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Many engineering products have more than one failure mode and the evolution of each mode can be monitored by measuring a performance characteristic (PC). It is found that the underlying multi-dimensional degradation often occurs with inherent process stochasticity and heterogeneity across units, as well as dependency among PCs. To accommodate these features, in this paper, we propose a novel multivariate degradation model based on the inverse Gaussian process. The model incorporates random effects that are subject to a multivariate normal distribution to capture both the unit-wise variability and the PC-wise dependence. Built upon this structure, we obtain some mathematically tractable properties such as the joint and conditional distribution functions, which subsequently facilitate the future degradation prediction and lifetime estimation. An expectation-maximization algorithm is developed to infer the model parameters along with the validation tools for model checking. In addition, two simulation studies are performed to assess the performance of the inference method and to evaluate the effect of model misspecification. Finally, the application of the proposed methodology is demonstrated by two illustrative examples.

Objective: The purpose of this paper is to explored different distributions in conditional Value at Risk (VaR) modeling as an option in the Mexican market. Methodology: We estimate a GARCH model under the Gaussian, Normal Inverse Gaussian, Skew Generalized t and the Stable distribution assumption, then we implement the model in predicting one-day ahead VaR and finally we examine the performance among the four VaR models during a period of high volatility. Results: The backtesting result confirms that the stable-VaR approach outperforms the other models in the VaR’s prediction at 99% confidence level. Limitations: Although the VaR is a widely used risk measure is not a coherent risk measure, for this reason, a natural extension of our work should be to estimate the expected shortfall and this may produce different insights. Conclusions: Our findings reveal that models that consider some empirical characteristic of financial returns such as leptokurtic, volatility clustering and asymmetry improve the VaR predicting capacity. This finding is important in the search of more robust approaches for VaR estimates. Recepción: 09/08/2018 Aceptación: 31/10/2019

This letter deals with the adaptive detection problem in compound Gaussian sea clutter with inverse Gaussian texture when the mismatched signal occurs. In order to reject the mismatched signal, we introduce a fictitious signal to the null hypothesis based on the adaptive beamformer orthogonal rejection test (ABORT). We adopt the two-step generalized likelihood ratio test (GLRT) criterion and develop two selective detectors, i.e., the ABORT-like compound Gaussian with inverse Gaussian (A-IGCG) detector and ABORT-like maximum a posteriori (MAP) compound Gaussian with inverse Gaussian (AM-IGCG) detector. Both the proposed detectors are proved to be constant false alarm rate (CFAR) detectors. The several experiments that compare the proposed detectors with other selective detectors and compound Gaussian detectors are conducted by using two types of real sea data, i.e., IPIX 1998 and 2019 real sea data. The numerical results indicate that the novel detectors exhibit significant tradeoffs between performances of detection and selectivity property.

The high reliability and extended life of seawater hydraulic components is dependent on the tribological properties of key friction pairs, more and more attention has been given to the performance degradation and lifetime of the friction pairs. In this paper, the wear process of Tungsten carbide (WC) with 6%wt of Ni (WC–6Ni)/SiC friction pair is studied through the tribological test under seawater lubrication. Considering the random effects and the complex wear mechanism of WC-6Ni/SiC, an efficient Inverse Gaussian (IG) process supported lifetime prediction model is established by coupling the multiple stresses under seawater lubrication, and the parameters in the proposed IG process model are derived. Two actual cases are used to validate the proposed method. The prediction model and experimental results show the high accuracy in time-varying lifetime prediction of WC-6Ni/SiC, which is of great significance to the evaluation of friction pair and the maintenance for seawater hydraulic components.

This letter addresses the persymmetric adaptive detection problem of range-spread targets in compound Gaussian sea clutter. Based on the generalized likelihood ratio test (GLRT), Rao test, and Wald test, three novel compound Gaussian detectors are developed. Specifically, the sea clutter is modeled as compound Gaussian with inverse Gaussian distribution, and the detectors are derived by the two-step maximum a posterior (MAP) procedures. In the first step, we assume the clutter covariance matrix (CCM) and the inverse Gaussian texture are known and derive the proposed detectors' test statistics. In the second step, we use the persymmetric property and MAP criterion to estimate the CCM and inverse Gaussian texture parameters. Then the proposed detectors are proved to be constant false alarm rate (CFAR) detectors with respect to the real CCM. The simulation experiments are conducted by comparing the proposed detectors with their counterparts on both synthetic data and real sea clutter data. The numerical results indicate that the novel detectors exhibit better detection performance than their competitors.

In this paper, we study higher-order Riesz transforms associated with the inverse Gaussian measure given byπn/2ex2dxonℝn. We establishLpℝn,ex2dx-boundedness properties and obtain representations as principal values singular integrals for the higher-order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.

This paper introduces a new family of Generalised Hyper-Elliptical (GHE) distributions providing further generalisation of the generalised hyperbolic (GH) family of distributions, considered in Ignatieva and Landsman (2019) . The GHE family is constructed by mixing an elliptical distribution with a Generalised Inverse Gaussian (GIG) distribution. We present an innovative theoretical framework where a closed form expression for the tail conditional expectation (TCE) is derived for this new family of distributions. We demonstrate that the GHE family is especially suitable for heavy-tailed insurance losses data. Our theoretical TCE results are verified for two special cases, Laplace - GIG and Student-t - GIG mixtures. Both mixtures are shown to outperform the GH distribution, providing excellent fit to univariate and multivariate insurance losses data. The TCE risk measure computed for the GHE family of distributions provides a more conservative estimator of risk in the extreme tail, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from extreme losses. Our multivariate analysis allows to quantify correlated risks by means of the GHE family: the TCE of the portfolio is decomposed into individual components, representing individual risks in the aggregate loss.

The in-depth study of data, such as their precise characteristics or detection of anomalies, necessitates the estimation of the underlying probability density. The kernel estimation method, which consists of developing estimators based on special functions called kernels, is one of the most popular methods for reaching this aim. Despite the existence of numerous kernel-type estimators, there is still a high demand for new efficient kernel estimation strategies that can reveal hidden features in the data of important phenomena. In this regard, the development of modern distribution and their manageable implementation are beneficial. The Gumbel type 2 distribution is one of these modern distributions that has the feature of realizing a distributional compromise between the famous Frechet and Weibull distributions. Among others, it presents distributional functions with high levels of flexibility. With this in mind, we define the Gumbel type 2 kernel function, which uses some analytical properties of the Gumbel type 2 distribution to achieve the aim of efficiency. This claim is proved through the theoretical analysis of the corresponding bias, variance, and rate of convergence under the mean square error. The question of how to select the estimator’s underlying bandwidth in practice is addressed. Finally, the performance of the proposed kernel estimator is evaluated by using two real-world hydrology data sets of bimodal nature. For these data sets, it performs better visually than the Weibull, gamma 1, inverse Gaussian, beta prime, and Erlang kernel estimators.

This paper considers the parameter estimation for the inverse Gaussian regression model (IGRM) in the presence of multicollinearity. The inverse Gaussian modified ridge-type estimator (IGMRTE) is d...

The survival probability of a composite material subject to external stress is governed by the condition that the strength of the material must exceed the applied stress. There are several models p...

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Conventional reliability analysis techniques are focused on the occurrence of failures over time. However, in certain situations where the occurrence of failures is tiny or almost null, the estimation of the quantities that describe the failure process is compromised. In this context the degradation models were developed, which have as experimental data not the failure, but some quality characteristic attached to it. Degradation analysis can provide information about the components lifetime distribution without actually observing failures. In this thesis we proposed different methodologies for degradation data based on the inverse Gaussian distribution. Initially, we introduced the inverse Gaussian deterioration rate model for degradation data and a study of its asymptotic properties with simulated data. We then proposed an inverse Gaussian process model with frailty as a feasible tool to explore the influence of unobserved covariates, and a comparative study with the traditional inverse Gaussian process based on simulated data was made. We also presented a mixture inverse Gaussian process model in burn-in tests, whose main interest is to determine the burn-in time and the optimal cutoff point that screen out the weak units from the normal ones in a production row, and a misspecification study was carried out with the Wiener and gamma processes. Finally, we considered a more flexible model with a set of cutoff points, wherein the misclassification probabilities are obtained by the exact method with the bivariate inverse Gaussian distribution or an approximate method based on copula theory. The application of the methodology was based on three real datasets in the literature: the degradation of LASER components, locomotive wheels and cracks in metals. As técnicas convencionais de análise de confiabilidade são voltadas para a ocorrência de falhas ao longo do tempo. Contudo, em determinadas situações nas quais a ocorrência de falhas é pequena ou quase nula, a estimação das quantidades que descrevem os tempos de falha fica comprometida. Neste contexto foram desenvolvidos os modelos de degradação, que possuem como dado experimental não a falha, mas sim alguma característica mensurável a ela atrelada. A análise de degradação pode fornecer informações sobre a distribuição de vida dos componentes sem realmente observar falhas. Assim, nesta tese nós propusemos diferentes metodologias para dados de degradação baseados na distribuição gaussiana inversa. Inicialmente, nós introduzimos o modelo de taxa de deterioração gaussiana inversa para dados de degradação e um estudo de suas propriedades assintóticas com dados simulados. Em seguida, nós apresentamos um modelo de processo gaussiano inverso com fragilidade considerando que a fragilidade é uma boa ferramenta para explorar a influência de covariáveis não observadas, e um estudo comparativo com o processo gaussiano inverso usual baseado em dados simulados foi realizado. Também mostramos um modelo de mistura de processos gaussianos inversos em testes de burn-in, onde o principal interesse é determinar o tempo de burn-in e o ponto de corte ótimo para separar os itens bons dos itens ruins em uma linha de produção, e foi realizado um estudo de má especificação com os processos de Wiener e gamma. Por fim, nós consideramos um modelo mais flexível com um conjunto de pontos de corte, em que as probabilidades de má classificação são estimadas através do método exato com distribuição gaussiana inversa bivariada ou em um método aproximado baseado na teoria de cópulas. A aplicação da metodologia foi realizada com três conjuntos de dados reais de degradação de componentes de LASER, rodas de locomotivas e trincas em metais.

The inverse Gaussian regression (IGR) model parameters are generally estimated using the maximum likelihood (ML) estimation method. Since the multicollinearity problem exists among the explanatory ...

In this article, normal inverse Gaussian (NIG) autoregressive model is introduced. The parameters of the model are estimated using expectation maximization (EM) algorithm. The efficacy of the EM algorithm is shown using simulated and real-world financial data. It is shown that NIG autoregressive model fit very well on the considered financial data and hence could be useful in modelling of various real-life time-series data.

PurposeThis article considers Inverse Gaussian distribution as the basic lifetime model for the test units. The unknown model parameters are estimated using the method of moments, the method of maximum likelihood and Bayesian methods. As part of maximum likelihood analysis, this article employs an expectation-maximization algorithm to simplify numerical computation. Subsequently, Bayesian estimates are obtained using the Metropolis–Hastings algorithm. This article then presents the design of optimal censoring schemes using a design criterion that deals with the precision of a particular system lifetime quantile. The optimal censoring schemes are obtained after taking into account budget constraints.Design/methodology/approachThis article first presents classical and Bayesian statistical inference for Progressive Type-I Interval censored data. Subsequently, this article considers the design of optimal Progressive Type-I Interval censoring schemes after incorporating budget constraints.FindingsA real dataset is analyzed to demonstrate the methods developed in this article. The adequacy of the lifetime model is ensured using a simulation-based goodness-of-fit test. Furthermore, the performance of various estimators is studied using a detailed simulation experiment. It is observed that the maximum likelihood estimator relatively outperforms the method of moment estimator. Furthermore, the posterior median fares better among Bayesian estimators even in the absence of any subjective information. Furthermore, it is observed that the budget constraints have real implications on the optimal design of censoring schemes.Originality/valueThe proposed methodology may be used for analyzing any Progressive Type-I Interval Censored data for any lifetime model. The methodology adopted to obtain the optimal censoring schemes may be particularly useful for reliability engineers in real-life applications.

This paper deals with developing sequential procedures for estimating the mean of an inverse Gaussian (IG) distribution when the population coefficient of variation (CV) is known. We consider the minimum risk and bounded risk point estimation problems respectively. Moreover, we make use of a weighted squared-error loss function and aim to control the associated risk functions. Instead of the usual estimator, i.e., the sample mean, Searls J. Amer. Stat. Assoc. 50, 1225–1226 (1964) estimator is utilized for the purpose of estimation. Second-order approximations are also obtained under both estimation set-ups. We establish that Searls’ estimator dominates the usual estimator (sample mean) under proposed sequential sampling procedures. An extensive simulation analysis is carried out to validate the theoretical findings and real data illustrations are also provided to show the practical utility of our proposed sequential stopping strategies.

In this paper the running average of a subordinator with a tempered stable distribution is considered. We investigate a family of previously unexplored infinite-activity subordinators induced by the probability distribution of the running average process and determine their jump intensity measures. Special cases including gamma processes and inverse Gaussian processes are discussed. Then we derive easily implementable formulas for the distribution functions, cumulants, and moments, as well as provide explicit estimates for their asymptotic behaviors. Numerical experiments are conducted for illustrating the applicability and efficiency of the proposed formulas. Two important extensions of the running average process and its equi-distributed subordinator are examined with concrete applications to structural degradation modeling and financial derivatives pricing, where their advantages relative to several existing models are highlighted together with the mention of Euler discretization and compound Poisson approximation techniques.

This paper describes a comprehensive survival analysis for the inverse Gaussian distribution employing Bayesian and Fiducial approaches. It focuses on making inferences on the inverse Gaussian (IG) parameters μ and λ and the average remaining time of censored units. A flexible Gibbs sampling approach applicable in the presence of censoring is discussed and illustrations with Type II, progressive Type II, and random rightly censored observations are included. The analyses are performed using both simulated IG data and empirical data examples. Further, the bootstrap comparisons are made between the Bayesian and Fiducial estimates. It is concluded that the shape parameter ( [Image: see text] ) of the inverse Gaussian distribution has the most impact on the two analyses, Bayesian vs. Fiducial, and so does the size of censoring in data to a lesser extent. Overall, both these approaches are effective in estimating IG parameters and the average remaining lifetime. The suggested Gibbs sampler allowed a great deal of flexibility in implementation for all types of censoring considered.

We study the distribution of confirmation times of Bitcoin transactions, conditional on the size of the current memory pool. We argue that the time until a Bitcoin transaction is confirmed resembles the time to ruin in a corresponding Cramer-Lundberg process. This well-studied model gives mathematical insights in the mempool behaviour over time. Specifically, for situations where one chooses a fee, such that the total size of incoming transactions with higher fee is close to the total size of transactions leaving the mempool (heavy traffic), a diffusion approximation leads to an inverse Gaussian distribution for the confirmation times. The results of this paper are particularly interesting for users that want to make a Bitcoin transaction during heavy-traffic situations, as evaluation of the well-known inverse Gaussian distribution is computationally straightforward.

In this work, we discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials. In particular, we discuss the cumulant-generating function and associated...

The optimal design and performance monitoring of wind farms depend on the precise assessment of spatial and temporal distribution of wind speed. The aim of this research is to investigate the appropriateness of nine popular probability distribution models (exponential, gamma, generalised extreme value, inverse Gaussian, Kumaraswamy, log-logistic, lognormal, Nakagami, and Weibull) for the assessment of wind speed distribution (WSD) at 10 sites situated at topographically distinct locations in Tamil Nadu, India, based on 39 years of data. The results suggest that a single distribution cannot produce best fit for all the stations. On an individual level, the generalised extreme value distribution provided the most suitable fit for majority of the stations, followed by the Kumaraswamy distribution. The Kumaraswamy distribution has performed well even if the WSD of the station is negatively skewed. Hence, based on the ranking and performance consistency, the Kumaraswamy distribution can be preferred irrespective of the topographical heterogeneity of the stations.

This paper considers the estimation of parameters for the inverse Gaussian regression model in the presence of multicollinearity. To mitigate this issue, we propose the inverse Gaussian James–Stein estimator (IGJSE) and compare its performance with other estimation methods i.e. maximum likelihood estimator (MLE), ridge and Liu estimators. Theoretical comparisons between the usual MLE, ridge, Liu and IGJSE are derived using matrix mean − squared error and scalar mean squared error criterions. The statistical properties of these estimators are systematically scrutinized via a Monte Carlo simulation. The findings of the simulation study clearly demonstrate that the proposed IGJSE showed a consistent behaviour as compared to the other estimation methods in all the evaluated situations. Finally, we evaluate the performance of IGJSE with the help of real example. The simulation and real example results show that the performance of IGJSE is better as compared to other estimation methods with correlated explanatory variables.

Different sets of dry spell length such as complete series, monthly maximum, seasonal maximum, and annual maximum are applied and modeled with different probability distribution functions (such as Gumbel Max, generalized extreme value, Log-Logistic, generalized logistic, inverse Gaussian, Log-Pearson 3, generalized Pareto) to recognize in which duration, dry spells cause drought. The drought situation and temporal analysis in the North of Iraq region were done using the SPI index and by software of DrinC at a time scale of 3.6 and 12 months. Because of applicability, availability of data and the aim of the study, SPI is selected to analyze the dry spells in this study. Based on the maximum length of the available statistical period, the statistics for the years 1980 to 2019 were used from nine meteorological stations for analysis. The results of the study showed the severity of drought during the study period which related to dry spells. The results of this research confirm the variation of drought occurrence with varying degrees in different time and different dry spells condition in Iraq.

Laser-generated focused ultrasound (LGFU) transducers used for ultrasound therapy commonly have large diameters (6–15 mm), but smaller lateral dimensions (

Shared frailty models are often used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor (frailty) and baseline hazard function which is common to all individuals. There are certain as- sumptions about the baseline distribution and distribution of frailty. Mostly assumption of gamma distribution is considered for frailty distribution. To compare the results with gamma frailty model, we introduce three shared frailty models with generalized exponential as baseline distribution. The other three shared frailty models are inverse Gaussian shared frailty model, compound Poisson shared frailty model and compound negative binomial shared frailty model. We t these models to a real life bivariate survival data set of McGilchrist and Aisbett (1991) related to kidney infection using Markov Chain Monte Carlo (MCMC) technique. Model comparison is made using Bayesian model selection criteria and a better model is suggested for the data.

This article models the dynamics of age-specific incremental mortality as a stochastic process in which the drift rate can be simply and effectively modeled as the average annual improvement rate of a group time trend for all ages and the distribution of residuals can be fitted by one of the Gaussian distribution and four non-Gaussian distributions (Student t, jump diffusion, variance gamma, and normal inverse Gaussian). We use the one-factor copula model with six distributions for the factors (normal–normal, normal–Student t, Student t–normal, Student t–Student t, skewed t–normal, and skewed t–Student t) to capture the inter-age mortality dependence. We then construct three annuity portfolios (Barbell, Ladder, and Bullet) with equal portfolio value (total net single premium) and portfolio mortality duration but different portfolio mortality convexities. Finally, we apply our model to managing longevity risk by an approximation to the change in the portfolio value in response to a proportional or constant change in the force of mortality, and by estimating Value at Risk for the three annuity portfolios.

Considering that power consumption (PC) is an extremely important indicator in digital circuit design, lower PC has always been our pursuit. PC and power supply voltage are positively correlated, and in this case, we must reduce the operating voltage of the circuit. However, as the voltage continues to decrease, various secondary effects and process variations become increasingly influential, making the delay distribution and its statistical characteristics more difficult to predict. In this paper, an inverse Gaussian distribution is used to model the propagation delay. Taking into account the local process variation, the multi-input delay analytical expression is derived according to the sub-threshold current formula to accurately predict the distribution and statistical characteristics of the delay, and the delay is obtained by calculation instead of Monte Carlo simulation, which greatly reduces the simulation time. The accuracy of the delay expression and delay distribution have been tested under 22 nm FDSOI technology and good results were obtained with operating voltages from 0.20 V to 0.30 V, in which the mean error of the delay is approx. 1.5%, the variance error is approx. 4.3%, and the error of the cumulative distribution function is approx. 2%.

Poisson regression is used to model enumeration data such as data on the number of DHF cases. This model has the assumption that is fulfilled is the average and the variance must have the same value or it is called the equidispersion. But this assumption is not fulfilled because the data on the number of dengue cases experienced violations of this assumption. The violation is that the average value is smaller than the variance value or it is called overdispersion. This results in incorrect conclusions because the prediction standard error is underestimated. The way to prevent this is by combining the Poisson distribution and discrete or continuous distribution, this combination is called Mixed Poisson Distribution. Researchers use one of the Mixed Poisson methods, namely Inverse Gaussian Poisson Regression (PIG) because the method is used when the data is overdispersed and the parameters are known or close form on the likelihood function. Based on the results of the study, it is known that the height of the area is a factor that significantly influences DHF cases in South Sulawesi.

Degradation tests have been used for the purpose of assessing reliability information. In this paper, we consider a degradation test design problem under a statistical model mis-specification scena...

Purpose The purpose of this study is to describe precisely the wind speed regime and characteristics of a runway of an International Airport, the north-western part of Turkey. Design methodology approach Three different probability distributions, namely, Inverse Gaussian (IG), widely used two-parameter Weibull and Rayleigh distributions in the literature, are used to represent wind regime and characteristics of the runway. The parameters of each distribution are estimated by the pattern search (PS)-based heuristic algorithm. The results are compared with the other three methods-based numerical computation, including maximum-likelihood method, moment method (MoM) and power density method, respectively. To evaluate the fitting performance of the proposed method, several statistical goodness tests including the mostly used root mean square error (RMSE) and chi-squared (X2) are conducted. Findings In the light of the statistical goodness tests, the results of the IG-based PS attain better performance than the classical Weibull and Rayleigh functions. Both the RMSE and X2 values achieved by the IG-based PS method lower than that of Weibull and Rayleigh distributions. It exhibits a better fitting performance with 0.0074 for RMSE and 0.58 × 10−4 for X2 for probability density function (PDF) in 2012 and with RMSE of 0.0084 and X2 of 0.74 × 10−4 for PDF in 2013. As regard the cumulative density function of the measured wind data, the best results are found to be Weibull-based PS with RMSE of 0.0175 and X2 of 3.25 × 10−4 in 2012. However, Weibull-based MoM shows more excellent ability in 2013, with RMSE of 0.0166 and X2 of 2.94 × 10−4. Consequently, it is considered that the results of this study confirm that IG-based PS with the lowest error value can a good choice to model more accurately and characterize the wind speed profile of the airport. Practical implications This paper presents a realistic point of view regarding the wind regime and characteristics of an airport. This study may cast the light on researchers, policymakers, policy analysts and airport designers intending to investigate the wind profile of a runway at the airport in the world and also provide a significant pathway on how to determine the wind distribution of the runway. Originality value Instead of the well-known Weibull distribution for the representing of wind distribution in the literature, in this paper, IG distribution is used. Furthermore, the suitability of IG to represent the wind distribution is validated when compared with two-parameter Weibull and Rayleigh distributions. Besides, the performance and efficiency of PS have been evaluated by comparing it with other methods.

Several researchers have addressed the problem of testing the homogeneity of the scale parameters of several independent inverse Gaussian distributions based on the likelihood ratio test. However, only approximations of the distribution function of the test statistic are available in the literature. In this note, we present the exact distribution of the likelihood ratio test statistic for testing the equality of the scale parameters of several independent inverse Gaussian populations in a closed form. To this end, we apply the Mellin inverse transform and the Jacobi polynomial expansion to the moments of the likelihood ratio test statistic. We also propose an approximate method based on the Jacobi polynomial expansion. Finally, we apply an accurate numerical method, which is based on the inverse of characteristic function, to obtain a near-exact approximation of the likelihood ratio test statistic distribution. The proposed methods are illustrated via numerical and real data examples.

Applications of misspecified models in the field of survival analysis particularly frailty models may result in poor generalization and biases. Since gamma and inverse Gaussian distributions are often used interchangeably as frailty distributions for heterogeneous survival data, clear distinction between them is necessary. Based on closed form expressions of unconditional hazards and survival functions, in this paper we compare the effectiveness of gamma and inverse Gaussian distributions for frailty term in modeling survival data for heterogeneous populations. Different baseline hazards were considered including exponential, Weibull and Gompertz. We derived the closed form expressions for unconditional hazards and survival functions under each baseline distribution for both gamma and inverse Gaussian frailty models. Both graphical and extensive statistical simulation approaches are applied to compare the models. For the inference purpose, real data concerning the East Coast Fever (ECF) transmission dynamics is applied. General overview from the graphical analysis and results from both real and synthetic data indicate that gamma distribution under the Gompertz and Weibull baseline hazards is better compared to inverse Gaussian in modeling survival data for a heterogeneous population. Simulation, graphical and inferential analyses were done using appropriate packages in R language.

This paper deals with developing an improved accelerated sequential procedure to estimate the unknown mean μ of an inverse Gaussian distribution, when the scale parameter λ also remains unknown. Th...

In this paper, we introduce a Topp-Leone Lindley (TLL) distribution by using Topp-Leone (TL) family. Mathematical properties of the TLL distribution are studied. Estimation of the unknown parameters is derived by the methods of maximum likelihood, least squares and maximum product spacings. The performance of the estimation methods is evaluated by means of Monte-Carlo simulation. In the application part of the research, we have used a long term measured wind speed data from ten stations in Inner Mongolia of China. The suitability of the TLL distribution and other six distributions (inverse Gaussian, Birnbaum-Saunders, power Lindley, weighted Lindley, Weibull, inverse Weibull) used to fit for wind speed data is evaluated based on root mean square error, coefficient of determination, the log-likelihood, Akaike information criterion (AIC), Bayesian information criterion (BIC), Kolmogorov-Smirnov (K–S) statistic and power density error. The results substantiate that TLL distribution is widely applicable at all selected stations. TLL distribution outperforms the others at five stations, ranks the second at three stations and classifies as the third in remaining two stations. Alternatively, Weighted Lindley distribution is the next best distribution for analyzing wind speed data in selected stations. It is stronger than others at three stations while ranking the second at five stations and fourth in two stations. Although this finding questions the accuracy of Weibull distribution in modeling wind speed data, it provides superior estimation of wind power density for the ten stations.

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampere property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampere equations in differential geometry.

We worked out a method in Maple environment to help understand the difficult transport processes in horizontal subsurface flow constructed wetlands filled with coarse gravel (HSFCW-C). With this process, the measured tracer results of the inner points of a HSFCW-C can be fitted more accurately than with the conventionally used distribution functions (Gaussian, Lognormal, Fick (Inverse Gaussian) and Gamma). This research outcome only applies for planted HSFCW-Cs. The outcome of the analysis shows that conventional solutions completely stirred series tank reactor (CSTR) model and convection-dispersion transport (CDT) model do not describe the internal transport processes with sufficient accuracy. This study may help us develop better process descriptions of very complex transport processes in HSFCW-Cs. Our results also revealed that the tracer response curves of planted HSFCW-C conservative inner points can be fitted well with Frechet distribution only if the response curve has one peak.

The phenomenon of wave set-up may substantially contribute to the formation of devastating coastal flooding in certain coastal areas. We study the appearance and properties of empirical probability density distributions of the occurrence of different set-up heights on an approximately 80 km long section of coastline near Tallinn in the Gulf of Finland, eastern Baltic Sea. The study area is often attacked by high waves propagating from various directions, and the typical approach angle of high waves varies considerably along the shore. The distributions in question are approximated by an exponential distribution with a quadratic polynomial as the exponent. Even though different segments of the study area have substantially different wave regimes, the leading term of this polynomial is usually small (between −0.005 and 0.005) and varies insignificantly along the study area. Consequently, the distribution of wave set-up heights substantially deviates from a Rayleigh or Weibull distribution (that usually reflect the distribution of different wave heights). In about three-quarters of the occasions, it is fairly well approximated by a standard exponential distribution. In about 25 % of the coastal segments, it qualitatively matches a Wald (inverse Gaussian) distribution. The Kolmogorov–Smirnov test (D value) indicates that the inverse Gaussian distribution systematically better matches the empirical probability distributions of set-up heights than the Weibull, exponential, or Gaussian distributions.

Control charts are commonly applied for monitoring and controlling the performance of the manufacturing process. Usually, control charts are designed based on the main quality characteristics variable. However, there exist numerous other variables which are highly associated with the main variable. Therefore, generalized linear model (GLM)-based control charts were used, which are capable of maintaining the relationship between variables and of monitoring an abrupt change in the process mean. This study is an effort to develop the Phase II GLM-based memory type control charts using the deviance residuals (DR) and Pearson residuals (PR) of inverse Gaussian (IG) regression model. For evaluation, a simulation study is designed, and the performance of the proposed control charts is compared with the counterpart memory less control charts and data-based control charts (excluding the effect of covariate) in terms of the run length properties. Based on the simulation study, it is concluded that the exponential weighted moving average (EWMA) type control charts have better detection ability as compared with Shewhart and cumulative sum (CUSUM) type control charts under the small or/and moderate shift sizes. Moreover, it is shown that utilizing covariate may lead to useful conclusions. Finally, the proposed monitoring methods is implemented on the dataset related to the yarn manufacturing industry to highlight the importance of the proposed control chart.

The 27 November 2019 Mw 6.0 earthquake that occurred in the southwestern part of the Hellenic Arc near Crete Island provided evidence of the high potential for strong earthquakes and active seismicity in the Hellenic Arc. In addition, tsunamis have been reported to occur for the region after major earthquakes in the historical past, so the seismic hazard of the Hellenic Arc should be evaluated in detail. The aim of this study is to evaluate the seismic hazard of the Hellenic Arc more reliably and accurately by estimating the conditional probabilities of a strong earthquake based on Weibull, gamma, log-normal, exponential, Rayleigh, and inverse Gaussian distribution models for the inter-event time of Mw ≥ 6.0 earthquakes that occurred between 1900 and 2019 in the study area. The fit between each model and the data was tested using four different test criteria, namely the log-likelihood value, Akaike information criterion, Bayesian information criteria, and Kolmogorov–Smirnov test. According to the results, the inverse Gaussian distribution was selected as the best, the log-normal distribution as the second best, the Weibull and gamma distributions as intermediate, and the Rayleigh and exponential distribution as the poorest and second poorest model, respectively. The conditional probability of an earthquake with magnitude Mw ≥ 6.0 is estimated to be higher than 0.70 according to all of the models used in this study for the base year te = 0 (te = 2015) and t > 5 years (t > 2020). Moreover, the results obtained based on the inverse Gaussian, exponential, log-normal, and Weibull distribution models are close to each other and are higher than 0.60 for te = 0 and t ≥ 3 years (t ≥ 2018). The outcomes of this study when using the different distribution models will contribute to assessments of the seismic as well as tsunami hazards for the region.

In this paper we propose semi-closed-form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options (TVOs) under affine GARCH models based on Gaussian and Inverse Gaussian distributions. We illustrate the advantage of the proposed analytic expressions by comparing them with those obtained from benchmark Monte-Carlo simulations. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.

For nearly 60 years, the lognormal distribution has been the most widely used function in the field of atmospheric science for characterizing atmospheric aerosol size distribution. We verify whether the three-parameter inverse Gaussian distribution (IGD) is a more suitable function than the lognormal distribution for characterizing aerosol size distribution. An attractive feature of IGD is that with it a new method of moments (MOM) can be established for resolving atmospheric aerosol dynamics which is described by a kinetic aerosol dynamics equation, i.e., inverse Gaussian distributed MOM (IGDMOM). The advantage of IGDMOM is that all of its moments can be analytically calculated using a closure moment function inherited from IGD. The precision and efficiency of IGDMOM are verified by comparing it with other recognizable methods in test cases of four representative atmospheric aerosol dynamics. Several key statistical quantities determining aerosol size distributions, including kth moments (k = 0, 1/3, 2/3, and 2), geometric standard deviation, skewness, and kurtosis, are evaluated. IGDMOM has higher precision than the lognormal MOM with nearly identical efficiency. The article provides a novel alternative to atmospheric scientists for solving kinetic aerosol dynamics equations.

In numerous application areas, when the response variable is continuous, positively skewed, and well fitted to the inverse Gaussian distribution, the inverse Gaussian regression model (IGRM) is an ...