Study of the bivariate normal distribution raises the full range of issues involving objective Bayesian inference, including the different types of objective priors (e.g., Jeffreys, invariant, reference, matching), the different modes of inference (e.g., Bayesian, frequentist, fiducial) and the criteria involved in deciding on optimal objective priors (e.g., ease of computation, frequentist performance, marginalization paradoxes). Summary recommendations as to optimal objective priors are made for a variety of inferences involving the bivariate normal distribution. In the course of the investigation, a variety of surprising results were found, including the availability of objective priors that yield exact frequentist inferences for many functions of the bivariate normal parameters, including the correlation coefficient.
The bivariate normal density with unit variance and correlation $ρ$ is well-known. We show that by integrating out $ρ$, the result is a function of the maximum norm. The Bayesian interpretation of this result is that if we put a uniform prior over $ρ$, then the marginal bivariate density depends only on the maximal magnitude of the variables. The square-shaped isodensity contour of this resulting marginal bivariate density can also be regarded as the equally-weighted mixture of bivariate normal distributions over all possible correlation coefficients. This density links to the Khintchine mixtu…
This paper develops some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, and the ratio of the conditional variance of one variable given the other to the marginal variance of the other variable. The criterion used is the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. The paper uses various matching criteria, namely, quantile matching, matching of distribution functions, highest posterior densit…
One major drawback of state-of-the-art artificial intelligence is its lack of explainability. One approach to solve the problem is taking causality into account. Causal mechanisms can be described by structural causal models. In this work, we propose a method for estimating bivariate structural causal models using a combination of normalising flows applied to density estimation and variational Gaussian process regression for post-nonlinear models. It facilitates causal discovery, i.e. distinguishing cause and effect, by either the independence of cause and residual or a likelihood ratio test. …
This paper provides two different novel approaches of slice sampling to estimate the parameters of absolute continuous Marshall-Olkin bivariate Pareto distribution with location and scale parameters. We carry out the bayesian analysis taking gamma prior for shape and scale parameters and truncated normal for location parameters. Credible intervals and coverage probabilities are also provided for all methods. A real-life data analysis is shown for illustrative purpose.
Statistical analyses of directional or angular data have applications in a variety of fields, such as geology, meteorology and bioinformatics. There is substantial literature on descriptive and inferential techniques for univariate angular data, with the bivariate (or more generally, multivariate) cases receiving more attention in recent years. More specifically, the bivariate wrapped normal, von Mises sine and von Mises cosine distributions, and mixtures thereof, have been proposed for practical use. However, there is a lack of software implementing these distributions and the associated infe…
We derive the rate of decay of the tail dependence of the bivariate skew normal distribution under the equal-skewness condition θ1 = θ2,= θ, say. The rate of convergence depends on whether θ > 0 or θ < 0. The latter case gives rate asymp- totically identical with the case θ = 0. The asymptotic behaviour of the quantile function for the univariate skew normal is part of the theoretical development.
A random variable is equi-dispersed if its mean equals its variance. A Poisson distribution is a classical example of this phenomenon. However, a less well-known fact is that the class of normal densities that are equi-dispersed constitutes a one parameter exponential family. In the present article our main focus is on univariate and bivariate models with equi-dispersed normal component distributions. We discuss distributional features of such models, explore inferential aspects and include an example of application of equi-dispersed models. Some related models are discused in Appendices.
Bivariate meta-analysis provides a useful framework for combining information across related studies and has been utilised to combine evidence from clinical studies to evaluate treatment efficacy on two outcomes. It has also been used to investigate surrogacy patterns between treatment effects on the surrogate endpoint and the final outcome. Surrogate endpoints play an important role in drug development when they can be used to measure treatment effect early compared to the final outcome and to predict clinical benefit or harm. The standard bivariate meta-analytic approach models the observed …
Model-based geostatistical design involves the selection of locations to collect data to minimise an expected loss function over a set of all possible locations. The loss function is specified to reflect the aim of data collection, which, for geostatistical studies, could be to minimise the prediction uncertainty at unobserved locations. In this paper, we propose a new approach to design such studies via a loss function derived through considering the entropy about the model predictions and the parameters of the model. The approach also includes a multivariate extension to generalised linear s…
Forecast ensembles are typically employed to account for prediction uncertainties in numerical weather prediction models. However, ensembles often exhibit biases and dispersion errors, thus they require statistical post-processing to improve their predictive performance. Two popular univariate post-processing models are the Bayesian model averaging (BMA) and the ensemble model output statistics (EMOS). In the last few years increased interest has emerged in developing multivariate post-processing models, incorporating dependencies between weather quantities, such as for example a bivariate dis…
The modelling of empirically observed data is commonly done using mixtures of probability distributions. In order to model angular data, directional probability distributions such as the bivariate von Mises (BVM) is typically used. The critical task involved in mixture modelling is to determine the optimal number of component probability distributions. We employ the Bayesian information-theoretic principle of minimum message length (MML) to distinguish mixture models by balancing the trade-off between the model's complexity and its goodness-of-fit to the data. We consider the problem of modell…
In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi (1969). This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post-model-selection or post-shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian lo…
The problem of estimating location (scale) parameters $θ_1$ and $θ_2$ of two distributions when the ordering between them is known apriori (say, $θ_1\leq θ_2$) has been extensively studied in the literature. Many of these studies are centered around deriving estimators that dominate the best location (scale) equivariant estimators, for the unrestricted case, by exploiting the prior information that $θ_1 \leq θ_2$. Several of these studies consider specific distributions such that the associated random variables are statistically independent. This paper considers a general bivariate model and g…
We derive the asymptotic rate of decay to zero of the tail dependence of the bivariate skew Variance Gamma (VG) distribution under the equal-skewness condition, as an explicit regularly varying function. Our development is in terms of a slightly more general bivariate skew Generalized Hyperbolic (GH) distribution. Our initial reduction of the bivariate problem to a univariate one is motivated by our earlier study of tail dependence rate for the bivariate skew normal distribution
Pearson's is the most common correlation statistic, used mainly in parametric settings. Most common among nonparametric correlation statistics are Spearman's and Kendall's. We show that for bivariate normal i.i.d. samples the pairwise asymptotic relative efficiency between these three statistics depends monotonically on the population correlation coefficient. This monotonicity is a corollary to a stronger result. The proofs rely on the use of l'Hospital-type rules for monotonicity patterns.
In this work, we derive some novel properties of the bimodal normal distribution. Some of its mathematical properties are examined. We provide a formal proof for the bimodality and assess identifiability. We then discuss the maximum likelihood estimates as well as the existence of these estimates, and also some asymptotic properties of the estimator of the parameter that controls the bimodality. A bivariate version of the BN distribution is derived and some characteristics such as covariance and correlation are analyzed. We study stationarity and ergodicity and a triangular array central limit…
This paper studies circular correlations for the bivariate von Mises sine and cosine distributions. These are two simple and appealing models for bivariate angular data with five parameters each that have interpretations comparable to those in the ordinary bivariate normal model. However, the variability and association of the angle pairs cannot be easily deduced from the model parameters unlike the bivariate normal. Thus to compute such summary measures, tools from circular statistics are needed. We derive analytic expressions and study the properties of the Jammalamadaka-Sarma and Fisher-Lee…
We perform a Bayesian analysis of the p-variate skew-t model, providing a new parameterization, a set of non-informative priors and a sampler specifically designed to explore the posterior density of the model parameters. Extensions, such as the multivariate regression model with skewed errors and the stochastic frontiers model, are easily accommodated. A novelty introduced in the paper is given by the extension of the bivariate skew-normal model given in Liseo & Parisi (2013) to a more realistic p-variate skew-t model. We also introduce the R package mvst, which allows to estimate the multiva…
We consider component-wise equivariant estimation of order restricted location/scale parameters of a general bivariate distribution under quite general conditions on underlying distributions and the loss function. This paper unifies various results in the literature dealing with sufficient conditions for finding improvments over arbitrary location/scale equivariant estimators. The usefulness of these results is illustrated through various examples. A simulation study is considered to compare risk performances of various estimators under bivariate normal and independent gamma probability models…
The present paper is a sequel to and generalization of Fung and Seneta (2016) whose main result gives the asymptotic behaviour as $ u \to 0^{+}$ of $λ_L(u) = P(X_1 \leq F_1^{-1}(u) | X_2 \leq F_2^{-1}(u)),$ when $\bf{X} \sim SN_2(\boldsymbolα, R)$ with $α_1 = α_2 = α,$ that is: for the bivariate skew normal distribution in the equi-skew case, where $R$ is the correlation matrix, with off-diagonal entries $ρ,$ and $F_i(x), i=1,2$ are the marginal cdf's of $\textbf{X}$. A paper of Beranger et al. (2017) enunciates an upper-tail version which does not contain the constraint $α_1=α_2= α$ but requi…
We introduce a multivariate hidden Markov model to jointly cluster time-series observations with different support, i.e. circular and linear. Relying on the general projected normal distribution, our approach allows for bimodal and/or skewed cluster-specific distributions for the circular variable. Furthermore, we relax the independence assumption between the circular and linear components observed at the same time. Such an assumption is generally used to alleviate the computational burden involved in the parameter estimation step, but it is hard to justify in empirical applications. We carry …
A random phenomenon may have two sources of random variation: an unstable identity and a set of external variation-generating factors. When only a single source is active, two mutually exclusive extreme scenarios may ensue that result in the exponential or the normal, the only truly univariate distributions. All other supposedly univariate random variation observed in nature is truly bivariate. In this article, we elaborate on this new paradigm for random variation and develop a general bivariate distribution to reflect it. It is shown that numerous current univariate distributions are special…
We study the problem of maximum likelihood estimation for general patterns of bivariate missing data for normal and multinomial random variables, under the assumption that the data is missing at random (MAR). For normal data, the score equations have nine complex solutions, at least one of which is real and statistically significant. Our computations suggest that the number of real solutions is related to whether or not the MAR assumption is satisfied. In the multinomial case, all solutions to the score equations are real and the number of real solutions grows exponentially in the number of st…
In brain oncology, it is routine to evaluate the progress or remission of the disease based on the differences between a pre-treatment and a post-treatment Positron Emission Tomography (PET) scan. Background adjustment is necessary to reduce confounding by tissue-dependent changes not related to the disease. When modeling the voxel intensities for the two scans as a bivariate Gaussian mixture, background adjustment translates into standardizing the mixture at each voxel, while tumor lesions present themselves as outliers to be detected. In this paper, we address the question of how to standard…
Consider a continuous random pair $(X,Y)$ whose dependence is characterized by an extreme-value copula with Pickands dependence function $A$. When the marginal distributions of $X$ and $Y$ are known, several consistent estimators of $A$ are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and Capéraà, Fougères and Genest [Biometrika 84 (1997) 567--577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of $X$ and $Y$ are unknown. Results on the limit behavior o…
For the univariate current status and, more generally, the interval censoring model, distribution theory has been developed for the maximum likelihood estimator (MLE) and smoothed maximum likelihood estimator (SMLE) of the unknown distribution function, see, e.g., [12], [7], [4], [5], [6], [10], [11] and [8]. For the bivariate current status and interval censoring models distribution theory of this type is still absent and even the rate at which we can expect reasonable estimators to converge is unknown. We define a purely discrete plug-in estimator of the distribution function which locally c…
Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of the form $P(Y\leq y \mid X >t) $ is challenging when $t$ is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the random vector $(X,Y)$, suitably normalized, given that $X$ is large. There already exists a wide literature on bivariate models for which this limiting distribution exists. In this paper, a statistical analysis of this problem is done. Estimators of the limiting distribution (which is assumed to exist) and the normalizing functions are provided, as …
We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function $h(u)$ as $u\to 0^+$ or $1^-$. This is focussed on important univariate distributions when $h(\cdot)$ has no simple closed form, with a view to assessing asymptotic rate of decay to zero of tail dependence in the context of bivariate copulas. The Introduction motivates the study in terms of the standard Normal. The Normal, Skew-Normal and Gamma are used as initial examples. Finally, we discuss approximation to the lower quantile of the Variance-…
Since the two seminal papers by Fisher (1915, 1921) were published, the test under a fixed value correlation coefficient null hypothesis for the bivariate normal distribution constitutes an important statistical problem. In the framework of asymptotic robust statistics, it remains being a topic of great interest to be investigated. For this and other tests, focused on paired correlated normal random samples, Rényi's pseudodistance estimators are proposed, their asymptotic distribution is established and an iterative algorithm is provided for their computation. From them the Wald-type test stat…
The main objective of this paper consists in creating a new class of copulae from various joint distributions occurring in connection with certain Brownian motion processes. We focus our attention on the distributions of univariate Brownian motions having a drift parameter and their maxima and on correlated bivariate Brownian motions by considering the maximum value of one of them. The copulae generated therefrom and their associated density functions are explicitly given as well as graphically represented.
In the context of large-scale multiple testing, hypotheses are often accompanied with certain prior information. In this paper, we present a single-index modulated (SIM) multiple testing procedure, which maintains control of the false discovery rate while incorporating prior information, by assuming the availability of a bivariate $p$-value, $(p_1,p_2)$, for each hypothesis, where $p_1$ is a preliminary $p$-value from prior information and $p_2$ is the primary $p$-value for the ultimate analysis. To find the optimal rejection region for the bivariate $p$-value, we propose a criteria based on t…
Motivated by the problem of testing tetrad constraints in factor analysis, we study the large-sample distribution of Wald statistics at parameter points at which the gradient of the tested constraint vanishes. When based on an asymptotically normal estimator, the Wald statistic converges to a rational function of a normal random vector. The rational function is determined by a homogeneous polynomial and a covariance matrix. For quadratic forms and bivariate monomials of arbitrary degree, we show unexpected relationships to chi-square distributions that explain conservative behavior of certain …
