A Generalization of the Poisson Distribution

A new generalization of the Poisson distribution, with two parameters λ₁ and λ₂, is obtained as a limiting form of the generalized negative binomial distribution. The variance of the distribution is greater than, equal to or smaller than the mean according as λ₂ is positive, zero or negative. The distribution gives a very close fit to supposedly binomial, Poisson and negative-binomial data and provides with a model suitable to most unimodel or reverse J-shaped distributions. Diagrams showing the variations in the form of the distribution for different values of λ₁ and λ₂ are given.     

A bayesian choice between poisson, binomial and negative binomial models

In this paper, we propose a Bayesian method for modelling count data by Poisson, binomial or negative binomial distributions. These three distributions have in common that the variance is, at most, a quadratic function of the mean. We use prior distributions on the variance function coefficients to consider simultaneously the three possible models and decide which one fits the data better. This approach sheds new light on the analysis of the Sibship data (Sokal and Rohlf, 1987). The Jeffreys-Lindley paradox is discussed through some illustrations.     

Bayesian analysis of overdispersed chromosome aberration data with the negative binomial model

The usual assumption of a Poisson model for the number of chromosome aberrations in controlled calibration experiments implies variance equal to the mean. However, it is known that chromosome aberration data from experiments involving high linear energy transfer radiations can be overdispersed, i.e. the variance is greater than the mean. Present methods for dealing with overdispersed chromosome data rely on frequentist statistical techniques. In this paper. the problem of overdispersion is considered from a Bayesian standpoint. The Bayes Factor is used to compare Poisson and negative binomial models for two previously published calibration data sets describing the induction of dicentric chromosome aberrations by high doses of neutrons. Posterior densities for the model parameters, which characterise dose response and overdispersion are calculated and graphed. Calibrative densities are derived for unknown neutron doses from hypothetical radiation accident data to deterimine the impact of different model assumptions on dose estimates. The main conclusion is that an initial assumption of a negative binomial model is the conservative approach to chromosome dosimetry for high LET radiations.     

Estimating the Fraction of Acceptable Product

Minimum variance unbiased estimates are given of the cumulative distribution function for the normal, binomial, Poisson, and negative exponential distributions. Although diverse areas of application might lead to interest in estimating the cumulative distribution function, most of the examples in this paper suppose that one is interested in the fraction of product meeting fixed specification limits.     

Sharper variance upper bound for unbiased estimation in inverse sampling

Sathe (1977) improved upon the bounds of Mikulski and Smith (1976) for the variance of inverse binomial estimator. We have obtained still better upper bounds. The betterment is numerically illustrated.     

A New Class of Topp–Leone Power Series Distributions with Reliability Application

Okasha et al. (J Failure Anal Prevention, 2017. doi: 10.1007/s11668-017-0263-x) introduced the novel Topp–Leone geometric distribution. Here, we introduce a class of distributions containing [32]’s distribution as a particular case. The class of distributions contains several important distributions, including the Topp–Leone geometric, Topp–Leone Poisson, Topp–Leone logarithmic, Topp–Leone binomial and Topp–Leone negative binomial distributions. We derive comprehensive mathematical properties of the class. We obtain closed form expressions for the density function, cumulative distribution function, survival and hazard rate functions, moments, mean residual lifetime, mean past lifetime, order statistics and moments of order statistics. The class is shown to be more flexible by reanalyzing the real data set in [32].     

Discrete uniform mixtures via posterior means

In this paper, we are concerned with identification of a discrete uniform mixture by the posterior mean. An exact formula for a prior distribution is given. Also some examples featuring negative binomial, negative hypergeometric and beta-Pascal distributions are provided.     

Estimating an endpoint with high-order moments

We present a new method for estimating the endpoint of a unidimensional sample when the distribution function decreases at a polynomial rate to zero in the neighborhood of the endpoint. The estimator is based on the use of high-order moments of the variable of interest. It is assumed that the order of the moments goes to infinity, and we give conditions on its rate of divergence to get the asymptotic normality of the estimator. The good performance of the estimator is illustrated on some finite sample situations.     

The LR-Chart: An Alternative to the MR-Chart

An individuals chart is proposed in which the mean level is estimated using the regression smoother LOWESS, and deviations from this mean estimate are plotted. This LOWESS-range (LR) chart yields an estimator of the process variance from stable measurements, which is almost as efficient as the estimator based on the sample standard deviation. The estimator often has lower mean squared-error than a moving-range-based estimator when applied to mean-shifted measurements.     

The beta exponential distribution

The exponential distribution is perhaps the most widely applied statistical distribution for problems in reliability. In this note, we introduce a generalization—referred to as the beta exponential distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the beta exponential distribution. We derive expressions for the moment generating function, characteristic function, the first four moments, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, the distribution of sums and ratios, and the asymptotic distribution of the extreme order statistics. We also discuss simulation issues, estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix. We hope that this generalization will attract wider applicability in reliability.     

Estimation of mean frequency and variance of ultrasonic Doppler signal by using second-order autoregressive model

In order to estimate the mean frequency and variance of the diagnostic ultrasound Doppler signal in the presence of clutter noise, a new estimator using a second-order autoregressive (AR) model, called the AR estimator, is proposed. The sampled signal that contains information of both the Doppler signal and clutter is described by the second-order AR model with two poles. The mean frequency and variance of a unidirectional Doppler signal can be estimated, respectively, from the phase and the magnitude of the pole, with larger phase between the two poles. If the clutter is not completely rejected, all conventional estimators, including the autocorrelation (AC) estimator, result in erroneous estimations for the mean frequency and variance of the Doppler signal, whereas the AR estimator gives an accurate estimation. In the absence of clutter, however, the performance of both the AC and AR estimators are similar. If the blood flows in both directions in a sample volume and the clutter is rejected to the extent that it no longer obscures the Doppler signal, the proposed method can estimate simultaneously the mean frequencies and variances of both the forward and reverse blood flows. The performance of the proposed AR estimator was compared with that of the AC estimator by computer simulations and experiments, and it was found that when the number of available sampled data is small, the AR estimator does not require the use of a clutter filter, which simplifies Doppler signal detection.     

16th International Conference on Production Research (ICPR-16) Manufacturing-Service-People-Sustainable Systems

Robust design uses the ordinary least squares method to obtain adequate response functions for the process mean and variance by assuming that experimental data are normally distributed and that there is no major contamination in the data set. Under these assumptions, the sample mean and variance are often used to estimate the process mean and variance. In practice, the above assumptions are not always satisfied. When these assumptions are violated, one can alternatively use the sample median and median absolute deviation to estimate the process mean and variance. However, the median and median absolute deviation both suffer from a lack of efficiency under the normal distribution, although they are fairly outlier-resistant. To remedy this problem, we propose new robust design methods based on a highly efficient and outlier-resistant estimator. Numerical studies substantiate the new methods developed and compare the performance of the proposed methods with the ordinary dual-response robust design.     

A new adaptive mean frequency estimator: application to constant variance color flow mapping

A class of adapted mean frequency estimators is proposed for color flow mapping. These estimators can be fitted to the specific characteristics of a given Doppler signal to optimize the compromise between the range of analysable frequencies and the variance of mean frequency estimation. A sub-optimal estimator is derived for real-time applications, and an adaptive criterion based on the Doppler signal variance is developed for color flow mapping applications. Its performance is compared to that of the usual correlation phase estimator on simulated Doppler signals and on synthetic Doppler images. An improvement in image quality is achieved, mainly for low signal-to-noise ratio Doppler signals.     

General control charts for attributes

Traditional Shewhart-type control charts ignore the skewness of the plotted statistic. Occasionally, the skewness is too large to be ignored, and in such cases the classical Shewhart chart ceases to deliver satisfactory performance. In this paper, we develop a general framework for constructing Shewhart-like control charts for attributes based on fitting a quantile function that preserves all first three moments of the plotted statistic. Furthermore, these moments enter explicitly into the formulae for calculating the limits. To enhance the accuracy of these limits the value of the skewness measure used in the calculations is inflated by 44%. This inflation rate delivers accurate control limits for diversely-shaped attribute distributions like the binomial, the Poisson, the geometric and the negative binomial. A new control chart for the M/M/S queueing model is developed and its performance evaluated.     

Properties of Squeezed Binomial States and Squeezed Negative Binomial States

Abstract

The effect of squeezing on binomial states and on negative binomial states is studied in terms of second-order correlation functions and quasi-probabilities of Wigner and Q-functions. The photon number distribution of these states is also discussed. The results presented for squeezed binomial (negative binomial) states may be useful for studying a transition from squeezed coherent states to squeezed number (quasi-thermal) states.     

James-Stein estimators for failure rates and probabilities

A basic concept in this paper is that a group of components under study represents a sample from a common source (e.g. a factory) that has a failure rate distribution, with distribution parameters originally unknown. This concept is used to obtain generalized James-Stein estimators for individual component parameters. Such ‘shrinkage estimators’ are more accurate than conventional point estimates. They are developed here for component failure rate parameters based on Poisson data, and for failure probabilities based on binomial data. Optimal weights and shrinking factors are obtained as well as formulae for estimating the mean value and the variance of the source population, and the mean values and the variances of individual component estimates. Close connections to the empirical Bayesian estimation are shown     

Postgrouped sampling method of estimation

A postgrouped sampling is considered for estimating the (finite or finite), population mean. Double sampling and an empirical-weighted estimator is used. Unbiasedness, variance and efficiency are considered. Its properties are discussed allowing the simple random sampling with replacement (SRSWR) design in the first phase, and in each stratum for the second phase. It is shown that for a fixed sample size in each postgroup, the variance of the proposed estimator with less prior information is asymptotically equivalent to the usual stratified estimator for fixed allocation. Some examples are provided for natural populations., The method is also extended to simple random sampling without replacement (SRSWOR) design in the first phase, and in each stratum for the second phase. Unbiased variance estimation is provided for both types of sampling designs.     

Memory-type control charts with multiple auxiliary information for process mean

Memory-type auxiliary-information-based (AIB) control charts are very effective in detecting small-to-moderate shifts in the process mean. In this study, we first develop a unique uniformly minimum variance unbiased estimator of the process mean that requires information on the study variable as well as on several correlated auxiliary variables. Then, based on this estimator, adaptive and nonadaptive CUSUM and EWMA charts are developed with either fixed or variable sampling interval for monitoring the process mean, namely, the multiple AIB (MAIB) charts. The proposed charts encompass existing charts with or without the auxiliary information. The run length characteristics of the proposed charts are computed with the Monte Carlo simulations when sampling from a multivariate normal distribution. Based on the run length comparisons, it is found that the MAIB charts are uniformly and substantially more sensitive than the AIB charts when monitoring the process mean. Real datasets are also considered to explain the implementation of the MAIB charts.