Simulating from polynomial-normal distributions

The family of polynomial-normal distributions includes a number of widely used distributions, such as the Gram–Charlier and Edgeworth distributions. In this note, we present three simple algorithms, (i) CDF Inversion, (ii) Acceptance–Rejection, (iii) and Ratio–of–Uniforms, for simulating variates from a polynomial-normal distribution. Details on the efficiency of the Acceptance–Rejection and the Ratio–of–Uniforms algorithms and a comparison across the various implementations are provided.     

Comparisons of various types of normality tests

Normality tests can be classified into tests based on chi-squared, moments, empirical distribution, spacings, regression and correlation and other special tests. This paper studies and compares the power of eight selected normality tests: the Shapiro–Wilk test, the Kolmogorov–Smirnov test, the Lilliefors test, the Cramer–von Mises test, the Anderson–Darling test, the D'Agostino–Pearson test, the Jarque–Bera test and chi-squared test. Power comparisons of these eight tests were obtained via the Monte Carlo simulation of sample data generated from alternative distributions that follow symmetric short-tailed, symmetric long-tailed and asymmetric distributions. Our simulation results show that for symmetric short-tailed distributions, D'Agostino and Shapiro–Wilk tests have better power. For symmetric long-tailed distributions, the power of Jarque–Bera and D'Agostino tests is quite comparable with the Shapiro–Wilk test. As for asymmetric distributions, the Shapiro–Wilk test is the most powerful test followed by the Anderson–Darling test.     

Analytic approach to coset–leader decoding leads to extension of the Welch–Berlekamp theorem

This paper provides a complete proof of the Welch–Berlekamp theorem on which the Welch–Berlekamp algorithm was founded. By introducing an analytic approach to coset–leader decoders for Reed–Solomon codes, the Welch–Berlekamp key-equation of error corrections is enlarged and a complete proof of the Welch–Berlekamp theorem is derived in a natural way, and the theorem is extended such that the BCH-bound constraint is moved.     

Chi–square tests for comparing vectors of proportions for several cluster samples

Test statistics are developed for comparing vectors of proportions obtained from several independent two–stage cluster samples. It is assumed that clusters are selected with probability proportional to size for each sample. Wald's general method of constructing quadratic forms is used to obtain a large sample chi–square test. More easily evaluted chi–square tests are derived from the Dirichlet–multinnomial model. Corresponding goodness–of–fit test for the Dirichlet–multinomial model are also derived.     

General cumulative Kullback–Leibler information

The cumulative residual Kullback–Leibler information is defined on the semi-infinite (non negative) interval. In this paper, we extend the cumulative residual Kullback–Leibler information to the whole real line and propose a general cumulative Kullback–Leibler information. We study its application to a test for normality in comparison with some competing test statistics based on the empirical distribution function including the well-known tests applied in practice like Kolmogorov–Smirnov, Cramer–von Mises, Anderson–Darling, and other existing tests.     

On the generalized maximum likelihood estimator of survival function under Koziol–Green model

In this paper, we derive the generalized maximum likelihood estimator (GMLE) of the survival function of a random variable for censored data under the Koziol–Green model. Its small sample properties are compared with those of Abdushukurov–Cheng–Lin (ACL), Kaplan–Meier, and Nelson–Aalen type estimators. The large sample analysis indicates that the GMLE, ACL, and Nelson–Aalen type estimators are asymptotically equivalent.     

A mixed GWMA–CUSUM control chart for monitoring the process mean

The memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.     

Equilibrium excess-of-loss reinsurance–investment strategy for a mean–variance insurer under stochastic volatility model

This article considers an optimal excess-of-loss reinsurance–investment problem for a mean–variance insurer, and aims to develop an equilibrium reinsurance–investment strategy. The surplus process is assumed to follow the classical Cramér–Lundberg model, and the insurer is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset. The market price of risk depends on a Markovian, affine-form and square-root stochastic factor process. Under the mean–variance criterion, equilibrium reinsurance–investment strategy and the corresponding equilibrium value function are derived by applying a game theoretic framework. Finally, numerical examples are presented to illustrate our results.     

An Empirical Analysis of Some Nonparametric Goodness-of-Fit Tests for Censored Data

In this article, we consider some nonparametric goodness-of-fit tests for right censored samples, viz., the modified Kolmogorov, Cramer–von Mises–Smirnov, Anderson–Darling, and Nikulin–Rao–Robson χ² tests. We also consider an approach based on a transformation of the original censored sample to a complete one and the subsequent application of classical goodness-of-fit tests to the pseudo-complete sample. We then compare these tests in terms of power in the case of Type II censored data along with the power of the Neyman–Pearson test, and draw some conclusions. Finally, we present an illustrative example.     

An entropy test for the Rayleigh distribution and power comparison

In this paper, a goodness-of-fit test is proposed for the Rayleigh distribution. This test is based on the Kullback–Leibler discrimination methodology proposed by Song [2002, Goodness of fit tests based on Kullback–Leibler discrimination, IEEE Trans. Inf. Theory 48(5), pp. 1103–1117]. The critical values and powers for some alternatives are obtained by simulation. The proposed test is compared with other tests, namely Kolmogorov–Smirnov, Kuiper, Cramer–von Mises, Watson and Anderson–Darling. The use of the proposed test is shown in a real example.     

Uncertainty estimation in heterogeneous capture–recapture count data

The Conway–Maxwell–Poisson estimator is considered in this paper as the population size estimator. The benefit of using the Conway–Maxwell–Poisson distribution is that it includes the Bernoulli, the Geometric and the Poisson distributions as special cases and, furthermore, allows for heterogeneity. Little emphasis is often placed on the variability associated with the population size estimate. This paper provides a deep and extensive comparison of bootstrap methods in the capture–recapture setting. It deals with the classical bootstrap approach using the true population size, the true bootstrap, and the classical bootstrap using the observed sample size, the reduced bootstrap. Furthermore, the imputed bootstrap, as well as approximating forms in terms of standard errors and confidence intervals for the population size, under the Conway–Maxwell–Poisson distribution, have been investigated and discussed. These methods are illustrated in a simulation study and in benchmark real data examples.     

Measures of biomarker dependence using a copula-based multivariate epsilon–skew–normal family of distributions

In this note we develop a new multivariate copula model based on epsilon–skew–normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon–skew–normal distribution.     

A simple test procedure in standardizing the power of Hosmer–Lemeshow test in large data sets

The Hosmer–Lemeshow (H–L) test is a widely used method when assessing the goodness-of-fit of a logistic regression model. However, the H–L test is sensitive to the sample sizes and the number of groups in H–L test. Cautions need to be taken for interpreting an H–L test with a large sample size. In this paper, we propose a simple test procedure to evaluate the model fit of logistic regression model with a large sample size, in which a bootstrap method is used and the test result is determined by the power of H–L test at the target sample size. Simulation studies show that the proposed method can effectively standardize the power of the H–L test under the pre-specified level of type I error. Application to the two datasets illustrates the usefulness of the proposed model.     

An extended fatigue life distribution

A five-parameter extended fatigue life model called the McDonald–Birnbaum–Saunders (McBS) distribution is proposed. It extends the Birnbaum–Saunders and beta Birnbaum–Saunders [G.M. Cordeiro and A.J. Lemonte, The β-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling. Comput. Statist. Data Anal. 55 (2011), pp. 1445–1461] distributions and also the new Kumaraswamy–Birnbaum–Saunders distribution. We obtain the ordinary moments, generating function, mean deviations and quantile function. The method of maximum likelihood is used to estimate the model parameters and its potentiality is illustrated with an application to a real fatigue data set. Further, we propose a new extended regression model based on the logarithm of the McBS distribution. This model can be very useful to the analysis of real data and could give more realistic fits than other special regression models.     

On Nelson–Aalen type estimation in the partial Koziol–Green model

In this paper, a Nelson–Aalen (NA) type estimator is derived and its sample properties are compared with the partial Abdushukurov–Cheng–Lin (PACL), generalized maximum likelihood (GMLE), and Kaplan–Meier (KM) estimators under the partial Koziol–Green model. These comparisons are made through Monto Carlo simulations under various sample sizes. The results indicate that the NA estimator always performs better than the KM estimator and is competitive with other estimators. Moreover, the PACL, GMLE, and NA estimators are shown to be asymptotically equivalent.     

Wilks’ factorization of the matrix-variate Dirichlet–Riesz distributions

The Riesz distributions on positive definite symmetric matrices are used to introduce a class of Dirichlet–Riesz distributions. In addition, several distributional properties are stated. Essentially, we show the relationship between the Dirichlet–Riesz distributions of the first kind and the second kind, respectively. We derive Wilks’ factorization of the matrix-variate Dirichlet–Riesz. Further, several results on the product of Riesz and beta–Riesz matrices with a set of Dirichlet–Riesz matrices of the first kind have been derived.     

Quantile estimation and comparing two independent groups with an approach based on percentile bootstrap

Although the most common approach for comparing two independent groups is on the basis of some measure of location, determination of the differences in the tails of the groups is often of interest. In this study, Harrell–Davis estimator, Sfakianakis–Verginis estimators and default quantile estimator of R are used in conjunction with a percentile bootstrap method with the aim of comparing two independent groups via the quantiles, and the relative efficiencies of Harrell–Davis and Sfakianakis–Verginis estimators are compared. General performance of Sfakianakis–Verginis estimators was much better than Harrell–Davis estimator in terms of both saving actual type I error and relative efficiency.     

Standards of Probability Sampling for Legal Evidence

Brief recognition is given to bicentennials of the death of James Stirling (1692–1770) and birth of Ferdinand Rudolph Hassler (1770–1843); to sesquicentennials of the publication of the 3rd definitive edition of Laplace's Théorie analytique des probabilités, and of the births of William Chauvenet (1820–1870) and Isaac Todhunter (1820–1884); to centennials of the births of Louis Bachelier (1870–1946) and Jean Perrin (1870–1942); and to the semicentennial of the initial publication (1920) of Metron, an International Review of Statistics.     

A generalised likelihood framework for partially observed capture–recapture–recovery models

We provide a closed form likelihood expression for multi-state capture–recapture–recovery data when the state of an individual may be only partially observed. The corresponding sufficient statistics are presented in addition to a matrix formulation which facilitates an efficient calculation of the likelihood. This likelihood framework provides a consistent and unified framework with many standard models applied to capture–recapture–recovery data as special cases.     

Inferring gene regulatory networks by an order independent algorithm using incomplete data sets

Analyzing incomplete data for inferring the structure of gene regulatory networks (GRNs) is a challenging task in bioinformatic. Bayesian network can be successfully used in this field. k-nearest neighbor, singular value decomposition (SVD)-based and multiple imputation by chained equations are three fundamental imputation methods to deal with missing values. Path consistency (PC) algorithm based on conditional mutual information (PCA–CMI) is a famous algorithm for inferring GRNs. This algorithm needs the data set to be complete. However, the problem is that PCA–CMI is not a stable algorithm and when applied on permuted gene orders, different networks are obtained. We propose an order independent algorithm, PCA–CMI–OI, for inferring GRNs. After imputation of missing data, the performances of PCA–CMI and PCA–CMI–OI are compared. Results show that networks constructed from data imputed by the SVD-based method and PCA–CMI–OI algorithm outperform other imputation methods and PCA–CMI. An undirected or partially directed network is resulted by PC-based algorithms. Mutual information test (MIT) score, which can deal with discrete data, is one of the famous methods for directing the edges of resulted networks. We also propose a new score, ConMIT, which is appropriate for analyzing continuous data. Results shows that the precision of directing the edges of skeleton is improved by applying the ConMIT score.