Delta-lognormal Distribution Research Articles

Updated fiducial distribution of parameters in the associated delta-lognormal population.

In this paper we consider a special kind of semicontinous distribution. We try to concern with the situation where the probability of zero observation is associated with the location and scale parameters in lognormal distribution. We first propose a goodness-of-fit test to ensure that the data can be fit by the associated delta-lognormal distribution. Then we define the updated fiducial distributions of the parameters and establish the results that the confidence interval has asymtotically correct level while the significance level of the hypothesis testing is also asymtotically correct. We propose an exact sampling method to sample from the updated fiducial distribution. It can be seen in our simulation study that the inference on the parameters is largely improved. A real data example is also used to illustrate our method.

Simultaneous Confidence Intervals for Ratios of the Percentiles of the Delta-Lognormal Distribution

Statistical percentiles are extremely important and valuable measurement tools in various contexts. They offer significant advantages when it comes to estimating a wide range of values across different fields, making them essential in statistical analysis and research. This importance motivates researchers to study the construction of simultaneous confidence intervals for ratios of percentiles of the delta-lognormal distribution. This research introduces four distinct methods for constructing such confidence intervals: the fiducial generalized confidence interval, the Bayesian-based highest posterior density credible interval using Jeffreys prior, Jeffreys rule prior, and Uniform prior. All methods are subjected to evaluation and comparison through coverage probabilities and expected lengths in Monte Carlo simulations. The simulation study shows that, overall, the Bayesian-based highest posterior density credible interval works better and more accurately than the fiducial generalized confidence interval. To further validate the findings of the simulation study, these methods are applied to real-world daily rainfall data collected from a river basin in Thailand.

Confidence intervals for ratio of means of delta-lognormal distributions based on left-censored data with application to rainfall data in Thailand.

Thailand is a country that is prone to both floods and droughts, and these natural disasters have significant impacts on the country's people, economy, and environment. Estimating rainfall is an important part of flood and drought prevention. Rainfall data typically contains both zero and positive observations, and the distribution of rainfall often follows the delta-lognormal distribution. However, it is important to note that rainfall data can be censored, meaning that some values may be missing or truncated. The interval estimator for the ratio of means will be useful when comparing the means of two samples. The purpose of this article was to compare the performance of several approaches for statistically analyzing left-censored data. The performance of the confidence intervals was evaluated using the coverage probability and average length, which were assessed through Monte Carlo simulation. The approaches examined included several variations of the generalized confidence interval, the Bayesian, the parametric bootstrap, and the method of variance estimates recovery approaches. For (ξ1, ξ2) = (0.10,0.10), simulations showed that the Bayesian approach would be a suitable choice for constructing the credible interval for the ratio of means of delta-lognormal distributions based on left-censored data. For (ξ1, ξ2) = (0.10,0.25), the parametric bootstrap approach was a strong alternative for constructing the confidence interval. However, the generalized confidence interval approach can be considered to construct the confidence when the sample sizes are increase. Practical applications demonstrating the use of these techniques on rainfall data showed that the confidence interval based on the generalized confidence interval approach covered the ratio of population means and had the smallest length. The proposed approaches' effectiveness was illustrated using daily rainfall datasets from the provinces of Chiang Rai and Chiang Mai in Thailand.

Confidence Intervals for the Difference and Ratio of Medians of the Delta-Lognormal Distribution

Confidence Intervals for the Difference and Ratio of Medians of the Delta-Lognormal Distribution

Bayesian Estimation for the Difference between Coefficients of Quartile Variation of Delta-Lognormal Distributions: An Application to Rainfall in Thailand

The coefficient of quartile variation is a valuable measure used to assess data dispersion when it deviates from a normal distribution or displays skewness. In this study, we focus specifically on the delta-lognormal distribution. The lognormal distribution is characterized by its asymmetrical nature and comprises exclusively positive values. However, when these values undergo a logarithmic transformation, they conform to a symmetrical (or normal) distribution. Consequently, this research aims to establish confidence intervals for the difference between coefficients of quartile variation within lognormal distributions incorporating zero values. We employ the Bayesian, generalized confidence interval, and fiducial generalized confidence interval methods to construct these intervals, involving data simulation using RStudio software. We evaluate the performance of these methods based on coverage probabilities and average lengths. Our findings indicate that the Bayesian method, employing Jeffreys’ prior, performs well in low variability, while the generalized confidence interval method is more suitable for higher variability. Therefore, we recommend using both approaches to construct confidence intervals for the difference between the coefficients of the quartile variation in lognormal distributions that include zero values. Furthermore, we apply these methods to rainfall data in Thailand to illustrate their alignment with actual and simulated data.

Confidence Intervals for Mean and Difference between Means of Delta-Lognormal Distributions Based on Left-Censored Data

A delta-lognormal distribution consists of zero and positive values. The positive values follow a lognormal distribution, which is an asymmetric distribution. It is well known that the logarithm of these values follows a normal distribution, which is a symmetric distribution. The delta-lognormal distribution is used in medical and environmental sciences. This study considers the challenges of constructing confidence intervals for the mean and difference between means of delta-lognormal distributions containing left-censored data and applies them to compare two daily rainfall average areas in Thailand. Three different approaches for constructing confidence intervals for the mean of the delta-lognormal distribution containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, and the parametric bootstrap approach, are developed. Moreover, four different approaches for constructing confidence intervals for the difference between means of delta-lognormal distributions containing left-censored data, based on the generalized confidence interval approach, the Bayesian approach, the parametric bootstrap approach, and the method of variance estimates recovery approach, are considered. The performance of the proposed confidence intervals is evaluated by Monte Carlo simulation. The simulation studies indicate that the Bayesian approach can be considered as an alternative to construct a credible interval for the mean of the delta-lognormal distribution containing left-censored data. Additionally, the generalized confidence interval and Bayesian approaches can be recommended as alternatives to estimate the confidence interval for the difference between means of delta-lognormal distributions containing left-censored data. All approaches are illustrated using the daily rainfall data from Chiang Mai and Lampang provinces in Thailand.

Confidence Interval Estimation for the Ratio of the Percentiles of Two Delta-Lognormal Distributions with Application to Rainfall Data

The log-normal distribution (skewed distribution or asymmetry distribution) is used to describe random variables comprising positive real values. It is well known that the logarithm values of these are normally distributed (symmetry distribution). Positively right-skewed data applicable to the log-normal distribution are frequently observed in the fields of environmental studies, biology, and medicine. The number of zero observations follows a binomial distribution. However, problems can arise in the analysis of data containing zero observations along with log-normally distributed data, for which the delta-lognormal distribution is often referred to for using the analysis of the data. In statistics, the percentile provides the relative standing of a numerical data point when compared to all of the others in a distribution with reference to the observations at or below it. In this study, estimates for the confidence interval for the ratio of the percentiles of two delta-lognormal distributions are constructed using fiducial generalized confidence interval approaches based on the fiducial quantity and the optimal generalized fiducial quantity, the Bayesian approach, and the parametric bootstrap method. As assessed by Monte Carlo simulations using the RStudio programming in terms of the coverage probability and the average length, the Bayesian approach performed quite well by providing adequate coverage probabilities along with the shortest average lengths in all of the scenarios tested. Daily rainfall data contain both zero and positive values. The daily rainfall data can usually be fitted to the delta-lognormal distribution. Their application to rainfall data is also provided to illustrate their efficacies with real data. The efficacy of the approach is used to compare two rainfall dispersion populations.

On comparing design-based estimation versus model-based prediction to assess the abundance of biological populations

Assessing the abundance of biological populations is a central technical challenge in ecology, whether fundamental or applied. Although the question faced is simple, it is actually a complicated topic that has produced a vast array of methods. The first step is always sampling the geographical space, ideally by means of a randomized selection process. In that case, with a frequentist interpretation of probability, abundance can be estimated by taking into account randomization, or predicted conditionally on the sample at hand, by specifying a statistical model. This leads to a choice between so-called design-based estimation and model-based prediction. The goal of this methodological article is to contribute to the understanding of fundamental notions regarding these two statistical frameworks by the targeted audience, mainly quantitative ecologists. For this purpose, we illustrate the comparison between design-based estimation in the case of simple random sampling without replacement (SRSWOR) and model-based prediction. As an example, we model count data with a delta-lognormal distribution and rely on uniformly minimum-variance unbiased estimators (UMVUEs) for the prediction of abundance. We investigate the robustness of the predictor by contaminating the delta-lognormal distribution using actual count data. Data from a survey concerning wintering populations in France of two wader species, namely, northern lapwing (Vanellus vanellus) and European golden plover (Pluvialis apricaria) serve as illustrative examples. By means of Monte Carlo simulations, we highlight the lack of robustness of the predictor based on the delta-lognormal distributional model, in terms of both actual bias and precision. We organize the discussion around the illustrative examples in the context of the sampling design, the model and the data.

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.

On evaluating the efficiency of the delta-lognormal mean estimator and predictor

A variable taking positive values from a lognormal distribution and null values with a given probability is distributed according to the so-called delta-lognormal distribution. Two situations arise depending on whether the data are regarded as a random sample from an infinite population (superpopulation) or from a finite population, itself considered as a random sample from a superpopulation. In the case of an infinite population, estimating the mean can be accomplished using a uniformly minimum-variance unbiased estimator (UMVUE). Likewise, the prediction of the mean in the case of a finite population may be based on the UMVUE. In both cases, one expects a gain in precision when taking into account the shape of the distribution by relying on the UMVUE rather than on the sample mean, which is a nonparametric estimator (or predictor).1.For the infinite population case, the relative efficiency results presented in this article are more complete and more accurate than those published so far.2.The article fills a gap regarding the question of relative efficiency in the case of a finite population.3.Calculations were performed using the exact expression for the variance of the UMVUE of the mean, expressed in terms of the confluent hypergeometric limit function.

Simultaneous confidence intervals for all pairwise differences between the coefficients of variation of rainfall series in Thailand

The delta-lognormal distribution is a combination of binomial and lognormal distributions, and so rainfall series that include zero and positive values conform to this distribution. The coefficient of variation is a good tool for measuring the dispersion of rainfall. Statistical estimation can be used not only to illustrate the dispersion of rainfall but also to describe the differences between rainfall dispersions from several areas simultaneously. Therefore, the purpose of this study is to construct simultaneous confidence intervals for all pairwise differences between the coefficients of variation of delta-lognormal distributions using three methods: fiducial generalized confidence interval, Bayesian, and the method of variance estimates recovery. Their performances were gauged by measuring their coverage probabilities together with their expected lengths via Monte Carlo simulation. The results indicate that the Bayesian credible interval using the Jeffreys’ rule prior outperformed the others in virtually all cases. Rainfall series from five regions in Thailand were used to demonstrate the efficacies of the proposed methods.

Bayesian confidence intervals for variance of delta-lognormal distribution with an application to rainfall dispersion

Bayesian confidence intervals for variance of delta-lognormal distribution with an application to rainfall dispersion

The Bayesian confidence intervals for measuring the difference between dispersions of rainfall in Thailand.

The coefficient of variation is often used to illustrate the variability of precipitation. Moreover, the difference of two independent coefficients of variation can describe the dissimilarity of rainfall from two areas or times. Several researches reported that the rainfall data has a delta-lognormal distribution. To estimate the dynamics of precipitation, confidence interval construction is another method of effectively statistical inference for the rainfall data. In this study, we propose confidence intervals for the difference of two independent coefficients of variation for two delta-lognormal distributions using the concept that include the fiducial generalized confidence interval, the Bayesian methods, and the standard bootstrap. The performance of the proposed methods was gauged in terms of the coverage probabilities and the expected lengths via Monte Carlo simulations. Simulation studies shown that the highest posterior density Bayesian using the Jeffreys’ Rule prior outperformed other methods in virtually cases except for the cases of large variance, for which the standard bootstrap was the best. The rainfall series from Songkhla, Thailand are used to illustrate the proposed confidence intervals.

A Bayesian approach to construct confidence intervals for comparing the rainfall dispersion in Thailand.

Natural disasters such as drought and flooding are the consequence of severe rainfall fluctuation, and rainfall amount data often contain both zero and positive observations, thus making them fit a delta-lognormal distribution. By way of comparison, rainfall dispersion may not be similar in enclosed regions if the topography and the drainage basin are different, so it can be evaluated by the ratio of variances. To estimate this, credible intervals using the highest posterior density based on the normal-gamma prior (HPD-NG) and the method of variance estimates recovery (MOVER) for the ratio of delta-lognormal variances are proposed. Monte Carlo simulation was used to assess the performance of the proposed methods in terms of coverage probability and relative average length. The results of the study reveal that HPD-NG performed very well and was able to meet the requirements in various situations, even with a large difference between the proportions of zeros. However, MOVER is the recommended method for equal small sample sizes. Natural rainfall datasets for the northern and northeastern regions of Thailand are used to illustrate the practical use of the proposed credible intervals.

Measuring the dispersion of rainfall using Bayesian confidence intervals for coefficient of variation of delta-lognormal distribution: a study from Thailand.

Since rainfall data series often contain zero values and thus follow a delta-lognormal distribution, the coefficient of variation is often used to illustrate the dispersion of rainfall in a number of areas and so is an important tool in statistical inference for a rainfall data series. Therefore, the aim in this paper is to establish new confidence intervals for a single coefficient of variation for delta-lognormal distributions using Bayesian methods based on the independent Jeffreys’, the Jeffreys’ Rule, and the uniform priors compared with the fiducial generalized confidence interval. The Bayesian methods are constructed with either equitailed confidence intervals or the highest posterior density interval. The performance of the proposed confidence intervals was evaluated using coverage probabilities and expected lengths via Monte Carlo simulations. The results indicate that the Bayesian equitailed confidence interval based on the independent Jeffreys’ prior outperformed the other methods. Rainfall data recorded in national parks in July 2015 and in precipitation stations in August 2018 in Nan province, Thailand are used to illustrate the efficacy of the proposed methods using a real-life dataset.

Determining Discharge Characteristics and Limits of Heavy Metals and Metalloids for Wastewater Treatment Plants (WWTPs) in China Based on Statistical Methods

Industrial wastewater and sewage are both important sources of heavy metals and metalloids in urban wastewater treatment plants (WWTPs). China has made great efforts to control heavy metal and metalloid pollution by setting discharge limits for WWTPs. There is, however, limited discharge data and no systematic methodology for the derivation of discharge limits. In this study, 14 heavy metals and metalloids (Hg, alkyl mercury, As, Cd, Cr, Cr6+, Pb, Ni, Be, Ag, Cu, Zn, Mn, Se) that are listed in the Discharge Standard of Pollutants for Municipal Wastewater Treatment Plant (GB 18918-2002) were selected for the analysis of discharge characteristics while using the supervised monitoring data from more than 800 WWTPs located in nine provinces in China. Of the 14 heavy metals and metalloids, all but alkyl mercury were detected in the discharge water. There was a high rate of detection of As, Cu, Mn, Se, and there were some samples that exceeded the standard concentrations of Cr, Cr6+, Pb, and Ni. Removal rates of Hg, As, Cd, Cr, Cr6+, Pb, Ni, Cu, Zn, Mn, and Se were higher than 40%, comparable to values from other countries. Hg and As were selected to analyze the influencing factors of effluent and derive discharge limits of WWTPs using a statistical method, because these two metals had more detected data than other metals. The study used supervised monitoring data from Zhejiang WWTPs with 99 for Hg and 112 for As. Based on the delta-lognormal distribution, the results showed that geographic location was significantly closely correlated with Hg (P = 0.027 < 0.05) and As (P ≈ 0 < 0.05) discharge concentrations, while size (for Hg P = 0.695 > 0.05, for As P = 0.088 > 0.05) and influent concentration (R2 < 0.5) were not. Derived Hg and As discharge limits suggest that it is necessary to establish stricter discharge limits for WWTPs, which is more consistent with the real-world situation in China. The study here comprehensively researches the discharge characteristics of heavy metals and metalloids in effluent of WWTPs in China, and developed for the first time in China heavy metals and metalloids discharge limits based on statistical methods. The results may inform special discharge limit settings for WWTPs in China.

Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data

Highly skewed and non-negative data can often be modeled by the delta-lognormal distribution in fisheries research. However, the coverage probabilities of extant interval estimation procedures are less satisfactory in small sample sizes and highly skewed data. We propose a heuristic method of estimating confidence intervals for the mean of the delta-lognormal distribution. This heuristic method is an estimation based on asymptotic generalized pivotal quantity to construct generalized confidence interval for the mean of the delta-lognormal distribution. Simulation results show that the proposed interval estimation procedure yields satisfactory coverage probabilities, expected interval lengths and reasonable relative biases. Finally, the proposed method is employed in red cod densities data for a demonstration.

Confidence intervals for the ratio of coefficients of variation of delta-lognormal distribution

Confidence intervals for the ratio of coefficients of variation of delta-lognormal distribution

Population structure, distribution and relative abundance of the blue shark (Prionace glauca) in the Caribbean Sea and adjacent waters of the North Atlantic

Catch and fishing effort data from the Venezuelan Pelagic Longline Observer Program (VPLOP) in the Caribbean Sea and adjacent waters of the Atlantic were used to investigate the population structure, distribution and relative abundance of the blue shark, Prionace glauca. A total of 959 blue sharks were sampled by trained observers during the period of January 1994 to December 2007. Fork length (FL) in males ranged from 79 to 355cm and from 97 to 349cm in females. Sexual segregation at the spatial and temporal scale was observed, males dominated January–June and during October–December across all areas; while females dominated during July–September with a higher magnitude in the Atlantic side (Guyana-Amazon area). Sex ratio for mature (>180cm FL) blue shark favored females in the Caribbean Sea from July to December, and in the Guyana-Amazon area in July–September; while immature blue sharks were dominated by males across all areas. Thirty-four pregnant females were caught in the Caribbean Sea and three in the Guyana-Amazon area. Pregnant females displayed a seasonal trend which increased from September through January, where >50% of the pregnant females occurred during November–January; of the pregnant females caught in the Caribbean Sea two had embryos close to the size-at-birth [>35cm total length (TL)]. The spatial distribution of blue sharks in the Caribbean Sea was marked by high concentrations around the offshore islands of Venezuela and the Netherland Antilles year round; while in the Guyana-Amazon area the high concentrations were more seasonally localized off northeastern coast of South America but diminished progressively and by the end of the year was absent in the area. Relative abundance of blue shark was estimated from the Venezuelan pelagic longline (VPL) fleet using a General Linear Model Approach with a delta lognormal model distribution to account for the high proportion of zero observations. The variability within vessel groups was partitioned into each vessel variance and the auto-correlation within a vessel was evaluated by using a variance–covariance matrix model. Diagnostic checks and deviance table analysis were performed to identify the set of model factors and interactions that adequately explained the observed variability. The results indicated that variability of catch rates within vessels is as large as the variability among vessel groups, where smaller and medium size vessels show higher catch rates of blue sharks in the VPL fleet. Standardized catch rates indicated in general a lower relative abundance in the most recent years followed by a slight recovery that requires caution due to the predicted high variance. Fishing activities can have uneven effects in the blue shark population due to its spatial and temporal segregation in the study area; potential changes in the fishing operations of the VPL fleet can have negative effects in the population structure of blue sharks in the Caribbean Sea and adjacent waters if habitat for blue sharks are increasingly used to target other valuable species (e.g., large tunas and swordfish).

Confidence intervals for the mean of the delta-lognormal distribution

Data that are skewed and contain a relatively high proportion of zeros can often be modelled using a delta-lognormal distribution. We consider three methods of calculating a 95% confidence interval for the mean of this distribution, and use simulation to compare the methods, across a range of realistic scenarios. The best method, in terms of coverage, is that based on the profile-likelihood. This gives error rates that are within 1% (lower limit) or 3% (upper limit) of the nominal level, unless the sample size is small and the level of skewness is moderate to high. Our results will also apply to the delta-lognormal linear model, when we wish to calculate a confidence interval for the expected value of the response variable, given the value of one or more explanatory variables. We illustrate the three methods using data on red cod densities, taken from a fisheries trawl survey in New Zealand.