A maximum likelihood estimation (MLE) based on records is obtained and a proper prior distribution to attain a Bayes estimation (both informative and non-informative) based on records for quadratic loss and squared error loss functions is also calculated. The study considers the shortest confidence interval and Highest Posterior Distribution confidence interval based on records, and using Mean Square Error MSE criteria for point estimation and length criteria for interval estimation, their appropriateness to each other is examined.

This paper deals with Bayesian estimations of scale parameter of the exponential distribution based on upper record range (R n ). This has been done in two steps; point and interval. In the first step the quadratic, squared error and absolute error, loss functions have been considered to obtain Bayesianpoint estimations. Also in the next step the shortest Bayes interval (Hight Posterior Density interval) and Bayes interval with equal tails based on upper record range have been found. Therefore, the Homotopy Perturbation Method (HPM) has been applied to obtain the limits of Hight Posterior Density intervals. Moreover, efforts have been made to meet the admissibility conditions for linear estimators based on upper record range of the form mR n +d by obtained Bayesian point estimations. So regarding the consideration of loss functions, the prior distribution between the conjunction family has been chosen to be able to produce the linear estimations from upper record range statistics. Finally, some numerical examples and simulations have been presented.

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A useful subfamily of the exponential family is considered. The ML estimation based on upper record values are calculated for the parameter, Cumulative Density Function, and Probability Density Function of the subfamily. The relationship between MLE based on record values and a random sample are discussed, along with some properties of these estimators, and its utility is shown for large samples.