New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel.

This paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The $ \nu- $monotonicity definitions, namely $ \nu- $(strictly) increasing and $ \nu- $(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with $ \nu- $monotonicity definitions, we find that the investigated discrete fractional operators will be $ \nu^2- $(strictly) increasing or $ \nu^2- $(strictly) decreasing in certain domains of the time scale $ \mathbb{N}_a: = \{a, a+1, \dots\} $. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.

Bayesian and Frequentist Inferences on a Type I Half-Logistic Odd Weibull Generator with Applications in Engineering.

In this article, we have proposed a new generalization of the odd Weibull-G family by consolidating two notable families of distributions. We have derived various mathematical properties of the proposed family, including quantile function, skewness, kurtosis, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves, probability weighted moments, moments of (reversed) residual lifetime, entropy and order statistics. After producing the general class, two of the corresponding parametric statistical models are outlined. The hazard rate function of the sub-models can take a variety of shapes such as increasing, decreasing, unimodal, and Bathtub shaped, for different values of the parameters. Furthermore, the sub-models of the introduced family are also capable of modelling symmetric and skewed data. The parameter estimation of the special models are discussed by numerous methods, namely, the maximum likelihood, simple least squares, weighted least squares, Cramér-von Mises, and Bayesian estimation. Under the Bayesian framework, we have used informative and non-informative priors to obtain Bayes estimates of unknown parameters with the squared error and generalized entropy loss functions. An extensive Monte Carlo simulation is conducted to assess the effectiveness of these estimation techniques. The applicability of two sub-models of the proposed family is illustrated by means of two real data sets.

A New Family of Continuous Probability Distributions.

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of "Farlie-Gumbel-Morgenstern copula", "the modified Farlie-Gumbel-Morgenstern copula", "the Clayton copula", and "the Renyi's entropy copula" are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family.

Analysis of Heterozygous BRCA1 5382ins Founder Mutation in a Cohort of Egyptian Breast Cancer Female Patients Using Pyrosequencing Technique.

BACKGROUND: Up to half of the heritable mutations in breast cancer (BC) are attributed to BRCA1 and BRCA2 genes. The mutation prevalence is variable based on ethnicity and may be influenced by founder mutations. The aim of this pilot study is to determine for the first time, the prevalence of BRCA1 5382insC founder mutation in a cohort of Egyptian familial breast cancer patients (FBC). METHODS: Female patients were selected to have familial type of breast cancer. Twenty healthy females were included as a control group. Peripheral blood samples were withdrawn from all studied females and were analyzed for BRCA1 5382insC founder mutation detection using pyrosequencing technique. RESULTS: Eighty Egyptian FBC females were eligible to be enrolled in the study with a mean age of 48.31 ± 10.97years.We found a BRCA1 5382insC mutation carrier frequency of 5% of total studied FBC patients (4 out of 80 patients) with 95% confidence interval (1.61-12.99). There was a high statistical significant difference between carriers and non-carriers concerning the number of affected family members by BC, (p=0.001).  Conclusion: BRCA1 5382insC founder mutation is not uncommon among Egyptian FBC females. The carrier frequency is comparable to that reported worldwide; however it is lower than those from previous Egyptian studies using different molecular techniques. The strong association between the mutation and the number of affected family members suggest wider screening of the mutation among high risk families using the reliable pyrosequencing technique.