In this paper, we study to approximate the homomorphisms and derivations for 3-variable Cauchy functional equations in RC-algebras and Lie RC-algebras by the fixed point method.

In this paper, an approximate tolerance interval is presented for the discrete size-biased Poisson-Lindley distribution. This approximate tolerance interval, is constructed based on large sample Wald confidence interval for the parameter of the size-biased Poisson-Lindley distribution. Then, coverage probabilities and expected widths of the proposed tolerance interval is considered. The results show that the coverage probabilities have a better performance for the small values of the parameter and are close to the nominal confidence level, and are conservative for the large values of the parameter. Finally, an applicable example is provided for illustrating approximate tolerance interval.

The aim of this paper is to propose a convex risk measure in the framework of fuzzy random theory and verify its advantage over the conventional variance approach. For this purpose, this paper defines the quadratic deviation (QD) of fuzzy random variable as the mathematical expectation of QDs of fuzzy variables. As a result, the new risk criterion essentially describes the variation of a fuzzy random variable around its expected value. For triangular and trapezoidal fuzzy random variables as well as their linear combinations, we establish the analytical expressions of their QDs, and obtain the desirable convexity about the analytical expressions with respect to critical parameters. To explore the practical value of the proposed QD, we apply it to a portfolio selection problem to quantify the investment risk, and develop three mean-QD models to find the optimal allocation of the fund in different risky securities. Due to the convexity of our QD, the original three mean-QD models can be turned into their equivalent convex parametric quadratic programming problems, which can be solved by conventional optimization methods. The computational results clearly demonstrate that our new QD significantly reduces the computational complexity that cannot be avoided when variance is used as a risk criterion. Finally, the numerical comparison between the proposed mean-QD model and mean-variance model is conducted to show the consistency between the optimal results in both techniques. Meanwhile, the comparison between the proposed QD, variance, spread, and second moment is made to summarize the similarities and differences between them, distinguish these four risk criteria and determine their respective application scopes in decision systems

Summary In this paper, Probability Density Function (PDF) for thickness of the shotcrete cover in tunnel temporary support is investigated. Based on collected data from a recently constructed tunnel in Iran, the proper PDF is suggested. To achieve this, three goodness of fit statistical methods are considered to find the appropriate distribution function. The results show that the Generalized Extreme Value distribution function has the best rank with respect to other distribution functions. Introduction One of the most important issues in the reliability analysis of the structures is proper selection of the distribution function for existing random variables. In tunnels, the problem of temporary supporting system involves many random variables including thickness of the shotcrete. In this research, based on the data obtained from an existing tunnel in Iran, the probability distribution function for the shotcrete thickness is determined by using curve fitting techniques. Methodology and Approaches Different distribution functions have been studied and compared based on three well-known goodness of fit tests, called Chi-Squared, Kolmogorov Smirnov and Anderson Darling. For this purpose, nine points in each section and 43 individual sections are considered in the tunnel route. It is reasonable an individual point in all sections is considered to obtain a distribution function. Also, 15 types of distribution functions are chosen to have an appropriate fit over the data. The EasyFit software is utilized for assessment and finding the appropriate distribution function using the three mentioned goodness of fit tests. Results and Conclusions According to the results, the Generalized Extreme Value Distribution is recommended for PDF of shotcrete thickness in tunnel supporting systems. It is noted that this distribution function has the best rank in all three goodness of fit tests. Although the results are obtained by using data from a specific tunnel project, they can be improved by considering more data from other projects. The distribution function can be used to obtain uncertainties of thickness of shotcrete in reliability analysis of tunnels.

This paper introduces a new two-stage randomized response model for estimating the mean of a sensitive quantitative random variable. The proposed model is obtained for both simple and stratified random sampling. The efficiency of the proposed estimator, under both sampling schemes, is investigated with respect to various estimators and it is found to be more efficient. Moreover, a new measure for evaluating the performance of any randomized response estimator is introduced. The measure considers the relative efficiency of the randomized response estimator, and the privacy protection it offers. The performance of the proposed estimator is examined using the new measure and it is found to have an overall better performance than its rival estimators. A real data example is also examined using the proposed model and various models from literature.

Bachground :While the main advantage of the confidence interval is that it enables more precise evaluations when the risk for the outcome of interest is related to the cluster size, the predicted confidence interval width demonstrates the degree of variability in the future data instead. Methods : We present a novel algorithm to create an intra‐cluster robust neighborhood confidence interval and width for each cluster to rank the widths from the narrowest to the widest width to determine each cluster's predicted variability and evaluate the corresponding observed values. An example was developed that assesses the finite‐sample behavior of this new method. Results : Robust neighborhood intra-cluster predicted CI width was obtained   for interpreting results  of  binary unequal sizes data. Narrow confidence intervals CI bounds suggest the results are not subjected to a high degree of random variations. Conclusions: Intra-cluster predicted robust neighborhood CI and its corresponding width is a useful instrument in binary outcome unequal cluster sizes data as a method of analysis.