Parameter estimation for the dirichlet-multinomial distribution using supplementary beta-binomial data

We develop estimates for the parameters of the Dirichlet-multinomial distribution (DMD) when there is insufficient data to obtain maximum likelihood or method of moment estimates known in the literature. We do, however, have supplemetary beta-binomial data pertaining to the marginals of the DMD, and use these data when estimating the DMD parameters. A real situation and data set are given where our estimates are applicable.     

Pseudo maximum likelihood estimation for the dirichlet-multinomial distribution

Pseudo maximum likelihood estimation (PML) for the Dirich-let-multinomial distribution is proposed and examined in this pa-per. The procedure is compared to that based on moments (MM) for its asymptotic relative efficiency (ARE) relative to the maximum likelihood estimate (ML). It is found that PML, requiring much less computational effort than ML and possessing considerably higher ARE than MM, constitutes a good compromise between ML and MM. PML is also found to have very high ARE when an estimate for the scale parameter in the Dirichlet-multinomial distribution is all that is needed.     

Approximate distribution and test of fit for the clustering effect in the dirichlet multinomial model

The Dirichlet-multinomial model is considered as a model for cluster sampling. The model assumes that the design's covariance matrix is a constant times the covariance under multinomial sampling. The use of this model requires estimating a parameter C, that measures the clustering effect. In this paper, a regression estimate for C is obtained. An approximate distribution of this estimator is obtained through the use of asymptotic techniques. A goodness of fit statistic for testing the fit of the Dirichlet Multinomial model is also obtained, based on those asymptotic techniques. These statistics provide a means of knowing when the data satisfy the model assumption. These results are used to analyze data concerning the authorship of Greek prose.     

Latent class models for ecological inference on voters transitions

This paper introduces some new models of ecological inference within the context of estimation of voter transitions across elections. In particular, we assume that voters of a given party in a given occasion may be split into two latent types: faithful voters, who will certainly vote again for the same party and movers, who will reconsider their choice. Our models allow for unobserved heterogeneity across polling stations both in the weights of the two latent classes within each party and also when modelling the choice of unfaithful voters. Different ways of modelling the unobserved heterogeneity are considered by exploiting properties of the Dirichlet-multinomial distribution and the Brown Payne model of voting transitions can be seen as a special case within the class of models presented here. We discuss pseudo-maximum likelihood estimation and present an application to recent elections in Italy.     

Hierarchical Models for Cross-Classified Overdispersed Multinomial Data

When a vector of sample proportions is not obtained through a simple random sampling, the covariance matrix for the sample vector can differ substantially from the one corresponding to the multinomial model (Wilson 1989). For example, clustering effects of subject effects in repeated-measure experiments can cause the variance of the observed proportions to be much larger than variances under the multinomial model. The phenomenon is generally referred to as overdispersion. Tallis (1962) proposed a model for identically distributed multinomials with a common measure of correlation and referred to it as the generalized multinomial model. This generalized multinomial model is extended in this article to account for overdispersion by allowing the vectors of proportions to vary according to a Dirichlet distribution. The generalized Dirichlet-multinomial model (as it is referred to here) allows for a second order of pairwise correlation among units, a type of assumption found reasonable in some biological data (Kupper and Haseman 1978) and introduced here to business data. An alternative derivation allowing for two kinds of variation is also considered. Asymptotic normal properties of parameter estimators are used to construct Wald statistics for testing hypotheses. The methods are illustrated with applications to performance evaluation monthly data and an integrated circuit yield analysis.     

Regression Analysis of Multivariate Fractional Data

The present article discusses alternative regression models and estimation methods for dealing with multivariate fractional response variables. Both conditional mean models, estimable by quasi-maximum likelihood, and fully parametric models (Dirichlet and Dirichlet-multinomial), estimable by maximum likelihood, are considered. A new parameterization is proposed for the parametric models, which accommodates the most common specifications for the conditional mean (e.g., multinomial logit, nested logit, random parameters logit, dogit). The text also discusses at some length the specification analysis of fractional regression models, proposing several tests that can be performed through artificial regressions. Finally, an extensive Monte Carlo study evaluates the finite sample properties of most of the estimators and tests considered.     

Gamma-minimax results for the class of unimodal priors

Olman and Shmundak proved 1985 that in estimating a bounded normal mean under squared error loss the Bayes estimator with respect to the uniform distribution on the parameter interval is gamma-minimax when the parameter interval is sufficiently small and the class of priors consists of all symmetric and unimodal distributions. Recently, one of the authors showed that this result remains valid for quite general families of distributions which satisfy some regularity conditions. In the present paper a generalization to the class of unimodal priors with fixed mode is derived. It is proved that the Bayes estimator with respect to a suitable mixture of two uniform distributions is gamma-minimax for sufficiently small parameter intervals. To that end appropriate characterizations of a saddle point in the corresponding statistical games are established. Some results of a numerical study are presented.     

Detection and estimation of structural change in heavy-tailed sequence

In this paper, bootstrap detection and ratio estimation are proposed to analysis mean change in heavy-tailed distribution. First, the test statistic is constructed into a ratio form on the CUSUM process. Then, the asymptotic distribution of test statistic is obtained and the consistency of the test is proved. To solve the problem that the null distribution of the test statistic contains unknown tail index, we present a bootstrap approximation method to determine the critical values of the null distribution. We also discuss how to estimate change point based on ratio method. The consistency and rate of convergence for the change-point estimator are established. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real data sets. Especially the simulation results of bootstrap test are better than those of another existing method.     

Nonparametric estimation of linear functionals of a bivariate distribution under univariate censoring

In the presence of univariate censoring, a class of nonparametric estimators is proposed for linear functionals of a bivariate distribution of paired failure times. The estimators are shown to be root-n consistent and asymptotically normal. An adjusted empirical log-likelihood ratio statistic is developed and proved to follow a chi-square distribution asymptotically. Two types of confidence intervals, based on the normal approximation method and the empirical likelihood method, respectively, are constructed to make inference about the linear functionals. Their performance is evaluated in several simulation studies and a real example.     

Model selection of the generalized von Mises distribution based on empirical mode decomposition with data analyses

This paper presents a method for selecting a distribution within the generalized von Mises (GvM) class. In this method, the logarithmic form of the GvM probability frequency function is regarded as the sum of a constant and several cosine functions with different frequencies. Based on the empirical mode decomposition (EMD) method, the corresponding logarithmic series is decomposed to several intrinsic mode functions (IMF) whose corresponding instantaneous frequencies (IF) are used to be the basis of the GvM model selection. The applications of the proposed method are illustrated using simulated circular data and real wind direction data. The results demonstrate that the method proposed here can provide a good choice for the GvM model selection.     

Statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data

In this article, the statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data is discussed. The estimation of the parameters for Gompertz distribution is obtained using maximum likelihood method (MLE) and Bayesian method under three different loss functions. We also proved the existence and uniqueness of the MLE. The one-sample Bayesian prediction intervals are obtained. The work is done for different values of the parameters. We apply the Monto Carlo simulation to compare the proposed methods, also an example is discussed to construct the Prediction intervals.     

Nonparametric Bayes methods for directional data

A model for directional data in q dimensions is studied. The data are assumed to arise from a distribution with a density on a sphere of q — 1 dimensions. The density is unimodal and rotationally symmetric, but otherwise of unknown form. The posterior distribution of the unknown mode (mean direction) is derived, and small-sample posterior inference is discussed. The posterior mean of the density is also given. A numerical method for evaluating posterior quantities based on sampling a Markov chain is introduced. This method is generally applicable to problems involving unknown monotone functions.     

Quasi Bayesian likelihood

If the form of the distribution of data is unknown, the Bayesian method fails in the parametric inference because there is no posterior distribution of the parameter. In this paper, a theoretical framework of Bayesian likelihood is introduced via the Hilbert space method, which is free of the distributions of data and the parameter. The posterior distribution and posterior score function based on given inner products are defined and, consequently, the quasi posterior distribution and quasi posterior score function are derived, respectively, as the projections of the posterior distribution and posterior score function onto the space spanned by given estimating functions. In the space spanned by data, particularly, an explicit representation for the quasi posterior score function is obtained, which can be derived as a projection of the true posterior score function onto this space. The methods of constructing conservative quasi posterior score and quasi posterior log-likelihood are proposed. Some examples are given to illustrate the theoretical results. As an application, the quasi posterior distribution functions are used to select variables for generalized linear models. It is proved that, for linear models, the variable selections via quasi posterior distribution functions are equivalent to the variable selections via the penalized residual sum of squares or regression sum of squares.     

Random weighting estimation of sampling distributions via importance resampling

This paper presents a new random weighting-based adaptive importance resampling method to estimate the sampling distribution of a statistic. A random weighting-based cross-entropy procedure is developed to iteratively calculate the optimal resampling probability weights by minimizing the Kullback-Leibler distance between the optimal importance resampling distribution and a family of parameterized distributions. Subsequently, the random weighting estimation of the sampling distribution is constructed from the obtained optimal importance resampling distribution. The convergence of the proposed method is rigorously proved. Simulation and experimental results demonstrate that the proposed method can effectively estimate the sampling distribution of a statistic.     

ADF模式中漂移项和趋势项检验量分布与Bootstrap检验研究

在推导ADF检验模式下趋势项和漂移项伪t检验量极限分布基础上,提出修正的系数检验量。研究表明,它们与DF检验模式下检验量具有相同的极限分布;构造漂移项和趋势项检验的Bootstrap实现方法并证明了有效性。将蒙特卡洛模拟技术与临界值检验方法进行对比,结果表明Bootstrap方法能够明显降低检验的水平扭曲,在检验功效方面也有一定优势。模拟也显示临界值检验的局限性和Bootstrap方法的稳健性。     

Kernel estimation of a distribution function

A distribution function is estimated by a kernel method with

a poinrwise mean squared error criterion at a point x. Relation- ships between the mean squared error, the point x, the sample size and the required kernel smoothing parazeter are investigated for several distributions treated by Azzaiini (1981). In particular it is noted that at a centre of symmetry or near a mode of the distribution the kernei method breaks down. Point- wise estimation of a distribution function is motivated as a more useful technique than a reference range for preliminary medical diagnosis.     

The Gamma–Kumaraswamy distribution: An alternative to Gamma distribution

In this article, a new generalization of the Kumaraswamy distribution, namely the Gamma–Kumaraswamy distribution, is defined and studied. Various properties of the Gamma–Kumaraswamy are obtained. The structural analysis of the distribution in this article includes limiting behavior, mode, quantiles, moments, skewness, kurtosis, Shannon’s entropy, and order statistics. The method of maximum likelihood estimation is proposed for estimating the model parameters. For illustrative purposes, two real datasets are analyzed as application of the Gamma–Kumaraswamy distribution.     

Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant random matrix

The Euler characteristic heuristic has been proposed as a method for approximating the upper tail probability of the maximum of a random field with smooth sample path. When the random field is Gaussian, this method is proved to be valid in the sense that the relative approximation error is exponentially smaller. However, very little is known about the validity of the method when the random field is non-Gaussian. In this paper, as a milestone to developing the general theory about the validity of the Euler characteristic heuristic, we examine the Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant non-Gaussian random matrix. In this particular example, if the probability density function of the random matrix converges to zero sufficiently fast at the boundary of its support, the approximation error of the Euler characteristic heuristic is proved to be small and the approximation is valid. Moreover, for several standard orthogonally invariant random matrices, the approximation formula for the distribution of the largest eigenvalue and its asymptotic error are obtained explicitly. Our formulas are practical enough for the purpose of numerical calculations.     

On Semiparametric Mode Regression Estimation

It has been found that, for a variety of probability distributions, there is a surprising linear relation between mode, mean, and median. In this article, the relation between mode, mean, and median regression functions is assumed to follow a simple parametric model. We propose a semiparametric conditional mode (mode regression) estimation for an unknown (unimodal) conditional distribution function in the context of regression model, so that any m-step-ahead mean and median forecasts can then be substituted into the resultant model to deliver m-step-ahead mode prediction. In the semiparametric model, Least Squared Estimator (LSEs) for the model parameters and the simultaneous estimation of the unknown mean and median regression functions by the local linear kernel method are combined to infer about the parametric and nonparametric components of the proposed model. The asymptotic normality of these estimators is derived, and the asymptotic distribution of the parameter estimates is also given and is shown to follow usual parametric rates in spite of the presence of the nonparametric component in the model. These results are applied to obtain a data-based test for the dependence of mode regression over mean and median regression under a regression model.     

A characteristic function-based approach to approximate maximum likelihood estimation

The choice of the summary statistics in approximate maximum likelihood is often a crucial issue. We develop a criterion for choosing the most effective summary statistic and then focus on the empirical characteristic function. In the iid setting, the approximating posterior distribution converges to the approximate distribution of the parameters conditional upon the empirical characteristic function. Simulation experiments suggest that the method is often preferable to numerical maximum likelihood. In a time-series framework, no optimality result can be proved, but the simulations indicate that the method is effective in small samples.