Estimation of binomial parameters when both , are unknown

We revisit the classic problem of estimation of the binomial parameters when both parameters n,p are unknown. We start with a series of results that illustrate the fundamental difficulties in the problem. Specifically, we establish lack of unbiased estimates for essentially any functions of just n or just p. We also quantify just how badly biased the sample maximum is as an estimator of n. Then, we motivate and present two new estimators of n. One is a new moment estimate and the other is a bias correction of the sample maximum. Both are easy to motivate, compute, and jackknife. The second estimate frequently beats most common estimates of n in the simulations, including the Carroll–Lombard estimate. This estimate is very promising. We end with a family of estimates for p; a specific one from the family is compared to the presently common estimate and the improvements in mean-squared error are often very significant. In all cases, the asymptotics are derived in one domain. Some other possible estimates such as a truncated MLE and empirical Bayes methods are briefly discussed.     

Empiric bayes estimation of binomial probability

An empirical Bayes estimator of a binomial parameter, based on orthogonal polynomials on (0,1), is introduced. The resulting estimator of the prior density is asymptotically optimal. The method allows one to combine Bayes and empiric Bayes methods with smoothing in a natural way.     

An information-theoretic approach to incorporating prior information in binomial sampling

The incorporation of prior information about θ, where θ is the success probability in a binomial sampling model, is an essential feature of Bayesian statistics. Methodology based on information-theoretic concepts is introduced which (a) quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and (b) allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information about θ which is embedded in any prior distribution. In effect, the most conservative prior distribution from a specified class (each member o f which carries the available prior information about θ) is that prior distribution within the class over which the likelihood function has the greatest average domination. The most conservative prior distributions from five different families of prior distributions over the interval (0,1) including the beta distribution are determined and compared for three situations: (1) no prior estimate of θ is available, (2) a prior point estimate or θ is available, and (3) a prior interval estimate of θ is available. The results of the comparisons not only advocate the use of the beta prior distribution in binomial sampling but also indicate which particular one to use in the three aforementioned situations.     

Posterior ranges of functions of parameters under priors with specified quantiles

Suppose some quantiles of the prior distribution of a nonnegative parameter θ are specified. Instead of eliciting just one prior density function, consider the class Γ of all the density functions compatible with the quantile specification. Given a likelihood function, find the posterior upper and lower bounds for the expected value of any real-valued function h(θ), as the density varies in Γ. Such a scheme agrees with a robust Bayesian viewpoint. Under mild regularity conditions about h(θ) and the likelihood, a procedure for finding bounds is derived and applied to an example, after transforming the given functional optimisation problems into finite-dimensional ones.     

Asymptotically optimum estimation of a probability in inverse binomial sampling under general loss functions

The optimum quality that can be asymptotically achieved in the estimation of a probability p using inverse binomial sampling is addressed. A general definition of quality is used in terms of the risk associated with a loss function that satisfies certain assumptions. It is shown that the limit superior of the risk for p asymptotically small has a minimum over all (possibly randomized) estimators. This minimum is achieved by certain non-randomized estimators. The model includes commonly used quality criteria as particular cases. Applications to the non-asymptotic regime are discussed considering specific loss functions, for which minimax estimators are derived.     

On confidence limits for the difference of two binomial parameters

In the present article we suggest two new methods for calculating approximate confidence limits for the differences of the two binomial parameters. Different methods for determining the confidence interval are compared.     

Some remarks on classical and bayesian reliability estimation of binomial and poisson distributions

The problems of estimating the reliability function and Pr{X₁+...+X_k ≤ Y} are considered. The random variables X’s and Y are assumed to follow binomial and Poisson distributions. Classical estimators available in the literature are discussed and Bayes estimators are derived. In order to obtain the estimators of these parametric functions, the basic role is played by the estimators of factorial moments of the two distributions.     

Maximum likelihood estimation of the parameters of the bivariate binomial distribution

We consider the problem of maximum likelihood estimation of the parameters of the bxvariate binomial distribution, In the statistical literature, this problem is solved when the observed sample is available in the form of a 2x2 contingency table, that is, with all four cell fre quencies given,, The present paper provides a solution for this problem when only the marginal totals of the 2x2 table are observed, which is the natural set-up in a bivariate sampling situation.. Thus, based on a sample [(X_i,Y_i:), i = 1, …, k] from a bivariate binomial population, we derive maximum likelihood (ML) estimators for the two marginal parameters p₁,p₂: and the covariance parameter p₁₁: It. turns out that the ML estimators for P₁: and P₂: are expressed explicitly in terms of the sample values, whereas the ML estimator for p₁₁: can only be obtained numerically by iterative methods Two nu merical illustrations are also presented     

Variance components estimation for the balanced random effects model under mixed prior distributions

For the balanced random effects models, when the variance components are correlated either naturally or through common prior structures, by assuming a mixed prior distribution for the variance components, we propose some new Bayesian estimators. To contrast and compare the new estimators with the minimum variance unbiased (MVUE) and restricted maximum likelihood estimators (RMLE), some simulation studies are also carried out. It turns out that the proposed estimators have smaller mean squared errors than the MVUE and RMLE.     

Bayesian estimation and case influence diagnostics for the zero-inflated negative binomial regression model

In recent years, there has been considerable interest in regression models based on zero-inflated distributions. These models are commonly encountered in many disciplines, such as medicine, public health, and environmental sciences, among others. The zero-inflated Poisson (ZIP) model has been typically considered for these types of problems. However, the ZIP model can fail if the non-zero counts are overdispersed in relation to the Poisson distribution, hence the zero-inflated negative binomial (ZINB) model may be more appropriate. In this paper, we present a Bayesian approach for fitting the ZINB regression model. This model considers that an observed zero may come from a point mass distribution at zero or from the negative binomial model. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. The approach can be easily implemented using standard Bayesian software, such as WinBUGS. The performance of the proposed method is evaluated with a simulation study. Further, a real data set is analyzed, where we show that ZINB regression models seems to fit the data better than the Poisson counterpart.     

Confidence interval estimation for negative binomial group distribution

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.     

Bayes estimators in the presence of a guess value

The paper deals with the problem of parameter estimation in the presence of a guess value and attempts to justify the use of Bayes estimators as an alternative to ordinary shrinkage estimators. Finally, certain Bayes estimators of exponential parameters are obtained under type II censoring, and these are compared with the corresponding MLEs and ordinary shrinkage estimators using a Monte Carlo study.     

An EM algorithm for estimating negative binomial parameters

An EM algorithm is proposed for computing estimates of parameters of the negative bi-nomial distribution; the algorithm does not involve further iterations in the M-step, in contrast with the one given in Schader & Schmid (1985). The approach can be applied to the corresponding problem in the logarithmic series distribution. The convergence of the proposed scheme is investigated by simulation, the observed Fisher information is derivedand numerical examples based on real data are presented.     

Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions

Block and Basu bivariate exponential distribution is one of the most popular absolute continuous bivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Statist Methodol. 2010;7:464–477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. In this paper, we consider the Bayesian inference of the unknown parameters of the BBBW distribution. The Bayes estimators are obtained with respect to the squared error loss function, and the prior distributions allow for prior dependence among the unknown parameters. Prior independence also can be obtained as a special case. It is observed that the Bayes estimators of the unknown parameters cannot be obtained in explicit forms. We propose to use the importance sampling technique to compute the Bayes estimates and also to construct the associated highest posterior density credible intervals. The analysis of two data sets has been performed for illustrative purposes. The performances of the proposed estimators are quite satisfactory. Finally, we generalize the results for the multivariate case.     

Optimal aggregation of input-output coefficients - a decision theoretic approach

We oresent a criterion fnr the optimal aggreation of industries in input-output analysis, when the aggregation is wade prior to data collection. This criterion is based on a model assuming -hat the input-output coefficients are used in. some decision process, and that they are unknown (or random) prior to data collection. We show that our criterion, based on decision theoretic considerations, differs considerably from traditional criteria for good aggregation. Our model is also applide as an example, to the Netherlands economy.     

Some inadmissibility results for estimating ordered uniform scale parameters

Independent random samples (of possibly unequal sizes) are drawn from k (≥2) uniform populations having unknown scale parameters μ₁,…,μ_k. The problem of componentwise estimation of ordered parameters is investigated. The loss function is assumed to be squared error and the cases of known and unknown ordering among μ₁,…,μ_k. are dealt with separately. Sufficient conditions for an estimator to be inadmissible are provided and as a consequence, many natural estimators are shown to be inadmissible, Better estimators are provided.     

Bayesian beta nonlinear models with constrained parameters to describe ruminal degradation kinetics

The models used to describe the kinetics of ruminal degradation are usually nonlinear models where the dependent variable is the proportion of degraded food. The method of least squares is the standard approach used to estimate the unknown parameters but this method can lead to unacceptable predictions. To solve this issue, a beta nonlinear model and the Bayesian perspective is proposed in this article. The application of standard methodologies to obtain prior distributions, such as the Jeffreys prior or the reference priors, involves serious difficulties here because this model is a nonlinear non-normal regression model, and the constrained parameters appear in the log-likelihood function through the Gamma function. This paper proposes an objective method to obtain the prior distribution, which can be applied to other models with similar complexity, can be easily implemented in OpenBUGS, and solves the problem of unacceptable predictions. The model is generalized to a larger class of models. The methodology was applied to real data with three models that were compared using the Deviance Information Criterion and the root mean square prediction error. A simulation study was performed to evaluate the coverage of the credible intervals.     

A Jackknifed estimators for the negative binomial regression model

Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators for estimating the ridge parameter k in the negative binomial (NB) regression have been proposed. The Jackknifed estimators are obtained to remedy the multicollinearity and reduce the bias. A simulation study is provided to evaluate the performance of estimators. Both mean squared error (MSE) and the percentage relative error (PRE) are considered as the performance criteria. The simulated result indicated that some of proposed Jackknifed estimators should be preferred to the ML method and ridge estimators to reduce MSE and bias.     

On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distributions is determined and compared for the situation when no prior information is available. The results of the comparison advocate the use of the Polya (beta-binomial) prior distribution in hypergeometric sampling.     

Convolution power kernels for density estimation

We propose a new type of non-parametric density estimators fitted to random variables with lower or upper-bounded support. To illustrate the method, we focus on nonnegative random variables. The estimators are constructed using kernels which are densities of empirical means of m i.i.d. nonnegative random variables with expectation 1. The exponent m   plays the role of the bandwidth. We study the pointwise mean square error and propose a pointwise adaptive estimator. The risk of the adaptive estimator satisfies an almost oracle inequality. A noteworthy result is that the adaptive rate is in correspondence with the smoothness properties of the unknown density as a function on (0,+∞)$(0,+\infty )$. The adaptive estimators are illustrated on simulated data. We compare our approach with the classical kernel estimators.