Lognormal Variables Research Articles

Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity

The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus being a good candidate for modeling the dependence between the two previously mentioned random variables. To model the claim frequency, a generalized linear model based on a mixed Poisson distribution -like for example, the Negative Binomial (NB), usually works. However, finding a distribution for the claim severity is not that easy. In practice, the Lognormal distribution fits well in many cases. Since the natural logarithm of a Lognormal variable is Normal distributed, this relation is generalised using the Box-Cox transformation to model the average claim severity. Therefore, we propose a bivariate Sarmanov model having as marginals a Negative Binomial and a Normal Generalized Linear Models (GLMs), also depending on the parameters of the Box-Cox transformation. We apply this model to the analysis of the frequency-severity bivariate distribution associated to a pay-as-you-drive motor insurance portfolio with explanatory telematic variables.

Finite element modeling of fretting wear in anisotropic composite coatings: Application to HVOF Cr3C2–NiCr coating

This paper presents a two-dimensional (2D) plane strain finite element model to simulate fretting wear in composite cermet coating. The coating considered in this investigation is High Velocity Oxy-Fuel (HVOF) sprayed Cr3C2–NiCr with 55% volume fraction of Cr3C2. The material microstructure is modelled using Voronoi tessellations with a log-normal variation of grain size. Moreover, the individual phases of the material in the coating were assigned randomly to resemble the microstructure from an actual SEM micrograph. The ceramic carbide phase is orthorhombic and the cubic matrix possesses a high anisotropy index. As a result, each grain was modelled with random orientation to account for material anisotropy. The RVE dimensions were chosen such that its elastic response represented the overall response of a poly-aggregate. In order to simulate debonding of the ceramic carbide phase from the matrix, cohesive elements were used at the grain boundaries. Damage mechanics was used to model degradation of cohesive elements resulting from repeated fretting cycles. A grain deletion algorithm was developed to simulate removal of material from fretting wear. The crack patterns predicted from the model match closely with the patterns observed in experimental studies on wear of HVOF Cr3C2–NiCr coating. The model also predicts carbide pullout, a major damage mechanism in HVOF Cr3C2–NiCr coating subjected to wear. Experiments were also conducted to evaluate and corroborate the wear rate of HVOF Cr3C2–NiCr coating. The wear rate from the model matches closely with experiments at a constant load and displacement amplitude. The results from the model were then extended to obtain a fretting wear map under a combination of various loads and displacement amplitudes.

An application of uncertainty analysis to rock mass properties characterization at porphyry copper mines

Deterministic methods have been widely used for estimation of rock mass properties. It is, however, accepted that there are some sources which impose uncertainties on rock mass properties. Spatial variability is one of the uncertainty sources that must be considered in design stage. In this study, a stochastic procedure is presented for characterization of rock mass properties at a porphyry copper mine in which the spatial variability is typically observed so that the uncertainty in rock mass properties plays a huge role in its quality. Intact rock data obtained during an enormous experimental campaign are statistically analyzed to ensure that they have been collected from a probabilistic society. Then, structural ratings of the rock masses and surface conditions of the structures have been mapped in the field. Finally, a particular stochastic code has been developed in matrix laboratory to evaluate the intrinsic uncertainty of obtained geomechanical properties at the depth of planned underground openings based on Hoek-Brown failure criterion. The results show that rock mass properties might not necessarily follow probability distribution functions of intact rock and joints properties. Moreover, gamma distribution function has been determined as the best fitted distribution function for rock mass uniaxial compressive strength. On the other hand, it was observed that rock mass tensile strength and its deformation modulus are lognormal variables.

A theorem on a product of lognormal variables and hybrid models for children’s exposure to soil contaminants

This study developed hybrid Bayesian models to investigate the modeling process for children’s exposure to soil contaminants, which involves the intrinsic uncertainty of the exposure model, people’s judgments regarding random variables, and limited data resources. A hybrid Bayesian p-box was constructed, which was facilitated by a multiple integral dimensionality reduction (MIDR) theorem. The results indicated that exposure frequency (EF) dominated the exposure dose. The hybrid Bayesian p-box for the Frequentist-Bayesian (F–B) model at the 95th percentile of the simulated average daily dose (ADD) values corresponded to a 4.40 order-of-magnitude difference between the upper and lower bounds of the p-box. This considerable uncertainty was magnified by the combination of the highest posterior density (HPD) regions for three groups of the distribution parameters. For the Interior-Bayesian (I–B) hybrid model, the uncertainty of the outcomes, namely, [1.75 × 10−8, 2.18 × 10−8] mg kg−1d−1, was limited by the HPD regions for only one parameter unless the hyperparameters for the variables’ distributions were further evaluated. It was concluded that the hybrid models could provide a novel understanding of the complexity of the exposure modeling process compared to the traditional modeling method.

The Lognormal Characteristic Function

A simple expression for the characteristic function of the lognormal distribution has eluded researchers. This is useful for computing the sum of lognormal variables, either with each other or with other statistical variables. In this paper, we provide a simple formula for the characteristic function.

Generalized log-normal chain-ladder

ABSTRACT We propose an asymptotic theory for distribution forecasting from the log-normal chain-ladder model. The theory overcomes the difficulty of convoluting log-normal variables and takes estimation error into account. The results differ from that of the over-dispersed Poisson model and from the chain-ladder-based bootstrap. We embed the log-normal chain-ladder model in a class of infinitely divisible distributions called the generalized log-normal chain-ladder model. The asymptotic theory uses small σ asymptotics where the dimension of the reserving triangle is kept fixed while the standard deviation is assumed to decrease. The resulting asymptotic forecast distributions follow t distributions. The theory is supported by simulations and an empirical application.

Best predictors in logarithmic distance between positive random variables

Abstract The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the L 2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L 1 norm. Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝ n in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean. That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.

Statistical Multiplexing Gain Analysis of Processing Resources in Centralized Radio Access Networks

The next generation of wireless networks faces the challenges of the explosion of mobile data traffic, the associated power consumption, and operation cost. The centralized radio access networks (Centralized RANs) architectures have been proposed to reduce the power consumption and the network operating cost. By integrating many distributed base stations' processing resources in a processing pool and sharing processing resources on demand, the overall required processing resources for the Centralized RAN can be reduced compared to the conventional RAN. This can be measured by the statistical multiplexing gain (SMG). However, most of the SMG analysis only considered the temporal traffic distribution which is not suitable for the current mobile networks. In this paper, we analyze the SMG of processing resources based on a temporal-spatial joint traffic distribution model, which considers the mobile data traffic distribution both in the time and space domains. Based on this model, we derive a formula for the SMG and also a closed-form approximation for that when the spatial traffic distribution is lognormal distribution. The theoretical analysis and simulation results show that the SMG increases with the service threshold ratio P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sub> , but the growth trend of SMG for different area types is not always the same. We also find that the traffic distribution parameters, such as the standard deviation of the lognormal distribution variable's natural logarithm, have a significant influence on the SMG.

The Distribution of the Average of Log-Normal Variables and Exact Pricing of the Arithmetic Asian Options: A Simple, Closed-Form Formula

We overcome a long-standing obstacle in statistics. In doing so, we show that the distribution of the sum of log-normal variables is log-normal. Furthermore, we offer a breakthrough result in finance. In doing so, we introduce a simple, exact and explicit formula for pricing the arithmetic Asian options. The pricing formula is as simple as the classical Black-Scholes formula.

Stochastic cohesive law of concrete based on energy consumption equivalence

A method for determining the stochastic cohesive law of concrete was developed using a parallel spring model. This method is based on the energy consumption equivalence between stochastic damage and fracture and assumes that the micro-spring failure strain is a lognormal stochastic variable. The stochastic cohesive laws of dam concrete and wet-screening concrete with different maximum aggregate diameters were obtained. The fracture processes of notched three-point bending beams were calculated, and the characteristic lengths of the dam and wet-screening concrete were suggested as 6–8 times the maximum aggregate diameter.

Stringent sampling inspection plans using upper envelopes for cell counts

A new type of stringent sampling inspection plan is proposed for cell counts generally in line with a Poisson or a mixed-Poisson type of distribution. The proposed plan is based on setting an upper envelope estimated from the sample counts which should remain below the given microbiological limit m. We used Monte-Carlo simulation methodology to evaluate the proposed plans and showed that the lot acceptance criterion using the upper envelope improves consumer protection in general. Both two and three class attribute plans are investigated for establishing the upper envelope. Our study reveals that the proposed plans can offer better consumer protection and economy when compared to both two and three class attribute plans. It is also demonstrated that a slightly stringent plan can be found in place of lognormal type variables plans. The limitations of the proposed new plans, and a discussion on alternative stringent rules for batch acceptance are also provided. The new plan was also validated using the APC cell counts observed for 2000 milk powder batches. Our conclusion is that the proposed stringent plans can improve protection to the consumer, particularly for exported food products.

The spatial empirical Bayes predictor of the small area mean for a lognormal variable of interest and spatially correlated random effects

The standard small area estimator, the empirical best linear unbiased predictor (EBLUP), estimates small area parameters by way of linear mixed models. The EBLUP assumes normal and independent random small area effects as well as normal and independent random sampling errors. Under these assumptions, the variable of interest also follows a normal distribution. In practice, however, the above assumptions are often violated. The variable of interest is often non-normal and highly skewed, and the small areas are frequently spatially dependent. In this paper, we propose the spatial empirical Bayes predictor (SEBP) of the small area mean of a positively skewed variable of interest in the presence of spatial dependence among the random small area effects. We assume that the variable of interest follows a normal distribution after a log transformation and that its log transform is linked to some auxiliary variables by a nested error regression model. The SEBP is derived under the log-transformed nested error regression model. By way of simulation, we show that compared to its alternatives, i.e., the direct estimator which is solely based on the survey data for the small area under study, the EBLUP which does not take into account spatial dependence and skewness, the empirical Bayes predictor which takes into account skewness but not spatial dependence among the small areas, the SEBP has the smallest average relative bias and average relative root-mean-squared error for various combinations—though not all—of skewness and spatial correlation.

On the efficient simulation of the left-tail of the sum of correlated log-normal variates

Abstract The sum of log-normal variates is encountered in many challenging applications such as performance analysis of wireless communication systems and financial engineering. Several approximation methods have been reported in the literature. However, these methods are not accurate in the tail regions. These regions are of primordial interest as small probability values have to be evaluated with high precision. Variance reduction techniques are known to yield accurate, yet efficient, estimates of small probability values. Most of the existing approaches have focused on estimating the right-tail of the sum of log-normal random variables (RVs). Here, we instead consider the left-tail of the sum of correlated log-normal variates with Gaussian copula, under a mild assumption on the covariance matrix. We propose an estimator combining an existing mean-shifting importance sampling approach with a control variate technique. This estimator has an asymptotically vanishing relative error, which represents a major finding in the context of the left-tail simulation of the sum of log-normal RVs. Finally, we perform simulations to evaluate the performances of the proposed estimator in comparison with existing ones.

A Bayesian approach to inference on the variance of lognormal data

Krishnamoorthy, Mathewand Ramachandran (2006) developed a method to draw inference on the mean and variance of one or more lognormal distributions. Their method was based on generalised confidence intervals (frequentist methods). In this article we focus on the variance of the lognormal distribution and implement a Bayesian approach and obtain credibility intervals to compare the performance of four different non-informative prior distributions. This is done by means of various Monte Carlo simulation studies as well as practical examples. The accuracy (coverage) and efficiency (interval length) of some of the Bayesian priors, particularly for the highest posterior density (HPD) credibility intervals will be illustrated in these simulation studies and examples. It can be observed that the frequentist approach is equivalent to the Bayesian approach, when using the Independence Jeffreys prior. Even so, the Bayesian approach offers some additional benefits, namely, through the calculation of the HPD intervals. Hypothesis testing and practical applications are also presented. Further results comparing various estimators of the lognormal variance are derived and evaluated. The usefulness of the Bayesian approach is also illustrated in its ability to easily modify the method to account for the possibility of zero-valued observations. This is something for which there is (to our knowledge) currently no frequentist method available and serves to highlight the usefulness of the Bayesian approach.

Seepage analysis of earth dams considering spatial variability of hydraulic parameters

To investigate the influence of spatial variability of hydraulic parameters on the flow in earth dams, saturated-unsaturated seepage is numerically simulated combining Monte Carlo simulation and random field theory. The van Genuchten model is adopted to represent soil-water characteristic curve (SWCC). The SWCC fitting parameters (a, n) and the saturated hydraulic conductivity parameter (ks) are considered as lognormal random fields. An approach of logarithmic translation is used for generating lognormal variables with lower limits greater than zero. The influence of the coefficient of variation (COV) and autocorrelation distance of hydraulic parameters on the flow rate is studied and compared. Sensitivity analysis indicates that the COVs of SWCC parameter n and saturated hydraulic conductivity ks have a larger effect on the flow rate than that of SWCC parameter a. A larger horizontal autocorrelation distance corresponds to a larger mean and COV of the flow rate. Neglecting the spatial variability of hydraulic parameters leads to overestimation of the mean and COV of the flow rate in earth dams, which could lead to conservative design of dams.

Imputing the mean of a heteroskedastic log-normal missing variable: A unified approach to ratio imputation

Imputing the mean of a heteroskedastic log-normal missing variable: A unified approach to ratio imputation

On the Efficient Simulation of Outage Probability in a Log-Normal Fading Environment

The outage probability (OP) of the signal-to-interference-plus-noise ratio (SINR) is an important metric that is used to evaluate the performance of wireless systems. One difficulty toward assessing the OP is that, in realistic scenarios, closed-form expressions cannot be derived. This is, for instance, the case of the Log-normal environment, in which evaluating the OP of the SINR amounts to computing the probability that a sum of correlated Log-normal variates exceeds a given threshold. Since such a probability does not admit a closed-form expression, it has thus far been evaluated by several approximation techniques, the accuracies of which are not guaranteed in the region of small OPs. For these regions, simulation techniques based on variance reduction algorithms are a good alternative, being quick and highly accurate for estimating rare event probabilities. This constitutes the major motivation behind this paper. More specifically, we propose a generalized hybrid importance sampling scheme, based on a combination of a mean shifting and a covariance matrix scaling, to evaluate the OP of the SINR in a Log-normal environment. We further our analysis by providing a detailed study of two particular cases. Finally, the performance of these techniques is performed both theoretically and through various simulation results.

New Approach to Estimating the Standard Deviations of Lognormal Cost Variables in the Monte Carlo Analysis of Construction Risks

If soundly conducted, risk assessment could yield considerable savings for project investors. Monte Carlo Simulation (MCS) has been widely embraced by risk management guides as an instrumental tool for this purpose. This research aims to develop a new method to improve the rigor of MCS by establishing the link between parameter estimation and assessment of individual risk sources. The method is validated by virtue of its predictive power for the likelihood of a project being successful in securing investors. Eight Taiwanese sewerage Build-Operate-Transfer projects are investigated. Compared to the discounted cash flow approach, this new method can provide a more accurate prediction using the expert’s assessment as input of financial impact and occurrence likelihood of individual risks. This finding furnishes solid empirical evidence for the value MCS might add to project appraisal.

Analysis of a Stochastic Model for Bacterial Growth and the Lognormality of the Cell-Size Distribution

This paper theoretically analyzes a phenomenological stochastic model for bacterial growth. This model comprises cell division and the linear growth of cells, where growth rates and cell cycles are drawn from lognormal distributions. We find that the cell size is expressed as a sum of independent lognormal variables. We show numerically that the quality of the lognormal approximation greatly depends on the distributions of the growth rate and cell cycle. Furthermore, we show that actual parameters of the growth rate and cell cycle take values that give a good lognormal approximation; thus, the experimental cell-size distribution is in good agreement with a lognormal distribution.

Tail behavior of sums and differences of log-normal random variables

We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to depend on the correlation between the components, and the explicit solution is found by solving a tractable quadratic optimization problem. These results can be used either to approximate the probability of tail events directly, or to construct variance reduction procedures to estimate these probabilities by Monte Carlo methods. In particular, we propose an efficient importance sampling estimator for the left tail of the distribution function of the sum of log-normal variables. As a corollary of the tail asymptotics, we compute the asymptotics of the conditional law of a Gaussian random vector given a linear combination of exponentials of its components. In risk management applications, this finding can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one.