Learning Bayesian network structure from data has attracted a great deal of research in recent years. It is shown that finding the optimal network is an NP-hard problem when data is complete. This problem gets worse when data is incomplete i.e. contains missing Values and/or hidden variables. Generally, there are two cases of learning Bayesian networks from incomplete data: in a known structure, and unknown structure. In this paper, we try to find the best Parameters for a known structure by introducing the “effective Parameter”, in a way that the likelihood of the network structure given the completed data being maximized. This approach can be attached to any algorithm such as SEM (structural expectation maximization) that needs the best Parameters to be known to reach the optimal Bayesian network structure. We prove that the proposed method gains the optimal Parameters with respect to the likelihood function. Results of applying the proposed method to some known Bayesian networks show the speed of the proposed method compared to the well-known methods.

The use of the two-Parameter exponential distribution model in fitting survival and reliability analysis data in the presence of censored Random data has recently attracted the attention of a large number of authors. Considering the importance of the model, its Parameter estimation is discussed using the method of moment, maximum likelihood and shrinkage estimation. To present the interval shrinkage estimator, it is first proved that the moment estimators are asymptotically unbiased and the interval shrinkage estimator performs better compared to other estimators. Finally, using two real data sets and statistical criteria, the goodness of fit of the model is compared with censored Random data based on Parameter estimation methods.

In this article a spatial model is presented for extreme Values with marginal generalized extreme value (GEV) distribution. The spatial model would be able to capture the multi-scale spatial dependencies. The small scale dependencies in this model is modeled by means of copula function and then in a hierarchical manner a Random field is related to location Parameters of marginal GEV distributions in order to account for large scale dependencies. Bayesian inference of presented model is accomplished by offered Markov chain Monte Carlo (MCMC) design, which consisted of Gibbs sampler, Random walk Metropolis-Hastings and adaptive independence sampler algorithms. In proposed MCMC design the vector of location Parameters is updated simultaneously based on devised multivariate proposal distribution. Also, we attain Bayesian spatial prediction by approximation of the predictive distribution. Finally, the estimation of model Parameters and possibilities for capturing and separation of multi-scale spatial dependencies are investigated in a simulation example and analysis of wind speed extremes.

Objective: Due to worldwide variations, reference Values of urinary calcium to creatinine ratio in pediatric population are not yet well established. To determine normal Values for urinary calcium to creatinine ratio and its relation to urinary sodium or potassium, a descriptive (correlation type) study was conducted in 7 to 12 years old healthy children in Urmia, Iran.Methods: Primary school children were divided into two sectors and 7 clusters (4 cluster school boys and 3 school girls). The subjects were Randomly selected. Random, non–fasting morning urine samples were obtained from 364 healthy children aged 7 to 12 years during fall 2005 and immediately sent to laboratory to determine urine calcium (Uca), creatinine(Cr), sodium (Na) and potassium (K). For data analysis, mean and 95th percentile of UCa/Cr and UNa/K were used. Pearson test was used to determine any relationship between UCa/Cr and UNa/K Values. For comparison of UCa/Cr and UNa/K Values between males and females, Mann- Withny test was used.Findings: A total number of 364 children were enrolled in the study. There were 208 (57.1%) males and 156 (42.9%) females. The mean and 95th Percentile for UCa/Cr was 0.11 (0.10 and 0.24 respectively. The mean and 95th percentiles for UNa/K were 2.30 (1.42 and 5.21 respectively. There was no significant difference in UCa/Cr and UNa/K between two sexes (P>0.05). We found a weak relationship between UCa/Cr and UNa/K (P<0.01).Conclusion: UCa/Cr value may differ according to geographic location. For screening purposes, reference Values should be determined in each geographic location.

Introduction A general censoring scheme called progressive Type-II right censoring has been considered. The removal plan can be fixed or Random, chosen according to a discrete probability distribution. In many practical problems, not only does an experiment process determines inevitably to use Random removals, but also a fixed removal assumption may be cumbersome to analyze some results of statistical inference. The scenario of Random removals has been introduced by Yuen and Tse (1996) under the Weibull lifetime distribution and the discrete Uniform distribution for Random removals. Tse et al. (2000) discussed Binomial removals even though the Parameter p enormously impressed the experiment time, and the Uniform and Binomial distributions were independent of the lifetime distribution. The limitations mentioned above motivate us to propose a new method for determining removals based on the failure times. Material and Methods Let the lifetimes of the n units placed on the life-test be distributed as two-Parameter Weibull distribution. The proposed Random removals use the relationship between the Weibull and Exponential and are based on two approaches: the normalized spacings with Random and fixed coefficients according to progressively Type-II censored order statistics from the Exponential distribution. Wherein the time distance between consecutive failure times depends on the type of lifetime distribution and the number of units that will be removed after each failure are proportional to a root function of the difference between two last failure times divided by the time of the first failure. The joint probability mass functions of Random removals are also derived. The estimations of Parameters are derived using different estimation procedures such as the maximum likelihood, maximum product spacing, and least-squares methods. The proposed Random removal schemes are compared to the discrete Uniform and the Binomial removal schemes via a Monte Carlo simulation study in terms of their biases, root mean squared errors of estimators, expected total test times and the Ratio of the Expected Experiment Time (REET) Values. Finally, an innovative technique is introduced for deriving progressive type II censoring samples from a real data set. Results and Discussion From comparing the REET Values, it is evident that a slight reduction in expected experiment time occurs when a large number of units are tested for lifetimes under Uniform and Binomial distributions with a considerable probability, p, especially for cases with decreasing failure rate ,> 1. Although the Binomial distribution with p < 0: 5 has relatively acceptable performance, two proposed approaches have smaller REET Values, which decreases significantly as the sample size n increases. However, binomial removals perform better than uniform removals in terms of E(Xm: m: n). Still, the expected test time depends very much on the value of removal probability p. Conclusion It is shown that the expected total time under the Random coefficients has the most negligible value concerning other approaches and reduces the expected full time on the test.

In this paper, we study the estimation problems for the two-Parameter exponentiated Gumbel distribution based on lower record Values. An exact confidence interval and an exact joint confidence region for the Parameters are constructed. A simulation study is conducted to study the performance of the proposed confidence interval and region. Finally, a numerical example with real data set is given to illustrate the proposed procedures.

Spatial generalized linear mixed models are used commonly for modelling non-Gaussian discrete spatial responses. We present an algorithm for Parameter estimation of the models using Laplace approximation of likelihood function. In these models, the spatial correlation structure of data is carried out by Random effects or latent variables. In most spatial analysis, it is assumed that Random effects have Gaussian distribution, but the assumption is questionable. This assumption is replaced in the present work, using a skew Gaussian distribution for the latent variables, which is more flexible and includes Gaussian distribution. We examine the proposed method using a real discrete data set.

We deal with a Sazonov space (X: real separable) valued symmetric alpha stable Random measure Phi with independent increments on the measurable space (R-k, B(R-k)). A pair (k, mu), called here a control pair, for which k : X x R-k -> R+, mu a positive measure on (R-k, B(R-k)), is introduced. It is proved that the law of Phi is governed by a control pair; and every control pair will induce such Phi. Moreover, k is unique for a given mu. Our derivations are based on the Generalized Bochner Theorem and the Radon-Nikodym Theorem for vector measures.