In this paper, a new class of estimators namely Constrained Bayes Estimators are obtained under Balanced Loss Function (BLF) and Weighted Balanced Loss Function (WBLF) using a ``Bayesian solution". The Constrained Bayes Estimators are calculated for the natural parameter of one-parameter exponential families of distributions. A common approach to the prior uncertainty in Bayesian analysis is to choose a class $Gamma$ of prior distributions and look for an optimal decision within the class $Gamma$. This is known as robust Bayesian methodology. Among several methods of choosing the optimal rules in the context of the robust Bayes method, we discuss obtaining Posterior Regret Constrained Gamma-Minimax (PRCGM) rule under Squared Error Loss and then employing the ``Bayesian solution", we obtain the optimal rules under BLF and WBLF.

Robust linear regression is one of the most popular problems in the robust statistics community. The parameters of this method are often estimated via least trimmed squares, which minimizes the sum of the k smallest squared residuals. So, the estimation method in contrast to the common least squares estimation method is very computationally expensive. The main idea of this paper is to propose a new estimation method in partial linear models based on minimizing the sum of the k smallest squared residuals which determines the set of outlier point and provides robust estimators. In this regard, first, difference based method in estimation parameters of partial linear models is introduced. Then the method of obtaining robust difference based estimators in partial linear models is introduced which is based on solving an optimization problem minimizing the sum of the k smallest squared residuals. This method can identify outliers. The simulated example and applied numerical example with real data found the proposed robust difference based estimators in the paper produce highly accurate results in compare to the common difference based estimators in partial linear models.

In this paper we propose a new two-parameters distribution, which is an extension of the Lindley distribution with increasing and bathtub-shaped failure rate, called as the Lindley-logarithmic (LL) distribution. The new distribution is obtained by compounding Lindley (L) and Logarithmic distributions. We obtain several properties of the new distribution such as its probability density function, its failure rate functions, quantiles and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented in this paper. At the end, in order to show the flexibility and potentiality of this new class, some series of real data is used to fit.

In some applied problems we need to choose a population from the given populations and estimate the parameter of the selected population. Suppose k random samples are chosen from k populations with proportional hazard rate model or proportional reversed hazard rate model. According to a specified selection rule, it is desired to estimate the parameter of the best (worst) selected population. In this paper, under the entropy loss function we obtain the  uniformly minimum risk unbiased (UMRU) estimator of  the parameters of the selected population, and derived sufficient conditions for minimaxity of a given estimator. Then we find the class of admissible and inadmissible linear estimators of the parameters of the selected population and determine the class of dominators of a given estimator. We show that the UMRU estimator is inadmissible and compare the obtained estimators by plotting their risk functions.

Consider a repairable system which starts operating at t=0. Once the system fails, it is immediately replaced by another one of the same type or it is repaired and back to its working functions. In this paper, the system's activity is studied from t>0 for a fixed period of time w. Different replacement policies are considered. In each cases, for a fixed period of time w, the probability model and likelihood function of repair process, say window censored, are obtained. The obtained results depend on the lifetime distribution of the original system, so, expression for the maximum likelihood estimator and Fisher information are derived, by assuming the lifetime follows an exponential distribution.

Penalized estimators for estimating regression parameters have been considered by many authors for many decades. Penalized regression with rectangular norm is one of the mainly used since it does variable selection and estimating parameters, simultaneously. In this paper, we propose some new estimators by employing uncertain prior information on parameters. Superiority of the proposed shrinkage estimators over the least absoluate and shrinkage operator (LASSO) estimator is demonstrated via a Monte Carlo study. The prediction rate of the proposed estimators compared to the LASSO estimator is also studied in the US State Facts and Figures dataset.

Usually, we estimate the unknown parameter by observing a random sample and using the usual methods of estimation such as maximum likelihood method. In some situations, we have information about the real parameter in the form of a guess. In these cases, one may shrink the maximum likelihood or other estimators towards a guess value and construct a shrinkage estimator. In this paper, we study the behavior of a Bayes shrinkage estimator for the scale parameter of exponential distribution based on censored samples under an asymmetric and scale invariant loss function. To do this, we propose a Bayes shrinkage estimator and compute the relative efficiency between this estimator and the best linear estimator within a subclass with respect to sample size, hyperparameters of the prior distribution and the vicinity of the guess and real parameter. Also, the obtained results are extended to Weibull and Rayleigh lifetime distributions.

Nowadays, the use of various censorship methods has become widespread in industrial and clinical tests. Type I and Type II progressive censoring are two types of these censors. The use of these censors also has some disadvantages. This article tries to reduce the defects of the type I progressive censoring by making some change to progressive censorship. Considering the number and the time of the withdrawals as a random variable, this is done. First, Type I, Type II progressive censoring and two of their generalizations are introduced. Then, we introduce the new censoring based on the Type I progressive censoring and its probability density function. Also, some of its special cases will be explained and a few related theorems are brought. Finally, the simulation algorithm is brought and for comparison of introduced censorship against the traditional censorships a simulation study was done.

In this paper, based on generalized order statistics the Bayesian and maximum liklihood estimations of the parameters, the reliability and the hazard functions of Gompertz distribution are investigated. Specializations to Bayesian and maximum liklihood estimators, some lifetime parameters of progressive II censoring and record values are obtained. Also by using two real data sets and simulated data accurations of different estimates of the parameters are compared. Next the Bayesian and maximum liklihood estimates of the Gompertz distribution are compared with Weibull and Lomax distrtibutions.