Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 539860, 21 pages
http://dx.doi.org/10.1155/2010/539860
Research Article

Estimation of the Parameters of the Reversed Generalized Logistic Distribution with Progressive Censoring Data

1Faculty of Computers and Informatics, Zagazig University, Zagazig 44519, Egypt
2Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 30 October 2010; Revised 20 December 2010; Accepted 23 December 2010

Academic Editor: Attila Gilányi

Copyright © 2010 Z. A. Abo-Eleneen and E. M. Nigm. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The reversed generalized logistic (RGL) distributions are very useful classes of densities as they posses a wide range of indices of skewness and kurtosis. This paper considers the estimation problem for the parameters of the RGL distribution based on progressive Type II censoring. The maximum likelihood method for RGL distribution yields equations that have to be solved numerically, even when the complete sample is available. By approximating the likelihood equations, we obtain explicit estimators which are in approximation to the MLEs. Using these approximate estimators as starting values, we obtain the MLEs using iterative method. We examine numerically MLEs estimators and the approximate estimators and show that the approximation provides estimators that are almost as efficient as MLEs. Also we show that the value of the MLEs decreases as the value of the shape parameter increases. An exact confidence interval and an exact joint confidence region for the parameters are constructed. Numerical example is presented in the methods proposed in this paper.

1. Introduction

There are many scenarios in life-testing and reliability experiments whose units are lost or removed from experimentation before failure. The loss may occur un intentionally, or it may have been designed so in the study. In many situations, however, the removal of units prior to failure is preplanned in order to provide saving in terms of time and cost associated with testing. There are types of censored test. Type I and Type II have been investigated extensively by many authors (see, e.g., [13]). A generalization of type II censoring is progressive Type II censoring. Under this scheme, units are placed in test at time zero. Immediately following the first failure, surviving units are removed from the test at random. Then, Immediately following the second failure, surviving units are removed from the test at random. This process continues until, at the time of the observed failure, the remanding are all removed from the experiment. In this censoring scheme, are all prefixed. However, in some particle situations, the size may occur at random (see [4]). Note that if , then which corresponds to the type II censoring. If , then which corresponds to the complete sample.

The statistical inference on the parameters of lifetime distribution under progressive Type II censoring has been studied by many authors such as Cohen [5], Mann [1], Viveros and Balakrishnan [6], and Balakrishnan and Sandhu [7]. Viveros and Balakrishnan has discussed inference for the Weiblull and exponential distributions under progressive Type II censoring and derived explicit expression for the best linear unbiased estimator (BLUE) of the parameters of the one- and two-parameter exponential distribution. Balakrishnan et al. [8] have discussed the inference from the extreme value distribution under progressive Type II censored sample. Wu [9, 10] obtained estimation results concerning a progressive Type II censored sample from two-parameter Weibuul distribution and Pareto distribution, respectively. Balakrishnan et al. [11] have examined numerically the bias and the mean square error of the MLEs based on a progressive Type II censored sample from a Gaussian distribution. Point and interval estimation for the parameters of the logistic distribution based on progressive Type II censored samples are obtained by Balakrishnan and Kannon [12]. Arturo [13] have investigated the estimation problem of exponential parameters, on the basis of general progressive Type II censored sample. Balakrishnan et al. [14] have discussed the inference from the extreme value distribution under progressive Type II censored sample. Nigm and Aboeleneen [15] have obtained the maximum likelihood estimators of the location and scale parameters of inverse Weibull distribution and observed Fisher information matrix based on progressive Type II censored samples and other inference. Patel [16] has obtained MLE for exponential model with changing failure rates based on two-stage progressively multiply type II censored samples. Recently Gajjar and Patel [17] studied the estimation for a mixture of exponential distributions based on progressively type II censored sample. The reversed generalized Logistic (RGL) distribution are very useful classes of densities as they posses a wide range of indices of skewness and kurtosis. Therefore, an important application of RGL distribution is its use in studying robustness of estimators and tests. Balakrishnan and Leung [18] present two real data examples for the usefulness of RGL distribution, one about oxygen concentration and another about resistance of automobile paint. In both examples the authors choose by eye and verify the validity of this assumption by - plots. Possible applications of this distribution in bioassays as a dose-response curve are discussed and illustrated with some examples by El-Said et al. [19]. This paper considers RGL model and discuss inference based on progressive Type II censored samples. In Section 2, we discuss the maximum likelihood estimators (MLEs) of the location and scale parameters of RGL distribution when the shape parameter is known and we provide an approximation of the maximum likelihood function that leads to explicit estimators. In Section 3, the observed Fisher information matrix is obtained. In Section 4, we obtain MLEs for shape and scale parameters of RGL distribution. We also obtained an exact confidence interval for the scale parameter and an exact joint region for the scale and shape parameters are constructed. A numerical example to show the usefulness of our results is provided in Section 5. Results of simulation study conducted to evaluate the performance of these approximate estimators to the MLEs, in terms of both bias and mean squared error, also to show how the change in the value of shape parameter is after the estimators, are provided in Section 6.

2. Location-Scale Parameters Estimation When the Shape Parameter Is Known

Assume the failure time distribution to be the RGL distribution with probability density function (pdf) and the corresponding cumulative distribution function (cdf) is given by With and , we have a standard RGL distribution with pdf and cdf and the corresponding survival distribution function is given by respectively.

With , we get the Logistic distribution with pdf and cdf, respectively, and the corresponding cumulative distribution function (cdf) is given by Let be a progressively type II censored sample from (2.1) with censoring scheme . The likelihood function based on the progressive Type II censored sample is given by where The likelihood function may be rewritten as where .

Using the relation , the log-likelihood function is then given by form (2.10), we derive the likelihood equation for and as Equation (2.11) may be written, respectively, as where is the cdf of the Logistic distribution defined in (2.6). Equation (2.12) do not provide explicit solution for the parameters and have to be solved numerically to obtain the MLEs of the two-parameter. Hence it may be needed to develop approximation solutions to MLEs equations which yields explicit solutions to the parameters. These explicit solution may be a good starting value for the iterative solution for the MLEs equations. Several Approximate solutions for MLEs have been discussed in the book by Tiku et al. [20]. In this text we approximate the Logistic cdf by expanding it in Taylor series around . From the book of Balakrishnan and Aggarwala [21], it is known that where is the order statistics from a progressively Type-II censored sample from the uniform distribution.

So, we have and hence where and is given by the book of Balakrishnan and Aggarwala [21], For the logistic distribution, is easily seen as By expanding around and keeping only the first two terms, we have where Using the above expression, we obtain the approximate MLEs equations as We may write (2.20) as which yields the estimator of as where

Also (2.21) may be written as

Now replacing in (2.25) by , we have

It is easy to see that the last four terms vanish, so we got the quadratic equation of as or where Hence, we have Since , only one root is admissible, and hence the approximate MLE of is given by the approximate MLEs are thus given explicitly by (2.23) and (2.31).

Remark 2.1. When the shape parameter the approximate MlEs estimators of and are identical to those derived in Balakrishnan and Kannan [12] for the case of Logistic Distribution under progressive Type II censoring.

3. Observed Fisher Information Matrix

We need to compute the asymptotic variances-covariance matrix. In this section, we derive the observed Fisher information matrix for the full and approximate equations.

Now, we derive the observed Fisher information matrix for the Likelihood equation (2.11). We have Similarly, from the approximate likelihood equations, we obtain from (2.20) and (2.21) Now, let The observed information matrix can be inverted to obtain the asymptotic variance-covariance matrix of the estimators as where

4. Scale and Shape Parameters Estimation When the Location Parameter Is Known

The maximum likelihood estimators (MLEs) for the scale and shape parameters of the RGL distribution based on progressive Type II censoring are derived. An exact confidence interval for the scale parameter and an exact joint confidence region for the scale and shape parameters are investigated also. The pdf in (2.1) for the shape and scale parameters may be written as and the associated cumulative distribution function (cdf) is given by The likelihood function is given by where and .

The log-likelihood function is then given by

Hence, we have the likelihood equations for and as The MLEs and can be obtained by solving the likelihood equations. Equation (4.7) yields the MLe of to be

Equation (4.6), in conjunction with the MLE of in (4.8), reduces to Since (4.9) can not be solved analytically for , some numerical methods such as Newton's method must be employed.

One can obtained an approximate MLEs estimators and by approximating the logistic cdf as we did in Section 2.

In the rest of this section, an exact confidence interval for the scale parameter and an exact joint confidence region for the scale and shape parameters are investigated. Let denote a progressive type II censored sample from a two-parameter RLG distribution, with censoring scheme . Further, Let , . It can be seen that is a progressive Type II censored order statistic from an exponential distribution with mean 1.

Let us consider the following transformation:

Thomas and Wilson [22] established that the generalized spacing as defined above are independent and identically distributed as an exponential distribution with mean 1. Hence, has a Chi-square distribution with 2 degree of freedom and has a Chi-square distribution with degree of freedom. Now, let

It is easy to see that has an distribution with and 2 degree for freedom and has a Chi-square distribution with 2 degree of freedom. Also and are independent.

Lemma 4.1. Suppose that . Let where , , then, is strictly decreasing function in for any . Furthermore, if , the equation has a unique solution for any .

Proof. The proof is similar to the proof of Lemma 1 in Wu [9].

The following theorem gives an exact confidence interval for the parameter .

Theorem 4.2. Suppose that , , are the order statistics of a progressively type II right censored sample of size , from a two-parameter RGL distribution, with censoring scheme , then confidence interval for the scale parameter is given by where and is the solution of for the equation

Proof. The proof is similar to the proof of Theorem 1 in Wu [9].

Let denote the percentile of Chi-square distribution with the right-tail probability and degree of freedom. An exact joint confidence region for the parameters and is given by the following theorem.

Theorem 4.3. Suppose that , , are the order statistics of a progressively type II right censored sample of size , from a two-parameter RGL distribution, with censoring scheme , then confidence region for the shape and scale parameters and is given by the following inequalities: where and is the solution of for the equation

Proof. The proof is similar to the proof of Theorem 2 in Wu [9].

5. Illustrative Example

To illustrate the methods of inference proposed in this paper, the following example is discussed.

Example 5.1. A sample of size of a progressively Type II censored data giving the log-time to breakdown of an insulating fluid tested at 34 kilovolts from RGL distribution was randomly selected from the observation in Nelson's [23, Table 5.1], refer to Table 10 in this paper, as utilized earlier by Viveros and Balakrishnan [6] and studied by Balakrishnan and Kannan [12] as a data came from logistic distribution.
For this example,
(1) We obtain the MLEs of and and the approximated MLEs of and , for different value of the shape parameters as Table 1 shows.
Table 1 provides the values of MLEs of and and the approximated values of and , for , 1.5, 2, and 3. The entries were computed using (2.12) for the MLEs values and (2.23) and (2.31) for the approximated values and MATHEMATICA.
Note that for , the RGL distribution encompasses to the logistic distribution as a further special case moreover, these values agree with the reported MLEs and the approximated MLEs in Balakrishnan and Kannan [12] for the logistic distribution.
(2) We can determine numerically the values of the MLEs and by solving (4.8) and (4.9) numerically to be [0.2248] and [0.6445], respectively. By Theorem 4.2 and using Mathematica Package we can get a confidence interval for the scale parameter as , where the percentile Furthermore, to obtain a joint confidence interval for the scale parameter and the shape parameter , we need the following percentiles: By Theorem 4.3, a joint confidence interval for parameter and determined by the following inequalities:
In connecting to the previous work, Lawless [2, pages 147–154, 533–539] used the conditional method to handle data from complete and Type II right censored samples and suggested it for progressively Type II censored samples. Viveros and Balakrishnan [6] extend the conditional method to obtain conditional inference under progressively Type II censored samples. In compassion between the conditional method and the unconditional method used in this paper, it is important to mention here that the intervals obtained by Viveros and Balakrishnan [6] are conditional in that they are based on the specific progressive censored sample, while the interval and region we have obtained here are unconditional in that the above percentiles given by Theorems 1 and 2 can be used to obtain confidence interval and confidence region for any other progressively type censored sample with same scheme, that is, same . Moreover the conditional approach of Viveros and Balakrishnan [6] requires intensive numerical integration computationals for determining percentage points while the method used here does not require any numerical integration.

tab1
Table 1: MLEs of estimators and the approximated MLEs of estimators for , and .

6. Simulation Results

In this section, we introduce a simulation study to compare the performance of the approximate MLEs estimators with the MLEs and also to shows the effect of the change of the value of the shape parameter on the values of MLEs estimators and the approximate MLEs estimators. A progressively Type-II censored samples from the standard RGL distribution were generated by using the algorithm presented in Balakrishnan and Sandhu [24] according to the following steps.(1)Generate independent random variables .(2)For given values of the progressive censoring scheme , set ,  .(3)Set , ; then is a progressive Type II censored sample of size m from .(4)Finally, set for , where is the inverse cdf of the standard RGL distribution.

We computed MLEs estimators and the approximate MLEs estimators (table entries) according to the following steps.(1)For given values of the progressive censoring scheme and , generate A progressively Type-II censored samples from the standard RGL distribution using the above algorithm.(2)Calculate the approximate MLEs estimators from (2.23) and (2.31).(3)Calculate variances and covariances matrix of the approximate MLEs estimators from (3.2).(4)The MLEs estimators of the parameters were obtained by solving equation (2.12) numerically by using the approximate MLEs computed from (2.11) as starting values for the numerical iterations.(5)Calculate variances and covariances matrix of MLEs estimators from (3.1).(6)Steps (1)–(5) are repeated 10,000 times, and the average of the MLEs estimators, approximate MLEs estimators and variances and covariances matrix are caudated.(7)The simulation, were carried out for sample size and 20 and for each , and . Furthermore we provide a different choices of the effective sample , and different progressive censoring schemes in each case including the two extreme censoring schemes of and .

The results of the MLEs estimators and variances and covariances matrix of MLEs estimators are displayed in Tables 1, 3, 5, and 7 and The results of the approximate MLEs estimators and variances and covariances matrix of the approximate MLEs estimators are displayed in Tables 2, 4, 6, and 8.

tab2
Table 2: Average, variances, and covariance of the MLEs when the shape parameter .
tab3
Table 3: Average, variances, and covariance of the approximate MLEs when the shape parameter .
tab4
Table 4: Average, variances, and covariance of the MLEs when the shape parameter .
tab5
Table 5: Average, variances, and covariance of the approximate MLEs when the shape parameter .
tab6
Table 6: Average, variances, and covariance of the MLEs when the shape parameter .
tab7
Table 7: Average, variances, and covariance of the approximate MLEs when the shape parameter .
tab8
Table 8: Average, variances, and covariance of the MLEs when the shape parameter .
tab9
Table 9: Average, variances, and covariance of the approximate MLEs when the shape parameter .
tab10
Table 10: Progressively censored sample generated by Nelson [23].

From Tables 18, we observe that the approximate estimators and the MLEs estimators are seems to be identical in terms of bias and variance. For all sample size and censoring schemes, the approximate estimators are almost as efficient as the MLEs. The values in Tables 1 and 2 where are close to the results reported by Balakrishnan and Kannan [12, Table 1]. For all choices, the censoring scheme , provide the smallest bias and variance for the estimates. Both bias and variance of the estimators decreases significantly as the effective sample proportion increases.

Finally the tables shows that for all sample size and censoring schemes, both the bias and variance of the MLEs estimators decreases significantly by increasing the value of shape parameter .

7. Conclusion

In this paper, we considers the estimation problem for the parameters of the RGL distribution based on progressive Type II censoring. We have obtained explicit estimators which are approximation to the MLEs of the location and scale parameters. Also obtained An exact confidence interval and an exact joint confidence region for the scale and shape parameters beside the MLEs.

In addition, simulation study shows that the approximation estimators are almost as efficient as the MLEs. Also shows that the value of the MLEs decreases as the value of the shape parameter increases.

Acknowledgments

The author would like to express deep thanks to the Editor-in-Chief and the referees for their helpful comments and suggestions which led to a considerable improvement in the presentation of this paper.

References

  1. N. R. Mann, “Best linear invariant estimation for Weibull parameters under progressive censoring,” Technometrics, vol. 13, pp. 521–533, 1971. View at Google Scholar · View at Zentralblatt MATH
  2. J. F. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York, NY, USA, 1982.
  3. W. Q. Meeker and L. A. Escobar, Statistical Methods for Reliability Data, John Wiley & Sons, New York, NY, USA, 1998.
  4. S. K. Tse and H. K. Yuen, “Expected experiment times for the Wiebull distribution under progressive censoring with random removals,” Journal of Applied Statistics, vol. 25, pp. 75–83, 1998. View at Google Scholar
  5. A. C. Cohen, Jr., “Progressively censored samples in life testing,” Technometrics, vol. 5, pp. 327–339, 1963. View at Google Scholar · View at Zentralblatt MATH
  6. R. Viveros and N. Balakrishnan, “Interval estimation of parameters of life from progressively censored data,” Technometrics, vol. 36, no. 1, pp. 84–91, 1994. View at Google Scholar · View at Zentralblatt MATH
  7. N. Balakrishnan and R. A. Sandhu, “Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive type-II censored samples,” Sankhya. Series B, vol. 58, no. 1, pp. 1–9, 1996. View at Google Scholar · View at Zentralblatt MATH
  8. N. Balakrishnan, N. Kannan, C. T. Lin, and H. K. T. Ng, “Inference for the extreme value distribution based on progressive Type-II censored samples,” IEEE Transactions on Reliability, vol. 58, pp. 1–9, 2002. View at Google Scholar
  9. S. J. Wu, “Estimations of the parameters of the Weibull distribution with progressively censored data,” Journal of the Japan Statistical Society, vol. 32, no. 2, pp. 155–163, 2002. View at Google Scholar · View at Zentralblatt MATH
  10. S. J. Wu, “Estimation for the two-parameter Pareto distribution under progressive censoring with uniform removals,” Journal of Statistical Computation and Simulation, vol. 73, no. 2, pp. 125–134, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. N. Balakrishnan, N. Kannan, C. T. Lin, and H. K. T. Ng, “Point and interval estimation for the normal distribution based on progressive type-II censored samples,” IEEE Transactions on Reliability, vol. 52, no. 1, pp. 90–95, 2003. View at Google Scholar
  12. N. Balakrishnan and N. Kannan, “Point and interval estimation for parameters of the Logistic distribution based on progressively censored samples,” in Handbook of Statistics, N. Balakrishnan and C. R. Rao, Eds., vol. 20, pp. 431–456, Elsevier, Amsterdam, The Netherlands, 2001. View at Google Scholar
  13. A. J. Fernández, “On estimating exponential parameters with general type II progressive censoring,” Journal of Statistical Planning and Inference, vol. 121, no. 1, pp. 135–147, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. N. Balakrishnan, N. Kannan, C. T. Lin, and S. J. S. Wu, “Inference for the extreme value distribution under progressive type-II censoring,” Journal of Statistical Computation and Simulation, vol. 74, no. 1, pp. 25–45, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. E. M. Nigm and Z. A. Abo-eleneen, “Estimation of the parameters of the inverse Weibull distribution with progressive censoring data,” Journal of Applied Statistical Science, vol. 15, no. 1, pp. 39–47, 2007. View at Google Scholar
  16. M. N. Patel, “MLE for exponential model with changing failure rates based on two-stage progressively multiply type II censored samples,” Journal of Probability and Statistical Science, vol. 4, no. 2, pp. 221–232, 2006. View at Google Scholar
  17. K. A. Gajjar and M. N. Patel, “Estimation for a mixture of exponential distributions based on progressively type-II censored sample,” International Journal of Agriculteral Statistical Sciences, vol. 4, no. 1, pp. 169–176, 2008. View at Google Scholar
  18. N. Balakrishnan and M. Y. Leung, “Means, variances and covariances of order statistics, BLUEs for the type I generalized logistic distribution, and some applications,” Communications in Statistics. Simulation and Computation, vol. 17, no. 1, pp. 51–84, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  19. M. A. El-Saidi, B. Dimitrov, and S. Chukova, “Some moment properties and limit theorems of the reversed generalized logistic distribution with applications,” Communications in Statistics. Theory and Methods, vol. 25, no. 3, pp. 609–630, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. M. L. Tiku, W. Y. Tan, and N. Balakrishnan, Robust Inference, vol. 71 of Statistics: Textbooks and Monographs, Marcel Dekker, New York, NY, USA, 1988.
  21. N. Balakrishnan and R. Aggarwala, Progressive Censoring Theory, Methods, and Application, Statistics for Industry and Technology, Birkhäuser, Boston, Mass, USA, 2000.
  22. D. R. Thomas and W. M. Wilson, “Linear order statistic estimation for the two parameters Wiebull and extreme value distributions from type-II progressively censored samples,” Technometrics, vol. 14, pp. 670–691, 1972. View at Google Scholar
  23. W. Nelson, Applied Life Data Analysis, John Wiley & Sons, New York, NY, USA, 1982.
  24. N. Balakrishnan and R. A. Sandhu, “A simple simulational algorithm for generating progressive type-II censored samples,” The American Statistician, vol. 49, pp. 229–230, 1995. View at Google Scholar