Research Article | Open Access

Volume 2016 |Article ID 8727951 | 12 pages | https://doi.org/10.1155/2016/8727951

# Different Estimation Procedures for the Parameters of the Extended Exponential Geometric Distribution for Medical Data

Accepted03 Jul 2016
Published04 Aug 2016

#### Abstract

We have considered different estimation procedures for the unknown parameters of the extended exponential geometric distribution. We introduce different types of estimators such as the maximum likelihood, method of moments, modified moments, L-moments, ordinary and weighted least squares, percentile, maximum product of spacings, and minimum distance estimators. The different estimators are compared by using extensive numerical simulations. We discovered that the maximum product of spacings estimator has the smallest mean square errors and mean relative estimates, nearest to one, for both parameters, proving to be the most efficient method compared to other methods. Combining these results with the good properties of the method such as consistency, asymptotic efficiency, normality, and invariance we conclude that the maximum product of spacings estimator is the best one for estimating the parameters of the extended exponential geometric distribution in comparison with its competitors. For the sake of illustration, we apply our proposed methodology in two important data sets, demonstrating that the EEG distribution is a simple alternative to be used for lifetime data.

#### 1. Introduction

Many researches are interested in search distributions which can be used to describe real data sets. Generalizations of the standard exponential distribution have been introduced in the literature for this purpose, such as Gamma, Weibull, and Generalized Exponential distribution . Another useful generalization is known as extended exponential geometric distribution. Initially, the development of such distribution was made by Adamidis and Loukas  proposing exponential geometric distribution with two parameters, in which the hazard function could be decreasing. In a further paper, Adamidis et al.  explored extended exponential geometric (EEG) distribution. Let be a random variable representing a lifetime data, with extended exponential geometric (EEG) distribution; its probability density function (PDF) is given byfor all , , and . One of its peculiarities is that its hazard function can be increasing or decreasing, depending on the values of its parameters, giving great flexibility of fit for real applications.

This model arises naturally in competing risks scenarios. Let , where M is a random variable with geometrical distribution and are independent of M and are assumed to be independent and identically distributed according to exponential distribution; then the random variable has EEG distribution with , also known as exponential geometric (EG) distribution . Considering the same assumptions and , the random variable has EEG distribution with , also known as Complementary Exponential Geometric distribution . Due to its importance, some generalizations of the EEG distribution have been proposed, such as the Beta exponential geometric distribution , Exponentiated Exponential-Geometric distribution , Complementary Exponentiated Exponential Geometric distribution , and Generalized Exponential Geometric distribution .

Despite the fact that EEG distribution has good flexibility, a few estimation procedures have been proposed in the literature. Adamidis et al.  derived the maximum likelihood estimators (MLE) for the unknown parameters of the EEG distribution. Ramos et al.  developed a Bayesian analysis under noninformative priors. However, considering the frequentist approach, it is well known that, usually, for small samples, the MLE does not perform well. In this paper, we proposed nine new estimators for the parameters of the EEG distribution, which are given considering the following estimation procedures: the method of moments, modified moments, ordinary least squares, weighted least squares, L-moments, percentile, maximum product of spacings, Cramer-von Mises type minimum distance, and Anderson-Darling estimator.

The main aim of this paper is twofold. First, it aims to develop a guideline for choosing the most efficient estimators among ten different estimation procedures for the EEG distribution, which would be of interest to applied statisticians. Second, it aims to demonstrate that the EEG distribution is a simple alternative to be used in applications in medicine.

The originality of this study comes from the fact that, for the EEG distribution and considering the frequentist approach, only the MLE has been presented in the literature. The performances of the different estimation methods are compared using extensive numerical simulations. Additionally, these results are analogous for the exponential geometric distribution and the Complementary Exponential Geometric distribution. Related studies for other distributions can be found in Gupta and Kundu , Mazucheli et al. , Teimouri et al. , and Dey et al. .

The paper is organized as follows. In Section 2, we discuss some properties of the EEG distribution. In Section 3, we present ten estimation procedures for the parameters of our proposed model. In Section 4, a simulation study is presented in order to identify the most efficient estimators. In Section 5, we apply our proposed methodology in two real data sets. Some final comments are presented in Section 6.

#### 2. Extended Exponential Geometric Distribution

Let be a random variable with density function (1); the distribution function is given by

The survival and hazard functions of distribution is given, respectively, byThe hazard function (3) is decreasing for , is constant for , and is monotonically increasing when . Figure 1 presents different forms for the density and hazard functions for the EEG distribution considering different values of and .

For the random variable with EEG distribution, the moment generating function  is given byfor , where , for , and is known as Lerch transcendental function . Note that the Laplace transform of the EEG distribution can be easily obtained from the relation . The raw moments of the EEG distribution arefor . After some algebraic manipulation, the mean and variance of the EEG distribution are given, respectively, bywhere is the dilogarithm function given byThe mode and the median of the EEG distribution are

From Marshall and Olkin , we have the following inequality:where .

Shannon’s Entropy from EEG distribution , which played a central role as a measure of the uncertainty associated with a random variable, is given by

#### 3. Methods of Estimation

In this section, we discuss ten different estimation methods to obtain the estimates of the parameters and of the EEG distribution.

##### 3.1. Maximum Likelihood Estimation

Among the statistical inference methods, the maximum likelihood method is widely used due its desirable properties including consistency, asymptotic efficiency, and invariance. Under the maximum likelihood method, the estimators are obtained from maximizing the likelihood function (see e.g., ). Let be a random sample such that ; the likelihood function from (1) is given byThe logarithm of the likelihood function (11) is given byFrom and , we get the likelihood equationswhose solutions provide the maximum likelihood estimates, hereafter, and . Numerical methods such as Newton-Raphson are required to find the solution of the nonlinear system.

For large sample sizes, the obtained estimators are not biased and asymptotically efficient. The MLE estimates are asymptotically normally distributed with joint bivariate normal distribution given bywhere is the Fisher information matrix given by (see )

##### 3.2. Moments Estimators

The method of moments is one of the oldest procedures used for estimating parameters in statistical models. The moment estimators (ME) of the EEG distribution can be obtained by equating the first two theoretical moments,with the sample moments and , respectively. After some algebraic manipulation, the estimate for can be obtained by solvingNote that, by substituting in (18), the estimate for can be obtained by solving

Therefore, we firstly compute and, by substituting such estimate in (19), the estimate is obtained.

##### 3.3. Method of Modified Moments

A simple modification can be made in the method of moments for estimating the parameters of the EEG distribution. To obtain the moment estimators (MME), consider thatNote that the population coefficient of variation given byis independent of the scale parameter . So the estimate for can be obtained by solving the nonlinear equationand, by substituting in (23), the estimate for can be obtained by solving

##### 3.4. Percentile Estimators

The percentile estimator, originally suggested by Kao [18, 19], is a statistical method used to estimate the unknown parameters by comparing the sample points with the theoretical ones. This method has been widely used in distributions that have the quantile function in a closed form, such as the Weibull distribution and the Generalized Exponential distribution. For the EEG distribution, the quantile function is given by

The percentile estimates (PCE), and , can be obtained by minimizing with respect to and , where denotes some estimate of . The estimates of and can also be obtained by solving the following nonlinear equations:respectively. In this paper, we consider . However, several estimators of can be used instead (see ).

##### 3.5. L-Moments Estimators

Hosking  proposed an alternative method of estimation analogous to conventional moments, namely, L-moments estimators. These estimators are obtained by equating the sample L-moments with the population L-moments. Hosking  states that the L-moment estimators are more robust than the usual moment estimators and are also relatively robust to the effects of outliers and reasonably efficient when compared to the MLE for some distributions.

For the EEG distribution, the L-moments estimators (LME) can be obtained by equating the first two sample L-moments with the corresponding population L-moments. The first two sample L-moments are and the first two population L-moments arewhere is given in (25). After some algebraic manipulations, the estimate for can be obtained by solving the nonlinear equationNote that, by substituting in (30), the estimate for can be obtained by solving

##### 3.6. Ordinary and Weighted Least Squares Estimates

Let denote the order statistics (we assume the same notation for the next subsections) of the random sample of size from a distribution function . The least square estimators (LSE) and can be obtained by minimizingwith respect to and , where is given by (2). Equivalently, they can be obtained by solving the following nonlinear equations:where

The weighted least squares estimates (WLSE), and , can be obtained by minimizingThese estimates can also be obtained by solving the following nonlinear equations:

##### 3.7. Method of Maximum Product of Spacings

The maximum product of spacings (MPS) method is a powerful alternative to MLE for the estimation of the unknown parameters of continuous univariate distributions. Proposed by Cheng and Amin [22, 23], this method was also independently developed by Ranneby  as approximation to the Kullback-Leibler measure of information.

Let , for , be the uniform spacings of a random sample from the EEG distribution, where and Clearly . The maximum product of spacings estimates, and , are obtained by maximizing the geometric mean of the spacings,with respect to and , or, equivalently, by maximizing the logarithm of the geometric mean of sample spacings:

The estimates and of the parameters and can be obtained by solving the following nonlinear equationswhere and are given in (35).

Note that if we get . Therefore, the MPS estimators are sensitive to closely spaced observations, especially ties. When the ties are due to multiple observations, should be replaced by the corresponding likelihood , since .

Cheng and Amin  proved desirable properties of the MPS such as asymptotic efficiency and invariance; they also proved that the consistency of maximum product of spacings estimators holds under much more general conditions than for maximum likelihood estimators. The authors also present an interesting proof that the MPS estimates converge asymptotically to the ML estimates. Therefore, for the EEG distribution, the MPS estimators are asymptotically normally distributed (see  for more details) with joint bivariate normal distribution given bywhere is the Fisher information matrix.

##### 3.8. The Cramer-von Mises Minimum Distance Estimators

The Cramer-von Mises estimator (CME) is a type of minimum distance estimators (also called maximum goodness-of-fit estimators) which is based on the difference between the estimate of the cumulative distribution function and the empirical distribution function (see, [26, 27]).

MacDonald  motivates the choice of Cramer-von Mises type minimum distance estimators providing empirical evidence that the bias of the estimator is smaller than the other minimum distance estimators. The Cramer-von Mises estimates, and , are obtained by minimizingwith respect to and . These estimates can also be obtained by solving the following nonlinear equations:where and are given in (35).

##### 3.9. Methods of Anderson-Darling

Another type of minimum distance estimators is based on Anderson-Darling statistic (see ) and is known as the Anderson-Darling estimator (ADE). The Anderson-Darling estimates, and , of the parameters and are obtained by minimizing, with respect to and , the functionThese estimates can also be obtained by solving the following nonlinear equations:where and are in (35).

#### 4. Simulation Study

In this section, we develop a simulation study via Monte Carlo methods. The main goal of these simulations is to compare the efficiency of the different estimation methods for the parameters of the EEG distribution. The following procedure was adopted:(1)Set the sample size and the vector of parameter values .(2)Generate values of with size .(3)Using the values obtained in step (2), compute and via MLE, ME, MME, LSE, WLSE, PCE, MPS, CME, and ADE.(4)Repeat steps (2) and (3) times.(5)Using and , compute the mean relative estimates (MRE) and the mean square errors (MSE) , .

We expect that, considering this approach, the MREs are closer to one with smaller MSEs. The results were computed using the software R (R Core Development Team). The seed used to generate the random values was 2015. The chosen values to perform this procedure were , , and . The values of were selected to allow, respectively, the decreasing and increasing shape in the hazard function. Another motivation comes from the fact that, for , we have analogous results for the exponential geometric distribution  and, for , the results are analogous for the Complementary Exponential Geometric distribution .

Figures 2 and 3 present the MREs and MSEs for the estimates of for simulated samples considering different values of . The horizontal lines in Figures 2 and 3 correspond to MREs and MSEs being, respectively, one and zero.

It is worth noting that we only considered the samples in which all estimation procedures had converged, getting at the end simulated samples for different values of . Figure 4 presents the proportion of failure from each method.

Based on these figures, the MSEs of all estimates tend to zero for large and also, as expected, the values of MREs tend to one; that is, the estimates are asymptotically unbiased for the parameters. The ME and the CME estimators have, respectively, the largest MREs and MSEs among all the considered estimators. The percentile and the LSE estimators have, respectively, the largest proportion of failure for estimating the parameters of the EEG distribution.

The MPS estimators have the smallest MSEs and the MREs nearest to one for both parameters proving to be the most efficient procedure for estimating the unknown parameters. Moreover, the MPS estimators have good theoretical properties  such as consistency, asymptotic efficiency, normality, and invariance. Therefore, we conclude that the MPS estimators should be used for estimating the parameters of the EEG distribution.

#### 5. Applications

In this section, we considered two real data sets. The first one is presented by Boag  and is related to the ages (in months) of 18 patients who died from other causes than cancer. The second data set is presented by Silva  and refers to the serum-reversal time (in days) of children born to HIV-infected mothers who did not receive anti-HIV treatment (Table 4).

In Section 4, our simulation study indicated that the MPS estimators should be used for estimating the parameters of the EEG distribution. Initially, we compared the estimates obtained from the different procedures with the MPS estimator in terms of MREs. Then, we compared the results obtained from the EEG distribution fitted by the MPS estimators with some common lifetime models, such as Weibull, Gamma, Lognormal, and Generalized Exponential distributions.

The Kolmogorov-Smirnov (KS) test is considered to check the goodness of fit. This procedure is based on the KS statistic , where is the supremum of the set of distances, is the empirical distribution function, and is cumulative distribution function. In this case, we test the null hypothesis that the data comes from , and, with significance level of , we will reject the null hypothesis if value is smaller than . As discrimination criterion method, we considered the AIC (Akaike Information Criteria), AICc (Corrected Akaike Information Criteria), HQIC (Hannan-Quinn Information Criteria), and the CAIC (Consistent Akaike Information Criteria) computed, respectively, by , , , and , where k is the number of parameters to be fitted and is the estimate of . Given a set of candidate models for t, the preferred model is the one which provides the minimum values.

##### 5.1. Boag Data Set

Table 1 presents the data set related to the ages (in months) of 18 patients who died from other causes than cancer extracted from Boag , which considered the Lognormal distribution to describe such data.

 0.3 4 7.4 15.5 23.4 46 46 51 65 68 83 88 96 110 111 112 132 162

Considering the MPS estimators, we obtain and and and . In Table 2, we compared the estimates obtained from the different procedures with the MPS estimator in terms of MREs.

 MLE ME MME LME LSE WLSE PCE CME ADE 0.8033 0.7057 0.7397 0.8650 1.0365 0.9689 0.8192 0.8966 1.0112 0.6281 0.4883 0.5537 0.8226 0.9630 0.8908 0.6186 0.7219 1.0222

Table 2 confirmed the results obtained from our simulation study, in which for small sample sizes the obtained results may differ depending on the estimation procedure. For example, considering the method of moments, the estimate for is smaller than . Table 3 presents the results from KS test ( value), AIC, AICc, HQIC, and CAIC, for the EEG distribution adjusted by the MPS procedure and for different probability distributions. In Figure 5, we have the survival function adjusted by different distributions and nonparametric survival estimator.

 Test EEG Weibull Gamma Lognormal GE KS 0.9303 0.5185 0.3361 0.0561 0.3156 AIC 189.809 191.447 191.801 201.695 191.798 AICc 190.609 192.247 192.601 202.495 192.598 HQIC 190.054 191.692 192.047 201.940 192.044 CAIC 193.589 195.227 195.582 205.475 195.579
 2 2 2 5 9 9 19 32 32 46 50 56 56 78 91 95 106 129 129 148 149 156 175 176 191 192 204 209 211 225 229 230 238 254 271 274 276 290 291 292 297 297 322 334 334 334 344 346 353 353 359 365 366 367 370 378 378 382 382 385 398 400 402 414 422 424 428 434 435 440 443 446 448 448 451 454 459 460 461 473 480 481 484 487 493 497 498 502 511 511 513 514 516 521 524 526 537 538 541 543 544 544 545 549 551 553 553 554 556 559 571 576 577 578 582 588 590 596 609 610 615 619 626 627 648 653 678 680 687 696 729 744 748 777 847 848 867 874 894 901 907 974 1021

Comparing the empirical survival function with the adjusted distributions, a better fit for the EEG distribution among the chosen models can be observed. This result is confirmed from AIC, AICc, HQIC, and CAIC, since EEG distribution has the minimum values and values returned from the KS test are greater than the chosen models. From our proposed methodology, we observe that the extended exponential geometric distribution has superior fit among the chosen models. In this case, each of the causes of the death can be described by exponential distribution; since the lifetime associated with a particular risk is not observable (latent variables), we observe only the maximum lifetime () value among all risks, where the number of causes follows geometric distribution.

##### 5.2. Children Exposed to the Vertical Transmission of HIV

The data set related to the serum-reversal time (in days) of children born to HIV-infected mothers is presented in Table 3.

Considering the MPS estimators, we obtain and and and . In Table 5, we compared the estimates obtained from the different procedures with the MPS estimator in terms of MREs.

 MLE ME MME LME LSE WLSE PCE CME ADE 0.9922 0.9726 0.9774 1.0007 0.9481 0.9587 0.9580 0.9352 1.0259 0.9910 1.0169 1.0341 1.1184 0.8039 0.8684 0.9253 0.7682 1.1007

From Table 5, we observed that for large samples sizes the estimates are very closer independently of the chosen method. Moreover, due to the large sample size, the MPS estimates and ML estimates are almost the same; such theoretical result is well supported by Cheng and Amin . In Table 6, we have the results from KS test ( value), AIC, AICc, HQIC, and CAIC, for different probability distributions. Figure 6 presents the survival function adjusted by different distributions and nonparametric survival estimator.

 Test EEG Weibull Gamma Lognormal GE KS 0.6413 0.0067 0.0000 0.0000 0.0000 AIC 1950.82 1981.31 2001.92 2088.90 2005.43 AICc 1950.91 1981.39 2002.01 2088.98 2005.52 HQIC 1953.23 1983.71 2004.33 2091.31 2007.84 CAIC 1958.75 1989.23 2009.85 2096.82 2013.36

Comparing the empirical survival function with the adjusted distributions, a better fit for the extended exponential geometric distribution among the chosen models can be observed. This result is confirmed from AIC, AICc, HQIC, and CAIC, since EEG distribution has the minimum values among the chosen models. Moreover, considering a significance level of , the EEG distribution was the only model in which values returned from the KS test were greater than .

#### 6. Conclusions

In this paper, we derived and compared, via intensive simulation study, the estimations of the parameters of the EEG distribution using ten estimation methods. Most importantly, from our simulations, we discovered that the estimates are asymptotically unbiased for the parameters regardless of the estimation method. However, while the ME and CME estimators have, respectively, the largest MREs and MSEs among all the considered estimators, the MPS estimator has the smallest MSEs and the MREs nearest to one, for both parameters, proving to be the most efficient method compared to others for estimating the unknown parameters. As a final advise, combining these results with the good properties of the method such as consistency, asymptotic efficiency, normality, and invariance, we conclude that the MPS estimator is the best one for estimating the parameters of the EEG distribution in comparison with its competitors. Finally, we apply our proposed methodology in two important data sets, demonstrating that the EEG distribution is a simple alternative to be used for lifetime applications.

#### Competing Interests

The authors declare that they have no competing interests.

1. R. D. Gupta and D. Kundu, “Generalized exponential distributions,” Australian & New Zealand Journal of Statistics, vol. 41, no. 2, pp. 173–188, 1999. View at: Publisher Site | Google Scholar | MathSciNet
2. K. Adamidis and S. Loukas, “A lifetime distribution with decreasing failure rate,” Statistics and Probability Letters, vol. 39, no. 1, pp. 35–42, 1998.
3. K. Adamidis, T. Dimitrakopoulou, and S. Loukas, “On an extension of the exponential-geometric distribution,” Statistics & Probability Letters, vol. 73, no. 3, pp. 259–269, 2005. View at: Publisher Site | Google Scholar
4. F. Louzada, M. Roman, and V. G. Cancho, “The complementary exponential geometric distribution: model, properties, and a comparison with its counterpart,” Computational Statistics & Data Analysis, vol. 55, no. 8, pp. 2516–2524, 2011. View at: Google Scholar
5. M. Nassar and N. Nada, “A new generalization of the exponential-geometric distribution,” Journal of Statistics: Advances in Theory and Applications, vol. 7, pp. 25–48, 2012. View at: Google Scholar
6. F. Louzada, V. A. A. Marchi, and M. Roman, “The exponentiated exponential-geometric distribution: a distribution with decreasing, increasing and unimodal failure rate,” Statistics, vol. 48, no. 1, pp. 167–181, 2014. View at: Publisher Site | Google Scholar
7. F. Louzada, V. Marchi, and J. Carpenter, “The complementary exponentiated exponential geometric lifetime distribution,” Journal of Probability and Statistics, vol. 2013, Article ID 502159, 10 pages, 2013. View at: Publisher Site | Google Scholar
8. H. Bidram, J. Behboodian, and M. Towhidi, “A new generalized exponential geometric distribution,” Communications in Statistics—Theory and Methods, vol. 42, no. 3, pp. 528–542, 2013. View at: Publisher Site | Google Scholar | MathSciNet
9. P. L. Ramos, F. A. Moala, and J. A. Achcar, “Objective priors for estimation of extended exponential geometric distribution,” Journal of Modern Applied Statistical Methods, vol. 13, no. 2, pp. 226–243, 2014. View at: Google Scholar
10. R. D. Gupta and D. Kundu, “Generalized exponential distribution: different method of estimations,” Journal of Statistical Computation and Simulation, vol. 69, no. 4, pp. 315–337, 2001. View at: Publisher Site | Google Scholar | MathSciNet
11. J. Mazucheli, F. Louzada, and M. E. Ghitany, “Comparison of estimation methods for the parameters of the weighted Lindley distribution,” Applied Mathematics and Computation, vol. 220, no. 1, pp. 463–471, 2013. View at: Publisher Site | Google Scholar | MathSciNet
12. M. Teimouri, S. M. Hoseini, and S. Nadarajah, “Comparison of estimation methods for the Weibull distribution,” Statistics, vol. 47, no. 1, pp. 93–109, 2013. View at: Publisher Site | Google Scholar | MathSciNet
13. S. Dey, T. Dey, and D. Kundu, “Two-parameter Rayleigh distribution: different methods of estimation,” American Journal of Mathematical and Management Sciences, vol. 33, no. 1, pp. 55–74, 2014. View at: Publisher Site | Google Scholar
14. P. Kitidamrongsuk, Discriminating between the Marshall-Olkin exponential distribution and the gamma distribution [Doctor of Philosophy (Statistics)], National Institute of Development Administration, Bangkok, Thailand, 2010.
15. A. Erdelyi, W. Maguns, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York, NY, USA, 1953.
16. A. W. Marshall and I. Olkin, “A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families,” Biometrika, vol. 84, no. 3, pp. 641–652, 1997.
17. G. Casella and R. Berger, Statistical Inference, Duxbury, Belmont, Calif, USA, 2nd edition, 2002.
18. J. H. K. Kao, “Computer methods for estimating weibull parameters in reliability studies,” IRE Transactions on Reliability and Quality Control, vol. 13, pp. 15–22, 1958. View at: Publisher Site | Google Scholar
19. J. H. K. Kao, “A graphical estimation of mixed Weibull parameters in life-testing of electron tubes,” Technometrics, vol. 1, no. 4, pp. 389–407, 1959. View at: Publisher Site | Google Scholar
20. N. R. Mann, R. E. Schafer, and N. D. Singpurwalla, Methods for Statistical Analysis of Reliability and Life Data, John Wiley & Sons, New York, NY, USA, 1974. View at: MathSciNet
21. J. R. Hosking, “L-moments: analysis and estimation of distributions using linear combinations of order statistics,” Journal of the Royal Statistical Society, Series B—Methodological, vol. 52, no. 1, pp. 105–124, 1990. View at: Google Scholar | MathSciNet
22. R. C. H. Cheng and N. A. K. Amin, “Maximum product of spacings estimation with application to the lognormal distributions,” Math Report 79-1, Department of Mathematics, UWIST, Cardiff, UK, 1979. View at: Google Scholar
23. R. C. H. Cheng and N. A. K. Amin, “Estimating parameters in continuous univariate distributions with a shifted origin,” Journal of the Royal Statistical Society, Series B: Methodological, vol. 45, no. 3, pp. 394–403, 1983. View at: Google Scholar
24. B. Ranneby, “The maximum spacing method: an estimation method related to the maximum likelihood method,” Scandinavian Journal of Statistics, vol. 11, no. 2, pp. 93–112, 1984. View at: Google Scholar | MathSciNet
25. R. C. H. Cheng and M. A. Stephens, “A goodness-of-fit test using Moran's statistic with estimated parameters,” Biometrika, vol. 76, no. 2, pp. 385–392, 1989.
26. R. D'Agostino and M. Stephens, Goodness-of-Fit Techniques, Marcel Dekker, New York, NY, USA, 1986.
27. A. Luceño, “Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators,” Computational Statistics & Data Analysis, vol. 51, no. 2, pp. 904–917, 2006. View at: Publisher Site | Google Scholar | MathSciNet
28. P. D. M. Macdonald, “Comment on ‘an estimation procedure for mixtures of distributions’ by Choi and Bulgren,” Journal of the Royal Statistical Society, Series B: Methodological, vol. 33, no. 2, pp. 326–329, 1971. View at: Google Scholar
29. J. W. Boag, “Maximum likelihood estimates of the proportion of patients cured by cancer therapy,” Journal of the Royal Statistical Society, Series B, vol. 11, pp. 15–53, 1949. View at: Google Scholar
30. A. N. F. Silva, Estudo evolutivo das crianças expostas ao HIV e notificadas pelo Núcleo de Vigilância Epidemiológica do HCFMRP-USP [Dissertação de Mestrado], Faculdade de Medicina de Ribeirão Preto, Universidade de São Paulo, São Paulo, Brazil, 2004.

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