gamma distributions are widely used in applied fields due to its flexibility of accommodating right-skewed data. Although inference methods for a single gamma mean have been well studied, research on the common mean of several gamma populations are sparse. This paper addresses the problem of confidence interval estimation of the common mean of several gamma populations using the concept of generalized inference and the method of variance estimates recovery (MOVER). Simulation studies demonstrate that several proposed approaches can provide confidence intervals with satisfying coverage probabilities even at small sample sizes. The proposed methods are illustrated using two examples.

Robust statistical inference for matched win statistics.

As alternatives to the time-to-first-event analysis of composite endpoints, the win statistics, that is, the net benefit, the win ratio, and the win odds have been proposed to assess treatment effects, using a hierarchy of prioritized component outcomes based on clinical relevance or severity. Whether we are using paired organs of a human body or pair-matching patients by risk profiles or propensity scores, we can leverage the level of granularity of matched win statistics to assess the treatment effect. However, inference for the matched win statistics (net benefit, win ratio, and win odds)-quantities related to proportions-is either not available or unsatisfactory, especially in samples of small to moderate size or when the proportion of wins (or losses) is near 0 or 1. In this paper, we present methods to address these limitations. First, we introduce a different statistic to test for the null hypothesis of no treatment effect and provided a sample size formula. Then, we use the method of variance estimates recovery to derive reliable, boundary-respecting confidence intervals for the matched net benefit, win ratio, and win odds. Finally, a simulation study demonstrates the performance of the proposed methods. We illustrate the proposed methods with two data examples.

Bayesian Confidence Interval for Ratio of the Coefficients of Variation of Normal Distributions: A Practical Approach in Civil Engineering

The coefficient of variation (CV) is a useful statistical tool for measuring the relative variability between multiple populations, while the ratio of CVs can be used to compare the dispersion. In statistics, the Bayesian approach is fundamentally different from the classical approach. For the Bayesian approach, the parameter is a quantity whose variation is described by a probability distribution. The probability distribution is called the prior distribution, which is based on the experimenter’s belief. The prior distribution is updated with sample information. This updating is done with the use of Bayes’ rule. For the classical approach, the parameter is quantity and an unknown value, but the parameter is fixed. Moreover, the parameter is based on the observed values in the sample. Herein, we develop a Bayesian approach to construct the confidence interval for the ratio of CVs of two normal distributions. Moreover, the efficacy of the Bayesian approach is compared with two existing classical approaches: the generalised confidence interval (GCI) and the method of variance estimates recovery (MOVER) approaches. A Monte Carlo simulation was used to compute the coverage probability (CP) and average length (AL) of three confidence intervals. The results of a simulation study indicate that the Bayesian approach performed better in terms of the CP and AL. Finally, the Bayesian and two classical approaches were applied to analyse real data to illustrate their efficacy. In this study, the application of these approaches for use in classical civil engineering topics is targeted. Two real data, which are used in the present study, are the compressive strength data for the investigated mixes at 7 and 28 days, as well as the PM2.5 air quality data of two stations in Chiang Mai province, Thailand. The Bayesian confidence intervals are better than the other confidence intervals for the ratio of CVs of normal distributions. Doi: 10.28991/CEJ-SP2021-07-010 Full Text: PDF

Bootstrap Confidence Intervals for Common Signal-to-noise Ratio of Two-parameter Exponential Distributions

Signal-to-noise ratio (SNR) is a reciprocal of coefffficient of variation. The SNR is a measure of mean relative to the variability. Confidence procedures for common SNR of two-parameter exponential distributions were developed using generalized confidence interval (GCI) approach, large sample (LS) approach, adjusted method of variance estimates recovery (Adjusted MOVER) approach, and bootstrap approaches based on standard bootstrap (SB) and parametric bootstrap (PB). The performances of all approaches are measured by coverage probability and average length. Simulation studies show that all approaches have the coverage probabilities below the nominal confidence level of 0.95 when the common SNR is negative value. However, the coverage probabilities of all approaches are greater than the nominal confidence level of 0.95 when the common SNR is positive value. Moreover, the LS and AM approaches are the conservative confidence intervals. In addition, the GCI and PB approaches provide the confidence intervals with coverage probabilities close to the nominal confidence level of 0.95 when the sample sizes are large and the common SNR is positive value. The GCI and PB approaches are recommended to estimate the confidence intervals for the common SNR of two-parameter exponential distributions. Finally, all proposed approaches are employed in the data of the survival days of lung cancer patients for a demonstration.

Interval estimation for the difference of two correlated gamma means: a generalized inference method and hybrid methods

Gamma distribution plays an important role in applied fields due to its flexibility of accommodating right-skewed data. Although inference methods for single gamma mean and for difference of two independent gamma means have been well studied, inference methods for difference of two correlated gamma means are sparse. This paper considers the problem of interval estimation for the difference of two correlated gamma means. We propose several inference methods including a method based on the concepts of generalized inference and two hybrid methods based on the method of variance estimates recovery. An extensive Monte Carlo simulation study was conducted to assess the performance of the proposed methods in terms of coverage probabilities and average lengths of the estimated intervals. Two real data examples from medical and engineering studies are analyzed using the proposed methods.

Bayesian computation for the common coefficient of variation of delta-lognormal distributions with application to common rainfall dispersion in Thailand.

Rainfall fluctuation makes precipitation and flood prediction difficult. The coefficient of variation can be used to measure rainfall dispersion to produce information for predicting future rainfall, thereby mitigating future disasters. Rainfall data usually consist of positive and true zero values that correspond to a delta-lognormal distribution. Therefore, the coefficient of variation of delta-lognormal distribution is appropriate to measure the rainfall dispersion more than lognormal distribution. In particular, the measurement of the dispersion of precipitation from several areas can be determined by measuring the common coefficient of variation in the rainfall from those areas together. Herein, we compose confidence intervals for the common coefficient of variation of delta-lognormal distributions by employing the fiducial generalized confidence interval, equal-tailed Bayesian credible intervals incorporating the independent Jeffreys or uniform priors, and the method of variance estimates recovery. A combination of the coverage probabilities and expected lengths of the proposed methods obtained via a Monte Carlo simulation study were used to compare their performances. The results show that the equal-tailed Bayesian based on the independent Jeffreys prior was suitable. In addition, it can be used the equal-tailed Bayesian based on the uniform prior as an alternative. The efficacies of the proposed confidence intervals are demonstrated via applying them to analyze daily rainfall datasets from Nan, Thailand.

A test for the Behrens–Fisher problem based on the method of variance estimates recovery

A practical solution based on the method of variance estimates recovery is proposed for the Behrens–Fisher problem. Unlike the classical solution, the proposed test is not based on the central limit theorem and, therefore, it can control Type I error well for small or moderate sample sizes. In this sense, it performs better than the most commonly used Welch's approximate t-test, which may be liberal for small or moderate sample sizes. On the other hand, the powers of the proposed test appears to be close to that of the Welch's approximate t-test and better than those of the Fisher’s solution and the generalized p-value solution; the latter two have no fixed p-value and need to simulate. The proposed test has a meaningful interpretation within and not just across experiments. It can be a competitive alternative solution.

MOVER confidence intervals for a difference or ratio effect parameter under stratified sampling.

Stratification is commonly employed in clinical trials to reduce the chance covariate imbalances and increase the precision of the treatment effect estimate. We propose a general framework for constructing the confidence interval (CI) for a difference or ratio effect parameter under stratified sampling by the method of variance estimates recovery (MOVER). We consider the additive variance and additive CI approaches for the difference, in which either the CI for the weighted difference, or the CI for the weighted effect in each group, or the variance for the weighted difference is calculated as the weighted sum of the corresponding stratum-specific statistics. The CI for the ratio is derived by the Fieller and log-ratio methods. The weights can be random quantities under the assumption of a constant effect across strata, but this assumption is not needed for fixed weights. These methods can be easily applied to different endpoints in that they require only the point estimate, CI, and variance estimate for the measure of interest in each group across strata. The methods are illustrated with two real examples. In one example, we derive the MOVER CIs for the risk difference and risk ratio for binary outcomes. In the other example, we compare the restricted mean survival time and milestone survival in stratified analysis of time-to-event outcomes. Simulations show that the proposed MOVER CIs generally outperform the standard large sample CIs, and that the additive CI approach performs better than the additive variance approach. Sample SAS code is provided in the Supplementary Material.