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

Background: The signal-to-noise ratio (SNR) is recognized as an index of measurements reproducibility. We derive the maximum likelihood estimators of SNR and discuss confidence interval construction on the difference between two correlated SNRs when the readings are from bivariate normal and bivariate lognormal distribution. We use the Pearsons system of curves to approximate the difference between the two estimates and use the bootstrap methods to validate the approximate distributions of the statistic of interest. Methods: The paper uses the delta method to find the first four central moments, and hence the skewness and kurtosis which are important in the determination of the parameters of the Pearsons distribution. Results: The approach is illustrated in two examples;one from veterinary microbiology and food safety data and the other on data from clinical medicine. We derived the four central moments of the target statistics, together with the bootstrap method to evaluate the parameters of Pearsons distribution. The fitted Pearsons curves of Types I and II were recommended based on the available data. The R-codes are also provided to be readily used by the readers.

The Cox-Aalen Models as Framework for Construction of Bivariate Probability Distributions, Universal Representation

Starting with the Aalen （1989） version of Cox （1972） ＇regression model＇ we show the method for construction of ＂any＂ joint survival function given marginal survival functions. Basically, however, we restrict ourselves to model positive stochastic dependences only with the general assumption that the underlying two marginal random variables are centered on the set of nonnegative real values. With only these assumptions we obtain nice general characterization of bivariate probability distributions that may play similar role as the copula methodology. Examples of reliability and biomedical applications are given.

Coordinate-Momentum Intermediate Representation and Marginal Distributions of Quantum Mechanical Bivariate Normal Distribution

We introduce bivariate normal distribution operator for state vector [ψ） and find that its marginal distribution leads to one-dimensional normal distribution corresponding to the measurement probability ｜λ,v〈x｜.ψ〉｜^2, where ｜x〉λ,v is the coordinate-momentum intermediate representation. As a by-product, the one-dimensional normal distribution in statistics can be explained as a Radon transform of two-dimensional Gaussian function.

The Bivariate Transmuted Family of Distributions:Theory and Applications

The bivariate distributions are useful in simultaneous modeling of two random variables.These distributions provide a way to model models.The bivariate families of distributions are not much widely explored and in this article a new family of bivariate distributions is proposed.The new family will extend the univariate transmuted family of distributions and will be helpful in modeling complex joint phenomenon.Statistical properties of the new family of distributions are explored which include marginal and conditional distributions,conditional moments,product and ratio moments,bivariate reliability and bivariate hazard rate functions.The maximum likelihood estimation(MLE)for parameters of the family is also carried out.The proposed bivariate family of distributions is studied for the Weibull baseline distributions giving rise to bivariate transmuted Weibull(BTW)distribution.The new bivariate transmuted Weibull distribution is explored in detail.Statistical properties of the new BTW distribution are studied which include the marginal and conditional distributions,product,ratio and conditional momenst.The hazard rate function of the BTW distribution is obtained.Parameter estimation of the BTW distribution is also done.Finally,real data application of the BTW distribution is given.It is observed that the proposed BTW distribution is a suitable fit for the data used.

STATISTICAL INFERENCE FOR A BIVARIATE EXPONENTIAL DISTRIBUTION BASED ON GROUPED DATA

Consider the bivariate exponential distribution due to Marshall and Olkin[2], whose survival function is F(x, g) = exp[-λ1x-λ2y-λ12 max(x, y)] (x 0,y 0)with unknown Parameters λ1 > 0, λ2 > 0 and λ12 0.Based on grouped data, a newestimator for λ1, λ2 and λ12 is derived and its asymptotic properties are discussed.Besides, some test procedures of equal marginals and independence are given. Asimulation result is given, too.

Generalized Wigner Operator and Bivariate Normal Distribution in p-q Phase Space

We introduce a kind of generalized Wigner operator,whose normally ordered form can lead to the bivariatenormal distribution in p-q phase space.While this bivariate normal distribution corresponds to the pure vacuum state inthe generalized Wigner function phase space,it corresponds to a mixed state in the usual Wigner function phase space.

Bivariate Beta–Inverse Weibull Distribution:Theory and Applications

Probability distributions have been in use for modeling of random phenomenon in various areas of life.Generalization of probability distributions has been the area of interest of several authors in the recent years.Several situations arise where joint modeling of two random phenomenon is required.In such cases the bivariate distributions are needed.Development of the bivariate distributions necessitates certain conditions,in a field where few work has been performed.This paper deals with a bivariate beta-inverse Weibull distribution.The marginal and conditional distributions from the proposed distribution have been obtained.Expansions for the joint and conditional density functions for the proposed distribution have been obtained.The properties,including product,marginal and conditional moments,joint moment generating function and joint hazard rate function of the proposed bivariate distribution have been studied.Numerical study for the dependence function has been implemented to see the effect of various parameters on the dependence of variables.Estimation of the parameters of the proposed bivariate distribution has been done by using the maximum likelihood method of estimation.Simulation and real data application of the distribution are presented.

Modelling joint distribution of tree diameter and height using Frank and Plackett copulas

Bivariate distribution models are veritable tools for improving forest stand volume estimations.Their accuracy depends on the method of construction.To-date,most bivariate distributions in forestry have been constructed either with normal or Plackett copulas.In this study,the accuracy of the Frank copula for constructing bivariate distributions was assessed.The effectiveness of Frank and Plackett copulas were evaluated on seven distribution models using data from temperate and tropical forests.The bivariate distributions include:Burr III,Burr XII,Logit-Logistic,Log-Logistic,generalized Weibull,Weibull and Kumaraswamy.Maximum likelihood was used to fit the models to the joint distribution of diameter and height data of Pinus pinaster(184 plots),Pinus radiata(96 plots),Eucalyptus camaldulensis(85 plots)and Gmelina arborea(60 plots).Models were evaluated based on negative log-likelihood(-ΛΛ).The result show that Frank-based models were more suitable in describing the joint distribution of diameter and height than most of their Plackett-based counterparts.The bivariate Burr III distributions had the overall best performance.The Frank copula is therefore recommended for the construction of more useful bivariate distributions in forestry.

Metocean Design Parameter Estimation for Fixed Platform Based on Copula Functions

Considering the dependent relationship among wave height,wind speed,and current velocity,we construct novel trivariate joint probability distributions via Archimedean copula functions.Total 30-year data of wave height,wind speed,and current velocity in the Bohai Sea are hindcast and sampled for case study.Four kinds of distributions,namely,Gumbel distribution,lognormal distribution,Weibull distribution,and Pearson Type III distribution,are candidate models for marginal distributions of wave height,wind speed,and current velocity.The Pearson Type III distribution is selected as the optimal model.Bivariate and trivariate probability distributions of these environmental conditions are established based on four bivariate and trivariate Archimedean copulas,namely,Clayton,Frank,Gumbel-Hougaard,and Ali-Mikhail-Haq copulas.These joint probability models can maximize marginal information and the dependence among the three variables.The design return values of these three variables can be obtained by three methods:univariate probability,conditional probability,and joint probability.The joint return periods of different load combinations are estimated by the proposed models.Platform responses(including base shear,overturning moment,and deck displacement) are further calculated.For the same return period,the design values of wave height,wind speed,and current velocity obtained by the conditional and joint probability models are much smaller than those by univariate probability.Considering the dependence among variables,the multivariate probability distributions provide close design parameters to actual sea state for ocean platform design.

Dependence Rayleigh competing risks model with generalized censored data

The inference for the dependent competing risks model is studied and the dependent structure of failure causes is modeled by a Marshall-Olkin bivariate Rayleigh distribution. Under generalized progressive hybrid censoring(GPHC), maximum likelihood estimates are established and the confidence intervals are constructed based on the asymptotic theory. Bayesian estimates and the highest posterior density credible intervals are obtained by using Gibbs sampling. Simulation and a real life electrical appliances data set are used for practical illustration.

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.

A Note on the Relationship between the Pearson Product-Moment and the Spearman Rank-Based Coefficients of Correlation

This note derives the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation for the bivariate normal distribution. This new derivation shows the relationship between the two correlation coefficients through an infinite cosine series. A computationally efficient algorithm is also provided to estimate the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation. The algorithm can be implemented with relative ease using current modern mathematical or statistical software programming languages e.g. R, SAS, Mathematica, Fortran, et al. The algorithm is also available from the author of this article.

The finite-time ruin probability in the presence of Sarmanov dependent financial and insurance risks

Consider a discrete-time insurance risk model. Within period i, i≥ 1, Xi and Yi denote the net insurance loss and the stochastic discount factor of an insurer, respectively. Assume that {（Xi, Yi）, i≥1） form a sequence of independent and identically distributed random vectors following a common bivariate Sarmanov distribution. In the presence of heavy-tailed net insurance losses, an asymptotic formula is derived for the finite-time ruin probability.

二项-二维对数正态分布及其在极端海况预测中的应用

Extreme value analysis is an indispensable method to predict the probability of marine disasters and calculate the design conditions of marine engineering.The rationality of extreme value analysis can be easily affected by the lack of sample data.The peaks over threshold(POT)method and compound extreme value distribution(CEVD)theory are effective methods to expand samples,but they still rely on long-term sea state data.To construct a probabilistic model using shortterm sea state data instead of the traditional annual maximum series(AMS),the binomial-bivariate log-normal CEVD(BBLCED)model is established in this thesis.The model not only considers the frequency of the extreme sea state,but it also reflects the correlation between different sea state elements(wave height and wave period)and reduces the requirement for the length of the data series.The model is applied to the calculation of design wave elements in a certain area of the Yellow Sea.The results indicate that the BBLCED model has good stability and fitting effect,which is close to the probability prediction results obtained from the long-term data,and reasonably reflects the probability distribution characteristics of the extreme sea state.The model can provide a reliable basis for coastal engineering design under the condition of a lack of marine data.Hence,it is suitable for extreme value prediction and calculation in the field of disaster prevention and reduction.

Reliability of a Multicomponent Stress-strength Model Based on a Bivariate Kumaraswamy Distribution with Censored Data

In this paper,we consider a system which has k statistically independent and identically distributed strength components and each component is constructed by a pair of statistically dependent elements with doubly type-II censored scheme.These elements(X1,Y1),(X2,Y2),…,(Xk,Yk)follow a bivariate Kumaraswamy distribution and each element is exposed to a common random stress T which follows a Kumaraswamy distribution.The system is regarded as operating only if at least s out of k(1≤s≤k)strength variables exceed the random stress.The multicomponent reliability of the system is given by Rs,k=P(at least s of the(Z1,…,Zk)exceed T)where Zi=min(Xi,Yi),i=1,…,k.The Bayes estimates of Rs,k have been developed by using the Markov Chain Monte Carlo methods due to the lack of explicit forms.The uniformly minimum variance unbiased and exact Bayes estimates of Rs,k are obtained analytically when the common second shape parameter is known.The asymptotic confidence interval and the highest probability density credible interval are constructed for Rs,k.The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.

Copula-Based Bivariate Flood Frequency Analysis in a Changing Climate——A Case Study in the Huai River Basin, China

Copula-based bivariate frequency analysis can be used to investigate the changes in flood characteristics in the Huai River Basin that could be caused by climate change. The univariate distributions of historical flood peak, maximum 3-day and 7-day volumes in 1961-2000 and future values in 2061-2100 projected from two GCMs（CSIRO-MK3.5 and CCCma-CGCM3.1） under A2, A1 B and B1 emission scenarios are analyzed and compared. Then, bivariate distributions of peaks and volumes are constructed based on the copula method and possible changes in joint return periods are characterized. Results indicate that the Clayton copula is more appropriate for historical and CCCma-CGCM3.1 simulating flood variables, while that of Frank and Gumbel are better fitted to CSIRO-MK3.5 simulations. The variations of univariate and bivariate return periods reveal that flood characteristics may be more sensitive to different GCMs than different emission scenarios. Between the two GCMs, CSIRO-MK3.5 evidently predicts much more severe flood conditions in future, especially under B1 scenario, whereas CCCma-CGCM3.1 generally suggests contrary changing signals. This study corroborates that copulas can serve as a viable and flexible tool to connect univariate marginal distributions of flood variables and quantify the associated risks, which may provide useful information for risk-based flood control.

OPTIMAL ALLOCATION FOR ESTIMATING THE CORRELATION COEFFICIENT OF MORGENSTERN TYPE BIVARIATE EXPONENTIAL DISTRIBUTION BY RANKED SET SAMPLING WITH CONCOMITANT VARIABLE

Ranked set sample is applicable whenever ranking of a set of sample units can be done easily by a judgement method of the study variable or of the auxiliary variable. This paper considers ranked set sample based on the auxiliary variable X which is correlated with the study variable Y, where （X, Y） follows Morgenstern type bivariate exponential distribution. The authors discuss the optional allocation for unbiased estimators of the correlation coefficient p of the random variables X and Y when the auxiliary variable X is used for ranking the sample units and the study variable Y is measured for estimating the correlation coefficient. This paper first gives a class of unbiased estimators of p when the mean 0 of the study variable Y is known and obtains an essentially complete subclass of this class. Further, the optimal allocation of the unbiased estimators is found in this subclass and is proved to be Bayes, admissible, and minimax. Finally, the unbiased estimator of p under the optimal allocation in the case of known θ is reformed for estimating p in the case of unknown θ, and the reformed estimator is shown to be strongly consistent.