Bivariate Lognormal Distribution Research Articles

Using Pearson’s System of Curves to Approximate the Distributions of the Difference between Two Correlated Estimates of Signal-to-Noise Ratios: The Cases of Bivariate Normal and Bivariate Lognormal Distributions

Using Pearson’s System of Curves to Approximate the Distributions of the Difference between Two Correlated Estimates of Signal-to-Noise Ratios: The Cases of Bivariate Normal and Bivariate Lognormal Distributions

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.

Interval estimation of the common mean and difference of medians for a bivariate lognormal distribution

This paper considers the inference on the common mean and medians of a bivariate lognormal distribution. We propose point and interval estimators of the common mean in closed form. We derive confidence intervals of the common mean using the maximum likelihood method, generalized variable approach, method of variance estimate recovery and highest posterior density. The generalized confidence interval of the common mean under missing observations is also obtained. We obtain confidence limits for the difference of medians using first three techniques of interval estimation of the common mean and two bootstrap approaches. The coverage probability and expected length are used to assess the performance of the intervals. The proposed methodologies are exemplified through real-life data sets related to immunology, road accidents and vaccination programmes.

New bivariate and multivariate log-normal distributions as models for insurance data

The body of most multivariate financial data sets can be well modeled by log-normal distributions. Yet not many multivariate log-normal distributions are available in the literature. In this paper, we propose many new multivariate log-normal distributions. An application to an insurance data set is given.

Exact Statistical Distribution and Correlation of Human Height and Weight: Analysis and Experimental Confirmation

The statistical relationship between human height and weight is of especial importance to clinical medicine, epidemiology, and the biology of human development. Yet, after more than a century of anthropometric measurements and analyses, there has been no consensus on this relationship. The purpose of this article is to provide a definitive statistical distribution function from which all desired statistics (probabilities, moments, and correlation functions) can be determined. The statistical analysis reported in this article provides strong evidence that height and weight in a diverse population of healthy adults constitute correlated bivariate lognormal random variables. This conclusion is supported by a battery of independent tests comparing empirical values of 1) probability density patterns, 2) linear and higher order correlation coefficients, 3) statistical and hyperstatistics moments up to 6th order, and 4) distance correlation (dCor) values to corresponding theoretical quantities: 1) predicted by the lognormal distribution and 2) simulated by use of appropriate random number generators. Furthermore, calculation of the conditional expectation of weight, given height, yields a theoretical power law that specifies conditions under which body mass index (BMI) can be a valid proxy of obesity. The consistency of the empirical data from a large, diverse anthropometric survey partitioned by gender with the predictions of a correlated bivariate lognormal distribution was found to be so extensive and close as to suggest that this outcome is not coincidental or approximate, but may be a consequence of some underlying biophysical mechanism.

Modeling bivariate geyser eruption system with covariate-adjusted recurrent event process

ABSTRACT Geyser eruption is one of the most popular signature attractions at the Yellowstone National Park. The interdependence of geyser eruptions and impacts of covariates are of interest to researchers in geyser studies. In this paper, we propose a parametric covariate-adjusted recurrent event model for estimating the eruption gap time. We describe a general bivariate recurrent event process, where a bivariate lognormal distribution and a Gumbel copula with different marginal distributions are used to model an interdependent dual-type event system. The maximum likelihood approach is used to estimate model parameters. The proposed method is applied to analyzing the Yellowstone geyser eruption data for a bivariate geyser system and offers a deeper understanding of the event occurrence mechanism of individual events as well as the system as a whole. A comprehensive simulation study is conducted to evaluate the performance of the proposed method.

A Bayesian Approach for Determining the Relationship Between Various Elongate Mineral Particles (EMPs) Definitions.

A variety of dimensions (lengths and widths) of elongate mineral particles (EMPs) have been proposed as being related to health effects. In this paper, we develop a mathematical approach for deriving numerical conversion factors (CFs) between these EMP exposure metrics and applied it to the Minnesota Taconite Health Worker study which contains 196 different job exposure groups (28 similar exposure groups times 7 taconite mines). This approach comprises four steps: for each group (i) obtain EMP dimension information using ISO-TEM 10312/13794 analysis; (ii) use bivariate lognormal distribution to characterize overall EMP size distribution; (iii) use a Bayesian approach to facilitate the formation of the bivariate lognormal distribution; (iv) derive conversion factors between any pair of EMP definitions. The final CFs allow the creation of job exposure matrices (JEMs) for alternative EMP metrics using existing EMP exposures already characterized according to the National Institute of Occupational Safety and Health (NIOSH)-defined EMP exposure metric (length >5 µm with an aspect ratio ≥3.0). The relationships between the NIOSH EMP and other EMP definitions provide the basis of classification of workers into JEMs based on alternate definitions of EMP for epidemiological studies of mesothelioma, lung cancer, and non-malignant respiratory disease.

Probabilistic prediction of earthquake by bivariate distribution

The bivariate lognormal distribution (BLD) is proposed, as a model probabilistic, for joint distribution of earthquake prediction. In the recent decades, the probabilistic earthquake prediction using statistical analysis has become the cornerstone of research works in earthquake engineering field. In the case of earthquake prediction, all probability models have been established by distribution functions other than bivariate distribution functions. There is no denial of the fact that earthquake is absolutely considered as a multivariate phenomenon. Hence, comprehensive analysis of these variables needs joint probabilistic investigations. In this paper, the probability model of earthquake prediction has been studied by bivariate lognormal distribution function. For this aim, the parameters of BLD were estimated by maximum likelihood method. Then, the distribution function is established based upon two variables including earthquake magnitude and recurrence time. In this research, Tehran city, as one of the world’s most populous cities, was considered for the case study due to the fact that severe earthquakes have occurred in the past decades. The capability of using the probabilistic model for Tehran was obtained by Kolmogrov–Smirnov (KS) test. Overall, it was found that the most likely earthquake of Tehran may occur between 10 and 15 years from the last earthquake with magnitude between 6.6 and 6.8.

How variability in interfacial properties results in tougher bonded composite joints by triggering bridging

Toughening strategies are needed to design high performance secondary bonded joints. Although adhesively bonded joints present a promising solution for cost-saving assembly, they are however among the most vulnerable parts in composite structures. Due to the inherent heterogeneity of composite substrates and the infeasibility of ideal surface pretreatments, the bonding properties of composite joints feature a spatial variability that can result in a wide variety of debonding phenomenology, such as crack tip transfer, crack bifurcation or ligament bridging. In the present work, we apply a cohesive zone model with spatially stochastic interfacial properties to model the physical behaviors that take place during debonding of an adhesively bonded composite joint. The joint distribution of the two essential bonding properties, namely, the critical separating stress σm and the strain energy release rate Gc, follows a bivariate lognormal probability distribution. The proposed model captures the mechanisms of crack-tip transfer and the failure of bridging adhesive ligaments between the composite adherends, as well as the associated toughening of joints. We find that the extrinsic dissipation attributed to extra debonding areas and plastic-damage mechanisms of ligaments may be one of the primary causes for the thickness effect of bondline on the fracture toughness, commonly observed for adhesively bonded joints. Through a parametric study, we discuss the effect of variability in interfacial properties on the enhancement of toughness of bonded composite joints. The outcomes of this work may provide insights into the fundamental failure mechanisms of secondary bonded composite joints.

Simultaneous numerical representation of soil microsite production and consumption of carbon dioxide, methane, and nitrous oxide using probability distribution functions.

Production and consumption of nitrous oxide (N2 O), methane (CH4 ), and carbon dioxide (CO2 ) are affected by complex interactions of temperature, moisture, and substrate supply, which are further complicated by spatial heterogeneity of the soil matrix. This microsite heterogeneity is often invoked to explain non-normal distributions of greenhouse gas (GHG) fluxes, also known as hot spots and hot moments. To advance numerical simulation of these belowground processes, we expanded the Dual Arrhenius and Michaelis-Menten model, to apply it consistently for all three GHGs with respect to the biophysical processes of production, consumption, and diffusion within the soil, including the contrasting effects of oxygen (O2 ) as substrate or inhibitor for each process. High-frequency chamber-based measurements of all three GHGs at the Howland Forest (ME, USA) were used to parameterize the model using a multiple constraint approach. The area under a soil chamber is partitioned according to a bivariate log-normal probability distribution function (PDF) of carbon and water content across a range of microsites, which leads to a PDF of heterotrophic respiration and O2 consumption among microsites. Linking microsite consumption of O2 with a diffusion model generates a broad range of microsite concentrations of O2 , which then determines the PDF of microsites that produce or consume CH4 and N2 O, such that a range of microsites occurs with both positive and negative signs for net CH4 and N2 O flux. Results demonstrate that it is numerically feasible for microsites of N2 O reduction and CH4 oxidation to co-occur under a single chamber, thus explaining occasional measurement of simultaneous uptake of both gases. Simultaneous simulation of all three GHGs in a parsimonious modeling framework is challenging, but it increases confidence that agreement between simulations and measurements is based on skillful numerical representation of processes across a heterogeneous environment.

A Nifty mean chart method based on median ranked set sampling design

In the case of contamination for skewed distributions, the modified Shewhart, modified weighted variance, and modified skewness correction methods are newly introduced by Karagoz (Hacet J Math Stat 47(1):223–242, 2018). In this study, we propose to modify these methods by considering simple random sampling (SRS), ranked set sampling (RSS) and median ranked set sampling (MRSS) designs under the contaminated type I Marshall–Olkin bivariate Weibull and lognormal distributions. These bivariate distributions are chosen since they can represent a wide variety of shapes from nearly symmetric to highly skewed. We evaluate the performance of proposed modified methods based on different ranked set sampling designs by using Monte Carlo Simulation. The type I risks of $${\bar{X}}$$ charts for existing and newly proposed modified methods by using SRS, RSS and MRSS designs in the case of contamination for these distributions are obtained via simulation study. The proposed modified methods using RSS and MRSS designs for the $${\bar{X}}$$ chart can be a favorable substitute in process monitoring when the distribution is highly skewed and contaminated.

Statistics of progress variable and mixture fraction gradients in an open turbulent jet spray flame

The statistical behaviours of the variance, covariance and gradients of reaction progress variable c, and mixture fraction ξ have been analysed in an ethanol turbulent spray jet flame based on the results obtained from a carrier-phase three-dimensional Direct Numerical Simulation (DNS) dataset. It has been observed that the Favre Probability Density Functions (PDFs) of c and ξ extracted from the DNS data are correctly estimated by a standard β-PDF. Furthermore, the log-normal distribution has been found to accurately represent |∇ξ|, but small discrepancies were found at the head of the distribution. By contrast, |∇c| was found to be only marginally represented by a log-normal distribution with non-negligible departures from the DNS data at both the head and the tail of the distribution. It has also been found that ∇c and ∇ξ show predominant collinear alignment throughout the flame brush. Finally, the joint-PDF of |∇c| and |∇ξ| (i.e. P|∇c|,|∇ξ|) has been compared with the product of the marginal PDFs of |∇c| and |∇ξ| (i.e. P|∇c|×P|∇ξ|) extracted from DNS (i.e. assuming statistical independence) and a good agreement has been observed. The bivariate log-normal distributions with and without correlation have also been considered and fail to capture the DNS trend, which stems from the difficulties of approximating P|∇c| and P|∇ξ| with log-normal PDFs.

Statistics of reaction progress variable and mixture fraction gradients of a pulverised coal jet flame using Direct Numerical Simulation data

The statistical behaviour of the variances, covariance and gradients of the reaction progress variable (c), and the mixture fraction (ξ) have been analysed in a pulverised coal jet flame using a three-dimensional carrier phase direct numerical simulation (DNS) dataset. It has been observed that the Favre-probability density functions (PDFs) of c and ξ can be parametrised by the standard β function. Furthermore, the log-normal distribution has been found to accurately represent |∇c| and |∇ξ|. It is also found that ∇c and ∇ξ remain aligned throughout the flame brush. Finally the joint PDF of |∇c| and |∇ξ| has been compared with the product of the PDF of |∇c| and PDF of |∇ξ| extracted from carrier phase DNS, and it has been found that |∇c| and |∇ξ| are not statistically independent in the case investigated.The bivariate log-normal distributions with and without correlation have also been considered, and the former is found to be in better agreement with the carrier phase DNS data.

Probabilistic Aggregation of Uncertain Geological Resources

Commodities such as oil and gas occur in isolated reservoirs or accumulations, more generically called basic units here. To understand a study area’s economic potential and to craft plans for exploration and development, resource analysts often aggregate (sum, accumulate) basic unit magnitudes in distinct spatial subsets of the study area and then appraise the total area’s potential by summing these intermediate sums. In a probabilistic approach, magnitudes are modeled as random variables. Some have asked, “Do different methods of partitioning basic units into subsets lead to different probability distributions for the sum of all basic unit magnitudes?” Any method of aggregation of basic unit magnitudes which obeys the rules of probability leads to the same probability distribution of the sum of all unit magnitudes as that computed by direct summation of all basic unit magnitudes. A Monte Carlo simulation of a synthetic example in which the magnitude of resource in each unit is marginally lognormal and pairwise correlations among basic unit magnitudes are specified illustrates key features of probabilistic aggregation. The joint distribution of certain pairs of aggregates are closely approximated by a bivariate lognormal distribution.

Optimal Retention for a Quota-Share Reinsurance

The Indonesian Financial Services Authority (OJK) has instructed all insurance providers in Indonesia to apply a mandatory tariff for property insurance. The tariff has to be uniformly applied and the rule of set the maximum and minimum premium rates for protection against losses. Furthermore, the OJK issued the new rule regarding self-retention and domestic reinsurance. Insurance companies are obliged to have and implement self-retention for each risk in accordance with the self-retention limits. Fluctuations of total premium income and claims may lead the insurance company cannot fulfil the obligation to the insured, thus the company needs to conduct reinsurance. Reinsurance helps protect insurers against unforeseen or extraordinary losses by allowing them to spread their risks. Because reinsurer chargers premium to the insurance company, a properly calculated optimal retention would be nearly as high as the insurer financial ability. This paper is aimed at determining optimal retentions indicated by the risk measure Value at Risk (VaR), Expected Shortfall (ES) and Minimum Variance (MV). Here we use the expectation premium principle which minimizes individual risks based on their quota share reinsurance. Regarding to the data in an insurance property, we use a bivariate lognormal distribution to obtain VaR, ES and MV, and a bivariate exponential distribution to obtain MV. The bivariate distributions are required to derive the conditional probability of the amount of claim occurs given the benefit has occurred.

Bivariate Log-Normal Distribution of Stimulated Matrix Permeability and Block Size in Fractured Reservoirs: Proposing New Multilinear-Flow Regime for Transient-State Production

Summary This paper investigates the impacts of varied stimulated matrix permeability and matrix-block size on pressure behaviors and flow regimes of hydraulically fractured reservoirs using bivariate log-normal distribution. The main objective is assembling the variance in these two parameters to the analytical models of pressure and pressure derivative considering different porous-media petrophysical properties, reservoir configurations, and hydraulic-fracture (HF) characteristics. The motivation is eliminating the long-run discretization treatment in the porous media required by applying analytical models to describe the variance in the previously discussed parameters with the distance between HFs. Several analytical models for pressure response were generated in this study for hydraulically fractured reservoirs with rectangular-shaped drainage areas. These models take into account the change in stimulated matrix permeability from the maximum value close to the HF face to the minimum value at the so-called no-flow boundary between fractures. They also consider the change in the matrix-block size, corresponding to the change in the induced-fracture density (number of fractures per foot of length), from the minimum value close to the fracture face to the maximum value at the no-flow boundary. Bivariate log-normal distribution was used to describe the change in the stimulated matrix permeability and matrix-block size. The formations of interest are composed of stimulated reservoir volume (SRV), where the matrix is stimulated by the fracturing process, and unstimulated reservoir volume (USRV), where the stimulation process does not affect the matrix. The outcomes of this study can be summarized as Generating new analytical models for pressure and pressure derivative in hydraulically fractured reservoirs that consider the change in stimulated matrix-block size and permeability using bivariate log-normal distribution Understanding the effect of using the probability-density function (PDF) of matrix-block size and permeability in the pressure distribution of different reservoirs Observing the new multilinear-flow regime that develops at intermediate production time and represents several simultaneous linear-flow regimes inside HFs, SRV, and USRV Developing analytical models for the new multilinear-flow regime Studying the effects of petrophysical properties of HFs, induced fractures, and matrix as well as reservoir size and configuration on pressure behavior The most interesting points in this study are The applicability of bivariate log-normal distribution for describing the variance and nonuniform distribution of matrix-block size and permeability. The large variance in the matrix-block size and permeability causes significant decrease in wellbore-pressure drop. Small value of standard deviation of matrix-block size and permeability indicates the possibilities for significant decrease in wellbore pressure drop. The means of matrix-block size and permeability may not have significant effects on reservoir-pressure distribution. The new multilinear-flow regime is characterized by a one-eighth slope on the pressure-derivative curve and is seen always after HF linear flow and before boundary-dominated flow regime. Multilinear-flow regime develops to bilinear-flow regime with a one-quarter slope for uniform distribution of equal matrix-block size and permeability.

Log-normal Distribution Type Symmetry Model for Square Contingency Tables with Ordered Categories

For the analysis of square contingency tables with the same row and column ordinal classications, this article proposes a new model which indicates that the log-ratios of symmetric cell probabilities are proportional to the difference between log-row category and log-column category. The proposed model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate log-normal distribution. Also, this article gives the decomposition of the symmetry model using the proposed model with the orthogonality of test statistics. Examples are given. The simulation studies based on bivariate log-normal distribution are given.

New mean charts for bivariate asymmetric distributions using different ranked set sampling designs

ABSTRACTThe purpose of this study is to construct the mean control chart limits based on Shewhart, weighted variance and skewness correction methods using simple random sampling, ranked set sampling, median ranked set sampling and neoteric ranked set sampling designs. Type I Marshall-Olkin bivariate Weibull and bivariate lognormal distributions are used in the proposed mean control charts. The performance of the proposed control charts based on neoteric ranked set sampling designs is compared with their counterparts in ranked set sampling, median ranked set sampling and simple random sampling by Monte Carlo simulation. Simulation results revealed that the proposed mean charts perform much better than simple random sampling design.

Statistical Analysis of the Reaction Progress Variable and Mixture Fraction Gradients in Flames Propagating into Droplet Mist: A Direct Numerical Simulation Analysis

ABSTRACTThe statistics of reaction progress variable, , and mixture fraction, , and their gradients (i.e., and ) in flames propagating in droplet mist, where the fuel was supplied in the form of monodisperse droplets, have been analyzed for different values of turbulent velocity fluctuations (), droplet equivalence ratios (, and droplet diameters ( based on three-dimensional direct numerical simulations (DNS) in a canonical configuration under decaying turbulence. The combustion process in the gaseous phase has been found to take place predominantly in fuel-lean mode, even for . The probability of finding fuel-lean mixture increases with increasing initial droplet diameter due to slower evaporation of larger droplets. It has been shown that the joint probability density function (i.e., joint PDF) of and (i.e., ), cannot be approximated in terms of discrete delta functions throughout the flame brush for the cases considered here. Furthermore, the magnitude of cannot be adequately approximated by the product of marginal PDFs of , and variable, (i.e., ). The statistical properties of the Favre probability density functions (Favre-PDFs) of the mixture fraction, , and oxidizer-based reaction progress variable, , have been analyzed at several locations across the flame brush and a -function distribution has been found to capture the Favre-PDFs of and obtained from the DNS data. Furthermore, a log-normal distribution has been shown to capture the qualitative behaviors of the PDFs of the gradient of the mixture fraction and the gradient of the reaction progress variable, and , respectively, but discrepancies between the log-normal distribution and the DNS data were observed at the tails of PDFs. In addition, the interrelation between and was examined in terms of the PDFs of the cosine of the angle between them (i.e., and it was observed that most droplet cases exhibited much greater likelihood of positive values of than negative values. Finally, the joint PDF of and , , has been compared with that of P( (i.e., assuming statistical independence of and ) and a good level of agreement has been obtained. The bivariate log-normal distribution has been considered both assuming correlation between and and assuming no correlation for the purpose of modeling , and the variant with no correlation has been found to be more successful in capturing qualitative behavior of although quantitative discrepancies have been observed due to inaccuracies involved in parameterizing P( and P(by log-normal distributions.

Study of vertical breakwater reliability based on copulas

The reliability of a vertical breakwater is calculated using direct integration methods based on joint density functions. The horizontal and uplifting wave forces on the vertical breakwater can be well fitted by the lognormal and the Gumbel distributions, respectively. The joint distribution of the horizontal and uplifting wave forces is analyzed using different probabilistic distributions, including the bivariate logistic Gumbel distribution, the bivariate lognormal distribution, and three bivariate Archimedean copulas functions constructed with different marginal distributions simultaneously. We use the fully nested copulas to construct multivariate distributions taking into account related variables. Different goodness fitting tests are carried out to determine the best bivariate copula model for wave forces on a vertical breakwater. We show that a bivariate model constructed by Frank copula gives the best reliability analysis, using marginal distributions of Gumbel and lognormal to account for uplifting pressure and horizontal wave force on a vertical breakwater, respectively. The results show that failure probability of the vertical breakwater calculated by multivariate density function is comparable to those by the Joint Committee on Structural Safety methods. As copulas are suitable for constructing a bivariate or multivariate joint distribution, they have great potential in reliability analysis for other coastal structures.