Some Investigations about the Properties of Maximum Likelihood Estimations Based on Lower Record Values for a Sub-Family of the Exponential Family

Here, in this paper it has been considered a sub family of exponential family. Maximum likelihood estimations (MLE) for the parameter of this family, probability density function, and cumulative density function based on a sample and based on lower record values have been obtained. It has been considered Mean Square Error (MSE) as a criterion for determining which is better in different situations. Additionally, it has been proved some theories about the relations between MLE based on lower record values and based on a random sample. Also, some interesting asymptotically properties for these estimations have been shown during some theories.