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

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., [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, 𝑅₁ surviving units are removed from the test at random. Then, Immediately following the second failure, 𝑅₂ surviving units are removed from the test at random. This process continues until, at the time of the 𝑚th observed failure, the remanding 𝑅_𝑚=𝑛−𝑅₁−𝑅₂−⋯−𝑅_𝑚−1−𝑚 are all removed from the experiment. In this censoring scheme, 𝑅₁,𝑅₂,…,𝑅_𝑚 are all prefixed. However, in some particle situations, the size 𝑅_𝑖 may occur at random (see [4]). Note that if 𝑅₁=𝑅₂=⋯−𝑅_𝑚−1=0, then 𝑅_𝑚=𝑛−𝑚 which corresponds to the type II censoring. If 𝑅₁=𝑅₂=⋯=𝑅_𝑚=0, 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 [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. [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)𝑓=𝜙𝑥,𝜇,𝜎𝜎exp𝑥−𝜇𝜎1+exp𝑥−𝜇𝜎^−(𝜙+1),𝑥>𝜇,𝜎>0,𝜙>0, (2.1) and the corresponding cumulative distribution function (cdf) is given by𝐹𝑥,𝜇,𝜎=1−1+exp𝑥−𝜇𝜎^−𝜙,𝑥>𝜇,𝜎>0,𝜙>0. (2.2) With 𝜇=0 and 𝜎=1, we have a standard RGL distribution with pdf and cdf 𝑓(𝑥,0,1)=𝜙exp(𝑥)1+exp(𝑥)^−(𝜙+1),𝑥>0,𝜙>0, (2.3) and the corresponding survival distribution function is given by 1−𝐹(𝑥,0,1)=1+exp(𝑥)^−𝜙,𝜙>0, (2.4) respectively.

With 𝜙=1, we get the Logistic distribution with pdf and cdf, respectively,𝑔=1𝑥,𝜇,𝜎𝜎exp𝑥−𝜇𝜎1+exp𝑥−𝜇𝜎⁻²,𝑥>𝜇,𝜎>0, (2.5) and the corresponding cumulative distribution function (cdf) is given by1−𝐺𝑥,𝜇,𝜎=1+exp𝑥−𝜇𝜎⁻¹,𝑥>𝜇,𝜎>0. (2.6) 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 (𝑅₁,𝑅₂,…𝑅_𝑚). The likelihood function based on the progressive Type II censored sample is given by𝐿𝜇,𝜎=𝑘^𝑚_𝑖=1𝑓𝑥_{𝑖∶𝑚∶𝑛},𝜇,𝜎1−𝐹𝑥_{𝑖∶𝑚∶𝑛},𝜇,𝜎^𝑅𝑖, (2.7) where 𝑘=𝑛𝑛−𝑅₁−1𝑛−𝑅₁−𝑅₂−2⋯𝑛−𝑅₁−𝑅₂−𝑅₃⋯−𝑅_𝑚−1−𝑚+1. (2.8) The likelihood function may be rewritten as𝐿𝜇,𝜎=𝑘^𝑚_𝑖=1𝑓𝑦_{𝑖∶𝑚∶𝑛}1−𝐹𝑦_{𝑖∶𝑚∶𝑛}^𝑅𝑖, (2.9) 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 by𝐿^∗=log𝑘+𝑚log𝜙−𝑚log𝜎+^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}−^𝑚_𝑖=1log1+exp𝑦_{𝑖∶𝑚∶𝑛}+^𝑚_𝑖=1𝑅_𝑖+1log1−𝐹𝑦_{𝑖∶𝑚∶𝑛}, (2.10) form (2.10), we derive the likelihood equation for 𝜇 and 𝜎 as𝜕𝐿^∗=𝜕𝜇−𝑚𝜎+1𝜎^𝑚_𝑖=1exp𝑦_{𝑖∶𝑚∶𝑛}1+exp𝑦_{𝑖∶𝑚∶𝑛}+𝜙𝜎^𝑚_𝑖=1𝑅_𝑖𝑦+1exp_{𝑖∶𝑚∶𝑛}1+exp𝑦_{𝑖∶𝑚∶𝑛},𝜕𝐿^∗𝑚𝜕𝜎=−𝜎−1𝜎^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}+1𝜎^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}exp𝑦_{𝑖∶𝑚∶𝑛}1+exp𝑦_{𝑖∶𝑚∶𝑛}+𝜙𝜎^𝑚_𝑖=1𝑅_𝑖𝑦+1^{𝑖∶𝑚∶𝑛}exp𝑦_{𝑖∶𝑚∶𝑛}1+exp𝑦_{𝑖∶𝑚∶𝑛}. (2.11) Equation (2.11) may be written, respectively, as𝜕𝐿^∗=𝜕𝜇−𝑚𝜎+1𝜎^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝐺𝑦_{𝑖∶𝑚∶𝑛},𝜕𝐿^∗=𝜕𝜎−𝑚𝜎−1𝜎^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}+1𝜎^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}𝐺𝑦_{𝑖∶𝑚∶𝑛}, (2.12) 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. Several Approximate solutions for MLEs have been discussed in the book by Tiku et al. [20]. In this text we approximate the Logistic cdf 𝐺(𝑦_{𝑖∶𝑚∶𝑛}) by expanding it in Taylor series around 𝐸(𝑦_{𝑖∶𝑚∶𝑛})=𝛾_{𝑖∶𝑚∶𝑛}. From the book of Balakrishnan and Aggarwala [21], it is known that 𝐺𝑦_{𝑖∶𝑚∶𝑛}=𝑈_{𝑖∶𝑚∶𝑛}, (2.13) where 𝑈_{𝑖∶𝑚∶𝑛} is the 𝑖th order statistics from a progressively Type-II censored sample from the uniform 𝑈(0,1) distribution.

So, we have 𝑦_{𝑖∶𝑚∶𝑛}=𝐺⁻¹𝑈_{𝑖∶𝑚∶𝑛}, (2.14) and hence 𝜈_{𝑖∶𝑚∶𝑛}=𝐸𝑦_{𝑖∶𝑚∶𝑛}=𝐺⁻¹𝛼_{𝑖∶𝑚∶𝑛}, (2.15) where 𝛼_{𝑖∶𝑚∶𝑛}=𝐸(𝑈_{𝑖∶𝑚∶𝑛}) and is given by the book of Balakrishnan and Aggarwala [21], 𝛼_{𝑖∶𝑚∶𝑛}=1−^𝑚_{𝑗=𝑚−𝑖+1}𝑗+𝑅_{𝑚−𝑗+1}+⋯+𝑅_𝑚𝑗+1+𝑅_{𝑚−𝑗+1}+⋯+𝑅_𝑚,𝑖=1,…,𝑚. (2.16) For the logistic distribution, 𝐺⁻¹(⋅) is easily seen as 𝐺⁽⁻¹⁾(𝑢)=ln1−𝑢𝑢. (2.17) By expanding 𝐺(𝑦_{𝑖∶𝑚∶𝑛}) around 𝜈_{𝑖∶𝑚∶𝑛} and keeping only the first two terms, we have𝐺𝑦_{𝑖∶𝑚∶𝑛}=𝐺𝜈_{𝑖∶𝑚∶𝑛}+𝑦_{𝑖∶𝑚∶𝑛}−𝜈_{𝑖∶𝑚∶𝑛}𝐺̀(𝑦_{𝑖∶𝑚∶𝑛})_{𝑦𝑖∶𝑚∶𝑛=𝜈𝑖∶𝑚∶𝑛}=𝛾_{𝑖∶𝑚∶𝑛}+𝛿_{𝑖∶𝑚∶𝑛}𝑦_{𝑖∶𝑚∶𝑛}, (2.18) where𝛾_{𝑖∶𝑚∶𝑛}=𝐺𝜈_{𝑖∶𝑚∶𝑛}−𝜈_{𝑖∶𝑚∶𝑛}𝑔𝜈_{𝑖∶𝑚∶𝑛},𝛿_{𝑖∶𝑚∶𝑛}=𝑔𝜈_{𝑖∶𝑚∶𝑛}≥0,𝑖=1,…,𝑚. (2.19) Using the above expression, we obtain the approximate MLEs equations as𝜕𝐿^∗𝜕𝜇=−𝑚+^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_{𝑖∶𝑚∶𝑛}+𝛿_{𝑖∶𝑚∶𝑛}𝑦_{𝑖∶𝑚∶𝑛}=0,𝜕𝐿^∗𝜕𝜎=−𝑚−^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}+^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}𝛾_{𝑖∶𝑚∶𝑛}+𝛿_{𝑖∶𝑚∶𝑛}𝑦_{𝑖∶𝑚∶𝑛}=0. (2.20)(2.21) We may write (2.20) as−𝑚+^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_{𝑖∶𝑚∶𝑛}+^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}𝑥_{𝑖∶𝑚∶𝑛}−𝜇𝜎=0, (2.22) which yields the estimator of 𝜇 as ∑𝜇=^𝑚𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}𝑥_{𝑖∶𝑚∶𝑛}∑^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}+𝜎∑^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_{𝑖∶𝑚∶𝑛}−𝑚∑^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}=𝐾+𝐿𝜎, (2.23) where ∑𝐾=^𝑚𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}𝑥_{𝑖∶𝑚∶𝑛}∑^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛},∑𝐿=^𝑚𝑖=1𝑅_𝑖+1𝜙+1𝛾_{𝑖∶𝑚∶𝑛}−𝑚∑^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}. (2.24)

Also (2.21) may be written as−𝑚−^𝑚_𝑖=1𝑥_{𝑖∶𝑚∶𝑛}−𝜇𝜎+^𝑚_𝑖=1𝑅_𝑖𝑥+1𝜙+1^{𝑖∶𝑚∶𝑛}−𝜇𝜎𝛾_{𝑖∶𝑚∶𝑛}+^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}𝑥_{𝑖∶𝑚∶𝑛}−𝜇𝜎²=0. (2.25)

Now replacing 𝜇 in (2.25) by 𝑘+𝐿𝜎, we have1−𝑚+𝜎^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_{𝑖∶𝑚∶𝑛}−1𝑥_{𝑖∶𝑚∶𝑛}+1−𝑘𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_𝑖𝑥_{𝑖∶𝑚∶𝑛}−𝑘²+𝑚𝐿−𝐿^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_{𝑖∶𝑚∶𝑛}+𝐿^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}−2𝐿𝜎^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛}𝑥_{𝑖∶𝑚∶𝑛}−𝑘=0. (2.26)

It is easy to see that the last four terms vanish, so we got the quadratic equation of 𝜎 as𝑚𝜎²−^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_𝑖−1𝑥_{𝑖∶𝑚∶𝑛}−𝑘−^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_𝑖𝑥_{𝑖∶𝑚∶𝑛}−𝑘²=0, (2.27) or 𝑚𝜎²−𝐴𝜎−𝐵=0, (2.28) where 𝐴=^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_𝑖−1𝑥_{𝑖∶𝑚∶𝑛}−𝑘,𝐵=^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_𝑖𝑥_{𝑖∶𝑚∶𝑛}−𝑘²>0. (2.29) Hence, we have 𝜎=𝐴±𝐴²+4𝑚𝐵.2𝑚 (2.30) Since 𝐵≥0, only one root is admissible, and hence the approximate MLE of 𝜎 is given by𝜎=𝐴+𝐴²+4𝑚𝐵,2𝑚 (2.31) 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.

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𝜕²𝐿^∗𝜕𝜇²1=−𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑔𝑦_{𝑖∶𝑚∶𝑛},𝜕²𝐿^∗𝜕𝜎²=𝑚𝜎²+1𝜎^2𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}−1𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}𝐺𝑦_{𝑖∶𝑚∶𝑛}−^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}²𝑔𝑦_{𝑖∶𝑚∶𝑛},𝜕²𝐿^∗=𝑚𝜕𝜎𝜕𝜇𝜎²−1𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝐺𝑦_{𝑖∶𝑚∶𝑛}−1𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}𝑔𝑦_{𝑖∶𝑚∶𝑛}. (3.1) Similarly, from the approximate likelihood equations, we obtain from (2.20) and (2.21)𝜕²𝐿^∗𝜕𝜇²1=−𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛿_{𝑖∶𝑚∶𝑛},𝜕²𝐿^∗𝜕𝜎²=𝑚𝜎²+1𝜎^2𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}−1𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}𝛾_{𝑖∶𝑚∶𝑛}+𝛿_{𝑖∶𝑚∶𝑛}𝑦_{𝑖∶𝑚∶𝑛}−^𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}²𝛿_{𝑖∶𝑚∶𝑛},𝜕²𝐿^∗=𝑚𝜕𝜎𝜕𝜇𝜎²−1𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝛾_{𝑖∶𝑚∶𝑛}+𝛿_{𝑖∶𝑚∶𝑛}𝑦_{𝑖∶𝑚∶𝑛}−1𝜎^2𝑚_𝑖=1𝑅_𝑖+1𝜙+1𝑦_{𝑖∶𝑚∶𝑛}²𝛿_{𝑖∶𝑚∶𝑛}. (3.2) Now, let −𝜕²𝐿^∗𝜕𝜇²=𝑉₁𝜎²𝜕,−²𝐿^∗𝜕𝜎²=𝑉₂𝜎²𝜕,−²𝐿^∗=𝑉𝜕𝜎𝜕𝜇𝜎³². (3.3) The observed information matrix can be inverted to obtain the asymptotic variance-covariance matrix of the estimators as1𝜎²𝑉₁𝑉₂𝑉₂𝑉₃⁻¹=𝜎²𝑉₁₁𝑉₁₂𝑉₁₂𝑉₂₂, (3.4) where 𝑉₁₁=𝑉₃𝑉₁𝑉₃−𝑉²₂,𝑉₁₂=𝑉₂𝑉₁𝑉₃−𝑉²₂,𝑉₂₂=𝑉₁𝑉₁𝑉₃−𝑉²₁. (3.5)

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 as𝑓()=𝜙𝑥,𝜎𝜎𝑥exp𝜎𝑥1+exp𝜎^−(𝜙+1),𝜎>0,𝜙>0, (4.1) and the associated cumulative distribution function (cdf) is given by𝐹()𝑥𝑥,𝜎=1−1+exp𝜎^−𝜙,𝜎>0,𝛼>0. (4.2) The likelihood function is given by𝐿₁(𝑐,𝜎)=𝑘^𝑚_𝑖=1𝑓𝑥_{𝑖∶𝑚∶𝑛},𝜎1−𝐹𝑥_{𝑖∶𝑚∶𝑛},𝜎^𝑅𝑖, (4.3) where 𝑘=𝑛𝑛−𝑅₁−1𝑛−𝑅₁−𝑅₂−2⋯𝑛−𝑅₁−𝑅₂−𝑅₃⋯−𝑅_𝑚−1−𝑚+1, (4.4) and 𝑦_{𝑖∶𝑚∶𝑛}=𝑥_{𝑖∶𝑚∶𝑛}/𝜎.

The log-likelihood function is then given by 𝐿^∗₁=log𝑘+𝑚log𝜙−𝑚log𝜎+^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}−^𝑚_𝑖=1log1+exp𝑦_{𝑖∶𝑚∶𝑛}+^𝑚_𝑖=1𝑅_𝑖+1log1−𝐹𝑦_{𝑖∶𝑚∶𝑛}. (4.5)

Hence, we have the likelihood equations for 𝜙 and 𝜎 as𝜕𝐿^∗₁𝑚𝜕𝜎=−𝜎−^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}−1𝜎^𝑚_𝑖=1𝑦_{𝑖∶𝑚∶𝑛}exp𝑦_{𝑖∶𝑚∶𝑛}1+exp𝑦_{𝑖∶𝑚∶𝑛}+𝜙𝜎^𝑚_𝑖=1𝑅_𝑖𝑦+1^{𝑖∶𝑚∶𝑛}exp𝑦_{𝑖∶𝑚∶𝑛}1+exp𝑦_{𝑖∶𝑚∶𝑛},𝜕𝐿^∗₁𝑚𝜕𝜙=−𝜙+^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑦_{𝑖∶𝑚∶𝑛}. (4.6)(4.7) The MLEs 𝜙 and 𝜎 can be obtained by solving the likelihood equations. Equation (4.7) yields the MLe of 𝜙 to be𝑚𝜙=∑^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑥_{𝑖∶𝑚∶𝑛}/𝜎. (4.8)

Equation (4.6), in conjunction with the MLE of 𝜙 in (4.8), reduces to𝑚−^𝑚_𝑖=1𝑥_{𝑖∶𝑚∶𝑛}−^𝑚_𝑖=1𝑥_{𝑖∶𝑚∶𝑛}𝐺𝑥_{𝑖∶𝑚∶𝑛}𝜎+𝑚∑^𝑚_𝑖=1𝑅_𝑖+1𝑥_{𝑖∶𝑚∶𝑛}𝐺𝑥_{𝑖∶𝑚∶𝑛}/𝜎∑^𝑚_𝑖=1𝑅_𝑖+1ln1+exp𝑥_{𝑖∶𝑚∶𝑛}/𝜎=0. (4.9) Since (4.9) can not be solved analytically for 𝜎, some numerical methods such as Newton's method must be employed.

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 𝑅₁,𝑅₂,…,𝑅_𝑚. Further, Let 𝑌_𝑖=log(1+exp(𝑋_{𝑖∶𝑚∶𝑛}/𝜎))^𝜙, 𝑖=1,2,…,𝑚. It can be seen that 𝑌₁<𝑌₂<⋯<𝑌_𝑚 is a progressive Type II censored order statistic from an exponential distribution with mean 1.

Let us consider the following transformation:𝑍₁=𝑛𝑌₁𝑍₂=𝑛−𝑅₁−1𝑌₁−𝑌₂⋮𝑍_𝑚=𝑛−𝑅₁−𝑅₂−⋯−𝑅_𝑚−1−𝑚+1𝑌_𝑚−𝑌_𝑚−1. (4.10)

Thomas and Wilson [22] established that the generalized spacing 𝑍₁,𝑍₂,…,𝑍_𝑚 as defined above are independent and identically distributed as an exponential distribution with mean 1. Hence, 𝑊₁=2𝑍₁=2𝑛𝑌₁ (4.11) has a Chi-square distribution with 2 degree of freedom and𝑊₂=2^𝑚_𝑖=1𝑍_𝑖=2^𝑚_𝑖=1𝑅_𝑖+1𝑌_𝑖−𝑛𝑌₁ (4.12) has a Chi-square distribution with 2𝑚−2 degree of freedom. Now, let𝑊𝑈=²(𝑚−1)𝑊₁=∑^𝑚_𝑖=1𝑅_𝑖+1𝑌_𝑖−𝑛𝑌₁𝑛(𝑚−1)𝑌₁,𝑉=𝑊₁+𝑊₂=(𝑚−1)𝑊₁=2^𝑚_𝑖=1𝑅_𝑖+1𝑌_𝑖. (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.

Lemma 4.1. Suppose that 0<𝑌₁<𝑌₂<⋯<𝑌_𝑚. Let 𝑈(𝜎)=∑^𝑚_𝑖=1𝑅_𝑖+1𝑌_𝑖−𝑛𝑌₁𝑛(𝑚−1)𝑌₁, (4.14) where 𝑌_𝑖=log(1+exp(𝑋_{𝑖∶𝑚∶𝑛}/𝜎))^𝜙, 𝑖=1,2,…,𝑚, then, 𝑈(𝜎) is strictly decreasing function in 𝜎 for any 𝜎>0. Furthermore, if 𝑡>0, the equation 𝑈(𝜎)=𝑡 has a unique solution for any 𝜎>0.

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 (𝑅₁,𝑅₂,…𝑅_𝑚), then 100(1−𝛼)% confidence interval for the scale parameter 𝜎 is given by ℎ𝑋₁,…,𝑋_𝑛,𝐹_𝛼/2(2𝑚−2,2),ℎ𝑋₁,…,𝑋_𝑛,𝐹_1−𝛼/2(2𝑚−2,2), (4.15) where 0<𝛼<1 and ℎ(𝑋₁,…,𝑋_𝑛,𝑡) is the solution of 𝜎 for the equation ∑^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑌_{𝑖∶𝑚∶𝑛}−𝑛log1+exp𝑌_{1∶𝑚∶𝑛}𝑛(𝑚−1)log1+exp𝑌_{1∶𝑚∶𝑛}=𝑡. (4.16)

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

Let 𝜒²_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 (𝑅₁,𝑅₂,…𝑅_𝑚), then 100(1−𝛼)% confidence region for the shape and scale parameters 𝜙 and 𝜎 is given by the following inequalities: ℎ𝑋₁,…,𝑋_𝑛,𝐹_{√(1−1−𝛼)/2}(2𝑚−2,2)<𝜎<ℎ𝑋₁,…,𝑋_𝑛,𝐹_{√(1+1−𝛼)/2}(,𝜒2𝑚−2,2)_{2√(1+1−𝛼)/2(2𝑚)}2∑^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑋_{𝑖∶𝑚∶𝑛}𝜒/𝜎<𝜙<^{2√(1−1−𝛼)/2(2𝑚)}2∑^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑋_{𝑖∶𝑚∶𝑛},/𝜎 (4.17) where 0<𝛼<1 and ℎ(𝑋₁,…,𝑋_𝑛,𝑡) is the solution of 𝜎 for the equation ∑^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑋_{𝑖∶𝑚∶𝑛}/𝜎−𝑛log1+exp𝑦_{1∶𝑚∶𝑛}𝑛(𝑚−1)log1+exp𝑋_{𝑖∶𝑚∶𝑛}/𝜎=𝑡. (4.18)

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 𝑚=8 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 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.
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 percentile𝐹_0.025(14,2)=39.4265,𝐹_0.975(14,2)=0.2059. (5.1) Furthermore, to obtain a 95% joint confidence interval for the scale parameter 𝜎 and the shape parameter 𝜙, we need the following percentiles: 𝐹_0.025(14,2)=39.4265,𝐹_0.975(14,2)𝜒=0.2059,_20.0127(16)=31.2069,𝜒²_0.9873(16)=6.0684. (5.2) By Theorem 4.3, a 95% joint confidence interval for parameter 𝜎 and 𝜙 determined by the following inequalities: 0.6200<𝜎<2.3043,6.06842∑^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑋_{𝑖∶𝑚∶𝑛}/𝜎<𝜙<31.20692∑^𝑚_𝑖=1𝑅_𝑖+1log1+exp𝑋_{𝑖∶𝑚∶𝑛},/𝜎 (5.3)
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,𝑅₁,𝑅₂,…,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.

Table 1 

MLEs of estimators and the approximated MLEs of estimators for 𝜙=1,1.5,2, and 3.

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 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.(1)Generate 𝑚 independent 𝑈(0,1) random variables 𝑊₁,𝑊₂,…,𝑊_𝑚.(2)For given values of the progressive censoring scheme 𝑅_𝐼,𝑅₂,…,𝑅_𝑚, set 𝑉=𝑊^{1/𝑖+𝑅𝑚+𝑅𝑚−1+⋯+𝑅𝑚−𝑖+1}_𝑖,  𝑖=1,2,…,𝑚.(3)Set 𝑈=1−(𝑉_𝑚𝑉_𝑚−1,…,𝑉_{𝑚−𝑖+1}), 𝑖=1,2,…,𝑚; then 𝑈₁,𝑈₂,…,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.

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 𝑛,𝑚,𝑅_𝐼,𝑅₂,…,𝑅_𝑚 and 𝜙, 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 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.

Table 2 

Average, variances, and covariance of the MLEs when the shape parameter 𝜙=1.

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.

Table 7 

Average, variances, and covariance of the approximate MLEs when the shape parameter 𝜙=2.

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.

Table 10 

Progressively censored sample generated by Nelson [23].

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 (𝑅₁=𝑛−𝑚, 𝑅₂=⋯=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.

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.