Bivariate Normal Distribution Research Articles

Value-at-Risk Effectiveness: A High-Frequency Data Approach with Semi-Heavy Tails

In the broader landscape of cryptocurrency risk management, this study delves into the nuanced estimation of Value-at-Risk (VaR) for a uniformly weighted portfolio of cryptocurrencies, employing the bivariate Normal Inverse Gaussian distribution renowned for its semi-heavy tails. Utilizing high-frequency data spanning between 1 January 2017 and 25 October 2022, with a primary focus on Bitcoin and Ethereum, our research seeks to accentuate the resilience of VaR methodology as a paramount risk assessment tool. The essence of our investigation lies in advancing the comprehension of VaR accuracy by quantitatively comparing the observed returns of both cryptocurrencies with their corresponding estimated values, with a central theme being the endorsement of the Normal Inverse Gaussian distribution as a potent model for risk measurement, particularly in the domain of high-frequency data. To bolster the statistical reliability of our results, we adopt a forward test methodology, showcasing not only a contribution to the evolution of risk assessment techniques in Finance but also underscoring the practicality of sophisticated distributional models in econometrics. Our findings not only contribute to the refinement of risk assessment methods but also highlight the applicability of such models in precisely modeling and forecasting financial risk within the dynamic realm of cryptocurrencies, epitomized by the case study of Bitcoin and Ethereum.

A simple approach to probabilistic CPTu-based geotechnical stratigraphic profiling

Geotechnical profile delineation is an important aspect of the geotechnical engineering design process. The Cone Penetration Test (CPTu) is widely used for this purpose, as it is simple to classify soils based on their different responses to the cone. However, CPTu-based delineations typically involve engineering interpretation, as they are affected by the classification scheme applied, by inherent soil variability and by measurements error. A semi-automated CPTu data treatment tool is here presented aiming to simplify the soil delineation task, adding consistency and transparency to the involved engineering interpretation process and facilitating the transference of information between the different parties involved in design. The tool fits bivariate normal distributions to interpreted CPTu data according to some user specified criteria that quantify estimated classification and measurement uncertainties. Classification makes use of conventional class boundaries –here taken from Robertson, 1990 chart-, which are applied with user-specified refinement. Thin layers are consolidated into adjacent ones to further simplify the resulting profile. The operation of the proposed algorithm is illustrated by interpreting CPTu records from a deltaic sedimentary environment. The outputs of this simple method may be used as input for more advanced probabilistic methodologies, helping thus to bridge the gap between qualitative practice and quantitative research methods.

Iteratively Reweighted Least Squares Method for Estimating Polyserial and Polychoric Correlation Coefficients

An iteratively reweighted least squares (IRLS) method is proposed for the estimation of polyserial and polychoric correlation coefficients in this article. It calculates the slopes in a series of weighted linear regression models fitting on conditional expected values. For polyserial correlation, conditional expectations of the latent predictor is derived from the observed ordinal categorical variable, and the regression coefficient is obtained with the weighted least squares method. In estimating polychoric correlation coefficient, conditional expectations of the response variable and the predictor are updated in turns. Standard errors of the estimators are obtained using the delta method based on data summaries instead of the whole data. Conditional univariate normal distribution is exploited and a single integral is numerically evaluated in the proposed algorithm, comparing to the double integral computed numerically based on the bivariate normal distribution in the traditional maximum likelihood (ML) approaches. This renders the new algorithm very fast in the estimation of both polyserial and polychoric correlation coefficients. Thorough simulation studies are conducted to compare the performances of the proposed method with the classical ML methods. Real data analyses illustrate the advantage of the new method in computing speed. Supplementary materials for this article are available online.

Product of bi-dimensional VAR(1) model components. An application to the cost of electricity load prediction errors

Abstract The multi-dimensional vector autoregressive (VAR) time series is often used to model the impulse-response functions of macroeconomics variables. However, in some economical applications, the variable of main interest is the product of time series describing market variables, like e.g. the cost, being the product of price and volume. In this paper, we analyze the product of the bi-dimensional VAR(1) model components. For the introduced time series, we derive general formulas for the autocovariance function and study its properties for different cases of cross-dependence between the VAR(1) model components. The theoretical results are then illustrated in the simulation study for two types of bivariate distributions of the residual series, namely the Gaussian and Student’s t. The obtained results are applied for the electricity market case study, in which we show that the additional cost of balancing load prediction errors prior to delivery can be well described by time series being the product of the VAR(1) model components with the bivariate normal inverse Gaussian distribution.

Generalized Cramér’s coefficient via f-divergence for contingency tables

AbstractVarious measures in two-way contingency table analysis have been proposed to express the strength of association between row and column variables in contingency tables. Tomizawa et al. (2004) proposed more general measures, including Cramér’s coefficient, using the power-divergence. In this paper, we propose measures using the f-divergence that has a wider class than the power-divergence. Unlike statistical hypothesis tests, these measures provide quantification of the association structure in contingency tables. The contribution of our study is proving that a measure applying a function that satisfies the condition of the f-divergence has desirable properties for measuring the strength of association in contingency tables. With this contribution, we can easily construct a new measure using a divergence that has essential properties for the analyst. For example, we conducted numerical experiments with a measure applying the $$\theta$$ θ -divergence. Furthermore, we can give further interpretation of the association between the row and column variables in the contingency table, which could not be obtained with the conventional one. We also show a relationship between our proposed measures and the correlation coefficient in a bivariate normal distribution of latent variables in the contingency tables.

Using spatial point process models, clustering and space partitioning to reconfigure fire stations layout

AbstractFire stations (FS) are typically non-uniformly distributed across space, and their service area is, in general, defined based on administrative boundaries. Since the location of FS may considerably influence the readiness and the effectiveness of the provided services, national and regional governments need research-based information to adequately plan where to establish firefighting facilities. In this study, we propose a method to reconfigure the fire stations layout using spatial point process models, clustering and space partitioning. First, modelling fire intensity variation across space through a point process model enables to replicate the process independently by simulation. Subsequently, for each simulation, the k-means algorithm is used to define a siting location, minimizing the total within distance between the fire occurrences and the new position. This method allows to obtain a set of locations from which the respective distribution is inferred. Assuming a bivariate normal spatial distribution, we further define confidence siting regions. Ultimately, new FS service areas are defined by Voronoi tessellation. To exemplify the application of the method, we apply it to reconfigure the fire station layout at Aveiro, Portugal.

Thurstonian model for the four-interval oddity task

The four-interval oddity task is an extension of the Triangle method, which is the three-interval oddity task and is one of the most broadly used conventional sensory discrimination methods. It is a special case of the m-interval oddity task or the m-alternative odd-man-out task where m = 4. It is also a special situation of the unspecified ‘M + N’ with M = 3 and N = 1. Some new forms of the analytical psychometric function for the four-interval oddity task based on a normal distribution function, a bivariate normal distribution function, and a non-central F-distribution function are provided originally in this paper. The performance of the four-interval oddity task in both difference testing and similarity/equivalence testing is explored and compared with some unspecified forced-choice sensory discrimination methods including the Triangle. Tables and R codes are presented and provided for estimations of the probability of correct response, Pc, Thurstonian discriminal distance δ or d´, and the B value for estimating the variance of d´ for the four-interval oddity task.

A first-order Stein characterization for absolutely continuous bivariate distributions

A random variable X has a standard normal distribution if and only if for any continuous and piecewise continuously differentiable function f such that the expectations exist. This first-order characterizing equation, called the Stein identity, has been extended to other univariate distributions. For the multivariate normal distribution, a number of Stein identities have already been developed, all of them second order equations. In this study, we developed a new Stein characterization for the bivariate normal distribution. Unlike many existing multivariate versions in the literature, ours is a system of first-order equations which has the univariate Stein identity as a special case. We also constructed a generalized Stein characterization for other absolutely continuous bivariate distributions. Finally, we illustrated how this Stein characterization looks like for some known absolutely continuous bivariate distributions.

Bayesian semiparametric joint model of multivariate longitudinal and survival data with dependent censoring.

We consider a novel class of semiparametric joint models for multivariate longitudinal and survival data with dependent censoring. In these models, unknown-fashion cumulative baseline hazard functions are fitted by a novel class of penalized-splines (P-splines) with linear constraints. The dependence between the failure time of interest and censoring time is accommodated by a normal transformation model, where both nonparametric marginal survival function and censoring function are transformed to standard normal random variables with bivariate normal joint distribution. Based on a hybrid algorithm together with the Metropolis-Hastings algorithm within the Gibbs sampler, we propose a feasible Bayesian method to simultaneously estimate unknown parameters of interest, and to fit baseline survival and censoring functions. Intensive simulation studies are conducted to assess the performance of the proposed method. The use of the proposed method is also illustrated in the analysis of a data set from the International Breast Cancer Study Group.

A Truncated Mixture Transition Model for Interval-Valued Time Series

Abstract We propose a model for interval-valued time series that specifies the conditional joint distribution of the upper and lower bounds as a mixture of truncated bivariate normal distributions. It preserves the interval natural order and provides great flexibility on capturing potential conditional heteroscedasticity and non-Gaussian features. The standard expectation maximization (EM) algorithm applied to truncated mixtures does not provide a closed-form solution in the M step. A new EM algorithm solves this problem. The model applied to the interval-valued IBM daily stock returns exhibits superior performance over competing models in-sample and out-of-sample evaluation. A trading strategy showcases the usefulness of our approach.

Unveiling the Elemental Composition and Categorization of Sikkim Hot-Spring Water with Laser-Induced Breakdown Spectroscopy and Advanced Analytical Techniques

The research investigates the elemental composition of hot-spring water in Sikkim, specifically in Yumthang, Tarum, and Reshi, using laser-induced breakdown spectroscopy. This technique involves exciting the sample with a high-energy laser pulse, causing it to emit light that can be analyzed to determine its elemental composition. The results of the study indicate that the hot-spring water in Sikkim is of meteoric origin, meaning it originated from precipitation. To understand the abundance of elements in the hot-spring water samples, the Sulphur/Nitrogen, Chlorine/Nitrogen, and Sulphur/Chlorine ratios were calculated using the ratiometric method. These ratios were then used to classify the samples using univariate and multivariate statistical tools. The box plot was used for the univariate analysis, while the bivariate normal distribution method was used for the multivariate analysis. We have also discussed the limitations of overlapping emission peaks on the classification of the samples. The study found that the abundance of nitrogen in the samples played a significant role in their classification. Principal component analysis was also employed to classify the samples. This technique involves identifying the most significant variables contributing to the variation in the data and using them to classify the samples. Overall, the study provides important insights into the elemental composition of hot-spring water in Sikkim and highlights the usefulness of laser-induced breakdown spectroscopy in analyzing such samples.

Systematic Review and Meta-Analysis of the Use of Serum Leucine-Rich Alpha-2 Glycoprotein to Assess Crohn's Disease Activity.

Although fecal calprotectin is the most widely used biomarker for assessing Crohn's disease activity, serum leucine-rich alpha-2 glycoprotein has recently attracted attention, especially in Japan. Here we performed a systematic review and meta-analysis of serum leucine-rich alpha-2 glycoprotein to obtain evidence for its accuracy in assessing Crohn's disease activity. On February 1, 2023, we performed searches of PubMed, Web of Science, and CENTRAL. The Prospero number is CRD42023396034. The primary outcomes were the sensitivity and specificity of serum leucine-rich alpha-2 glycoprotein for assessing Crohn's disease activity. We used a bivariate generalized linear mixed model, assuming a binomial distribution at the test level and a bivariate normal distribution at the between-test level. We selected 9 studies involving 797 individuals in our systematic review. Regarding the primary outcomes, the synthesized sensitivity and specificity of serum leucine-rich alpha-2 glycoprotein were 77.0% (95% confidence interval, 67.8% to 84.2%) and 81.1% (95% confidence interval, 72.6% to 87.4%), respectively. The area under the curve was 0.86, and the partial area under the curve was 0.78. Regarding between-study heterogeneity, both the I2 value by Zhou and Dendukuri approach and the I2 value by Holling sample size-adjusted approaches were 0%. Our systematic review and meta-analysis of serum leucine-rich alpha-2 glycoprotein demonstrated its accuracy in assessing Crohn's disease activity. Further studies are needed to demonstrate its clinical utility and clinical validity.

A phase II multivariate EWMA chart for monitoring multi-dimensional ratios of process means with individual observations

In recent years, there has been a resurgence in the development of control charts for monitoring the mean of the ratio of two correlated variables. However, most of the existing research has focused on the univariate mean of the ratio of two correlated variables under the assumption that the process follows a bivariate normal distribution. Furthermore, most of the existing research utilize biased estimators of the mean of the ratio of two correlated variables to develop control charts. More importantly, in certain applications, critical quality characteristics to be closely monitored are actually the ratios of the means of correlated variables, in that the process is considered to be stable as long as the ratios of the means of correlated variables remain constant at given levels, regardless of how each variable changes. We are thus motivated in this study to develop an exponentially weighted moving average (EWMA) based Phase II control chart for monitoring multi-dimensional ratios of the means of correlated variables.In this study, we provide a general framework for estimating parameters and control limits which is applicable without having to assume that the process follows a multivariate normal distribution. The performance of the proposed chart is evaluated under different multivariate distributions. Finally, a real data application of the proposed chart is presented to illustrate the practicality of the proposed chart.

Probabilistic generic transformation model between two rock mass properties: specific fracture energy and P-wave velocity

This study proposed a probabilistic generic transformation model between two rock mass properties, specific fracture energy, Ev, and P-wave velocity, VP. To build the transformation model, 12 pairwise data sets of Ev and VP were collected from six different construction sites involving construction of mountain tunnels in Japan. This database is labeled as “RockMass/2/350”. A probabilistic transformation model was built based on a bivariate standard normal distribution with these 350 data points. The model is generic, because it is based on a variety of sites. The performance of the constructed transformation model was evaluated through a cross-validation. It was found that 98.2% of the validation data fell within the computed 95% confidence interval of the model estimation, and this result provides a preliminary validation of the probabilistic transformation model. Unlike existing deterministic transformation models for estimating VP from Ev, the proposed model can explicitly evaluate the transformation uncertainty with a quantitative metric such as a percentile. For practical application, a 3D model of the spatial distribution for Young’s modulus, E, was visualized based on the proposed transformation model. Since the proposed model is probabilistic, it can provide the spatial distribution for percentiles of E values. The constructed 3D model presented in this paper can be directly used as an input data for finite element or finite difference analysis, and probabilistic evaluation of excavation simulation is feasible based on the proposed probabilistic model. The quantitative information on such uncertainty can be useful in decision-making for tunnel constructions, such as selection of a cautious characteristic value.

Bayesian joint modeling of patient-reported longitudinal data on frequency and duration of migraine

In this methodological study, we address the joint modeling of longitudinal data on the frequency and duration migraine attacks collected from patients in a clinical study in which patients were repeatedly asked at each hospital visit to report the number of days of migraine attacks they had in the last $30$ days and the corresponding average duration of attacks. In our motivating data set, the migraine frequency outcome is a count variable inflated at multiples of $5$ and $10$ days, whereas the migraine duration outcome is reported entirely in discrete hours, including $0$ for non-migraine days and inflated at multiples of $12$ hours. In our study, we propose a joint modeling approach that models each migraine outcome by a multiple inflated negative binomial model with random effects and assumes a bivariate normal distribution for the random effects. We estimate the model parameters under Bayesian inference. We examine the performance of the proposed joint model using a Monte Carlo simulation study and compare its performance with a separate modeling approach in which each longitudinal count outcome is modeled separately. Finally, we present the results of the analysis of migraine data.

Moment ratio inequality of bivariate Gaussian distribution and three-dimensional Gaussian product inequality

We prove the three-dimensional Gaussian product inequality (GPI)E[X12X22m2X32m3]≥E[X12]E[X22m2]E[X32m3] for any centered Gaussian random vector (X1,X2,X3) and m2,m3∈N. We discover a novel inequality for the moment ratio |E[X22m2+1X32m3+1]|E[X22m2X32m3], which implies the 3D-GPI. The interplay between computing and hard analysis plays a crucial role in the proofs.

Utilization of El Niño–Southern Oscillation projected by climate models in improvement of seasonal precipitation predictability

AbstractImproved seasonal precipitation forecasts can enable more effective water resource management decisions in a number of sectors, including municipal supply, agriculture, hydropower generation, and tourism. This study develops an effective straightforward statistical approach to enhance the quality of seasonal precipitation forecasts through the utilization of El Niño–Southern Oscillation (ENSO) information projected by Coupled General Climate Models (CGCMs). A stochastic weather generation (WG) model is developed to predict seasonal precipitation condition on ENSO condition. The WG model links a nonhomogeneous Markov Chain representing ENSO occurrence model to a bivariate normal distribution for seasonal precipitation conditioned on ENSO phase. Two verification metrics are suggested to measure the degree of predictability of raw, calibrated and climatological seasonal precipitation forecasts over northwest Costa Rica as a case study. Results indicate the potential to narrow the uncertainty of seasonal precipitation forecasts by incorporating CGCMs ENSO cycle information. Precipitation during the late part of the wet season (LS) has more predictability than precipitation in the early part of the wet season (ES). In addition, the degree of predictability decreases with an increase in lead time for a given forecast. A lead time of 1 year maintains a moderate level of predictability likely to support tangible benefits to various decision‐making processes.

Receiver operating characteristic analysis for evaluating a proper experimental condition of eddy current tests under a low signal-to-noise ratio

This study proposed a method to evaluate proper experimental conditions of eddy current tests under a low signal-to-noise ratio. This method was applied to quantify the detection reliability of the eddy current test to detect corrosion pits on the surface of austenitic stainless-steel weld cladding from the perspectives of probability of detection and probability of false alarm. The fluctuations of flaw signal and noise were described using truncated bivariate normal distributions of a two-dimensional vector composed of the real and imaginary components of a signal. The probabilities of detection and false alarms were formalized as cumulative distributions of the estimated distributions of the maximum flaw signal and the noise exceeding a decision threshold. The receiver operating characteristic curves were created by coupling the probabilities of detection and false alarms at various thresholds. The area under the curve was used to quantify the detection reliability, simultaneously considering the probabilities of detection and false alarms. The results obtained in this study support the applicability of this method in quantitatively evaluating the detection reliability of a non-destructive inspection under a low signal-to-noise ratio.

Comparison via Simulation of Association Coefficients Calculated between Categorical Variables

Aim: The aim of this study was to compare the robustness of some association coefficients used to determine the relationships between categorical variables under different experimental conditions. Material and Methods: A simulation study was conducted where random numbers were generated from a bivariate standard normal distribution with correlations of 0.5 and 0.9. Sample sizes were set at 30, 50, 100, 150 and 200. Random numbers were equally spaced and coded as 3×3, 4×4 and 5×5 cross-tabulations, respectively. The robustness of Pearson’s, Spearman's rank, Kendall's tau-b, Kendall's tau-c, Goodman-Kruskal’s gamma and Somer's d coefficients were compared under different experimental conditions consisting of combinations of specified population correlation degrees, table dimensions and sample sizes. Results: The Goodman-Kruskal’s gamma coefficient gave the closest result to the relationship levels set at the beginning of the study in all experimental conditions. However, after a certain level, it was negatively affected by the increase in table dimension and sample size. Kendall's tau-b and tau-c coefficients were furthest from the actual degree of the association. Spearman's rank correlation was more robust than Kendall's tau-b, Kendall's tau-c and Somer's d coefficients. Conclusion: The results of the study showed that the dimension of the contingency tables and sample size were effective factors in the robustness of association coefficients for categorical variables. Therefore, researchers should consider the table dimension and sample size as well as the type of variable when selecting the association coefficient to be calculated.

Bivariate joint models for survival and change of cognitive function.

Changes in cognitive function over time are of interest in ageing research. A joint model is constructed to investigate. Generally, cognitive function is measured through more than one test, and the test scores are integers. The aim is to investigate two test scores and use an extension of a bivariate binomial distribution to define a new joint model. This bivariate distribution model the correlation between the two test scores. To deal with attrition due to death, the Weibull hazard model and the Gompertz hazard model are used. A shared random-effects model is constructed, and the random effects are assumed to follow a bivariate normal distribution. It is shown how to incorporate random effects that link the bivariate longitudinal model and the survival model. The joint model is applied to the English Longitudinal Study of Ageing data.