Maximum likelihood estimation for N-mixture models.

Multinomial -mixture models are commonly used to fit data from a removal sampling protocol. If the mixing distribution is negative binomial, the distribution of the counts does not appear to have been identified, and practitioners approximate the requisite likelihood by placing an upper bound on the embedded infinite sum. In this paper, the distribution which underpins the multinomial -mixture model with a negative binomial mixing distribution is shown to belong to the broad class of multivariate negative binomial distributions. Specifically, the likelihood can be expressed in closed form as the product of conditional and marginal likelihoods and the information matrix shown to be block diagonal. As a consequence, the nature of the maximum likelihood estimates of the unknown parameters and their attendant standard errors can be examined and tests of the hypothesis of the Poisson against the negative binomial mixing distribution formulated. In addition, appropriate multinomial -mixture models for data sets which include zero site totals can also be constructed. Two illustrative examples are provided.

BackgroundThe performance of the Beta Binomial (BB) model is compared with several existing models for mapping the EORTC QLQ-C30 (QLQ-C30) on to the EQ-5D-3L using data from lung cancer trials.MethodsData from 2 separate non small cell lung cancer clinical trials (TOPICAL and SOCCAR) are used to develop and validate the BB model. Comparisons with Linear, TOBIT, Quantile, Quadratic and CLAD models are carried out. The mean prediction error, R2, proportion predicted outside the valid range, clinical interpretation of coefficients, model fit and estimation of Quality Adjusted Life Years (QALY) are reported and compared. Monte-Carlo simulation is also used.ResultsThe Beta-Binomial regression model performed ‘best’ among all models. For TOPICAL and SOCCAR trials, respectively, residual mean square error (RMSE) was 0.09 and 0.11; R2 was 0.75 and 0.71; observed vs. predicted means were 0.612 vs. 0.608 and 0.750 vs. 0.749. Mean difference in QALY’s (observed vs. predicted) were 0.051 vs. 0.053 and 0.164 vs. 0.162 for TOPICAL and SOCCAR respectively. Models tested on independent data show simulated 95% confidence from the BB model containing the observed mean more often (77% and 59% for TOPICAL and SOCCAR respectively) compared to the other models. All algorithms over-predict at poorer health states but the BB model was relatively better, particularly for the SOCCAR data.ConclusionThe BB model may offer superior predictive properties amongst mapping algorithms considered and may be more useful when predicting EQ-5D-3L at poorer health states. We recommend the algorithm derived from the TOPICAL data due to better predictive properties and less uncertainty.Electronic supplementary materialThe online version of this article (doi:10.1186/s12955-014-0163-7) contains supplementary material, which is available to authorized users.

Two generalized hypergeometric distributions are identified as mixed binomial distributions by conditional specification. Both distributions show profiles that are not possible in other mixed binomial distributions such as the beta-binomial distribution. A simulation study illustrates that beta-binomial distribution is more precise to fit data with usual profiles but the two distributions presented can improve the capability of fitting data in other less common scenes.

This paper introduced a new mixed distribution, namely the negative binomial-generalized exponential (NB-GE) distribution, which it obtained by mixing the negative binomial (NB) distribution with a generalized exponential (GE) distribution. The NB-GE distribution offers the advantage of being able to handle this kind of data sets, while still maintaining similar characteristics as the traditional negative binomial. The closed form and the factorial moment of the NB-GE distribution are derived. The NB-GE distribution appropriate when we find the one related to count data which have a large number of zeros and overdispersion in Poisson distribution. In addition, we present the basic properties of the new distribution such as mean, variance, skewness and kurtosis. Including, the negative binomial-exponential distributions are presented as special cases of this NB-GE distribution. Parameters estimation is also implemented using maximum likelihood method and the usefulness of the NB-GE distribution is illustrated by real data set. When using a goodness of fit test, the results show that the NB-GE distribution can provide a better fit than the NB distribution and Poisson distribution for count data that contain a large number of zeros.

Negative binomial group distribution was proposed in the literature which was motivated by inverse sampling when considering group inspection: products are inspected group by group, and the number of non-conforming items of a group is recorded only until the inspection of the whole group is finished. The non-conforming probability p of the population is thus the parameter of interest. In this paper, the confidence interval construction for this parameter is investigated. The common normal approximation and exact method are applied. To overcome the drawbacks of these commonly used methods, a composite method that is based on the confidence intervals of the negative binomial distribution is proposed, which benefits from the relationship between negative binomial distribution and negative binomial group distribution. Simulation studies are carried out to examine the performances of our methods. A real data example is also presented to illustrate the application of our method.

This chapter describes how spatial patterns (e.g. spatial pattern of Colorado potato beetle [Leptinotarsa decemlineata] on potato plants) in the field are related to observed frequency distributions of sample counts. Four probability distributions are discussed, which are tools for characterizing the observed frequency distributions of sample observations. These distributions are the Poisson distribution, the negative binomial distribution, the binomial distribution and the beta-binomial distribution. This chapter shows how these probability distributions can be helpful for assessing the properties of sampling plans for decision making, and explains how the parameters of these distributions can be fitted to data using the maximum likelihood method, and how the goodness of fit can be assessed with the X2 statistic.

The calibration of transfer functions is essential for accurate pavement performance predictions in the Pavement-ME design. Several studies have used the least square approach to calibrate these transfer functions. Least square is a widely used simplistic approach based on certain assumptions. Literature shows that these least square approach assumptions may not apply to the non-normal distributions. This study introduces a new methodology for calibrating the transverse cracking and international roughness index (IRI) models in rigid pavements using maximum likelihood estimation (MLE). Synthetic data for transverse cracking, with and without variability, are generated to illustrate the applicability of MLE using different known probability distributions (exponential, gamma, log-normal, and negative binomial). The approach uses measured data from the Michigan Department of Transportation's (MDOT) pavement management system (PMS) database for 70 jointed plain concrete pavement (JPCP) sections to calibrate and validate transfer functions. The MLE approach is combined with resampling techniques to improve the robustness of calibration coefficients. The results show that the MLE transverse cracking model using the gamma distribution consistently outperforms the least square for synthetic and observed data. For observed data, MLE estimates of parameters produced lower SSE and bias than least squares (e.g., for the transverse cracking model, the SSE values are 3.98 vs 4.02, and the bias values are 0.00 and -0.41). Although negative binomial distribution is the most suitable fit for the IRI model for MLE, the least square results are slightly better than MLE. The bias values are -0.312 and 0.000 for the MLE and least square methods. Overall, the findings indicate that MLE is a robust method for calibration, especially for non-normally distributed data such as transverse cracking.

This article presents the results of a simulation study of the estimation of a common odds ratio in multiple-contingency tables when the data are correlated within clusters. The correlation is modeled by the beta-binomial distribution. The three objectives of the study were to determine (a) the variances, biases, and relative efficiencies of the beta-binomial maximum likelihood estimator (MLE), the binomial MLE, and the Mantel-Haenszel estimator of the common odds ratio; (b) the variances and biases of two estimators of the intracluster correlation coefficient, ρ; and (c) the effectiveness of adjusting variance estimates of the binomial and Mantel-Haenszel estimators using the two estimators of ρ. The simulation shows that increasing the size of the intracluster correlation coefficient increases the variance and the positive bias of all three estimators of the odds ratio. The Mantel-Haenszel estimator displays less variance and bias than the MLE except for large values of Ψ and ρ. An unexpected b...

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.

Mixture models are versatile tools that are used extensively in many fields, including operations, marketing, and econometrics. The main challenge in estimating mixture models is that the mixing distribution is often unknown, and imposing a priori parametric assumptions can lead to model misspecification issues. In this paper, we propose a new methodology for nonparametric estimation of the mixing distribution of a mixture of logit models. We formulate the likelihood-based estimation problem as a constrained convex program and apply the conditional gradient (also known as Frank–Wolfe) algorithm to solve this convex program. We show that our method iteratively generates the support of the mixing distribution and the mixing proportions. Theoretically, we establish the sublinear convergence rate of our estimator and characterize the structure of the recovered mixing distribution. Empirically, we test our approach on real-world datasets. We show that it outperforms the standard expectation-maximization (EM) benchmark on speed (16 times faster), in-sample fit (up to 24% reduction in the log-likelihood loss), and predictive (average 28% reduction in standard error metrics) and decision accuracies (extracts around 23% more revenue). On synthetic data, we show that our estimator is robust to different ground-truth mixing distributions and can also account for endogeneity. This paper was accepted by Serguei Netessine, operations management.

Chapter 7 - Modeling Abundance Using Multinomial N-Mixture Models

If X 1, X 2, … is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial, Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

This article presents the results of a simulation study of the estimation of a common odds ratio in multiple-contingency tables when the data are correlated within clusters. The correlation is modeled by the beta-binomial distribution. The three objectives of the study were to determine (a) the variances, biases, and relative efficiencies of the beta-binomial maximum likelihood estimator (MLE), the binomial MLE, and the Mantel-Haenszel estimator of the common odds ratio; (b) the variances and biases of two estimators of the intracluster correlation coefficient, ρ; and (c) the effectiveness of adjusting variance estimates of the binomial and Mantel-Haenszel estimators using the two estimators of ρ. The simulation shows that increasing the size of the intracluster correlation coefficient increases the variance and the positive bias of all three estimators of the odds ratio. The Mantel-Haenszel estimator displays less variance and bias than the MLE except for large values of Ψ and ρ. An unexpected b...

In this paper we present smooth goodness of fit tests for testing the mixture distribution of a sequence of i.i.d. random variables. We consider mixture models when the mixing distribution is known. We adapt a Schwarz’s criteria initiated by Ledwina (J Am Stat Assoc 89:1000–1005, 1994) and inspired by the Neyman (Skandinavian Aktuarial 20:149–199, 1937) smooth test procedure. A Monte Carlo study is provided in order to assess the performance of the test.

The simple step-stress model under type-II censoring based on Weibull lifetimes, which provides a more flexible model than the exponential model, is considered in this paper. For this model, the maximum likelihood estimates (MLE) of its parameters, as well as the corresponding observed Fisher information matrix, are derived. The likelihood equations do not lead to closed-form expressions for the MLE, and they need to be solved by using an iterative procedure, such as the Newton-Raphson method. We also present a simplified estimator, which is easier to compute, and hence is suitable to use as an initial estimate in the iterative process for the determination of the MLE. We then evaluate the bias, and mean square error of these estimates; and provide asymptotic, and bootstrap confidence intervals for the parameters of the Weibull simple step-stress model. Finally, the results are illustrated with some examples.