Trivariate Normal Distribution Research Articles

Latent Variable Modeling of Scientific Impact: Estimation of the Q Model Parameters with Structural Equation Models

Abstract Statistical modeling of scientific productivity and impact provides insights into bibliometric measures used also to quantify differences between individual scholars. The Q model decomposes the log-transformed impact of a published paper into a researcher capacity parameter and a random luck parameter. These two parameters are then modeled together with the log-transformed number of published papers (i.e., an indicator of productivity) by means of a trivariate normal distribution. In this work we propose a formulation of the Q model that can be estimated as a structural equation model. The Q model as a structural equation model allows to quantify the reliability of researchers’ Q parameter estimates, it can be extended to incorporate person covariates, and multivariate extensions of the Q model could also be estimated. We empirically illustrate our approach to estimate the Q model and also provide openly available code for R and Mplus. Peer Review https://www.webofscience.com/api/gateway/wos/peer-review/10.1162/qss_a_00313

Convergence of extremes from normal-skew-normal and minima and maxima from exchangeable trivariate normal distributions

In this article, the extreme-value distribution of the partial maximum for a random sequence of the normal-skew-normal (NSN) distribution, to the Gumbel distribution, is derived. The asymptotic distribution of the normalized maximum and its optimal convergence rate are determined using the optimal choice of normalizing constants. These results are applied to derive the extreme-value distributions of minimum and maximum for the exchangeable trivariate normal random vectors, and lifetimes of parallel and series systems having three dependent log-normal components.

Vulnerability of the Tukey M Robust Regression Method Against Multicollinearity

In this study, we investigate whether the Tukey M robust regression method provides a solution for the data sets suffering from multicollinearity problem. It is observed that high values of variance inflation factors (VIF) which is a sign of the multiple linear link among the explanatory variables, cannot be controlled by the robust methods which work through the residual values. The reason for this fact is that multicollinearity and high values of VIF which is a result of multicollinearity do not produce extreme residuals. For this reason, the robust methods cannot provide a solution for the high VIF problem. This fact is shown by an extensive simulation study. In the simulation study, the explanatory variables were derived from trivariate normal distribution for three different correlation values. In this study, we also used two real-life data examples and we observed that the results support the findings of the simulation study. For all these reasons, we can conclude that specialized methods should be utilized in the case of multicollinearity.

Computation of tolerance ellipses for bivariate and trivariate normal populations

We consider the computation of the critical constants c of a p-content -confidence tolerance ellipse for bivariate and trivariate normal populations which are probably the most used multivariate normal distributions in applications. There are only approximation methods available in the statistical literature for computing c. Without knowing the accurate value of c, it is impossible to assess the accuracy of the approximation methods. In this paper, a new method is given to compute c. Although the method is also based on Monte-Carlo simulation, it allows c to be computed as accurate as required if the number of simulation replications is sufficiently large. The R codes provided allow easy implementation of the method. Furthermore, the method allows the accuracy of the available approximation methods for bivariate and trivariate normal distributions to be assessed.

A procedure to develop a backbone ground-motion model: A case study for its implementation

The backbone modeling in ground-motion characterization (GMC) is a useful methodology to describe the epistemic uncertainty in median ground-motion predictions. The approach uses a backbone ground-motion model (GMM) and populates the GMC logic tree with the scaled and/or adjusted versions of the backbone GMM to capture the epistemic uncertainty in median ground motions. The scaling and/or adjustment should represent the specific features and uncertainties involved in source, path, and site effects at the target site. The identification of the backbone model requires different considerations specific to the nature of the ground-motion hazard problem. In this article, we present a scaled backbone modeling approach that considers the magnitude- and distance-scaling predictors as well as their correlation to address the epistemic uncertainty in median ground-motion predictions. This approach results in a trivariate normal distribution to fully define a range of epistemic uncertainty in a model sample space. The simultaneous consideration of magnitude and distance scaling while defining the epistemic uncertainty and the methodology followed for the simplified representation of trivariate normal distribution in ground-motion logic tree are the two important features in our procedure. We first present the proposed approach that is followed by a case study for Central and Eastern North America (CENA) stable continental region. The case study discusses the underlying assumptions and limitations of the proposed approach.

Efficient estimators of population mean in ranked set sampling scheme using two concomitant variables

In this article, we suggest an efficient class of difference-in-exponential ratio-type estimators having two concomitant variables under ranked set sampling (RSS) design. The bias and mean squared error (MSE) of the proposed estimators are obtained upto first degree of approximation. Comparison among the proposed and competitor estimators are made both theoretically and through simulation study. It turned out that when the variable of interest and the concomitant variables jointly follow a trivariate normal distributions, the proposed class of estimators of the population mean dominates all other competitor estimators. In addition, we support the result with a real data set.

Multivariate confidence region using quantile vectors

Multivariate confidence regions were defined using a chi-square distribution function under a normal as-sumption and were represented with ellipse and ellipsoid types of bivariate and trivariate normal distribution functions. In this work, an alternative confidence region using the multivariate quantile vectors is proposed to define the normal distribution as well as any other distributions. These lower and upper bounds could be ob-tained using quantile vectors, and then the appropriate region between two bounds is referred to as the quantile confidence region. It notes that the upper and lower bounds of the bivariate and trivariate quantile confidence regions are represented as a curve and surface shapes, respectively. The quantile confidence region is obtained for various types of distribution functions that are both symmetric and asymmetric distribution functions. Then, its coverage rate is also calculated and compared. Therefore, we conclude that the quantile confidence region will be useful for the analysis of multivariate data, since it is found to have better coverage rates, even for asymmetric distributions.

Generalized exponential-type ratio-cum ratio estimators of population mean in ranked set and stratified ranked set sampling

In this study, we propose generalized exponential type ratio-cum-ratio estimators of finite population mean using ranked Set Sampling (RSS) and stratified ranked set sampling (SRSS) schemes. The biases and Mean Square Errors (MSEs) of the proposed estimators are obtained up to first degree of approximation. Comparisons among the proposed and some existing estimators are made both theoretically and through simulation studies. Also, the estimators are compared in terms of relative bias (RB), relative mean square error (RRMSE) and percentage relative efficiency (PRE). It turned out that when the variable of interest and the auxiliary variables jointly follow a trivariate normal distribution, the proposed class of estimators of the population mean dominates all other competitor estimators.

A measure of departure from second-order marginal symmetry for multi-way tables with nominal categories

For multi-way tables, Bhapkar and Darroch (1990) gave the second-order marginal symmetry model. The present paper proposes a measure to represent the degree of departure from the second-order marginal symmetry model. The measure is expressed as the weighted sum of the Shannon entropy. The paper also gives the approximate confidence interval of the measure, and shows relationship between the measure and the trivariate normal distribution. Examples are given.

Early Clinical Development Planning via Biomarkers, Clinical Endpoints, and Simulation: A Case Study to Optimize for Phase 3 Dose Selection.

This study investigated a framework that leverages the relationship between biomarkers and a target clinical endpoint to optimize an early development plan. Different biomarker designs were assessed for proof of concept (PoC) and dose finding (DF) to improve phase 2b (Ph2b) design as well as phase 3 (Ph3) dose choice. A case study using a Bayesian trivariate normal distribution model for 2 biomarkers and a clinically relevant endpoint was utilized with simulation to assess performance characteristics. We found the following: (1) at typical sample sizes for early development trials, biomarkers appear useful for PoC but not for clinical endpoint DF; and (2) even with large amounts of prior information and near perfect correlation between biomarkers and clinical endpoints, Ph2b variability is only overcome by increased Ph2b sample sizes to improve Ph3 dose choice. For highly variable clinical endpoints, the fastest path should be to demonstrate PoC by biomarkers and then go directly to Ph2b to measure the target clinical endpoint.

Third Order Compound Option Valuation of Flexible Commodity Based Mining Enterprises

Flexibility in managerial decision making will alter the true value of real world projects. Standard actuarial practice for evaluating real-world projects such as commodity based mining operations rely upon Net Present Value methodology and in essence ignore any flexibility available to the operator to vary the project. Real Option analysis rectifies this to allow better evaluation of economic investment decisions by incorporating managerial flexibility into an option pricing framework. In this paper we extend the results of Konstandatos and Kyng (2012) to evaluate a multi-stage compound mining investment decision where the mining operators have the flexibility to delay project commencement as well as options to abandon production and to expand production to a new mining seam if conditions improve. We allow an independent abandonment of the expansion from the underlying project. We demonstrate that the flexibilities considered give rise to a third-order exotic compound structure, which are evaluated in terms of first, second and third order generalised compound option instruments (Konstandatos (2008)). Our novel representations of the project values contain generalizations of standard compound options and are interpretable as generalised call, call-on-call and call-on-call-on-call type options on the mined commodity price. We provide readily-implementable closed-form analytical formulae which are expressed in terms of the uni-variate, bi-variate and tri-variate Normal distribution functions.

Regression mean residual life of a system with three dependent components with normal lifetimes

In this paper, we use joint and conditional distributions of order statistics from a trivariate normal distribution to obtain closed expressions for reliability measures such as mean residual lifetime for systems with three dependent components with normal lifetimes.

Cost of Right-of-Way Acquisition: Recognizing the Impact of Condemnation via a Switching Regression Model

AbstractThe costs of acquiring parcels by condemnation are usually significantly higher than those of property acquired by negotiation, suggesting that right-of-way (R/W) acquisition costs may best be described by two different regression equations. This paper develops a switching regression model of acquisition cost to simultaneously predict the probability of whether a parcel will go to condemnation rather than be acquired through negotiation and the corresponding acquisition costs under these two regimes. The error terms of the selection equation and the two cost equations follow a trivariate normal distribution to reflect codependence on unobserved factors (such as a land owner’s tenacity or a site’s view value). When applied to the properties acquired across the state of Texas for transportation probjects between 2008 and 2011, the results of this switching regression model suggest that R/W appraisers and staff should pay special attention to parcels in commercial use involving a partial taking with ...

Modeling and one-sample Bayes inference for a new distribution for 3-D rotations motivated by a central limit

For modeling orientation data represented as 3 × 3 rotation matrices, we develop a wrapped trivariate normal distribution (wTND) under which random rotations have simple geometric construction as symmetric errors about a mean. While of interest in its own right, the wTND also provides simple and effective approximations to the isotropic Gaussian distribution on rotations, with some advantages over approximations based on other commonly used models. We develop non-informative Bayes inference for the wTND via Markov Chain Monte Carlo methods that allow straightforward computations in a model where maximum likelihood is undefined. Credible regions for model parameters (including a fixed 3× 3 mean rotation) are shown to possess good frequentist coverage properties. We illustrate the model and inference method with orientation data collected in texture analysis from materials science.

The evaluation of trivariate normal probabilities defined by linear inequalities

This paper considers the evaluation of probabilities which are defined by a set of linear inequalities of a trivariate normal distribution. It is shown that these probabilities can be evaluated by a one-dimensional numerical integration. The trivariate normal distribution can have any covariance matrix and any mean vector, and the probability can be defined by any number of one-sided and two-sided linear inequalities. This affords a practical and efficient method for the calculation of these probabilities which is superior to basic simulation methods. An application of this method to the analysis of pairwise comparisons of four treatment effects is discussed.

Jointly modeling the relationship between longitudinal and survival data subject to left truncation with applications to cystic fibrosis.

Numerous methods for joint analysis of longitudinal measures of a continuous outcome y and a time to event outcome T have recently been developed either to focus on the longitudinal data y while correcting for nonignorable dropout, to predict the survival outcome T using the longitudinal data y, or to examine the relationship between y and T. The motivating problem for our work is in joint modeling of the serial measurements of pulmonary function (FEV1% predicted) and survival in cystic fibrosis (CF) patients using registry data. Within the CF registry data, an additional complexity is that not all patients have been followed from birth; therefore, some patients have delayed entry into the study while others may have been missed completely, giving rise to a left truncated distribution. This paper shows in joint modeling situations where y and T are not independent, that it is necessary to account for this left truncation to obtain valid parameter estimates related to both survival and the longitudinal marker. We assume a linear random effects model for FEV1% predicted, where the random intercept and slope of FEV1% predicted, along with a specified transformation of the age at death follow a trivariate normal distribution. We develop an expectation-maximization algorithm for maximum likelihood estimation of parameters, which takes left truncation and right censoring of survival times into account. The methods are illustrated using simulation studies and using data from CF patients in a registry followed at Rainbow Babies and Children's Hospital, Cleveland, OH.

ASSESSING CONVERGENCE OF THE MARKOV CHAIN MONTE CARLO METHOD IN MULTIVARIATE CASE

The formal convergence diagnosis of the Markov Chain Monte Carlo (MCMC) is made using univariate and multivariate criteria. In 1998, a multivariate extension of the univariate criterion of multiple sequences was proposed. However, due to some problems of that multivariate criterion, an alternative form of calculation was proposed in addition to the two new alternatives for multivariate convergence criteria. In this study, two models were used, one related to time series with two interventions and ARMA (2, 2) error and another related to a trivariate normal distribution, considering three different cases for the covariance matrix. In both the cases, the Gibbs sampler and the proposed criteria to monitor the convergence were used. Results revealed the proposed criteria to be adequate, besides being easy to implement.

On the Emmenthal distribution of highly inclined asteroids

Highly inclined asteroids are objects with sin (i) > 0.3. Among highly inclined asteroids, we can distinguish between objects with inclinations smaller than that of the centre of the ν6= g − g6 secular resonance and objects at higher inclinations. Using the current mechanisms of dynamical mobility, it is not easy to increase the values of an asteroid with an initial small inclination to values higher than that of the centre of the ν6 resonance. The presence of highly inclined objects might therefore be related to the early phases of the Solar system. It has been observed that several dynamically stable regions are characterized by a very low number density of objects, unlike low-inclined bodies that tend to occupy all the dynamically viable regions. The distribution of asteroids at a high inclination in the domain of proper elements in dynamically stable regions resembles an Emmenthal cheese, with regions of low number density close to highly populated areas. While this phenomenon has been observed qualitatively in the past, no quantitative study has yet been carried out on the extent and long-term stability of these regions. In this paper, we identify two dynamically stable regions characterized by very low values of number density and permanence times of 100 Myr or more when the Yarkovsky force is considered. We show that the low number density of objects in these areas cannot be produced as a statistical fluctuation of any simple one-dimensional statistical distribution, such as the Poissonian, uniform and Gaussian distributions, or of a tri-dimensional distribution, such as the tri-variate normal distribution. The presence of unoccupied dynamically stable regions could indicate that the primordial asteroidal population might not have reached all available zones at high-i. This sets constraints on the scenarios for the early phases of the history of our Solar system.

Identification of parameters and the distribution of the minimum of the tri-variate normal

In this note, for tri-variate normal distributions, we consider the problem of the identification of parameters when only the distribution of the minimum is known.

Regression analysis using order statistics

In this article, we study the joint distribution of X and two linear combinations of order statistics, aTY(2) and bTY(2), where a = (a1, a2)T and b = (b1, b2)T are arbitrary vectors in R2 and Y(2) = (Y(1), Y(2))T is a vector of ordered statistics obtained from (Y1, Y2)T when (X, Y1, Y2)T follows a trivariate normal distribution with a positive definite covariance matrix. We show that this distribution belongs to the skew-normal family and hence our work is a generalization of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995) and Loperfido (Test 17:370–380, 2008).