Lognormal Means Research Articles

A burst storm from the repeating FRB 20200120E in an M81 globular cluster

ABSTRACT The repeating fast radio burst (FRB) source FRB 20200120E is exceptional because of its proximity and association with a globular cluster. Here we report 60 bursts detected with the Effelsberg telescope at 1.4 GHz. We observe large variations in the burst rate, and report the first FRB 20200120E ‘burst storm’, where the source suddenly became active and 53 bursts (fluence ≥0.04 Jy ms) occurred within only 40 min. We find no strict periodicity in the burst arrival times, nor any evidence for periodicity in the source’s activity between observations. The burst storm shows a steep energy distribution (power-law index α = 2.39 ± 0.12) and a bimodal wait-time distribution, with log-normal means of 0.94$^{+0.07}_{-0.06}$ s and 23.61$^{+3.06}_{-2.71}$ s. We attribute these wait-time distribution peaks to a characteristic event time-scale and pseudo-Poisson burst rate, respectively. The secondary wait-time peak at ∼1 s is ∼50 × longer than the ∼24 ms time-scale seen for both FRB 20121102A and FRB 20201124A – potentially indicating a larger emission region, or slower burst propagation. FRB 20200120E shows order-of-magnitude lower burst durations and luminosities compared with FRB 20121102A and FRB 20201124A. Lastly, in contrast to FRB 20121102A, which has observed dispersion measure (DM) variations of ΔDM > 1 pc cm−3 on month-to-year time-scales, we determine that FRB 20200120E’s DM has remained stable (ΔDM < 0.15 pc cm−3) over >10 months. Overall, the observational characteristics of FRB 20200120E deviate quantitatively from other active repeaters, but it is unclear whether it is qualitatively a different type of source.

Objective Bayesian analysis of the ratio of bivariate lognormal means

In this study, we consider the objective Bayesian analysis for the ratio of means in the bivariate lognormal distribution, in which properly assigning priors for the ratio of means is challenging due to the presence of nuisance parameters. As a result, we create probability matching priors and reference priors for the ratio of means. Jeffreys’ prior and the two group reference prior do not satisfy a first-order matching criterion. We also investigate the posterior distribution property for the general prior class, which includes Jeffreys’ prior, the reference prior, and the matching prior. Through a simulation study, we check the frequentist coverage probabilities and compare them with the generalized confidence interval method. It demonstrated that the proposed probability matching priors match very well with the target coverage probabilities even when the sample sizes are small in a frequentist sense. Two real examples are also provided.

Pretest and shrinkage estimators for log-normal means

Pretest and shrinkage estimators for log-normal means

An alternative method for estimating lognormal means

For a probabilistic model with positively skewed data, a lognormal distribution is one of the key distributions that play a critical role. Several lognormal models can be found in various areas, such as medical science, engineering, and finance. In this paper, we propose a new estimator for a lognormal mean and depict the performance of the proposed estimator in terms of the relative mean squared error (RMSE) compared with Shen’s estimator (Shen et al., 2006), which is considered the best estimator among the existing methods. The proposed estimator includes a tuning parameter. By finding the optimal value of the tuning parameter, we can improve the average performance of the proposed estimator over the typical range of σ2. The bias reduction of the proposed estimator tends to exceed the increased variance, and it results in a smaller RMSE than Shen’s estimator. A numerical study reveals that the proposed estimator has performance comparable with Shen’s estimator when σ2 is small and exhibits a meaningful decrease in the RMSE under moderate and large σ2 values.

On the Problem of Comparing the Means and Medians of Two Independent Lognormal Distributions

Inferences about the ratio of two lognormal means δ can depend on plausible values of ρ, the ratio of the normal standard deviations associated to these distributions. This aspect is not usually considered in most of the analyses carried out in some applied sciences. In this paper we propose a profile likelihood approach that allows the comparison of two independent lognormal data sets in a more exhaustive way. Inferences about δ, ρ and (δ, ρ) are jointly analyzed through a simple closed-form expression obtained for the profile likelihood function of the parameter vector (δ, ρ). A similar analysis is done for ψ and ρ, where ψ is the ratio of two lognormal medians, obtaining also a simple closed-form expression for the profile likelihood function of these parameters. These expressions allow us to construct likelihood contour plots that capture most of the information provided by the samples and become valuable to identify if a trade-off between the parameters under study occurs; in case of that, individual inferences should be analyzed carefully. A detailed series of Monte Carlo simulations are included; they illustrate the performance of profile likelihood and parametric bootstrap approaches, for different sample sizes and parameter values.

A revisit to testing the equality of means for several lognormal distributions

In this paper, we propose a new test statistic for testing the equality of several lognormal means based on Score statistic. Besides the traditional chi-square approximation to Score statistic, we use a parametric bootstrap method called computational approach test (CAT). We compare the proposed tests in terms of type I error rate and power under different number of groups and sample sizes to evaluate efficiency of these tests against the existing tests in the literature. Simulation results indicate that powers of these tests are outstandingly higher compared to their competitors under all the considered cases. Although the performance of chi-square approximation of Score statistic is insufficient in small sample cases for many problems, this approximation performs quite well for our problem even in some small sample sizes. Besides, we illustrate the considered methods using both numerical example and a real data example.

Objective Bayesian analysis using modified profile likelihood for the ratio of two log-normal means

Matching priors in principle provide a possible compromise between frequentist and Bayesian and default priors for routine use in Bayesian inference in that posterior probability of the parameter also provides an interpretation as confidence statements. Here we introduce a matching prior for the ratio of two log-normal means which is nontrivial because the derivation of matching prior requires the elicitation of a suitable orthogonal parameterization on the nuisance parameters and the computation of the marginal posterior distribution requires multidimensional integration over the nuisance parameter. Numerical integrations and approximation techniques could be used but they are difficult to use in general. Thus, we derive a matching prior based on a modification of the profile likelihood to avoid the elicitation of priors for the entire parameter and integration on the nuisance parameter. The proposed method is illustrated by real data examples and simulation studies under several configurations.

Testing the equality of several log-normal means based on a computational approach

ABSTRACTIn this study, we propose a new test based on a computational approach to test the equality of several log-normal means. We compare this test with some existing methods in terms of the type-I error rate and power using Monte Carlo simulations for varying values of number of groups and sample sizes. The simulation results indicate that the proposed test could be suggested as a good alternative for testing the equality of several log-normal means.

Data Representativeness: Issues and Solutions

In its control programmes on maximum residue level compliance and exposure assessments, EFSA requires the participating countries to submit results, from specific numbers of food item samples, analyzed in the countries. These data are used to obtain estimates such as the proportion of samples exceeding the maximum residue limits, and the mean and maximum residue concentration per food item to assess exposure. An important consideration is the design and analysis of the programmes. In this report, we combine elements of survey sampling methodology, and statistical modeling, as a benchmark framework for the programmes, starting from the translation of research questions into statistical problems, to the statistical analysis and interpretation. Particular focus is placed on the issues that could affect the representativeness of the data, and remedial procedures are proposed. For example, in the absence of information on the sampling design, a sensitivity analysis, across a range of designs, is proposed. On the other hand, weighted generalized linear mixed models, and generalized linear mixed models combining both conjugate and normal random effects, are proposed, to address selection bias. Likelihood-based analysis methods are also proposed to address missing and censored data problems. Suggestions for improvements in the design and analysis of the programmes are also identified and discussed. For instance, incorporation of stratified sampling methodology, in determining both the total number, and the allocation of samples to the participating countries, is proposed. All through the report, statistical analysis models which properly take into account the hierarchical (and thus correlated) structure in which the data are collected are proposed.

English

This paper presents the confidence interval for the ratio of means of lognormal distribution. We derived analytic expressions to find the coverage probability and the expected length of the proposed confidence interval. The lognormal distribution has been widely used for a skewed data in science, biology and economics. A ratio estimator is much attention in area of bioassay and bioequivalence. Recently, many researchers have been investigated this problem. For example, Lee and Lin (3) constructed the confidence interval for the normal means by using the generalized confidence interval and the generalized p-value proposed by (6). Later, Chen and Zhou (2) compared several methods for constructing the confidence interval for the ratio of lognormal means. They suggested a modified signed log-likelihood ratio approach which is the best among these confidence intervals. In this paper, we proposed to construct the confidence interval for the lognormal means when the coefficients of variation are known. Additionally, we derived analytic expressions to find its coverage probability and its expected length. Let   1 2 , ,..., , 1, 2, i i i i n X X X X i

Confidence intervals for a difference between lognormal means in cluster randomization trials.

Cluster randomization trials, in which intact social units are randomized to different interventions, have become popular in the last 25 years. Outcomes from these trials in many cases are positively skewed, following approximately lognormal distributions. When inference is focused on the difference between treatment arm arithmetic means, existent confidence interval procedures either make restricting assumptions or are complex to implement. We approach this problem by assuming log-transformed outcomes from each treatment arm follow a one-way random effects model. The treatment arm means are functions of multiple parameters for which separate confidence intervals are readily available, suggesting that the method of variance estimates recovery may be applied to obtain closed-form confidence intervals. A simulation study showed that this simple approach performs well in small sample sizes in terms of empirical coverage, relatively balanced tail errors, and interval widths as compared to existing methods. The methods are illustrated using data arising from a cluster randomization trial investigating a critical pathway for the treatment of community acquired pneumonia.

Comparing the mean vectors of two independent multivariate log-normal distributions

The multivariate log-normal distribution is a good candidate to describe data that are not only positive and skewed, but also contain many characteristic values. In this study, we apply the generalized variable method to compare the mean vectors of two independent multivariate log-normal populations that display heteroscedasticity. Two generalized pivotal quantities are derived for constructing the generalized confidence region and for testing the difference between two mean vectors. Simulation results indicate that the proposed procedures exhibit satisfactory performance regardless of the sample sizes and heteroscedasticity. The type I error rates obtained are consistent with expectations and the coverage probabilities are close to the nominal level when compared with the other method which is currently available. These features make the proposed method a worthy alternative for inferential analysis of problems involving multivariate log-normal means. The results are illustrated using three examples.

Estimation of reliability from a bivariate log-normal data

It has been established that the bivariate log-normal distribution is appropriate for modelling certain paired observations. In this paper, we have developed large-sample confidence intervals of the dependence and reliability R=P(X>Y) parameters from a bivariate log-normal distribution with equal log-normal means. The parameter R provides a general measure of difference between the two populations and has applications in many areas. The performance of these confidence intervals has been examined by extensive simulation studies. The results are illustrated with an example dealing with a quantitative assay problem.

Bayesian Estimation of Log-Normal Means with Finite Quadratic Expected Loss

The log-normal distribution is a popular model in biostatistics and other fields of statistics. Bayesian inference on the mean and median of the distribution is problematic because, for many popular choices of the prior for the variance (on the log-scale) parameter, the posterior distribution has no finite moments, leading to Bayes estimators with infinite expected loss for the most common choices of the loss function. We propose a generalized inverse Gaussian prior for the variance parameter, that leads to a log-generalized hyperbolic posterior, for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yield finite posterior moments of order r. We investigate the choice of prior parameters leading to Bayes estimators with optimal frequentist mean square error. For the estimation of the lognormal mean we show, using simulation, that the Bayes estimator under quadratic loss compares favorably in terms of frequentist mean square error to known estimators.

Modified method on the means for several log-normal distributions

Among statistical inferences, one of the main interests is drawing the inferences about the log-normal means since the log-normal distribution is a well-known candidate model for analyzing positive and right-skewed data. In the past, the researchers only focused on one or two log-normal populations or used the large sample theory or quadratic procedure to deal with several log-normal distributions. In this article, we focus on making inferences on several log-normal means based on the modification of the quadratic method, in which the researchers often used the vector of the generalized variables to deal with the means of the symmetric distributions. Simulation studies show that the quadratic method performs well only for symmetric distributions. However, the modified procedure fits both symmetric and skew distribution. The numerical results show that the proposed modified procedure can provide the confidence interval with coverage probabilities close to the nominal level and the hypothesis testing performed with satisfactory results.

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.

Testing Equality of Generalized Treatment Effects

Recently, the concept of generalized treatment effect, defined as P(X > Y) where X and Y denote continuous outcome variables for treatment arm and control arm, respectively, has been proposed as an appropriate measure of treatment effect in clinical trials with parallel design. Compared to the mean difference, the generalized treatment effect has many advantages; for example, it is a scaleless measure and it does not change under monotonic transformations. This article investigates the problem of testing equality of generalized treatment effects among several clinical trials. The proposed approach follows the same vein as the generalized variable method for testing equality of several log-normal means proposed by Li (2009). Numerical study demonstrates that the proposed test has excellent type I error control for clinical trials with small to medium sample sizes. Robustness study shows that the proposed method performs reasonably for categorical data.

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.

Simple confidence intervals for lognormal means and their differences with environmental applications

AbstractThe lognormal distribution has frequently been applied to approximate environmental data, with inference focusing on arithmetic means. Confidence interval estimation involving lognormal means in small to moderate sample sizes has received much attention over the years without a simple procedure in sight. We therefore propose a closed‐form procedure for constructing confidence intervals for a lognormal mean and a difference between two lognormal means. The advantage of our procedure is that it only requires confidence limits for a normal mean and variance. The results of a numerical study show that our method performs as well as the generalized confidence interval (GCI) approach, which relies completely on computer simulation. Two real datasets are used to illustrate the methodology. Copyright © 2008 John Wiley & Sons, Ltd.

Meta Analysis of Advanced Cancer Survival Data Using Lognormal Parametric Fitting: A Statistical Method to Identify Effective Treatment Protocols

We describe the use of a parametric lognormal model to calculate and compare survival statistics in the clinical treatment of advanced/metastatic pancreatic, breast and colon cancers. The fit using the lognormal model explained greater than 90% (R(2) ranged from 0.917 to 0.998 for a total of the 51 arms from published studies) of the variation in the cumulative survival statistics of patients treated for advanced cancers. A meta-analytic Q-test was performed to test whether there were significant differences between different studies. For all three cancer types, the Q-test showed highly significant differences between the survival arms (p<0.0001 for pancreatic, breast and colon cancers). The z-values expressed the difference of the average of lognormal means relative to each study in terms of deviation expressed in standard errors. The treatments that were most effective ranked with the highest z-value: Doxorubicin plus docetaxel for pancreatic cancer (z-value = 4.1); Capecitabine plus paclitaxel for breast cancer (z-value = 3); irinotecan, fluorouracil and folinate for colon cancer (z-value = 7.4).