This paper addresses the problem of Bayesian estimation of the parameters, reliability and hazard function in the context of record statistics values from the two-parameter Lomax distribution. The ML and the Bayes estimates based on records are derived for the two unknown parameters and the survival time parameters, reliability and hazard functions. The Bayes estimates are obtained based on conjugate prior for the scale parameter and discrete prior for the shape parameter of this model. This is done with respect to both symmetric loss function (squared error loss), and asymmetric loss function (linear-exponential (LINEX)) loss function. The maximum likelihood and the different Bayes estimates are compared via Monte Carlo simulation study. A practical example consisting of real record values including in the data from an accelerated test on insulating fluid reported by Nelson was used for illustration and comparison. Finally, Bayesian predictive density function, which is necessary to obtain bounds for predictive interval of future record is derived and discussed using a numerical example.

Let X1,X2, ...,Xr be the first r order statistics from a sample of size n from the generalized exponential distribution with shape parameter q. In this paper, we consider a Bayesian approach to predicting future order statistics based on the observed ordered data. The predictive densities are obtained and used to determine prediction intervals for unobserved order statistics for one-sample and two-sample prediction plans. A numerical study is conducted to illustrate the prediction procedures.

The objective of this study was to evaluate some environmental factors (sex, season and year of birth, birth type and dam age) on fleece weight, estimate of phenotypic and genetic parameters and trends of this trait on Cashmere goat of the southern Khorasan. GLM procedure of SAS software version 1.9 was used to assess the significant environmental effects on this trait. Records of 1455 kids, the progeny of 127 sires and 562 dams, collected between 1996 and 2012, were used to estimate phenotypic and genetic parameters. Results showed that Cashmere weight was significantly higher in males than females. Genetic analyses was carried out by MTGSAM program. The results of this study indicated that the effects of year of birth and sex at 5% and remaining effects at 1% were significant. To detecting of maternal additive genetic effect and maternal permanent environmental effect, six different animal models were fitted for fleece weight. The most appropriate model according to Akaikes information criterion was model 3with direct additive genetic and permanent environmental maternal effects. Direct and maternal heritability for first fleece weight were 0.49 and 0.09. The results indicated that fleece weight was influenced by maternal genetic effects. The genetic and phenotypic trends for this trait were estimated near to zero and insignificant. It seems, developing of selection index and removing of the environmental obstacles are major steps in the development of genetic and increase profitability in this Cashmere goat population.

Aim: This study aims to correct undercounts in cancer data before initiating a population-based cancer registry program, employing an innovative Bayesian methodology. Background: Underestimation is a widespread issue in cancer registries within developing countries. Methods: This secondary study utilized cancer registry data. We employed the Bayesian approach to correct undercounting in cancer data from 2005 to 2010, using the ratio of pathology to population-based data in the Golestan province as the initial value. Results: The results of this study showed that the lowest percentage of undercounting belonged to Khorasan Razavi province with an average of 21% and the highest percentage belonged to Sistan and Baluchestan province with an average of 38%. The average age-standardized incidence rate (ASR) for all provinces of the country except Golestan province was equal to 105. 72 (Confidence interval (CI) 95% 105. 35-106. 09) per 100, 000 and after Bayesian correction was 137. 17 (CI 95% 136. 74-137. 60) per 100, 000. In 2010 the amount of ASR before Bayesian correction was 100. 28 (CI 95% 124. 39-127. 09) per 100, 000 for women and 136. 49 (CI 95% 171. 20-174. 38) per 100, 000 for men. Also, after implementing the Bayesian correction, ASR increased to 125. 74 per 100, 000 for women and 172. 79 per 100, 000 for men. Conclusion: The study demonstrates the effectiveness of the Bayesian approach in correcting undercounting in cancer registries. By utilizing the Bayesian method, the average ASR after Bayesian correction with a 29. 74 percent change was 137. 17 per 100, 000. These corrected estimates provide more accurate information on cancer burden and can contribute to improved public health programs and policy evaluation. Furthermore, this research emphasizes the suitability of the Bayesian method for addressing underestimation in cancer registries. It also underscores its pivotal role in shaping the trajectory of future investigations in this field

The reliability function‎, ‎or the survival function at a specified time t‎, ‎denotes the proportion of products that remain operational beyond time t and continue to function‎. ‎This interpretation underscores the pivotal role of the survival function and its estimation in understanding lifetime phenomena‎. ‎This paper explores the estimation of the survival function for the inverse Lindley distribution based on lower records‎. ‎The estimation techniques encompass maximum likelihood and bootstrap methods‎. ‎Furthermore‎, ‎Bayesian approaches employing Metropolis-Hastings and importance sampling algorithms are employed‎. ‎In addition to deriving approximate confidence intervals using the delta method and percentile bootstrap intervals for the survival function‎, ‎Chen and Shao's shortest width credible intervals are also determined‎. ‎A comprehensive simulation study is presented to assess the effectiveness of both point and interval estimators‎. Finally‎, ‎an application of the results is given to a real data set.

Introduction Studying crime data has become one of the essential topics in the world due to its connection with human security. Analyzing this type of data can effectively prevent future crimes and identify spatial patterns and factors that facilitate the commission of crimes to control crime-prone areas. Most of the time, crime data has a spatio-temporal structure that causes the formation of different spatio-temporal patterns. Therefore, spatio-temporal monitoring of crime data is essential in identifying factors that cause crime and preventing crime. An important issue in many cities is related to crime events, and the spatio-temporal Bayesian approach leads to identifying crime patterns and hotspots. In Bayesian analysis of spatio-temporal crime data, there is no closed form for posterior distribution because of its non-Gaussian distribution and the existence of latent variables. In this case, we face challenges such as high dimensional parameters, extensive simulation and time-consuming computation in applying MCMC methods. Material and Methods In this paper, we apply INLA to analyze crime data in Colombia. To describe the above concepts, a three-stage hierarchical model is considered. The advantages of this method can be the estimation of criminal events at a specific time and location and exploring unusual patterns in places. Results and Discussion The Bayesian analysis of crime data is usually performed as Bayesian infer ence of pure spatial or temporal patterns. However, such spatial or temporal Bayesian analyses are not suitable for crime data. In this article, in a case study, Bayesian hierarchical spatio-temporal analysis of crime data in Colombia was discussed using the INLA approach, which considers spatio-temporal dependence and makes the model more flexible in detecting unusual patterns. Exploratory data analysis is also discussed, detecting areas with unusual behaviour over time. Four different models were fitted to the data, and the best model that includes spatio-temporal interaction was selected using the DIC criterion. The research results identify the most important centre of crime in the Kennedy area of Bogotá, , as well as the highest crime rate in the time frame. Then, hierarchical spatio-temporal Bayesian analysis of these data was done with the INLA approach. Conclusion The advantage of using this Bayesian approach is that it includes the effects of spatio-temporal correlation in the model and makes the model flexible in detecting areas with abnormal behaviour over time and in different places. For this purpose, four different models, including side effects and spatio-temporal combination, were fitted to the crime data. The best model, including the spatio-temporal interaction effect, was proposed using the deviance information criterion. The comprehensive and scientific comparison of the two Bayesian methods INLA and the MCMC algorithm in terms of accuracy, speed and even accessibility and convenient use for researchers requires independent scientific and practical research because, for example, the various methods of sampling in the MCMC algorithms and sometimes its different methods in INLA make it difficult to compare accuracy. How to use parallel calculations in the application of these two methods is also effective in comparing the speed, and simply comparing the outputs cannot express the advantage of one method over the other.

Descriptive statistics is the process of summarizing gathered raw data from a research and creating useful statistics, which help the better understanding of data. According to the types of variables, which consist of qualitative and quantitative variables, some descriptive statistics have been introduced. Frequency percentage is used in qualitative data, and mean, median, mode, standard deviation; standard error, variance, and range are some of the statistics which are used in quantitative data. In health sciences, the majority of continuous variables follow a normal distribution. skewness and kurtosis are two statistics which help to compare a given distribution with the normal distribution.