Multivariate Normal Distribution

Through the example of partial barrier options, we show that accuracy in the tail of the bivariate normal distribution is critical. We then propose a small change to a popular algorithm for the bivariate normal distribution in order to increase its accuracy.

Many spectral algorithms that are routinely applied to spectral imagery are based on the following models: statistical, linear mixture, and linear subspace. As a result, assumptions are made about the underlying distribution of the data such as multivariate normality or other geometric restrictions. Here we present a graph based model for spectral data that avoids these restrictive assumptions and apply graph based metrics to quantify certain aspects of the resulting graph. The construction of the spectral graph begins by connecting each pixel to its k-nearest neighbors with an undirected weighted edge. The weight of each edge corresponds to the spectral Euclidean distance between the adjacent pixels. The number of nearest neighbors, k, is chosen such that the graph is connected i.e., there is a path from each pixel x i to every other. This requirement ensures the existence of inter-cluster connections which will prove vital for our application to change detection. Once the graph is constructed, we calculate a metric called the Normalized Edge Volume (NEV) that describes the internal structural volume based on the vertex connectivity and weighted edges of the graph. Finally, we demonstrate a graph based change detection method that applies this metric.

Maximum multivariate cumulative sum (Max-MCUSUM) is one of the single control charts that plot single statistic as a representation of mean vector and covariance matrix. The Max-MCUSUM statistic has unknown specific distribution. The objective of this paper is to propose bootstrap-based Max-MCUSUM control chart for which reference value is predetermined in Phase I monitoring process. For various numbers of quality characteristics and correlation coefficients, the control limits estimated using bootstrap approach are presented in this paper. Furthermore, the average run lengths of bootstrap-based Max-MCUSUM control chart prove that the proposed control chart tends to be effective for monitoring the small shift in both mean and variance of a process. The illustrative examples are provided to demonstrate the applications of the proposed control chart for both simulation and real data.

An international portfolio allows simultaneous investment in both domestic and foreign markets. It hence has the potential for improved performance by exploiting a wider range of returns, and diversification benefits, than portfolios investing in just one market. However, to obtain the most efficient portfolios (along with the usual management of assets) the risks from currency fluctuations need good management, such as by using appropriate hedging. In this paper, we present a two-stage stochastic international portfolio optimisation model to find an optimal allocation for the combination of both assets and currency hedging positions. Our optimisation model allows a "currency overlay", or a deviation of currency exposure from asset exposure, to provide flexibility in hedging against, or in speculation using, currency exposure. The transaction costs associated with both trading and hedging are also included. To model the realistic dependence structure of the multivariate return distributions, a new scenario generation method, employing a regular-vine copula is developed. The use of vine copulas allows a better representation of the characteristics of returns, specifically, their non-normality and asymmetric dependencies. It hence improves the representation of the uncertainty underlying decisions needed for international portfolio optimisation problems. Efficient portfolios optimised with scenarios generated from the new vine-copula method are compared with the portfolios from a standard scenario generation method. Experimental results show that the proposed method, using realistic non-normal uncertainty, produces portfolios that give better risk-return reward than those from a standard scenario generation approach, using normal distributions. The difference in risk-return compensation is largest when the portfolios are constrained to require higher returns. The paper shows that it can be important to model the nonnormality in uncertainty, and not just assume normal distributions.

In this paper, a computational approach test (CAT) was proposed to test the equality of two multivariate normal mean vectors under heterogeneity of covariance matrices. The proposed test was compared with the other popular tests as well as their CAT versions in terms of estimated type I error rate and power. Simulation study shows that the proposed test and CAT versions of tests can be used as a good alternative test to test the equality of two multivariate normal mean vectors under heterogeneity of covariance matrices.

In this paper the authors are concerned with the application of conjoint measurement models to predict consumer choice of shopping centres. First, conjoint measurement models are discussed in the context of the development of spatial shopping-models. Next, the conceptual framework underlying the model and conjoint measurement are discussed. The second part of the paper describes an application of the methodology. Conjoint measurement is used to estimate consumer utility functions and a multivariate logit model as an approximation of the unit multivariate normal distribution is used to predict the probability that a consumer will choose a particular shopping centre. The results indicate that the methodology offers a potentially valuable approach to the modelling of spatial shopping-behaviour.

We examine the performance of a method of integrated population modelling for the joint analysis of different types of demographic data on individuals that exist in, and move between, different sites. The value of the approach is demonstrated by a simulation study which shows substantial improvement in parameter estimation when site-specific census data are combined with demographic data. The multivariate normal approximation to a multi-state mark-recapture likelihood is evaluated, and the performance of a diagonal variance-covariance matrix for the approximation is also examined. The work is motivated by a study of great cormorants. Analysis of the cormorant data suggests that breeders survive better than non-breeders, and also that probabilities of recruitment to breeding have been declining over time for all the colonies of the study. Supplementary material, including notes on the computation of standard errors and extended simulation results, are available online.

Lindley-Singpurwalla (1986)'s bivariate Pareto distribution is one of the most popular bivariate Pareto distribution. proposed a new bivariate Pareto distribution which also has Pareto marginals and it contains Lindley-Singpurwalla's bivariate Pareto model as a special case. It has several other interesting properties also. In this paper we re-visit Sankaran and Nair's bivariate Pareto model. We discuss several other new properties. The maximum likelihood estimators and two stage estimators are also investigated. We analyze two data sets for illustrative purposes. It is observed that this model can be used quite effectively to analyze competing risks data. Finally we propose some generalizations.

Let Z t = (Z1, . . . , Zp) be a p-variate Gaussian complex random variable. Let α = (n1, m1, . . . , np, mp) be a vector in N 2p and let ν(α) be the correspondent moment: We present conditions for nullity of the moment ν(α). Furthermore, we show the value of the non null moments for p = 1 and p = 2 and we conjecture the value for p > 2.

Challenges in evaluating nonlinear effects in multiple regression analyses include reliability, validity, multicollinearity, and dichotomization of continuous variables. While reliability and validity issues are solved by employing nonlinear structural equation modeling, multicollinearity remains a problem which may even be aggravated when using latent variable approaches. Further challenges of nonlinear latent analyses comprise the distribution of latent product terms, a problem especially relevant for approaches using maximum likelihood estimation methods based on multivariate normally distributed variables, and unbiased estimates of nonlinear effects under multicollinearity. The only methods that explicitly take the nonnormality of nonlinear latent models into account are latent moderated structural equations (LMS) and quasi-maximum likelihood (QML). In a small simulation study both methods yielded unbiased parameter estimates and correct estimates of standard errors for inferential statistics. The advantages and limitations of nonlinear structural equation modeling are discussed.

The purpose of this article is to account for informative sampling in fitting superpopulation model for multivariate observations, and in particular multivariate normal distribution, for longitudinal survey data. The idea behind the proposed approach is to extract the model holding for the sample data as a function of the model in the population and the first order inclusion probabilities, and then fit the sample model using maximum likelihood, pseudo maximum likelihood and estimating equations methods. As an application of the results, we fit the general linear model for longitudinal survey data under informative sampling using different covariance structures: the exponential correlation model, the uniform correlation model, and the random effect model, and using different conditional expectations of first order inclusion probabilities given the study variable. The main feature of the present estimators is their behaviours in terms of the informativeness parameters.

Multi-sensor data that track system operating behaviors are widely available nowadays from various engineering systems. Measurements from each sensor over time form a curve and can be viewed as functional data. Clustering of these multivariate functional curves is important for studying the operating patterns of systems. One complication in such applications is the possible presence of sensors whose data do not contain relevant information. Hence it is desirable for the clustering method to equip with an automatic sensor selection procedure. Motivated by a real engineering application, we propose a functional data clustering method that simultaneously removes noninformative sensors and groups functional curves into clusters using informative sensors. Functional principal component analysis is used to transform multivariate functional data into a coefficient matrix for data reduction. We then model the transformed data by a Gaussian mixture distribution to perform modelbased clustering with variable selection. Three types of penalties, the individual, variable, and group penalties, are considered to achieve automatic variable selection. Extensive simulations are conducted to assess the clustering and variable selection performance of the proposed methods. The application of the proposed methods to an engineering system with multiple sensors shows the promise of the methods and reveals interesting patterns in the sensor data.

Financial market prediction attracts immense interest among researchers nowadays due to rapid increase in the investments of financial markets in the last few decades. The stock market is one of the leading financial markets due to importance and interest of many stakeholders. With the development of machine learning techniques, the financial industry thrived with the enhancement of the forecasting ability. Probabilistic neural network (PNN) is a promising machine learning technique which can be used to forecast financial markets with a higher accuracy. A major limitation of PNN is the assumption of Gaussian distribution as the distribution of input variables which is violated with respect to financial data. The main objective of this study is to improve the standard PNN by incorporating a proper multivariate distribution as the joint distribution of input variables and addressing the multi-class imbalanced problem persisting in the directional prediction of the stock market. This m...

We consider maximum likelihood estimation with data from a bivariate Gaussian process with a separable exponential covariance model under fixed domain asymptotics. We first characterize the equivalence of Gaussian measures under this model. Then consistency and asymptotic normality for the maximum likelihood estimator of the microergodic parameters are established. A simulation study is presented in order to compare the finite sample behavior of the maximum likelihood estimator with the given asymptotic distribution.

Bootstrap methods are considered in the application of statistical process control because they can deal with unknown distributions and are easy to calculate using a personal computer. In this study we propose the use of bootstrap-t multivariate control technique on the minimax control chart. The technique takes care of correlated variables as well as the requirement of the distributional assumptions needed for the operation of the minimax control chart. The bootstrap-t technique provides the mean ˆB  of all the bootstrap estimators  is the estimate using the bootstrap sample and B is the number of bootstraps. The computation of the proposed bootstrap-t minimax statistic was performed on the values obtained from the bootstrap estimation. This method was used to determine the position of the four control limits of the minimax control chart. The bootstrap-t approach introduced to minimax multivariate control chart helps to detect shifts in the mean vector of a multivariate process and it overcomes the computational complexity of obtaining the distribution of multivariate data.

Minimax control chart uses the joint probability distribution of the maximum and minimum standardized sample means to obtain the control limits for monitoring purpose. However, the derivation of the joint probability distribution needed to obtain the minimax control limits is complex. In this paper the multivariate normal distribution is integrated numerically using Simpson's one third rule to obtain a non-linear polynomial (NLP) function. This NLP function is then substituted and solved numerically using Newton Raphson method to obtain the control limits for the minimax control chart. The approach helps to overcome the problem of obtaining the joint probability distribution needed for estimating the control limits of both the maximum and the minimum statistic for monitoring multivariate process.

We consider the problem of estimating the population probability distribution given a finite set of multivariate samples, using the maximum entropy approach. In strict keeping with Jaynes' original definition, our precise formulation of the problem considers contributions only from the smoothness of the estimated distribution (as measured by its entropy) and the loss functional associated with its goodness-of-fit to the sample data, and in particular does not make use of any additional constraints that cannot be justified from the sample data alone. By mapping the general multivariate problem to a tractable univariate one, we are able to write down exact expressions for the goodness-of-fit of an arbitrary multivariate distribution to any given set of samples using both the traditional likelihood-based approach and a rigorous information-theoretic approach, thus solving a long-standing problem. As a corollary we also give an exact solution to the 'forward problem' of determining the expected distributions of samples taken from a population with known probability distribution.

Le fait que la vraisemblance ne soit pas bornée dans les mélanges gaussiens est un handicap pratique et théorique. Utilisant la très faible hypothèse que chaque composante est d'effectif supérieur à la dimension de l'espace, nous proposons une borne aléatoire exacte très simple qui permet de contrôler les valeurs propres des matrices de covariances. Dans le cas univarié, l'estimateur du maximum de vraisemblance sous cette contrainte est convergeant, la preuve restant encore à établir dans le cas général. Cette stratégie est implémentée dans un algorithme EM et donne d'excellents résultats sur des données simulées.

The Simes inequality has received considerable attention recently because of its close connection to some important multiple hypothesis testing procedures. We revisit in this article an old result on this inequality to clarify and strengthen it and a recently proposed generalization of it to offer an alternative simpler proof.

The concept of False Discovery Rate (FDR), which is the expected proportion of false positives (Type I errors) among rejected hypotheses, has received increasing attention recently by researchers in multiple hypotheses testing. A similar measure involving false negatives (Type II errors), which we call the False Negatives Rate (FNR), is considered and an explicit formula of it is developed for a generalized step-up-step-down procedure in terms of probability distributions of ordered test statistics. It is shown that a step-down procedure can be used to control the FNR under certain conditions on the test statistics. Various FDR-controlling procedures that exist in a given multiple testing situation are further studied in terms of the FNR. In particular, an unbiasedness property is defined for an FDR-controlling multiple testing procedure by the inequality FDR + FNR ≤ 1. The FDR-controlling generalized step-up-step-down procedure considered in , in particular the Benjamini and Hochberg (1995) step-up procedure, is proved to be unbiased when the test statistics are independent and is conjectured to be so based on simulations when they are dependent. Also conjectured is the unbiasedness of the FDRcontrolling Benjamini and Liu (1999) step-down procedure with independent as well as dependent test statistics. Different FDR-controlling procedures are investigated based on simulated data from equi-correlated multivariate normals in terms of the quantity 1 -FNR -FDR that reflects the strength of unbiasedness.

In many applications of multiple hypothesis testing where more than one false rejection can be tolerated, procedures controlling error rates measuring at least k false rejections, instead of at least one, for some fixed k ≥ 1 can potentially increase the ability of a procedure to detect false null hypotheses. The k-FWER, a generalized version of the usual familywise error rate (FWER), is such an error rate that has recently been introduced in the literature and procedures controlling it have been proposed. A further generalization of a result on the k-FWER is provided in this article. In addition, an alternative and less conservative notion of error rate, the k-FDR, is introduced in the same spirit as the k-FWER by generalizing the usual false discovery rate (FDR). A k-FWER procedure is constructed given any set of increasing constants by utilizing the kth order joint null distributions of the p-values without assuming any specific form of dependence among all the p-values. Procedures controlling the k-FDR are also developed by using the kth order joint null distributions of the p-values, first assuming that the sets of null and nonnull p-values are mutually independent or they are jointly positively dependent in the sense of being multivariate totally positive of order two (MTP 2 ) and then discarding that assumption about the overall dependence among the p-values.

This paper shows an embedding of the manifold of multivariate normal densities with informative geometry into the manifold of definite positive matrices with the Siegel metric. This embedding allows us to obtain a general lower bound for the Rao distance, which is itself a distance, and we suggest employing it for statistical purposes, taking into account the similitude of the above related metrics. Furthermore, through this embedding, general statistical tests of hypothesis are derived, and some geometrical properties are studied too.

Given any mean zero, finite variance σ 2 random variable W , there exists a unique distribution on a variable W * such that EW f (W ) = σ 2 Ef (W * ) for all absolutely continuous functions f for which these expectations exist. This distributional 'zero bias' transformation of W to W * , of which the normal is the unique fixed point, was introduced in [9] to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.

Shrinkage estimation is a fundamental tool of modern statistics, pioneered by Charles Stein upon his discovery of the famous paradox involving the multivariate Gaussian. A large portion of the subsequent literature only considers the efficiency of shrinkage, and that of an associated procedure known as Stein's Unbiased Risk Estimate, or SURE, in the Gaussian setting of that original work. We investigate what extensions to the domain of validity of shrinkage and SURE can be made away from the Gaussian through the use of tools developed in the probabilistic area now known as Stein's method. We show that shrinkage is efficient away from the Gaussian under very mild conditions on the distribution of the noise. SURE is also proved to be adaptive under similar assumptions, and in particular in a way that retains the classical asymptotics of Pinsker's theorem. Notably, shrinkage and SURE are shown to be efficient under mild distributional assumptions, and particularly for general isotropic log-concave measures.

We consider the problem of finding a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown. Portnoy (2018) characterizes the scale-sign invariant rules and shows that the Hunt-Stein construction provides a randomized invariant rule that improves on any given randomized rule in the sense that it has greater minimal coverage among all procedures with a fixed expected length. Mathematical results here provide a specific mixture of two non-randomized invariant rules that achieve the minimax optimality. A multivariate confidence set based on a single observation vector is also developed.

Let CT,, x,), CT,, x2),..., CT,, XJ b e a sample from a multivariate normal distribution where r, are (unobservable) random variables and x, are random vectors in Rk. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {T,) is maximized by ranking according to the order of the best linear predictors (E(T,Ix,)}. Furthermore, it orderings are chosen according to linear functions {b'x,] then the conditional probability of correct order given (T, = t,; i = l,,.., n) is maximized when b'x, is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other than the best linear predictor may give a greater probability of correctly ordering (r,} if ((T,, xt)) are independent but not identically distributed, or if the distributions are not normal.

Under the assumption that neither the mean vector nor the variance- covariance matrix are known with certainty, the natural conjugate family of prior densities for the multivariate Normal process is identified. Prior-posterior and preposterior analysis is done assuming that the prior is in the natural conjugate family. A procedure is presented for obtaining non-degenerate joint posterior and preposterior distributions of all parameters even when the number of objective sample observations is less than the number of parameters of the process. 1 .

In this paper we introduce patchwork constructions for multivariate quasi-copulas. These results appear to be new since the kind of approach has been limited to either copulas or only bivariate quasi-copulas so far. It seems that the multivariate case is much more involved since we are able to prove that some of the known methods of bivariate constructions cannot be extended to higher dimensions. Our main result is to present the necessary and sufficient conditions both on the patch and the values of it for the desired multivariate quasi-copula to exist. We also give all possible solutions.

Abstact: A classification and grouping object into groups based on several factors often approached with diskriminan analysis. Process of grouping of discriminant analysis by build discriminant function of each group and count discriminant score of each object. Based on the characteristic of data, that are multivariate normal distribution, the mean between groups and variance covariance, it is discussed in the performance of the linear discriminant and quadratic analysis. Based on the result of analysis obtained for data with multivariate normal distribution, mean between groups significantly different and variance covariance homogeneous, misclassification of linear discriminant analysis is smaller than quadratic discriminant analysis, and otherwise for data with multivariate normal distribution, mean between groups significantly different and variance covariance not homogeneous, misclassification is smaller when using quadratic discriminant analysis than linear discriminant analysis.

Pairs trading under the copula approach is revisited in this paper. It is well known that financial returns arising from indices in markets may not follow the features of normal distribution and may exhibit asymmetry or fatter tails, in particular. Due to this, the Laplace distribution is employed in this work to fit the marginal distribution function, which will then be employed in a copula function. In fact, a multivariate copula function is constructed on two indices (based on the Laplace marginal distribution), enabling us to obtain the associated probabilities required for the process of pairs trade and creating an efficient tool for trading.

In this study, the multivariate gamma -gamma (G-G) distribution with exponential correlation is introduced and studied. Rapidly convergent infinite series representations are derived for the joint G -G probability density, cumulative distribution and moment generating functions. Based on these formulas, the performance of radio frequency wireless systems with diversity reception in the presence of composite multipath and shadow fading is analysed. Furthermore, the performance of single-input -multiple-output free space optical communication systems impaired by atmospheric turbulence is investigated. The correctness of the proposed analysis is demonstrated through various numerically evaluated results compared with equivalent computer simulation ones.

A very well-known traditional approach in discriminant analysis is to use some linear (or nonlinear) combination of measurement variables which can enhance class separability. For instance, a linear (or a quadratic) classifier finds the linear projection (or the quadratic function) of the measurement variables that will maximize the separation between the classes. These techniques are very useful in obtaining good lower dimensional view of class separability. Fisher's discriminant analysis, which is primarily motivated by the multivariate normal distribution, uses the first-and second-order moments of the training sample to build such classifiers. These estimates, however, are highly sensitive to outliers, and they are not reliable for heavy-tailed distributions. This paper investigates two distribution-free methods for linear classification, which are based on the notions of statistical depth functions. One of these classifiers is closely related to Tukey's half-space depth, while the other is based on the concept of regression depth. Both these methods can be generalized for constructing nonlinear surfaces to discriminate among competing classes. These depth-based methods assume some finite-dimensional parametric form of the discriminating surface and use the distributional geometry of the data cloud to build the classifier. We use a few simulated and real data sets to examine the performance of these discriminant analysis tools and study their asymptotic properties under appropriate regularity conditions.

In this paper we present a new multi-asset pricing model, which is built upon newly developed families of solvable multi-parameter single-asset diffusions with a nonlinear smile-shaped volatility and an affine drift. Our multi-asset pricing model arises by employing copula methods. In particular, all discounted single-asset price processes are modeled as martingale diffusions under a risk-neutral measure. The price processes are so-called UOU diffusions and they are each generated by combining a variable (Itô) transformation with a measure change performed on an underlying Ornstein-Uhlenbeck (Gaussian) process. Consequently, we exploit the use of a normal bridge copula for coupling the single-asset dynamics while reducing the distribution of the multi-asset price process to a multivariate normal distribution. Such an approach allows us to simulate multidimensional price paths in a precise and fast manner and hence to price path-dependent financial derivatives such as Asian-style and Bermudan options using the Monte Carlo method. We also demonstrate how to successfully calibrate our multi-asset pricing model by fitting respective equity option and asset market prices to the single-asset models and their return correlations (i.e. the copula function) using the least-square and maximum-likelihood estimation methods.

We present new extensions to a method for constructing several families of solvable one-dimensional time-homogeneous diffusions whose transition densities are obtainable in analytically closed-form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable diffusions that are generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families.

A very well-known traditional approach in discriminant analysis is to use some linear (or nonlinear) combination of measurement variables which can enhance class separability. For instance, a linear (or a quadratic) classifier finds the linear projection (or the quadratic function) of the measurement variables that will maximize the separation between the classes. These techniques are very useful in obtaining good lower dimensional view of class separability. Fisher's discriminant analysis, which is primarily motivated by the multivariate normal distribution, uses the first-and second-order moments of the training sample to build such classifiers. These estimates, however, are highly sensitive to outliers, and they are not reliable for heavy-tailed distributions. This paper investigates two distribution-free methods for linear classification, which are based on the notions of statistical depth functions. One of these classifiers is closely related to Tukey's half-space depth, while the other is based on the concept of regression depth. Both these methods can be generalized for constructing nonlinear surfaces to discriminate among competing classes. These depth-based methods assume some finite-dimensional parametric form of the discriminating surface and use the distributional geometry of the data cloud to build the classifier. We use a few simulated and real data sets to examine the performance of these discriminant analysis tools and study their asymptotic properties under appropriate regularity conditions.

In studies of morphology, methods for comparing amounts of variability are often important. Three different ways of utilizing determinants of covariance matrices for testing for surplus variability in a hypothesis sample compared to a reference sample are presented: an F-test based on standardized generalized variances, a parametric bootstrap based on draws on Wishart matrices, and a nonparametric bootstrap. The F-test based on standardized generalized variances and the Wishart-based bootstrap are applicable when multivariate normality can be assumed. These methods can be applied with only summary data available. However, the nonparametric bootstrap can be applied with multivariate nonnormally distributed data as well as multivariate normally distributed data, and small sample sizes. Therefore, this method is preferable when raw data are available. Three craniometric samples are used to present the methods. A Hungarian Zalava ´r sample and an Austrian Berg sample are compared to a Norwegian Oslo sample, the latter employed as reference sample. In agreement with a previous study, it is shown that the Zalava ´r sample does not represent surplus variability, whereas the Berg sample does represent such a surplus variability.

Comparing two measurement methods is vital in various fields, such as medical research, epidemiology, economics, and environmental studies, to determine whether a new measurement method can be used interchangeably with an existing one. Measurement error models (MEMs) are commonly used for this purpose, where the methods have different measuring scales. However, these models often assume normality, which can be problematic when dealing with skewed and heavy-tailed data. To address this issue, we propose the replicated measurement error model (RMEM) under scale mixtures of skew-normal (SMSN) distributions with different levels of skewness and heavy tails of the true covariate and error distributions. Our primary aim is to assess the extent of similarity and agreement between two measurement methods using this model. The expectation conditional maximization (ECM) approach is applied to fit the model. A simulation study is conducted to evaluate the effectiveness and robustness of the proposed methodology and is illustrated by analyzing systolic blood pressure data. The probability of agreement is used further to assess the agreement between the two measurement methods. The findings indicate that the proposed model effectively analyses replicated method comparison data with measurement errors, even when there are outliers, skewness, and heavy-tailedness.

Malaria research and mathematical models have mainly concentrated on malaria Plasmodium at the blood stage. This has left many questions concerning models of parasite dynamics in the liver and within the mosquito. These concerns are anticipated to keep scientists busy trying to understand the biology of the parasite for some more years to come. Thorough knowledge of parasite biology helps in designing appropriate drugs targeting particular stages of Plasmodium. To achieve this, there is need to study the transmission dynamics of malaria and the interaction between the infection in the liver, blood and mosquito using a mathematical model. In this study, a within-host mathematical model is proposed and considers the dynamics of P. falciparum malaria from the liver to the blood in the human host and then to the mosquito. Several techniques, including center manifold theory and sensitivity analysis are used to understand relevant features of the model dynamics like basic reproduction number, local and global stability of the disease-free equilibrium and conditions for existence of the endemic equilibrium. Results indicate that the infection rate of merozoites, the rate of sexual reproduction in gametocytes, burst size of both hepatocytes and erythrocytes are more sensitive parameters for the onset of the disease. However, a treatment strategy using highly effective drugs against such parameters can reduce on malaria progression and control the disease. Numerical simulations show that drugs with an efficacy above 90% boost healthy cells, reduce infected cells and clear parasites in human host. Therefore more needs to be done such as research in parasite biology and using highly effective drugs for treatment of malaria.

High-dimensional data, characterized by
datasets with many variables (dimension) relative to the
number of observations, is growing in prominence owing
to advances in data collection and storage capabilities. This
data type is widespread across various fields. While it
presents unique challenges and opportunities, robust
statistical approaches are crucial for effectively leveraging
the potential of high-dimensional data. In multivariate
statistical analysis, the likelihood ratio test (LRT) is
frequently used for evaluating the equality of two
covariance matrices. Nevertheless, the LRT lacks
definition in high-dimensional data contexts. This paper
aims to introduce a new test for ascertaining the equality of
two covariance matrices in high-dimensional data,
especially for datasets that follow a multivariate normal
distribution. The test (Tnew) proposed in this study utilizes
consistent estimators in quadratic and symmetric bilinear
forms. As the dimension and sample sizes approach infinity,
the asymptotic null distribution of the test converges to the
standard normal distribution. A simulation study was
conducted to assess the performance of the proposed test
compared to three existing tests. The existing tests were
proposed by Schott in 2007, Srivastava and Yanagihara in
2010, and Li and Chen in 2012. The focus was on type I
error rates and the test’s power under spherical and Toeplitz
covariance matrix structures. The simulation results
demonstrate that Tnew outperforms the tests proposed by
Srivastava and Yanagihara, as well as Li and Chen, in all
scenarios evaluated. Moreover, it performs comparably to
Schott’s test. Additionally, Tnew demonstrates remarkable
stability even when faced with alterations in the covariance
matrix structure. This robust performance of Tnew suggests
that it can serve as a reliable tool for statistical inference in
multivariate analyses, especially in high-dimensional
contexts. For practitioners, the use of the proposed test
could mean more accurate decision-making in scientific
research and policymaking, where precision and reliability
are paramount.

Transformed empirical processes (TEPs) have been used by the authors in a previous paper to construct consistent and selectively efficient goodness-of-fit tests of the Kolmogorov-Smirnov type. A straightforward application of the same ideas to the construction of tests of the Cramér-von Mises type with the same properties leads to cumbersome computations. This short note exhibits some of the inconvenients encountered, and introduces a new family of quadratic statistics of the Cramér-von Mises type, in order to circumvent the difficulties.

We now turn attention to statistical models in which the family F of possible pdfs for the observable X ∈ X are a k-dimensional parametric family F = {f(x | θ) : θ ∈ Θ} for some parameter space Θ ⊆ Rk and function f : X × Θ → R+. Examples include the Poisson distribution Po(θ) with Θ = R+ and the Be(α, β) distribution with θ = (α, β) ∈ Θ ⊂ R 2 +. Other examples include the univariate normal distribution No(μ, σ), with k = 2 and Θ = R×R+ with θ = (μ, σ ), and the pdimensional multivariate normal distribution No(μ,Σ) with k = p(p+ 3)/2-dimensional parameter θ = (μ,Σ), with mean vector μ ∈ Rp and p× p positive-definite covariance matrix Σ ∈ S +.

We now turn attention to statistical models in which the family F of possible pdfs for the observable X ∈ X are a k-dimensional parametric family F = {f(x | θ) : θ ∈ Θ} for some parameter space Θ ⊆ Rk and function f : X × Θ → R+. Examples include the Poisson distribution Po(θ) with Θ = R+ and the Be(α, β) distribution with θ = (α, β) ∈ Θ ⊂ R 2 +. Other examples include the univariate normal distribution No(μ, σ), with k = 2 and Θ = R×R+ with θ = (μ, σ ), and the pdimensional multivariate normal distribution No(μ,Σ) with k = p(p+ 3)/2-dimensional parameter θ = (μ,Σ), with mean vector μ ∈ Rp and p× p positive-definite covariance matrix Σ ∈ S +.