Extending the rank likelihood for semiparametric copula estimation

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

First we show that all genetic algorithms can be approximated by an algorithm which keeps the population in linkage equilibrium. Here the genetic population is given as a product of univariate marginal distributions. We describe a simple algorithm which keeps the population in linkage equilibrium. It is called the univariate marginal distribution algorithm (UMDA). Our main result is that UMDA transforms the discrete optimization problem into a continuous one defined by the average fitness W(p1, . . . , p n ) as a function of the univariate marginal distributions p i. For proportionate selection UMDA performs gradient ascent in the landscape defined by W(p). We derive a difference equation for p i which has already been proposed by Wright in population genetics. We show that UMDA solves difficult multimodal optimization problems. For functions with highly correlated variables UMDA has to be extended. The factorized distribution algorithm (FDA) uses a factorization into marginal and conditional distributions. For decomposable functions the optimal factorization can be explicitly computed. In general it has to be computed from the data. This is done by LFDA. It uses a Bayesian network to represent the distribution. Computing the network structure from the data is called learning in Bayesian network theory. The problem of finding a minimal structure which explains the data is discussed in detail. It is shown that the Bayesian information criterion is a good score for this problem.

Motivated by challenges in the analysis of biomedical data and observational studies, we develop statistical boosting for the general class of bivariate distributional copula regression with arbitrary marginal distributions, which is suited for binary, count, continuous or mixed outcomes. To arrive at a flexible model for the entire conditional distribution, not only the marginal distribution parameters but also the copula parameters are related to covariates through additive predictors. We suggest estimation by means of an adapted component-wise gradient boosting algorithm. A key benefit of boosting as opposed to classical likelihood or Bayesian estimation is the implicit data-driven variable selection mechanism as well as shrinkage. To the best of our knowledge, our implementation is the only one that combines a wide range of covariate effects, marginal distributions, copula functions, and implicit data-driven variable selection. We showcase the versatility of our approach to data from genetic epidemiology, healthcare utilization and childhood undernutrition. Our developments are implemented in the R package gamboostLSS, fostering transparent and reproducible research.

ABSTRACTA simple but efficient approach is proposed in this paper to construct the isotropic random field in (d ⩾ 2), whose univariate marginal distributions may be taken as any infinitely divisible distribution with finite variance. The three building blocks in our building tool box are a second-order Lévy process on the real line, a d-variate random vector uniformly distributed on the unit sphere, and a positive random variable that generates a Pólya-type function. The approach extends readily to the multivariate case and results in a vector random field in with isotropic direct covariance functions and with any specified infinitely divisible marginal distributions. A characterization of the turning bands simulation feature is also derived for the covariance matrix function of a Gaussian or elliptically contoured random field that is isotropic and mean square continuous in .

Left-censored linear regression models are quite popular models and have been deeply considered in the last three decades. In this paper, we consider a completely Bayesian approach for making a new Markov chain Monte Carlo (MCMC) algorithm with tractable full posteriors. Simulated consequences and real data analyses depict that the new Markov chain Monte Carlo algorithm has excellent mixing property and carry out very well than the present methods based on prediction accuracy.

BackgroundGait kinematics measured during equine gait analysis are typically evaluated by analysing (asymmetry‐based) discrete variables (eg, peak values) obtained from continuous kinematic signals (eg, timeseries of datapoints). However, when used for the assessment of complex cases of lameness, such as bilateral lameness, discrete variable analysis might overlook relevant functional adaptations.ObjectivesThe overall aim of this paper is to compare continuous and discrete data analysis techniques to evaluate kinematic gait adaptations to lameness.Study designMethod comparison.MethodsSixteen healthy Shetland ponies, enrolled in a research programme in which osteochondral defects were created on the medial trochlear ridges of both femurs, were used in this study. Kinematic data were collected at trot on a treadmill before and at 3 and 6 months after surgical intervention. Statistical parametric mapping and linear mixed models were used to compare kinematic variables between and within timepoints.ResultsBoth continuous and discrete data analyses identified changes in pelvis and forelimb kinematics. Discrete data analyses showed significant changes in hindlimb and back kinematics, where such differences were not found to be significant by continuous data analysis. In contrast, continuous data analysis provided additional information on the timing and duration of the differences found.Main limitationsA limited number of ponies were included.ConclusionsThe use of continuous data provides additional information regarding gait adaptations to bilateral lameness that is complementary to the analysis of discrete variables. The main advantage lies in the additional information regarding time dependence and duration of adaptations, which offers the opportunity to identify functional adaptations during all phases of the stride cycle, not just the events related to peak values.

Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function

Simultaneously testing a collection of null hypotheses about a data generating distribution based on a sample of independent and identically distributed observations is a fundamental and important statistical problem involving many applications. Methods based on marginal null distributions (i.e., marginal p-values) are attractive since the marginal p-values can be based on a user supplied choice of marginal null distributions and they are computationally trivial, but they, by necessity, are known to either be conservative or to rely on assumptions about the dependence structure between the test-statistics. Re-sampling based multiple testing (Westfall and Young, 1993) involves sampling from a joint null distribution of the test-statistics, and controlling (possibly in a, for example, step-down fashion) the user supplied type-I error rate under this joint null distribution for the test-statistics. A generally asymptotically valid null distribution avoiding the need for the subset pivotality condition for the vector of test-statistics was proposed in Pollard, van der Laan (2003) for null hypotheses about general real valued parameters. This null distribution was generalized in Dudoit, vanderLaan, Pollard (2004) to general null hypotheses and test-statistics. In ongoing recent work van der Laan, Hubbard (2005), we propose a new generally asymptotically valid null distribution for the test-statistics and a corresponding bootstrap estimate, whose marginal distributions are user supplied, and can thus be set equal to the (most powerful) marginal null distributions one would use in univariate testing to obtain a p-value. Previous proposed null distributions either relied on a restrictive subset pivotality condition (Westfall and Young) or did not guarantee this latter property (Dudoit, vanderLaan, Pollard, 2004). It is argued and illustrated that the resulting new re-sampling based multiple testing methods provide more accurate control of the wished Type-I error in finite samples and are more powerful. We establish formal results and investigate the practical performance of this methodology in a simulation and data analysis.

Improving Upon the Marginal Empirical Distribution Function When the Copula is Known

Suppose k-variate data are drawn from a mixture of two distributions, each having independent components. It is desired to estimate the univariate marginal distributions in each of the products, as well as the mixing proportion. This is the setting of two-class, fully parametrized latent models that has been proposed for estimating the distributions of medical test results when disease status is unavailable. The problem is one of inference in a mixture of distributions without training data, and until now it has been tackled only in a fully parametric setting. We investigate the possibility of using nonparametric methods. Of course, when k=1 the problem is not identifiable from a nonparametric viewpoint. We show that the problem is "almost" identifiable when k=2; there, the set of all possible representations can be expressed, in terms of any one of those representations, as a two-parameter family. Furthermore, it is proved that when $k\geq3$ the problem is nonparametrically identifiable under particularly mild regularity conditions. In this case we introduce root-n consistent nonparametric estimators of the 2k univariate marginal distributions and the mixing proportion. Finite-sample and asymptotic properties of the estimators are described.

In recent years, there has been an increasing interest in developing dose finding methods incorporating both efficacy and toxicity outcomes. It is reasonable to assume that efficacy and toxicity are associated; therefore, we need to model their stochastic dependence. Copula functions are very useful tools to model different kinds of dependence with arbitrary marginal distributions. We consider a binary efficacy-toxicity response with logit marginal distributions. Since the dose which maximizes the probability of efficacy without toxicity (P-optimal dose) changes depending on different copula functions, we propose a criterion which is useful for choosing between the rival copula models but also protects patients against doses that are far away from the P-optimal dose. The performance of this compromise criterion (called PKL) is illustrated for different choices of the parameter values.

Markov chainMonteCarlo (MCMC)methodshavebeenaround for almost as longasMonte Carlo techniques, even though their impact on statistics was not truly felt until the very early 1990s, except in the specialized fields of spatial statistics and image analysis, where those methods appeared earlier. The emergence of Markov based techniques in physics is a story that remains untold within this survey (see Landau and Binder, 2005). Also, we will not enter into a description of MCMC techniques, unless they have some historical link, as the remainder of this volume covers the technical aspects.Acomprehensive treatment with further references can also be found in Robert and Casella (2004). We will distinguish between the introduction of Metropolis-Hastings based algorithmsand those related to Gibbs sampling, since they each stem from radically different origins, even though their mathematical justification via Markov chain theory is the same. Tracing the development of Monte Carlo methods, we will also briefly mention what we might call the “second-generation MCMC revolution.” Starting in the mid to late 1990s, this includes the development of particle filters, reversible jump and perfect sampling, and concludes with more current work on population or sequential Monte Carlo and regeneration and the computing of “honest” standard errors. As mentioned above, the realization that Markov chains could be used in a wide varietyof situations only came (tomainstream statisticians)withGelfand and Smith (1990), despite earlier publications in the statistical literature such as Hastings (1970), Geman and Geman (1984), and Tanner and Wong (1987). Several reasons can be advanced: lack of computing machinery (think of the computers of 1970!), or background onMarkov chains, or hesitation to trust in the practicality of themethod. It thus required visionary researchers like Gelfand and Smith to convince the community, supported by papers that demonstrated, through a series of applications, that the method was easy to understand, easy to implement and practical (Gelfand et al., 1990, 1992; Smith and Gelfand, 1992; Wakefield et al., 1994). The rapid emergence of the dedicated BUGS (Bayesian inference using Gibbs sampling) software as early as 1991, when a paper on BUGS was presented at the Valencia meeting, was another compelling argument for adopting, at large, MCMC algorithms.∗Monte Carlo methods were born in Los Alamos, New Mexico, during World War II, eventually resulting in theMetropolis algorithm in the early 1950s. WhileMonte Carlo methods were in use by that time, MCMC was brought closer to statistical practicality by the work of Hastings in the 1970s. What can be reasonably seen as the first MCMC algorithm is what we now call theMetropolis algorithm, published by Metropolis et al. (1953). It emanates from the same group of scientists who produced the Monte Carlo method, namely the research scientists of Los Alamos, mostly physicists working on mathematical physics and the atomic bomb. MCMC algorithms therefore date back to the same time as the development of regular(MC only) Monte Carlo methods, which are usually traced to Ulam and von Neumann in the late 1940s. StanislawUlam associates the original ideawith an intractable combinatorial computation he attempted in 1946 (calculating the probability of winning at the solitaire card game). This ideawas enthusiastically adopted by John vonNeumann for implementationwith direct applications to neutron diffusion, the name “Monte Carlo” being suggested by Nicholas Metropolis. Eckhardt (1987) describes these early Monte Carlo developments, and Hitchcock (2003) gives a brief history of the Metropolis algorithm. These occurrences very closely coincide with the appearance of the very first general-purpose digital computer, the ENIAC,which came to life in February 1946, after three years of construction. The Monte Carlo method was set up by von Neumann, who was using it on thermonuclear and fission problems as early as 1947. That same year, Ulam and vonNeumann invented inversion andaccept-reject techniques (also recounted inEckhardt, 1987) to simulate from nonuniform distributions. Without computers, a rudimentary version invented by Fermi in the 1930s went unrecognized (Metropolis, 1987). Note also that, as early as 1949, a symposiumonMonteCarlowas supported byRand, theNational Bureau of Standards, and theOakRidge laboratory and thatMetropolis andUlam (1949) published the very first paper about the Monte Carlo method.

Model selection for discrete regular vine copulas

In survival analyses, infections at the catheter insertion site among kidney patients using portable dialysis machines pose a significant concern. Understanding the bivariate infection recurrence process is crucial for healthcare professionals to make informed decisions regarding infection management protocols. This knowledge enables the optimization of treatment strategies, reduction in complications associated with infection recurrence and improvement of patient outcomes. By analyzing the bivariate infection recurrence process in kidney patients undergoing portable dialysis, it becomes possible to predict the probability, timing, risk factors and treatment outcomes of infection recurrences. This information aids in identifying the likelihood of future infections, recognizing high-risk patients in need of close monitoring, and guiding the selection of appropriate treatment approaches. Limited bivariate distribution functions pose challenges in jointly modeling inter-correlated time between recurrences with different univariate marginal distributions. To address this, a Copula-based methodology is presented in this study. The methodology introduces the Kavya–Manoharan transformation family as the lifetime model for experimental units. The new bivariate models accurately measure dependence, demonstrate significant properties, and include special sub-models that leverage exponential, Weibull, and Pareto distributions as baseline distributions. Point and interval estimation techniques, such as maximum likelihood and Bayesian methods, where Bayesian estimation outperforms maximum likelihood estimation, are employed, and bootstrap confidence intervals are calculated. Numerical analysis is performed using the Markov chain Monte Carlo method. The proposed methodology’s applicability is demonstrated through the analysis of two real-world data-sets. The first data-set, focusing on infection and recurrence time in kidney patients, indicates that the Farlie–Gumbel–Morgenstern bivariate Kavya–Manoharan–Weibull (FGMBKM-W) distribution is the best bivariate model to fit the kidney infection data-set. The second data-set, specifically that related to UEFA Champions League Scores, reveals that the Clayton Kavya–Manoharan–Weibull (CBKM-W) distribution is the most suitable bivariate model for fitting the UEFA Champions League Scores. This analysis involves examining the time elapsed since the first goal kicks and the home team’s initial goal.

Osteoporosis (OP) is a common metabolic bone disease with high incidence, and is recognized as a major public health problem worldwide. It is essential to clarify the pathogenesis of the disease for improving the diagnosis, prevention and treatment of OP. The aim of this study was to clarify the pivotal pathways and hub genes in OP using Gibbs sampling. The gene expression profile datasets were obtained from Gene Expression Omnibus (GEO) database. The pathways were enriched by Kyoto Encyclopedia of Genes and Genomes (KEGG) with genes intersection ≥5 based on gene expression profile data. Then, the acquired pathways were converted into Markov chains (MC). Gibbs sampling was conducted to obtain a new MC. In addition, the average probabilities of each pathway in two states containing human mesenchymal stem cells (hMSC) _middle-aged and hMSC_elderly were calculated through Markov chain Monte Carlo (MCMC) algorithm. Moreover, gene expression variation was taken into account to adjust the probability. Pivotal pathways were identified under adjusted posterior value >0.8. Then, Gibbs sampling was implemented to find hub genes from pathways. There were 280 pathways determined by the gene intersection ≥5. Gibbs sampling identified two disturbed pathways (pathways in cancer and influenza A) and two hub genes (cyclin A1 and WNT2) under the adjusted probability >0.8. Gene expression analysis showed that all the disturbed pathways and hub genes had increased expression levels in hMSC_middle-aged samples compared with hMSC_elderly samples. We identified two pivotal pathways and two hub genes in OP using Gibbs sampling. The results contribute to the understanding of underlying pathogenesis and could be considered as potential biomarkers for the therapy of OP.