Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.     

Inferences on the lognormal mean for complete samples

In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be confusing if it is remembered that this implies that their equivalent confidence intervals are conservative. No implication of optimality is intended for these conservative procedures. It is emphasized that these are direct statistical inference methods for the lognormal mean, as opposed to the already well-known methods for the parameters of the underlying normal distribution. The method currently employed in MIL-STD-471A for statistical demonstration of MTTR is analyzed and compared to the new method in terms of asymptotic relative efficiency. The new methods are also compared to the optimum methods derived by Land (1971, 1973).     

Confidence intervals for the mean and a percentile based on zero-inflated lognormal data

The problems of estimating the mean and an upper percentile of a lognormal population with nonnegative values are considered. For estimating the mean of a such population based on data that include zeros, a simple confidence interval (CI) that is obtained by modifying Tian's [Inferences on the mean of zero-inflated lognormal data: the generalized variable approach. Stat Med. 2005;24:3223—3232] generalized CI, is proposed. A fiducial upper confidence limit (UCL) and a closed-form approximate UCL for an upper percentile are developed. Our simulation studies indicate that the proposed methods are very satisfactory in terms of coverage probability and precision, and better than existing methods for maintaining balanced tail error rates. The proposed CI and the UCL are simple and easy to calculate. All the methods considered are illustrated using samples of data involving airborne chlorine concentrations and data on diagnostic test costs.     

Performance of confidence interval tests for the ratio of two lognormal means applied to Weibull and gamma distribution data

Five estimation approaches have been developed to compute the confidence interval (CI) for the ratio of two lognormal means: (1) T, the CI based on the t-test procedure; (2) ML, a traditional maximum likelihood-based approach; (3) BT, a bootstrap approach; (4) R, the signed log-likelihood ratio statistic; and (5) R*, the modified signed log-likelihood ratio statistic. The purpose of this study was to assess the performance of these five approaches when applied to distributions other than lognormal distribution, for which they were derived. Performance was assessed in terms of average length and coverage probability of the CIs for each estimation approaches (i.e., T, ML, BT, R, and R*) when data followed a Weibull or gamma distribution. Four models were discussed in this study. In Model 1, the sample sizes and variances were equal within the two groups. In Model 2, the sample sizes were equal but variances were different within the two groups. In Model 3, the variances were different within the two groups and the larger variance was paired with the larger sample size. In Model 4, the variances were different within the two groups and the larger variance was paired with the smaller sample size. The results showed that when the variances of the two groups were equal, the t-test performed well, no matter what the underlying distribution was and how large the variances of the two groups were. The BT approach performed better than the others when the underlying distribution was not lognormal distribution, although it was inaccurate when the variances were large. The R* test did not perform well when the underlying distribution was Weibull or gamma distributed data, but it performed best when the data followed a lognormal distribution.     

The Effect of Mis-Specification Between the Lognormal and Weibull Distributions on the Interval Estimation of a Quantile for Complete Data

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.     

Construction of Simultaneous Confidence Intervals for Ratios of Means of Lognormal Distributions

For constructing simultaneous confidence intervals for ratios of means for lognormal distributions, two approaches using a two-step method of variance estimates recovery are proposed. The first approach proposes fiducial generalized confidence intervals (FGCIs) in the first step followed by the method of variance estimates recovery (MOVER) in the second step (FGCIs–MOVER). The second approach uses MOVER in the first and second steps (MOVER–MOVER). Performance of proposed approaches is compared with simultaneous fiducial generalized confidence intervals (SFGCIs). Monte Carlo simulation is used to evaluate the performance of these approaches in terms of coverage probability, average interval width, and time consumption.     

ROBUSTNESS PROPERTIES OF LOGNORMAL CONFIDENCE INTERVALS FOR LOGNORMAL AND GAMMA DISTRIBUTED DATA

ABSTRACT

The performances of six confidence intervals for estimating the arithmetic mean of a lognormal distribution are compared using simulated data. The first interval considered is based on an exact method and is recommended in U.S. EPA guidance documents for calculating upper confidence limits for contamination data. Two intervals are based on asymptotic properties due to the Central Limit Theorem, and the other three are based on transformations and maximum likelihood estimation. The effects of departures from lognormality on the performance of these intervals are also investigated. The gamma distribution is considered to represent departures from the lognormal distribution. The average width and coverage of each confidence interval is reported for varying mean, variance, and sample size. In the lognormal case, the exact interval gives good coverage, but for small sample sizes and large variances the confidence intervals are too wide. In these cases, an approximation that incorporates sampling variability of the sample variance tends to perform better. When the underlying distribution is a gamma distribution, the intervals based upon the Central Limit Theorem tend to perform better than those based upon lognormal assumptions.     

Tests of fit for a lognormal distribution

The problem of goodness of fit of a lognormal distribution is usually reduced to testing goodness of fit of the logarithmic data to a normal distribution. In this paper, new goodness-of-fit tests for a lognormal distribution are proposed. The new procedures make use of a characterization property of the lognormal distribution which states that the Kullback–Leibler measure of divergence between a probability density function (p.d.f) and its r-size weighted p.d.f is symmetric only for the lognormal distribution [Tzavelas G, Economou P. Characterization properties of the log-normal distribution obtained with the help of divergence measures. Stat Probab Lett. 2012;82(10):1837–1840]. A simulation study examines the performance of the new procedures in comparison with existing goodness-of-fit tests for the lognormal distribution. Finally, two well-known data sets are used to illustrate the methods developed.     

Analysis of over-dispersed count data with extra zeros using the Poisson log-skew-normal distribution

ABSTRACT

Mixed Poisson distributions are widely used in various applications of count data mainly when extra variation is present. This paper introduces an extension in terms of a mixed strategy to jointly deal with extra-Poisson variation and zero-inflated counts. In particular, we propose the Poisson log-skew-normal distribution which utilizes the log-skew-normal as a mixing prior and present its main properties. This is directly done through additional hierarchy level to the lognormal prior and includes the Poisson lognormal distribution as its special case. Two numerical methods are developed for the evaluation of associated likelihoods based on the Gauss–Hermite quadrature and the Lambert's W function. By conducting simulation studies, we show that the proposed distribution performs better than several commonly used distributions that allow for over-dispersion or zero inflation. The usefulness of the proposed distribution in empirical work is highlighted by the analysis of a real data set taken from health economics contexts.     

Bootstrap confidence intervals for the coefficient of quartile variation

ABSTRACT

In non-normal populations, it is more convenient to use the coefficient of quartile variation rather than the coefficient of variation. This study compares the percentile and t-bootstrap confidence intervals with Bonett's confidence interval for the quartile variation. We show that empirical coverage of the bootstrap confidence intervals is closer to the nominal coverage (0.95) for small sample sizes (n = 5, 6, 7, 8, 9, 10 and 15) for most distributions studied. Bootstrap confidence intervals also have smaller average width. Thus, we propose using bootstrap confidence intervals for the coefficient of quartile variation when the sample size is small.     

Some Further Issues Concerning Likelihood Inference for Left Truncated and Right Censored Lognormal Data

The maximum likelihood estimates (MLEs) of the parameters of a two-parameter lognormal distribution with left truncation and right censoring are developed through the Expectation Maximization (EM) algorithm. For comparative purpose, the MLEs are also obtained by the Newton–Raphson method. The asymptotic variance-covariance matrix of the MLEs is obtained by using the missing information principle, under the EM framework. Then, using asymptotic normality of the MLEs, asymptotic confidence intervals for the parameters are constructed. Asymptotic confidence intervals are also obtained using the estimated variance of the MLEs by the observed information matrix, and by using parametric bootstrap technique. Different confidence intervals are then compared in terms of coverage probabilities, through a Monte Carlo simulation study. A prediction problem concerning the future lifetime of a right censored unit is also considered. A numerical example is given to illustrate all the inferential methods developed here.     

Confidence intervals for coefficients of variation in two-parameter exponential distributions

This article examines confidence intervals for the single coefficient of variation and the difference of coefficients of variation in the two-parameter exponential distributions, using the method of variance of estimates recovery (MOVER), the generalized confidence interval (GCI), and the asymptotic confidence interval (ACI). In simulation, the results indicate that coverage probabilities of the GCI maintain the nominal level in general. The MOVER performs well in terms of coverage probability when data only consist of positive values, but it has wider expected length. The coverage probabilities of the ACI satisfy the target for large sample sizes. We also illustrate our confidence intervals using a real-world example in the area of medical science.     

Minimum variance unbiased estimators of the ratio of means of two lognormal variates

The minimum variance unbiased estimators from independent samples of the ratio of the means of two lognormal distributions with equal and unequal shape parameters are derived using a method due to Finney (1941). The like estimator for two gamma distributions of known shape is given. A numerical example from a recent cloud-seeding experiment is also given.     

Simultaneous Confidence Intervals for the Ratios of Marginal Means of a Multivariate Poisson Distribution

This article computes simultaneous confidence intervals for the ratios of marginal means of a multivariate Poisson distribution. For this, we propose a lognormal approximation technique and a bootstrap method. We demonstrate advantages of the proposed methods over existing ones through a simulation study. To illustrate their applicability to real-world problems, we apply the proposed methods to US data on infectious diseases.     

Likelihood inference for lognormal data with left truncation and right censoring with an illustration

The lognormal distribution is quite commonly used as a lifetime distribution. Data arising from life-testing and reliability studies are often left truncated and right censored. Here, the EM algorithm is used to estimate the parameters of the lognormal model based on left truncated and right censored data. The maximization step of the algorithm is carried out by two alternative methods, with one involving approximation using Taylor series expansion (leading to approximate maximum likelihood estimate) and the other based on the EM gradient algorithm (Lange, 1995). These two methods are compared based on Monte Carlo simulations. The Fisher scoring method for obtaining the maximum likelihood estimates shows a problem of convergence under this setup, except when the truncation percentage is small. The asymptotic variance-covariance matrix of the MLEs is derived by using the missing information principle (Louis, 1982), and then the asymptotic confidence intervals for scale and shape parameters are obtained and compared with corresponding bootstrap confidence intervals. Finally, some numerical examples are given to illustrate all the methods of inference developed here.     

Inference on System Reliability for Independent Series Components

In this article, we study inferences for reliability functions of the system having two components connected in series. Suppose that the lifetime of one component has a lognormal distribution. Lognormal, exponential, and weibull distributions are considered for the lifetime of the other component. Using the generalized inference approach, we obtain confidence intervals of our interested parameters with good coverage. Some frequentist properties in small-sample cases and large-sample cases are proved.     

An Availability System with General Repair Distribution: Statistical Inference

This article we study the statistical inferences of an availability system with imperfect coverage. The time-to-failure and time-to-repair of the active and standby components are assumed to be exponential and general distribution, respectively. Assume that the coverage factor is the same for an active-component failure as that for a standby-component failure. Firstly, we propose a consistent and asymptotically normal (CAN) estimator of availability for such repairable system. Based on the CAN estimator of the system availability, interval estimation and testing (hypothesis) are performed. To implement the simulation inference for the system availability, we adopt two repair-time distributions, such as lognormal and Weibull distribution, in which three types of Weibull distribution are considered according to the shape parameter β. The component holds the decreasing repair rate (DRR), constant repair rate (CRR), and increasing repair rate (IRR) if β < 1, β = 1, and β > 1, respectively. Finally, all simulation results are displayed by appropriate tables and curves for understanding performance of the statistical inference procedures presented in this article.     

Recommendations on the statistical analysis of the Comet assay

In 2010, the Statisticians in the Pharmaceutical Industry (PSI) Toxicology Special Interest Group met to discuss the design and analysis of the Comet assay. The Comet assay is one potential component of the package of safety studies required by regulatory bodies. As these studies usually involve a three-way nested experimental design and as the distribution of the measured response is usually either lognormal or lognormal plus a point mass at zero, the analysis is not straightforward. This has led to many different types of analysis being proposed in the literature, with several different methods applied within the pharmaceutical industry itself. This article summarises the PSI Toxicology Group's discussions and recommendations around these issues.     

Approximated non parametric confidence regions for the ratio of two percentiles

In the wood industry, it is common practice to compare in terms of the ratio of the same-strength properties for lumber of two different dimensions, grades, or species. Because United States lumber standards are given in terms of population fifth percentile, and strength problems arise from the weaker fifth percentile rather than the stronger mean, so the ratio should be expressed in terms of the fifth percentiles rather than the means of two strength distributions. Percentiles are estimated by order statistics. This paper assumes small samples to derive new non parametric methods such as percentile sign test and percentile Wilcoxon signed rank test, construct confidence intervals with covergage rate 1 – αx for single percentiles, and compute confidence regions for ratio of percentiles based on confidence intervals for single percentiles. Small 1 – αx is enough to obtain good coverage rates of confidence regions most of the time.     

Asymptotically Efficient Estimation of a Bivariate Gaussian–Weibull Distribution and an Introduction to the Associated Pseudo-truncated Weibull

Two important wood properties are stiffness (modulus of elasticity or MOE) and bending strength (modulus of rupture or MOR). In the past, MOE has often been modeled as a Gaussian and MOR as a lognormal or a two or three parameter Weibull. It is well known that MOE and MOR are positively correlated. To model the simultaneous behavior of MOE and MOR for the purposes of wood system reliability calculations, we introduce a bivariate Gaussian–Weibull distribution and the associated pseudo-truncated Weibull. We use asymptotically efficient likelihood methods to obtain an estimator of the parameter vector of the bivariate Gaussian–Weibull, and then obtain the asymptotic distribution of this estimator.