In some applications, the response variable assumes values in the unit interval. The standard linear regression model is not appropriate for modelling this type of data because the normality assumption is not met. Alternatively, the beta regression model has been introduced to analyze such observations. A beta distribution represents a flexible density family on (0, 1) interval that covers symmetric and skewed families. In this paper, a beta generalized linear mixed model with spatial random effect is proposed emphasizing on small values of the spatial range parameter and small sample sizes. Then some models with both fixed and varying precision parameter and different combinations of priors and sample sizes are discussed. Next, the Bayesian estimation of the model parameters is evaluated in an intensive simulation study. Selected priors improved the Bayesian estimation of the parameters, especially for small sample sizes and small values of range parameter. Finally, an application of the proposed model on data provided by Household Income and Expenditure Survey (HIES) of Tehran city is presented.

To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing as com-pared to simple models. This paper considers mixture of Topp-Leone distributions under classical and Bayesian perspective based on com-plete sample. The new distribution which exhibits decreasing and up-side down bathtub shaped density while the distribution has the ability to model lifetime data with decreasing, increasing and upside down bathtub shaped failure rates. We derive several properties of the new distribution such as moments, moment generating function, conditional moment, mean deviation, Bonferroni and Lorenz curves and the order statistics of the proposed distribution. Moreover, we estimate the pa-rameters of the model by using frequentist and Bayesian approaches. For Bayesian analysis, , ve loss functions, namely the squared error loss function (SELF), weighted squared error loss function (WSELF), mod-i, ed squared error loss function (MSELF), precautionary loss function (PLF), and K-loss function (KLF) and uniform as well as gamma pri-ors are considered to obtain the Bayes estimators and posterior risk of the unknown parameters of the model. Furthermore, credible intervals (CIs) and highest posterior density (HPD) intervals are also obtained. Monte Carlo simulation study is done to access the behavior of these estimators. For the illustrative purposes, a real-life application of the proposed distribution to a tensile strength data set is provided.

In order to perform Preventive Maintenance (PM), two approaches have evolved in the literature. The traditional approach is based on the use of statistical and reliability analysis of equipment failure. Under statistical-reliability (S-R)-based PM, the objective of achieving the minimum total cost is pursued by establishing fixed PM intervals, which are statistically optimal, at which a decision to replace or overhaul equipments or components is made. The second approach involves the use of sensor-based monitoring of equipment condition in order to predict occurrence of machine failure. Under condition-based (C-B) PM, intervals between PM works are no longer fixed, but are performed only “when needed”. It is obvious that Condition Based Maintenance (CBM) needs an on-line inspection and monitoring system that causes CBM to be expensive. Whenever this cost is infeasible, we can develop other methods to improve the performance of traditional (S-R)-based PM method. In this research, the concept of Bayesian inference was used. The time between machine failures was observed, and Bayesian inference is employed in (S-R)-based PM, it is tried to determine the optimal checkpoints.

In order to produce more flexible models in the reliability theory field, the Bayesian inference of 𝑚-component reliability model with the non-identical-component strengths for modified Weibull distribution under the progressive censoring scheme is considered. One of the key benefits is the generality of this model, so it includes some cases studied previously, such as multi-component stress-strength model with one and two non-identical-component and stress-strength models. In addition, the study of progressive censored data discussed in this paper is critical in many practical situations. The problem is considered in three cases: when the two common parameters for strengths and stress variables are unknown, known, and general. In each case, the approximation methods, such as the MCMC and Lindley’s approximation, are used to consider the -component stress-strength parameter. The Monte Carlo simulation study compares the performance of different methods—finally, a demonstration of how the proposed model may be utilized to analyse real data sets.