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

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 theMLEs using iterativemethod.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.


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., 1-3 .A generalization of type II censoring is progressive Type II censoring.Under this scheme, n units are placed in test at time zero.Immediately following the first failure, R 1 surviving units are removed from the test at random.

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Then, Immediately following the second failure, R 2 surviving units are removed from the test at random.This process continues until, at the time of the mth observed failure, the remanding m are all removed from the experiment.In this censoring scheme, R 1 , R 2 , . . ., R m are all prefixed.However, in some particle situations, the size R i may occur at random see 4 .Note that if then m n 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 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.

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 μ 0 and σ 1, we have a standard RGL distribution with pdf and cdf and the corresponding survival distribution function is given by respectively.
With φ 1, we get the Logistic distribution with pdf and cdf, respectively, and the corresponding cumulative distribution function cdf is given by Let X 1:m:n , X 2:m:n , . . ., X m:m:n be a progressively type II censored sample from 2.1 with censoring scheme R 1 , R 2 , . . .R m .The likelihood function based on the progressive Type II censored sample is given by where

2.8
The likelihood function may be rewritten as where y i:m:n x i:m:n − μ /σ.
Using the relation f y φ/σ exp y / 1 exp y 1 − F y , the log-likelihood function is then given by where G y 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.

2.16
For the logistic distribution, G −1 • is easily seen as

2.19
Using the above expression, we obtain the approximate MLEs equations as R i 1 φ 1 y i:m:n γ i:m:n δ i:m:n y i:m:n 0.

2.21
We may write 2.20 as International Journal of Mathematics and Mathematical Sciences which yields the estimator of μ as where

2.24
Also 2.21 may be written as

2.25
Now replacing μ in 2.25 by k Lσ, we have

2.26
It is easy to see that the last four terms vanish, so we got the quadratic equation of σ as where

2.29
Hence, we have Since B ≥ 0, only one root is admissible, and hence the approximate MLE of σ is given by the approximate MLEs are thus given explicitly by 2.23 and 2.31 .
Remark 2.1.When the shape parameter φ 1 the approximate MlEs estimators of μ and σ are identical to those derived in Balakrishnan and Kannan 12 for the case of Logistic Distribution under progressive Type II censoring.

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

3.1
Similarly, from the approximate likelihood equations, we obtain from 2.20 and 2.21

3.3
The observed information matrix can be inverted to obtain the asymptotic variancecovariance matrix of the estimators as where 3.5

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 y i:m:n x i:m:n /σ.The log-likelihood function is then given by

4.5
Hence, we have the likelihood equations for φ and σ as Since 4.9 can not be solved analytically for σ, some numerical methods such as Newton's method must be employed.

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One can obtained an approximate MLEs estimators φ and σ by approximating the logistic cdf G y i:m:n 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 X 1:m:n < X 2:m:n < • • • < X m:m:n denote a progressive type II censored sample from a two-parameter RLG distribution, with censoring scheme is a progressive Type II censored order statistic from an exponential distribution with mean 1.
Let us consider the following transformation:

4.10
Thomas and Wilson 22 established that the generalized spacing Z 1 , Z 2 , . . ., Z m 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 12 has a Chi-square distribution with 2m − 2 degree of freedom.Now, let

4.13
It is easy to see that U has an F distribution with 2m − 2 and 2 degree for freedom and V has a Chi-square distribution with 2 degree of freedom.Also U and V are independent.
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 σ.
Theorem 4.2.Suppose that X i:m:n , i 1, 2, . . ., m, are the order statistics of a progressively type II right censored sample of size n, from a two-parameter RGL distribution, with censoring scheme R 1 , R 2 , . . .R m , then 100 1 − α % confidence interval for the scale parameter σ is given by where 0 < α < 1 and h X 1 , . . ., X n , t is the solution of σ for the equation Proof.The proof is similar to the proof of Theorem 1 in Wu 9 .
Let χ 2 α δ denote the percentile of Chi-square distribution with the right-tail probability α and δ degree of freedom.An exact joint confidence region for the parameters φ and σ is given by the following theorem.Theorem 4.3.Suppose that X i:m:n , i 1, 2, . . ., m, are the order statistics of a progressively type II right censored sample of size n, from a two-parameter RGL distribution, with censoring scheme R 1 , R 2 , . . .R m , then 100 1 − α % confidence region for the shape and scale parameters φ and σ is given by the following inequalities: where 0 < α < 1 and h X 1 , . . ., X n , t is the solution of σ for the equation Proof.The proof is similar to the proof of Theorem 2 in Wu 9 .

Illustrative Example
To illustrate the methods of inference proposed in this paper, the following example is discussed.For this example, 1 We obtain the MLEs of μ and σ and the approximated MLEs of μ and σ, for different value of the shape parameters φ as Table 1 shows.
Table 1 provides the values of MLEs of μ and σ and the approximated values of μ and σ, for φ 1, 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 φ 1, 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 φ and σ 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 95% confidence interval for the scale parameter σ as 0.6200, 2.3043 , where the percentile F 0.025 14,2 39.4265, F 0.975 14,2 0.2059.Furthermore, to obtain a 95% joint confidence interval for the scale parameter σ and the shape parameter φ, we need the following percentiles:  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 m, n, R 1 , R 2 , . . ., R m .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.

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 φ 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.
2 For given values of the progressive censoring scheme The simulation, were carried out for sample size n 10 and 20 and for each φ 1, 1.5, 2, and 3. Furthermore we provide a different choices of the effective sample m, and different progressive censoring schemes in each case including the two extreme censoring schemes of 0, 0, . . ., n − m and n − m, 0, 0, . . ., 0 .
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.
From Tables 1-8, 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 φ 1 are close to the results reported by Balakrishnan and Kannan 12, Table 1 .For all choices, the censoring scheme R 1 n − m, R 2 • • • R m 0 provide the smallest bias and variance for the estimates.Both bias and variance of the estimators decreases significantly as the effective sample proportion m/n 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 φ.

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.

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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.

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 φ 2 by eye and verify the validity of this assumption by Q-Q 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.

Table 2 :
Average, variances, and covariance of the MLEs when the shape parameter φ 1.
randomly selected from the n 19 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.

Table 3 :
Average, variances, and covariance of the approximate MLEs when the shape parameter φ 1.

Table 4 :
Average, variances, and covariance of the MLEs when the shape parameter φ 1.5.

Table 5 :
Average, variances, and covariance of the approximate MLEs when the shape parameter φ 1.5.

Table 6 :
Average, variances, and covariance of the MLEs when the shape parameter φ 2.
2, . . ., m; then U 1 , U 2 , . . ., U m is a progressive Type II censored sample of size m from U 0, 1 .4 Finally, set X i F −1 U i for i 1, 2, . . ., m, where F −1 • is the inverse cdf of the standard RGL distribution.

Table 7 :
Average, variances, and covariance of the approximate MLEs when the shape parameter φ 2.We computed MLEs estimators and the approximate MLEs estimators table entries according to the following steps.1For given values of the progressive censoring scheme n, m, R I , R 2 , . . ., R m and φ, generate A progressively Type-II censored samples from the standard RGL distribution using the above algorithm.

Table 8 :
Average, variances, and covariance of the MLEs when the shape parameter φ 3.

Table 9 :
Average, variances, and covariance of the approximate MLEs when the shape parameter φ 3.