The best single-observational and two-observational percentile estimations in the exponentiated Weibull-geometric distribution compared with maximum likelihood and percentile estimations

In this research the best single-observation percentile estimation (BSPE) and best two-observation percentile estimation (BTPE), are introduced. Then theses estimators are obtained for probability density function and cumulative distribution function of the exponentiated Weibull-geometric (EWG) with increasing, decreasing, bathtub and unimodal shaped failure rate function. Finally, these estimators are compared with the maximum likelihood (ML) and percentile (PC) estimations using the Monte Carlo simulation and a real data set.


Introduction
The estimation of probability density function (PDF) and cumulative density function (CDF) of several lifetime distributions using the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU), percentile (PC), least squares (LS) and weighted least squares (WLS) estimators have been obtained and compared by researchers. A number of papers have been attempted to estimate the lifetime distribution parameters, for instance the estimation of pdf and cdf of the Pareto distribution by Dixit and Jabbari Nooghabi (2010), exponentiated Pareto distribution by Jabbari , exponentiated Gumbel distribution by Bagheri et al. (2013b), generalized Rayliegh distribution by Alizadeh et al. (2013) and generalized Poisson-exponential distribution by Bagheri et al. (2013a). Note that Menon (1963) and Zanakis and Mann (1982) estimated the parameters of Weibull distribution by best single-observation percentile estimation (BSPE) and best two-observation percentile estimation (BTPE), but in this research the PDF and CDF of the Exponentiated Weibull-Geometric (EWG) which is originally introduced by Mahmoudi and Shiran (2012) are obtained by BSPE and BTPE methods for One or Two known parameters and compared with the corresponding estimations found by PC and MLE procedures.
According to the structure in this paper, in Sects. 2 and 3, the BEPE, PCE, MLE and BTPE, PCE, MLE are obtained respectively. By using the Monte Carlo simulations, estimators were compared in Sect. 4, and the results for real data are provided in Sect. 5.

Calculating estimations when only one parameter is unknown
Let X 1 ; . . .; X n is a random sample with ordinal statistics of Y 1 ; . . .; Y n , of a distribution with the following probability density and cumulative distribution functions: Such that In this section, assuming that parameters b; c; h are known and parameter a is unknown, the BSPE, PCE and MLE of a are obtained.

Estimation of the BSP
If Y k is the p-th percentile (0\p\1) of distribution (2), then ½ is the greatest integer smaller than np. Therefore, a single-observation percentile estimation of a which is shown by a Ã is as follows: According to Dubey (1967, p. 122), a Ã has an asymptotic normal distribution with mean of a and variance of Þ . Now q is determined in a way that Var a Ã ð Þ is minimum, which in this case, solves the equation By an iterative method, q ¼ 0:1189 and finally, the optimal p is obtained by the following relation.
Therefore, the BSPE of a as shown byâ BSPE is determined as follows: Thus, the BSPE of functions (1) and (2) are obtained by the following relation, respectively.f

PCE
Let X 1 ; . . .; X n is a random sample distribution with CDF given in (2) with order statistics of Y 1 ; . . .; Y n , and p i is the Therefore, the PCEs of functions (1) and (2) are obtained as follows: For more details about the PCE method, see Kao (1958Kao ( , 1959 and Johnson et al. (1994). Mean square error (MSE) of percentile estimations of functions (1) and (2) is calculated by Monte Carlo simulation method of the sample mean.

MLE
According to a random sample of X 1 ; . . .; X n of distribution with the probability density function (1), the MLE of the parameter a, i.e.â MLE is obtained by: where replacingâ MLE by a in relations (1) and (2), The MLE of the probability density and cumulative distribution functions of EWG distribution can be obtained. Moreover, by Monte Carlo simulation method of the sample mean, the mean square error (MSE) of the MLE of functions (1) and (2) could be found.

Calculating estimators when two parameters are unknown
In this section, a random sample of size n from the pdf given in (1) is considered. We assume that the parameters c and b are unknown, and parameters a and h are known,.
Then the BTPE, PCE and MLE of c and b, for the pdf (1) and cdf (2) are obtained.

BTPE
Suppose X 1 ; . . .; X n is a random sample of distribution with cdf (2) with ordinal statistics of Y 1 ; . . .; Y n , and p i is the such that for two real values of p 1 and p 2 (0\p 1 \p 2 \1) and with the help of relation (4), a two-observational percentile estimation of c which is shown by c Ã can be obtained as follows: According to Dubey (1967, p. 122), c Ã has an asymptotic normal distribution with a mean of c and variance of Now, p Ã 1 and p Ã 2 should be determined in a way that Var c Ã ð Þ is minimized where, according to Dubey (1967, p. 122), p Ã 1 ¼ 0:16730679 and p Ã 2 ¼ 0:97366352. Therefore, calculating p 1 and p 2 with the help of (5), the BTPE of c which is shown byĉ BTPE is obtained as follows: In addition, for p 1 and p 2 (0\p 1 \p 2 \1), with the help of (3), a TPE of b which is shown by b Ã is obtained as follows: According to Dubey (1967, p. 122), b Ã has an asymptotic normal distribution with a mean of b and variance of Now, r 1 and r 2 should be determined in a way that Var b Ã ð Þ is minimized where, according to Dubey (1967, p. 122), r 1 ¼ 0:39777 and r 2 ¼ 0:82111. Therefore, Int J Syst Assur Eng Manag calculating p 1 and p 2 with the help of (6), the BTPE of b which is shown byb BTPE is obtained as follows: where replacingĉ BTPE andb BTPE in relations (1) and (2), the BTPE, for the pdf (1) and cdf (2), and MSE of these estimators can be achieved.

PCE
Let X 1 ; . . .; X n is a random sample of distribution with cdf (2) with ordinal statistics of Y 1 ; . . .; Y n , and p i is the per- Replacingĉ MLE andb BPTE by c and b in relations (1) and (2), the PCE of pdf and cdf of EWG distribution, and MSE of these estimators are obtained.

MLE
In this section, according to a random sample of X 1 ; . . .; X n from a distribution with pdf (1), the MLE of the parameters of c and b which are shown byĉ MLE andb MLE , respectively, are obtained by the help of a set of equations ¼ 0 and the Newton-Raphson numerical method. By replacing the c and b byĉ MLE andb BPTE in relations (1) and (2), the MLE of pdf and cdf of EWG distribution, and MSE of these estimators can be found.

Numerical experiments
In this section, a Monte Carlo simulation and a numerical example are presented to illustrate all the estimation methods described in the preceding sections.
Steps 1 to 3 were repeated 5000 times and the mean of MSE is obtained from 5000 times repetition was found. The optimal estimator is that one with a smallest Mean MSE. Comparing the results of simulations studies in Int J Syst Assur Eng Manag Tables 1, 2, 3, 4 show that the BSPE and the BTPE are the best. On the other hand based on a 1000 random samples simulated from the EWG distribution, Fig. 1 show the graphs of estimations of the pdf (1) for the estimation methods of the third section which is given in Table 5, which represents the superiority of the BTPE toward other estimates.

Application with real data set
In this section the BSPE, BTPE, PCE and MLE of pdf and cdf for the EWG distribution are computed and compared for a real data. The data is the waiting times (in minutes) of 100 bank customers collected by Ghitany et al. (2008) presented in the 'Appendix'. For known parameters b ¼ 5:16, c ¼ 0:55, h ¼ 0:95 based on MLE method, Table 6 shows the average (AV) and corresponding mean square error (MSE) of the BSPE, PCE, MLE of pdf (1), cdf (2). Comparing theses results show that the BSPE provides better fit to waiting time data. Also, For known parameters a ¼ 2:11, h ¼ 0:85 based on MLE method, Table 7 shows the average (AV) and corresponding mean square error (MSE) of the BSPE, PCE, MLE of pdf (1), cdf (2). Comparing theses results show that the BTPE provides better fit to waiting time data.

Conclusion
In this research, the pdf and the cdf of the four-parameter EWG distribution were determined using several methods. To do this task, we first assume for an unknown parameter the BSPE, PCE and MLE of these functions are obtained. Then for two unknown parameters the BTPE, PCE and MLE of these functions are found. Then Using the Monte Carlo simulation and real data set, it was shown that the BSPE and BTPE are better than the other estimators.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Estimated PDF BPTE Estimated PDF PCE Estimated PDF MLE Fig. 1 The graphs of estimations BTPE, PCE and MLE of the pdf (1)   Table 7 Estimate the average (AV) estimation and corresponding mean square error of pdf (1) and cdf (2) Method AV (