Parametric estimation of the number of classes in a population

This paper deals with the well-studied problem of how best to estimate the number of mutually exclusive and exhaustive classes in a population, based on a sample from it. Haas & Stokes review and provide non-parametric approaches, but there are associated difficulties especially for small sampling fractions and/or widely varying population class sizes. Sichel provided 'GIGP' methodology, for this problem and for other purposes; this paper utilizes the three-parameter GIGP distribution for this problem, and also for the estimation of the number of classes of size 1, as an alternative to the non-parametric approaches. Methodological and computational issues are considered, and examples indicate the potential for GIGP.     

On the estimation of the number of equally likely classes in a population

For a population with an unknown number M of equally likely classes, an estimator for M from random samples is given. The results are applied to the problem of determining the total amount of coinage in past civilizations.     

On the estimation of the most probable number in a serial dilution experiment

The purpose of this paper is to present some alternative estimates for the 'most probable number' of bacteria in a serial dilution experiment. These estimates are directed to be less biased than the ordinary maximum likelihood estimate. A numerical example illustrates the extent to which the variance and the mean square error of these estimates are generally less than those corresponding to the maximum likelihood estimate.     

On the estimation of the proportion of positives in a sequence of screening experiments

A metaanalytic estimator of the proportion of positives in a sequence of screening experiments is proposed. The distribution-free estimator is based on the empirical distribution of P-values from individual experiments, which is uniform under the global null hypotheses of no positives in the sequence of experiments performed. Under certain regularity conditions, the proportion of positives corresponds to the derivative of this distribution under the alternative hypothesis of the existence of some positives. The statistical properties of the estimator are established, including its bias, variance, and rate of convergence to normality. Optimal estimators with minimum mean squared error are also developed under specific alternative hypotheses. The application of the proposed methods is illustrated using data from a sequence of screening experiments with chemicals to determine their carcinogenic potential.     

Change-point estimation in a multinomial sequence and homogeneity of literary style

To help settle the debate around the authorship of Tirant lo Blanc, all the words in each chapter of that book are categorized according to their length and the appearance of various words is counted. The graphical exploration of the sequences of multinomial observations obtained reveals a clear single sudden change point that is consistently estimated to be between chapters 371 and 382 and might indicate a switch of author. Correspondence analysis indicates that at the end of the book the words tend to be longer and the frequency of various words changes significantly. By doing a cluster analysis of the multinomial observations, the evidence in favor of the existence of that stylistic boundary is strengthened, because the two clusters obtained match very closely the before and after change-point groups; only a few chapters at the end of the book appear to be misclassified by the change point.     

Limit laws for the cumulative number of ties for the maximum in a random sequence

Let {_Xn,n?1}$\{X_n,n?1\}$ be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation _Xn$X_n$ is a tie for the maximum if _Xn=max{_X1,…,_Xn-1}$X_n=\max \{X_1,\dots ,X_{n-1}\}$. In this paper, we obtain weak and strong laws of large numbers and central limit theorems for the cumulative number of ties for the maximum among the first n$n$ observations.     

Nonparametric estimation of the number of components of a superposition of renewal processes

Suppose all events occurring in an unknown number (ν)$(\nu )$ of iid renewal processes, with a common renewal distribution F  , are observed for a fixed time τ$\tau$, where both ν$\nu$ and F   are unknown. The individual processes are not known a priori, but for each event, the process that generated it is identified. For example, in software reliability application, the errors (or bugs) in a piece of software are not known a priori, but whenever the software fails, the error causing the failure is identified. We present a nonparametric method for estimating ν$\nu$ and investigate its properties. Our results show that the proposed estimator performs well in terms of bias and asymptotic normality, while the MLE of ν$\nu$ derived assuming that the common renewal distribution is exponential may be seriously biased if that assumption does not hold.     

Non-parametric testing for the number of change points in a sequence of independent random variables

A non-parametric procedure is derived for testing for the number of change points in a sequence of independent continuously distributed variables when there is no prior information available. The procedure is based on the Kruskal–Wallis test, which is maximized as a function of all possible places of the change points. The procedure consists of a sequence of non-parametric tests of nested hypotheses corresponding to a decreasing number of change points. The properties of this procedure are analyzed by Monte Carlo methods and compared to a parametric procedure for the case that the variables are exponentially distributed. The critical values are given for sample sizes up to 200.     

M-estimator-based robust estimation of the number of components of a superimposed sinusoidal signal model

In this paper, we consider the problem of estimating the number of components of a superimposed nonlinear sinusoids model of a signal in the presence of additive noise. We propose and provide a detailed empirical comparison of robust methods for estimation of the number of components. The proposed methods, which are robust modifications of the commonly used information theoretic criteria, are based on various M-estimator approaches and are robust with respect to outliers present in the data and heavy-tailed noise. The proposed methods are compared with the usual non-robust methods through extensive simulations under varied model scenarios. We also present real signal analysis of two speech signals to show the usefulness of the proposed methodology.     

Sparsistent and constansistent estimation of the varying-coefficient model with a diverging number of predictors

ABSTRACT

The varying-coefficient model is a flexible class of approaches that extends simple linear relationships between covariates and responses. Two related problems concerning these models are selecting relevant variables and determining non-varying coefficients among those relevant ones. In this paper we study the sparsistency and constansistency of the regularized estimation approach when the number of predictors diverges with the sample size. Here, constansistency refers to the desired property that the non-zero, non-varying coefficients are identified with probability tending to one.     

On the joint distribution of the random variables ‘number of inversions’ and ‘number of outstanding variables’ in a randomly arranged sequence

The bivariate probability distribution of the random variables [number of inversions] and [number of outstanding variables] in a sequence of n i.i.d. random variables is derived. As an application, the null covariance between the test statistics proposed by Mann and Brunk, respectively, for the ‘trend in location’ problem is obtained. It is shown that these test statistics are asymptotically uncorrelated under the null hypothesis.     

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.     

The particle filter based on random number searching algorithm for parameter estimation

This article addresses the issue of parameter estimation in linear system in the presence of Gaussian noises, under which the random number searching algorithm (LJ (Luus and Jaakola) algorithm) is combined with the Rao-Blackwellised particle filter (RBPF) algorithm. This yields the so-called RBPF algorithm based on LJ (RBPF-LJ). Unlike the mature alternatives of generic particle filter, the parameter particles of RBPF-LJ are set as random numbers that search in the parameter value scope, which is regulated based on the estimation result to track the changes of the unknown parameter. The contrasting simulations show that the proposed RBPF-LJ outperform the RBPF as well as the particle filter based on kernel smoothing contraction algorithm on the estimation of the dynamically linear or nonlinear parameter and it can obtain the similar estimation results on the static parameter if some coefficients are regulated.     

Full information maximum likelihood estimation in factor analysis with a large number of missing values

We consider the problem of full information maximum likelihood (FIML) estimation in factor analysis when a majority of the data values are missing. The expectation–maximization (EM) algorithm is often used to find the FIML estimates, in which the missing values on manifest variables are included in complete data. However, the ordinary EM algorithm has an extremely high computational cost. In this paper, we propose a new algorithm that is based on the EM algorithm but that efficiently computes the FIML estimates. A significant improvement in the computational speed is realized by not treating the missing values on manifest variables as a part of complete data. When there are many missing data values, it is not clear if the FIML procedure can achieve good estimation accuracy. In order to investigate this, we conduct Monte Carlo simulations under a wide variety of sample sizes.     

Random number generation and estimation with the bimodal asymmetric power-normal distribution

Recently, Bolfarine et al. [Bimodal symmetric-asymmetric power-normal families. Commun Statist Theory Methods. Forthcoming. doi:10.1080/03610926.2013.765475] introduced a bimodal asymmetric model having the normal and skew normal as special cases. Here, we prove a stochastic representation for their bimodal asymmetric model and use it to generate random numbers from that model. It is shown how the resulting algorithm can be seen as an improvement over the rejection method. We also discuss practical and numerical aspects regarding the estimation of the model parameters by maximum likelihood under simple random sampling. We show that a unique stationary point of the likelihood equations exists except when all observations have the same sign. However, the location-scale extension of the model usually presents two or more roots and this fact is illustrated here. The standard maximization routines available in the R system (Broyden–Fletcher–Goldfarb–Shanno (BFGS), Trust, Nelder–Mead) were considered in our implementations but exhibited similar performance. We show the usefulness of inspecting profile loglikelihoods as a method to obtain starting values for maximization and illustrate data analysis with the location-scale model in the presence of multiple roots. A simple Bayesian model is discussed in the context of a data set which presents a flat likelihood in the direction of the skewness parameter.     

A relative error-based estimation with an increasing number of parameters

The least product relative error (LPRE) estimator and test statistic to test linear hypotheses of regression parameters in the multiplicative regression model are studied when the number of covariate variables increases with the sample size. Some properties of the LPRE estimator and test statistic are obtained such as consistency, Bahadur presentation, and asymptotic distributions. Furthermore, we extend the LPRE to a more general relative error criterion and provide their statistical properties. Numerical studies including simulations and two real examples show that the proposed estimation performs well.