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In this article, the optimal model is determined for the family of models ‎$‎‎‎{E_r/M/1, rin N}‎$‎ with interarrival times with Erlang distribution‎‏ ‎and service times with exponential distribution‎. The method of choosing the optimal model is that first, a cost function is introduced, and then a new index is introduced according to the cost function and the ‎stationary‎ probability of the system called ‎‏‎‎$‎‎‎SER‎$‎.‎‏ A model with a larger ‎$‎SER‎$‎ index is ‎optimal.‎ Numerical analysis is also used to describe the method of determining optimal ‎model.

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Regression analysis using the method of least squares requires the establishment of basic assumptions. One of the problems of regression analysis in this way
faces major problems is the existence of collinearity among the regression variables. Many methods to solve the problems caused by the existence of the same have been introduced linearly. One of these methods is ridge regression. In this article, a new estimate for the ridge parameter using generalized maximum Tsallis entropy is presented and we call it the Ridge estimator of generalized maximum Tsallis entropy. For the cement dataset
Portland, which have strong collinearity and since 1332, different estimators have been presented for these data, this estimator is calculated and
We compare the generalized maximum Tsallis entropy ridge estimator, generalized maximum entropy ridge estimator and the least squares estimator.

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Risk means a situation in which there is a possibility of deviation from a predicted result. Insurance is one of the methods of risk exposure that leads to the transfer of all or part of the risk from the insured to the insurer. Insurance policies are usually classified into two types: personal and non-life (non-life) insurance. According to this classification, any insurance policy that deals with the health of the insured person or persons is a personal insurance policy, otherwise it will be a nonlife insurance policy. Many insurances in the insurance industry are classified as non-life insurances. Fire, automobile, engineering, shipping, oil and energy insurances are examples of these insurances. Explanation and calculation of three issues in risk models are very important: the ruin probability, the time of ruin and the amount of ruin. In this article, the main and well-known results that have been obtained so far in the field of non-life insurance; Emphasizing the possibility of ruin, it is given along with various examples. 

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Bootstrap is a computer-based resampling and statistical inference method that can provide an estimate for the uncertainty of distribution parameters and quantiles in a frequency analysis. In this article, in addition to calculating the bootstrap confidence interval for the quantiles with percentile bootstrap(BP), accelerated bias-corrected bootstrap(BCA) and t-bootstrap methods, calculating of the confidence interval is proposed using the bootstrap highest density method(HDI) for the probability distributions used in the hydrology data and we obtain the average length of the confidence interval as a criterion for evaluating the methods. To calculate the average length of the confidence interval with different methods, first, the best distribution among the widely used distributions is fitted to the original data, and the parameters of the fitted distribution are estimated by the maximum likelihood method, and from that we obtain the quantiles. Then we continue by repeating the simulated bootstrap samples until the probability of covering the real quantile reaches the nominal confidence level of 0.95. The simulation results show that the bootstrap highest density method gives the lowest average confidence interval length among all methods.
In previous studies, the probability of coverage with the same number of samples as the original sample and the number of bootstrap repetitions (for example, 1000) have been obtained, and finally the coverage probability closest to the nominal value of 0.95 was chosen as the optimal state, while in this article, for Avoiding a large number of bootstrap iterations, considering the number of different samples n = 20, 40, 60, ..., 160 that are simulated from the mother distribution, we continue the number of iterations of the bootstrap samples only until reaching the coverage probability of 0.95. Usually, two-parameter distributions require fewer samples than three-parameter distributions. In terms of the number of necessary repetitions (B) to reach the 95% confidence level, the t-bootstrap method usually required fewer repetitions, although the implement time of this method was longer in R software. The best distribution for the data used in this research is the two-parameter distribution of Frechet, which was selected as the best distribution in 80% of the stations, which can be used for similar studies that deal with the maximum annual precipitation values. According to the fitted distributions for the data of different stations, the two-parameter distributions always fit the data better than the three-parameter distributions. In general, the widest confidence intervals were obtained with BCA and tbootstrap methods, and the shortest confidence intervals were obtained with HDI and BP methods. Also, in terms of the distribution used and the length of the confidence interval, the two-parameter Gamma distribution has provided shorter confidence intervals and the three-parameter GEV and LP3 distributions have provided wider confidence intervals.
 

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Today, the diagnosis of diseases using artificial intelligence and machine learning algorithms are of great importance, because by using the data available in the study field of the desired disease, useful information and results can be obtained that reduce the occurrence of many deaths. Among these diseases, we can mention the diagnosis of diabetes, which has spread today due to the growth of urban life and the decrease in people's activity. So, it is very important to know whether a person is suffering from diabetes or not. In this article, the data set related to the information of people who have done the diabetes diagnosis test is used, this information is related to 520 people. People are classified into two groups based on whether their diabetes test result is positive or not, and Bayesian classification methods such as Bayesian Support Vector Machine, Naive Bayes, CNK and CatBoost ensemble classification method have been used to conclude which of these The methods can have a better ability to analyze the data and also to compare these methods use accuracy, precision, F1-score, recall, ROC diagram.

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The importance of statistics and its education in today’s world, which is full of information and data, is not hidden from
anyone. Statistical thinking is the main core of correct understanding of statistical concepts, data analysis and interpretation
of phenomena. With the aim of achieving a comprehensive definition of statistical thinking and determining its elements,
the present research has studied the researches of the last thirty years. This descriptive research has been carried out in a
qualitative metacomposite
method to provide an insight into the totality of existing studies. Based on the entry criteria,
123 researches were identified between 1990 and 2022, and finally, after screening, 22 researches were selected for detailed
review and analysis. According to the present metacomposite findings, the elements of statistical thinking are: 1) Being dataoriented:
paying attention to data, identifying the need for data, collecting and considering data, different representations of
data, and methods of converting them to each other. 2) Variability: Considering permanent changes in all phenomena. 3)
Statistical inference: paying attention to the types of sampling, reasoning and inference using statistical models, including the
use of statistical charts and generalizing the results from the sample to the population. [4) Analysis of the statistical context:
combining the statistical problem with the context.

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In ‎this ‎article, ‎queunig ‎model ‎‎$‎M/M/1‎$ ‎is ‎Considered, ‎in ‎which ‎the ‎innterarrival ‎of ‎customers ‎have ‎an ‎exponenial ‎disributon ‎with ‎the ‎parameter ‎‎$‎lambda‎$ ‎and ‎the ‎service ‎times‎ ‎have ‎an ‎exponenial ‎disributon with the ‎parameter ‎‎$‎mu‎$ ‎and ‎are ‎independent ‎of ‎the ‎interarrival ‎times.‎ ‎it ‎is ‎also ‎assumed ‎that ‎the ‎system ‎is ‎active ‎until ‎‎$‎T‎$‎. Then, under this stopping time Bayesian, ‎$‎E‎$‎-Bayesian and hierarchical Bayesian estima‎‏‎tion‎s of the traffic intensity parameter of this queuing model are obtained under the general entropy loss function and considering the gamma and erlang prior distributions for parameters ‎$‎lambda‎$ ‎and ‎‎$‎mu‎$‎, respicctively. Then, using numerical analysis and based on a new index, Bayesian, ‎$‎E‎$‎-Bayesian and hierarchical Bayesian estima‎‏‎tion‎s are compared.

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This paper explore some extropy properties of the lifetime of coherent systems with the assumption that the lifetime distribution of system components are independent and identically distributed. The presented results are obtained using the concept of system signature. To this aim, we first provide an expression for extropy of the lifetime of coherent systems. Then, stochastic extropy comparisons are discussed for coherent systems under the condition that both systems have the same characteristic structure. In cases where the number of system components is large or the system has a complex structure, it is difficult or time-consuming to obtain the exact extropy value of the system lifetime. Therefore, bounds are also obtained for extropy. In addition, a new criterion for selecting a preferred system based on relative extropy is proposed, which considers the lifetime of the desired system closest to the parallel system.

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Actuarial studies treat insurance losses as random variables, and appropriate probabilistic models are sought to model them. Since losses are evaluated in terms of a unitary amount, distributions with positive support are typically used to model them. However, in practice, losses are often bounded due to policyholder conditions, which must be considered when modeling. While this is not a problem for univariate cases, it becomes complicated for multivariate cases. Copulas can be helpful in such situations, but studying the correlation is crucial in the first step. Therefore, this paper addresses the problem of investigating the effect of restricted losses on correlation in multivariate cases.
The Pearson correlation coefficient is a widely used measure of linear correlation between variables. In this study, we examine the correlation between two random variables and investigate the estimator of the correlation coefficient. Furthermore, we analyze a real-world dataset from an Iranian insurance company, including losses due to physical damage and bodily injury as covered by third-party liability insurance.
Upper and lower limits for both the Pearson correlation coefficient and its estimator were derived. The Copula method was employed to obtain the bounds for the correlation parameters, while order statistics were used to obtain the bounds for the sample correlation coefficient. Furthermore, two methods were used to determine the correlation between physical damage and bodily injury, and the results were compared.
Our findings suggest that the commonly used upper and lower bounds of -1 and +1 for the Pearson correlation coefficient may not always apply to insurance losses. Instead, our analysis reveals that narrower bounds can be established for this measure in such cases. The results of this study provide important insights into modeling insurance losses in multivariate cases and have practical implications for risk management and pricing decisions in the insurance industry.
 

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Variance and entropy are distinct metrics that are commonly used to measure the uncertainty of random variables. While the variance shows how a random variable spreads more than expected, the entropy measure measures the uncertainty of an information approach, in other words, it measures the average amount of information of a random variable.

 For both uniform and normal distributions, variance is a ratio of power entropy. Finding such a monotonic relationship between variance and entropy for a larger class of these two distributions is very important and useful in signal processing, machine learning, information theory, probability and statistics, for example, it is used to reduce the errors of estimators and choose a strategy. gives, on average, the greatest or nearly greatest reduction in the entropy of the distribution of the target location, and the effectiveness of this method is tested using simulations with mining assay models. In this article, the upper bound of the variance for single-mode distributions whose tails are heavier than the tails of exponential distributions is created with the help of power entropy

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The statistical inference of the multi-component stress-strength parameter, $R_{s,k}$, is considered in the three-parameter Weibull distribution. The problem is studied in two cases. In the first case, assuming that the stress and strength variables have common shape and location parameters and non-common scale parameters and all these parameters are unknown, the maximum likelihood estimation and the Bayesian estimation of the parameter $‎R_{s,k}$ are investigated. In this case, as the Bayesian estimation does not have a closed form, it is approximated by two methods, Lindley and $mbox{MCMC}$. Also, asymptotic confidence intervals have been obtained. In the second case, assuming that the stress and strength variables have known common shape and location parameters and non-common and unknown scale parameters, the maximum likelihood estimation, the uniformly minimum variance unbiased estimators, the exact Bayesian estimation of the parameter $‎R_{s,k}$ and the asymptotic confidence interval is calculated. Finally, using Monte Carlo simulation, the performance of different estimators has been compared.

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Marshall-Olkin introduced a family of distributions which obtained by adding a parameter into other distributions. Santoz-Neto etal study an extended Weibull distribution. In this paper two Raiyle and Pareto extended weibull are studied under some momemts and Bayesian methods with some loss functions such as squared error, entropy, linex, squared error in logarithm and modified linex. Also the MCMC method are study for these two distributions.

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Circular data are measured in angles or directions. In many cases of sampling, instead of a random sample, we deal with a weighted model. In such sampling, observations are provided throughout with a positive function, weight function. This article deals with weight distributions in circular data. According to von Mises distrinution is the most widely used distribution for modeling circular data, maximum likelihood estimation of parameters in weighted von Mises distributions is investigated. In a simulation study, different weights are compared in the Van Mises circular distribution.

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In this article, logistic regression models are studied in which the response variables are two (or multiple) values and the explanatory variables (predictor or independent) are ordinary variables, but the errors have a vagueness nature in addition to being random. Based on this, we formulate the proposed model and determine the estimation of the coefficients for a case with only one explanatory variable using the method of least squares. In the end, we explain the results with an example.

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Insurance companies are modeled with mathematical and statistical models in terms of their random structure. In this paper, the individual risk model of insurance company with different interest rates in a period of time is considered and assumed that the interest rates have the probability transition matrix with finite and countable state. The finite and infinite time ruin probabilities are computed using the conditional probability on the first claim of density function. Moreover, the upper bounds for the infinite time ruin probability are obtained using the mathematical induction. In the numerical examples, the ruin probabilities for heavy tailed distributions are compared with the obtained probabilities in Bazyari (2022) for the classical individual risk model and also, the infinite time ruin probabilities for light tailed distributions are compared with Lundberg's inequality. The results show that the existence of interest rate with probability transition matrix and having finite state leads to decrease the ruin probabilities.

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‎The cost function and the system ‎s‎tationary‏‎ probability are two key criteria in the design of queuing systems. In this paper, the aim is to design a single server queuing models with infinite capacity, where the service times in the first model and the interarrival times in the second model are assumed to have an Erlang distribution. For this purpose, a new index based on the cost function and the system reliability probability is introduced, the larger of which indicates the optimality of the model. Several numerical examples and an applied example are presented to explain the computational details of the proposed method.

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‎Small and medium enterprises play a significant role in creating job opportunities‎, ‎gross domestic product‎, ‎increasing the production of domestic products and exports‎, ‎and are considered one of the key pillars of achieving the economic growth and development of countries‎. ‎The purpose of this article is to explain the position of SMEs and their contribution in employment‎, ‎economy‎, ‎export and access to banking resources in some selected countries and compare them with the data available in Iran‎. ‎For this purpose‎, ‎the total number of micro and small establishment in the private sector of the country (less than 50 employees) and the number of their persons employed have been estimated using the data of the labour force survey during 10 years period‎, ‎and their values‎ ​‎​are predicted for 1404 using the double exponential smoothing method‎. ‎The results indicate that the number of establishments with less than 50 employees in private sector increased during the period under review‎, ‎and by 1404‎, ‎its ‏‎count‎ will reach 4,596,855‎, ‎so that the share of the number of persons employed in these establishments is predicted to be 82.6 ‎percent‎ in the 1404‎. ‎However‎, ‎the evidence shows that the share of these enterprises in bank facilities over the past years was only 4.5 percent‎, ‎which is very low compared to other countries‎. ‎The findings of this research are helpful for the support programs of the government for SMEs‎.

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    We live in the information age, constantly surrounded by vast amounts of data from the world around us. To utilize this information effectively, it must be mathematically expressed and analyzed using statistics.
     Statistics play a crucial role in various fields, including text mining, which has recently garnered significant attention. Text mining is a research method used to identify patterns in texts, which can be in written, spoken, or visual forms.
      The applications of text mining are diverse, including text classification, clustering, web mining, sentiment analysis, and more. Text mining techniques are utilized to assign numerical values to textual data, enabling statistical analysis.
       Since working with data requires a solid foundation in statistics, statistical tools are employed in text analysis to make predictions, such as forecasting changes in stock prices or currency exchange rates based on current textual data. 
       By leveraging statistical methods, text mining can uncover, confirm, or refute the truths hidden within textual content. Today, this topic is widely used in machine learning. This paper aims to provide a basic understanding of statistical tools in text mining and demonstrates how these powerful tools can be used to analyze and interpret events.

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In this article, we propose using the L2 divergence to test the Exponentiality for random censored data. Then we compare the ability of the proposed test to detect the Exponential distribution by Monte Carlo simulation with other competing tests, including the Kolmogorov‐Smirnov, Anderson‐darling and Cramer von‐Mises tests, which are based on the empirical distribution function and the information criteria tests. The results of simulation studies show that the proposed test generally performs better than its competitor tests.