The EM ALGORITHM is a powerful tool and generic useful device in a variety of problems for maximum likelihood estimation with incomplete data which usually appears in practice. Here, the term ``incomplete" means a general state and in different situations it can mean different meanings, such as lost data, open source data, censored observations, etc. This paper introduces an application of the EM ALGORITHM in which the meaning of ``incomplete" data is non-precise or fuzzy observations. The proposed approach in this paper for estimating an unknown parameter in the parametric statistical model by maximizing the likelihood function based on fuzzy observations. Meanwhile, this article presents a case study in the electronics industry, which is an extension of a well-known example used in introductions to the EM ALGORITHM and focuses on the applicability of the ALGORITHM in a fuzzy environment. This paper can be useful for graduate students to understand the subject in fuzzy environment and moreover to use the EM ALGORITHM in more complex examples.

In Hierarchical Modulation (HM), the binary data is partitioned into a “ high-priority” (HP) and a “ lowpriority” (LP) bit stream that are separately and independently encoded before being mapped on non-uniformly spaced constellation points. Indeed, HM is a multi-level modulation scheme, proposed for enabling unequal error protection. The HP and LP streams vary in their susceptibility to noise. The basic HP signal is more robust, in other words, heavily protected against noise and interference, whereas the LP stream has much less robustness. In this paper, we propose an improved iterative receiver based on variational Bayesian expectation-maximization (VB-EM) scheme for semi-blind joint channel estimation and data detection based on bit-interleaved coded modulation (BICM), with the aim of improving the bit error rate (BER) of the noisy LP stream. Moreover, the proposed receiver allows us to reduce the pilot symbol overhead compared to the classical HM receivers based on pilot-assisted channel estimation.

The relation between single nucleotide polymorphisms (SNPs) and some diseases has been concerned by many researchers. Also the missing SNPs are quite common in genetic association studies. Hence, this article investigates the relation between existing SNPs in DNMT1 of human chromosome 19 with colorectal cancer. This article aims is to presents an imputation METHOD for missing SNPs not at random. In this case-control study, 100 patients suffering from colorectal cancer consulting with the Research Institute for Gastroenterology and Liver Disease of Shahid Beheshti University of Medical Sciences were considered as the case group and 100 other patients consulting with the same research institute were considered as the control group and the genetic test was applied in order to identify the genotype of the 6 SNPs of the DNMT1 of chromosome 19 for all the patients under investigation. The obtained data were analyzed using logistic regression, then a fraction of the data was eliminated both at random and not at random and the imputation was done through the EM ALGORITHM and the logistic regression coefficients variation before and after the imputation was compared. The results of this study implied that in both METHODs, at random and not at random missing SNPs, the estimation of the logistic regression coefficients after the imputation through EM ALGORITHM has a greater correspondence to the results obtained from the complete data in comparison with the METHOD of eliminating the missing values.

Discrete Phase-Type (DPH) distributions have one property that is not shared by Continuous Phase-Type (CPH) distributions, i. e., representing a deterministic value as a DPH random variable. This property distinguishes the application of DPH in stochastic modeling of real-life problems, such as stochastic scheduling, in which service time random variables should be compared with a deadline that is usually a constant value. In this paper, we consider a restricted class of DPH distributions, called Mixed Shifted Negative Binomial (MSNB), and show its exibility in producing a wide range of variances as well as its adequacy in  tting fat-tailed distributions. These properties render MSNB applicable to represent data on certain types of service time. Therefore, we adapt an Expectation-Maximization (EM) ALGORITHM to estimate the parameters of MSNB distributions that accurately  t trace data. To present the applicability of the proposed ALGORITHM, we use it to  t real operating room times and a set of benchmark traces generated from continuous distributions as case studies. Finally, we illustrate the e ciency of the proposed ALGORITHM by comparing its results with those of two existing ALGORITHMs in the literature. We conclude that our proposed ALGORITHM outperforms other DPH ALGORITHMs in  tting trace data and distributions.

In recent years, bivariate lifetime distributions are often used to model reli-ability and survival data. In this paper, we introduce a bivariate Burr III distribution, so that the marginals have Burr III distributions. It is observed that the joint probabil-ity density function, the joint cumulative distribution function and the joint survival distribution function can be expressed in compact forms. We suggest to use the ECM ALGORITHM to compute the maximum likelihood estimators of the unknown parame-ters. We report some simulation results and perform one data analysis for illustrative purposes.