Two Parameters Navoi Coffee Distribution Statistical Analysis

In this paper, we generalize the (?)P(?)a distribution from a single parameter to two parameters. For this distribution, we discuss the failure rate function and the features of its image as well as the mean residual life and the character of its corresponding graphs. Thereafter, we study the moment estimation, the interval estimation of the shape parameter and scale parameter. Comparing these estimates, we may have the following results:(1)We discuss the digital characteristics of (?)a distribution with two parameters, which includes the moment estimation and the maximum likelihood estimation (MLE) of the shape parameter and scale parameters in full sample. We discuss constrains of the sample for the existence for the moment estimation in different situations. Due to the complex expression for the maximum likelihood estimation (MLE), we discuss and prove the existence and uniqueness of the MLE in different cases. After converting the distribution into a shape-scale distribution, we focus on the order statistics and prove some related properties of it. In addition, the approximate maximum likelihood estimation (AMLE) and the best linear unbiased estimation (BLUE) are derived. When solving the best linear unbiased estimation (BLUE), we deduce the existence of the expectation and variance of the order statistics and derive the explicit expression by using the Gauss-Markov theorem. Finally, we compare these estimations by Monte-Carlo simulation. For the scale parameters, AMLE is much better than other estimations with respect to the mean square error. For the shape parameters, the moment estimation is best of all.(2)Base on the best linear unbiased estimation BLUE, we construct a pivotal quantity to obtain the interval estimation of the parameters. Meanwhile, we present the pivotal quintiles with different samples and different confidence level in one table.(3)For the type-Ⅱ sample, we discuss the BLUE and the approximate maximum likelihood estimation. We compare these three estimations by Monte-Carlo simulation. The BLUE is better than the AMLE for the estimation of the scale parameter. As for the shape parameter, the BLUE is also better than the AMLE.