Efficient Estimation of Semiparametric Multivariate Copula Models

Karmakar and Simonovic (2008)describe the methodology of assigning appropriate marginal distributions for three flood characteristics. It is found that the gamma distribution is best fitted for peak flow (P), and a nonparametric distribution from the orthonormal series method best fits to volume (V) and duration (D), based on the root mean square error, Akaike information criterion and Bayesian information criteria. In addition, the chi‐square test is performed to check the significance of fitness. In this paper, a methodology is developed to derive bivariate joint distributions of the flood characteristics using the concept of copulas, considering a set of parametric and nonparametric marginal distributions forP, VandDto mathematically model the correlated structure among them. In the conventional method of flood frequency analysis, the marginal distribution functions of peak flow, volume and duration are assumed to follow some specific parametric distribution function. The concept of copulas relaxes the restriction of traditional flood frequency analysis by selecting marginals from different families of probability distribution functions for flood characteristics. The present study performs a better selection of marginal distribution functions for flood characteristics by parametric and nonparametric estimation procedures, and demonstrates how the concept of copulas may be used for establishing a joint distribution function with mixed marginal distributions. The results obtained are useful for hydrologic design and planning purposes. The methodology is demonstrated with 70 years of stream flow data of Red River at Grand Forks of North Dakota, USA.

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.

Multivariate zero-inflated Bell distribution and its inference and applications

In this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution. This in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions. In particular, the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates (MLEs) or posterior means, with covariance matrix the inverse of the Fisher information. Additionally, the semiparametric and nonparametric MLEs for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter, with the same covariance matrix. The results are a semiparametric version and a nonparametric version of the parametric Bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric Bayesian inference and their frequentist counterparts.

In this paper, time series of length T are seen as frequency distributions. Each distribution is defined with respect to a statistical variable having T observed values. A methodological system based on Gini’s approach is put forward, so the statistical model through which time series are handled is a frequency distribution studied inside a linear system. In addition to the starting frequency distributions that are observed, other frequency distributions are treated. Thus, marginal distributions based on the notion of proportionality are introduced together with joint distributions. Both distributions are statistical models. A fundamental invariance property related to marginal distributions is made explicit in this research work, so one can focus on collections of marginal frequency distributions, identifying multiple frequency distributions. For this reason, the latter is studied via a tensor. As frequency distributions are practical realizations of nonparametric probability distributions over R, one passes from frequency distributions to discrete random variables. In this paper, a mathematical model that generates time series is put forward. It is a stochastic process based on subjective previsions of random variables. A subdivision of the exchangeability of variables of a statistical nature is shown, so a reinterpretation of principal component analysis that is based on the notion of proportionality also characterizes this research work.

Models of diameter and height distributions play an important role in forest mensuration and inventory, and many models and fitting methods are available for fitting these distributions separately. Very few models and methods appropriate for fitting the joint distribution of diameters and heights are available. The S BB and distributions obtained using Plackett's method are the only appropriate models published in the forestry literature to date. A bivariate joint distribution can be represented in terms of its underlying marginals and a copula function that models the dependency structure. A copula is a function that joins or “couples” a multivariate distribution function to its one-dimensional marginal distributions. The copula approach provides a general way of constructing bivariate distributions, including previously used bivariate models. The basics of the copula representation, the most useful copula functions, and maximum likelihood estimation (MLE) of the copula function are introduced. The emphasis of this article is on the choice of copula function(s) appropriate for modeling the joint distribution of tree diameters and heights. Using an empirical dataset of 102 Chinese fir (Cunninghamia lanceolata Lamb.) plantation plots, we fit each of the copula functions that we considered to each sample-plot joint distribution of tree diameters and heights. A two-stage MLE approach with a nonparametric (empirical) approach to the marginal distributions is adopted. In this case study the normal copula proves to be the best for modeling the joint distribution (dependence) of tree diameters and heights.

The paper in the first place reviews the information and the uncertainty measures of joint and marginal probability distributions of the sets and subsets of random events. Next it reminds on the relations of the unconditional and conditional entropy of joint and marginal distributions and their combinations. Then it elaborates the ways how these measures can be applied in the sea surface uncertainty evaluation, that is, how to gain the information from the wave properties visually observed and brought together in the global wave statistics. Furthermore, the examples offered demonstrate the computational procedures for the sea surface uncertainty evaluation and prospective usefulness of the information obtainable from the joint and marginal distributions of observed explicit and combined wave properties.

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.

Social scientists are frequently interested in how populations evolve over time. Creating poststratification weights for surveys, for example, requires information on the weighting variables’ joint distribution in the target population. Typically, however, population data are sparsely available across time periods. Even when population data are observed, the content and structure of the data—which variables are observed and whether their marginal or joint distributions are known—differ across time, in ways that preclude straightforward interpolation. As a consequence, survey weights are often based only on the small subset of auxiliary variables whose joint population distribution is observed regularly over time, and thus fail to take full advantage of auxiliary information. To address this problem, we develop a dynamic Bayesian ecological inference model for estimating multivariate categorical distributions from sparse, irregular, and noisy data on their marginal (or partially joint) distributions. Our approach combines (1) a Dirichlet sampling model for the observed margins conditional on the unobserved cell proportions; (2) a set of equations encoding the logical relationships among different population quantities; and (3) a Dirichlet transition model for the period-specific proportions that pools information across time periods. We illustrate this method by estimating annual U.S. phone-ownership rates by race and region based on population data irregularly available between 1930 and 1960. This approach may be useful in a wide variety of contexts where scholars wish to make dynamic ecological inferences about interior cells from marginal data. A new R package estsubpop implements the method.

Correlated pseudorandom variables with prescribed marginal distribution functions sometimes are required in simulation such as in Monte Carlo studies. In this paper, we present a general procedure and a simple but effective numerical approach to generating correlated random variables sampling sequence with prescribed marginal probability distribution functions and correlation coefficient matrix based on linear transformation-nonlinear transformation with Choesky factor. Some simulation results are reported. Simulation results show that the collections of random numbers generated by the presented procedure have desired correlations and pass the Kolmogorov-Smirnov non-parametric hypothesis test of specified marginal distribution. Some restrictions on the application of this method are discussed.

We consider the problem of inference on one of the two parameters of a probability distribution when we have some prior information on a nuisance parameter. When a prior probability distribution on this nuisance parameter is given, the marginal distribution is the classical tool to account for it. If the prior distribution is not given, but we have partial knowledge such as a fixed number of moments, we can use the maximum entropy principle to assign a prior law and thus go back to the previous case. In this work, we consider the case where we only know the median of the prior and propose a new tool for this case. This new inference tool looks like a marginal distribution. It is obtained by first remarking that the marginal distribution can be considered as the mean value of the original distribution with respect to the prior probability law of the nuisance parameter, and then, by using the median in place of the mean.

Recently, a procedure of impounding a given marginal distribution, with the auxiliary density function generated from the assumed form of the distribution for the concomitant of an extreme order statistic of a sample, has been developed to create a bivariate distribution. In this paper we consider a marginal Rayleigh distribution and impound it with the pdf of an extended form of Lindley distribution to generate a new bivariate distribution. Some important properties of this new bivariate model together with some characterization theorems have been also derived. Most commonly used reliability functions associated with the generated bivariate distribution as well have been obtained. The well-known maximum likelihood (ML) method of estimation is applied for the estimation of the parameters of the generated bivariate distribution. A simulation study is also carried out to illustrate the closeness of ML estimates with the true value of the parameters. A real-life bivariate data set is used to demonstrate the application of the results of this paper in generating the appropriate bivariate distributional model in an innovative manner.

Increasing attention has been paid to the use of non-Gaussian distributions as models of heterogeneity in sedimentary formations in recent years. In particular, the Levy-stable distribution has been shown to be a useful model of the distribution of the increments of data measured in well logs. Frequently, the width of this distribution follows a power–law type scaling with increment lag, thus suggesting a nonstationary, fractal, multivariate Levy distribution as a useful random field model. However, in this paper we show that it is very difficult to formulate a multivariate Levy distribution with any nontrivial spatial correlations that can be sampled from rigorously in large models. Conventional sequential simulation techniques require two properties to hold of a multivariate distribution in order to work: (1) the marginal distributions must be of relatively simple form, and (2) in the uncorrelated limit, the multivariate distribution must factor into a product of independent distributions. At least one of these properties will break down in a multivariate Levy distribution, depending on how it is formulated. This makes a rigorous derivation of a sequential simulation algorithm impossible. Nonetheless, many of the original observations that spurred the original interest in multivariate Levy distributions can be reproduced with a conventional normal scoring procedure. Secondly, an approximate formulation of a sequential simulation algorithm can adequately reproduce the Levy distributions of increments and fractal scaling frequently seen in real data.

A statistical analysis method for significant wave height and spectral peak frequency considering the random and time-varying effects based on copula function and Bayesian inference

Maximum likelihood estimation is used widely in classical statistics. However, except in a few cases, it does not have a closed form. Furthermore, it takes time to derive the maximum likelihood estimator (MLE) owing to the use of iterative methods such as Newton–Raphson. Nonetheless, this estimation method has several advantages, chief among them being the invariance property and asymptotic normality. Based on the first approximation to the solution of the likelihood equation, we obtain an estimator that has the same asymptotic behavior as the MLE for multivariate gamma distribution. The newly proposed estimator, denoted as , is also in closed form as long as the ‐consistent initial estimator is in the closed form. Hence, we develop some closed‐form ‐consistent estimators for multivariate gamma distribution to improve the small‐sample property. is an alternative to MLE and performs better compared to MLE in terms of computation time, especially for large datasets, and stability. For the bivariate gamma distribution, the is over 130 times faster than the MLE, and as the sample size increasing, the is over 200 times faster than the MLE. Owing to the instant calculation of the proposed estimator, it can be used in state–space modeling or real‐time processing models.