Background & Aim: Consider a sequence of independent Bernoulli trials with p denoting the probability of success at each trial. With this definition, the probability that the n th success proceed by r failures follows the negative binomial distribution (NB). NB model has been derived from two different forms. At first, the NB can be thought as a Poisson-gamma mixture. The second form of the NB can be derived as a full member of a single parameter exponential family distribution, and therefore considered as a GLM (generalized linear models).Methods & Materials: We have described a new generalized NB (GNB) distribution with three parameters a, b and k obtained as a compound form of the generalized Poisson and gamma distributions. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. The most important feature of this model is, its time dependent probabilities, and also it can be used for a variety of researches especially in the survival analysis.Results: This model has been illustrated with two datasets that are indirect measures of illness, along comparing the results of the fitting with NB. Results indicate too much satisfaction. Expected frequencies have been calculated for these data sets to show that the distribution provides a very satisfactory fit in different situations.Conclusion: Using GNB models allows analyzing very complex data. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. With k=0 the model becomes the ordinary NB and with a=1, it becomes a new model which we call it the generalized geometric distribution with two parameters. The most important feature of this model is its time-dependent probabilities.

Sir,Psoriasis aff ects 2-3% of world’s population.Tuberculosis is the 7th cause of death all over the world, one-third of population is affected by the disease. Romania, in 2010, occupied the 6th position, after Kazakhstan, Moldavia, Georgia, Kirgizstan, and Tajikistan.[3] In 2011, a large retrospective study was carried out on the incidence of present and past tuberculosis within 1236 psoriasis patients, not previously treated with biologics, over 2004-2011.

Historically, various methods were suggested for the estimation of Bernoulli and Binomial distributions parameter. One of the suggested methods is the Bayesian method, which is based on employing prior distribution. Their sound selection on parameter space play a crucial role in reducing posterior Bayesian estimator error. At times, large scale of the parametric changes on parameter space brings about an increase in error rate and enlargement of the comparison criteria .therefore, determining appropriate prior distribution on parameter space plays a key part in reducing comparison criteria. Accordingly, in this paper appropriately modified prior distribution was considered for Binomial distribution parameter and then, while extending certain conditions on prior distribution hyper parameters, an effective estimate under the title Expected Bayesian Estimate (AKA E-Bayes) would be proposed. At last, to evaluate the measurement methodologies utilized in this paper, under MSE criteria, extensive simulation studies were conducted, and the results were analyzed and above mentioned methods will be applied in a real example.

In order to facilitate the interpretation of raw scores, they are usually converted to scale scores. In some cases, these conversions are a series of nonlinear transformations that can affect the conditional standard error of measurement throughout the scale of score. Therefore, the purpose of this study was to introduce methods for calculating the conditional standard error of measurement based on the strong true score theory. Furthermore, comparison of normalized and equipercentile nonlinear transformations on the raw scores of the academic achievements of the graduates of mathematical sciences in 2014 and their effect on conditional standard error of measurement was also conducted. So, in order to achieve these purposes, we used a sample of 3943 high school graduates of Mathematics and Physics in 2014 who had participated in national university entrance examination in 2015 randomly selected by National Organization of Educational Testing. The conditional standard error of measurement under these transformations was estimated based on the binomial procedure of Brennan and Lee (1999) and Chang (2006) method based on the beta-binomial distribution. The results of this study indicated that the conditional standard error of measurement of the Chang was smoother than binomial procedure, but in both methods the estimated errors are larger for middle points and smaller for extreme points. Additionally, the conditional standard errors of measurement of equipercentile were always less than normalized tranformation, so the equipercentile method found to be better than normalized transformation.

The beta-binomial distribution is resulted when the probability of success per trial in the binomial distribution varies in successive trials and the mixing distribution is from the beta family. For experiments with binary outcomes, often it may happen that observations exhibit some extra binomial variation and occur in clusters. In such experiments the beta-binomial distribution can generally provide an adequate fit to the data. Here, we introduce an alternative when the mixing distribution is assumed to be from the log-Lindley family. The properties of this new model are explored and it is shown that similar to the beta-binomial distribution, the log-Lindley binomial distribution can also be applied in modeling clustered binary outcomes. An example with real experimental data from a developmental toxicity experiment is utilized to provide further illustration.

n this paper, we define a new Ma-Minda type class based on binomial distribution series. Our investigation will be focused on the coefficientsof the function f belonging to that class.

In this paper, using extended Weibull Marshall-Olkin-Nadarajah family of distributions, the exponential, modified Weibull, and Gompertz distributions are obtained, and density, survival, and hazard functions are simulated. Next, an algorithm is presented for the simulation of these distributions. For exponential case, Bayesian statistics under squared error, entropy Linex, squared error loss functions and modified Linex are calculated. Finally, the presented distributions are fitted to a real data set.

The estimation of unknown parameters of two-parameter Rayleigh distribution based on Type-II progressive censoring with binomial removals is studied. Maximum likelihood estimators of the parameters and their confidence intervals are derived. By applying Markov Chain Monte Carlo techniques, Bayes estimators, and corresponding highest posterior density confidence intervals of parameters are obtained. The expected time required to complete the life test under this censoring scheme is investigated. Monte Carlo simulations are performed to compare the performances of the different methods, and one data set is analyzed for illustrative purposes.