parameter estimation for a modified weibull distribution, for progressively type-ii censored samples

374 IEEE TRANSACTIONS ON RELIABILITY, VOL. 54, NO. 3, SEPTEMBER 2005

Parameter Estimation for a Modified WeibullDistribution, for Progressively Type-II

Censored SamplesH. K. T. Ng

Abstract—In this paper, the estimation of parameters basedon a progressively Type-II censored sample from a modifiedWeibull distribution is studied. The likelihood equations, and themaximum likelihood estimators are derived. The estimators basedon a least-squares fit of a multiple linear regression on a Weibullprobability paper plot are compared with the MLE via MonteCarlo simulations. The observed Fisher information matrix, aswell as the asymptotic variance-covariance matrix of the MLEare derived. Approximate confidence intervals for the parametersare constructed based on the -normal approximation to theasymptotic distribution of MLE, and log-transformed MLE. Thecoverage probabilities of the individual -normal-approximationconfidence intervals for the parameters are examined numerically.Some recommendations are made from the results of a MonteCarlo simulation study, and a numerical example is presented toillustrate all of the methods of inference developed here.

Index Terms—Interval estimation, lifetime data, maximum like-lihood estimation, Monte Carlo simulation, -normal approxima-tion, Weibull probability plot.

ACRONYMS1

c.s. censoring schemeCdf cumulative density functionPdf probability density functionLSRE least-square regression estimatorMLE maximum likelihood estimatorMWD modified Weibull distributionWPP Weibull Probability Paper

NOTATION

, , LSRE of parameters , ,, , MLE of parameters , ,

probability density functioncumulative density functionempirical cdfnumber of failures in a progressively censoredsample (effective sample size)sample sizenumber of surviving units censored at the time ofthe failure

Manuscript received June 12, 2003; revised August 1, 2004. Associate Editor:J. H. Lambert.

The author is with the Department of Statistical Science, Southern MethodistUniversity, Dallas, TX 75275 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TR.2005.853036

1The singular and plural of an acronym are always spelled the same.

progressively censored order statisticupper percentile of a -normal distributionwith mean 0, and standard deviation 1

I. INTRODUCTION

A three-parameter modified Weibull distribution (MWD) wasoriginally proposed by Lai, Xie, & Murthy [1] as an extensionof the Weibull model. This lifetime distribution is useful formodeling lifetime data with bathtub-shape or increasing hazard-rate function, which are common in reliability analysis. Lai, Xie,& Murthy [1] showed that a MWD fit data with a bathtub-shapedhazard-rate function better than competing models.

The cdf, and the pdf of a MWD random variable can bewritten as

(1.1)

and

(1.2)

respectively; where , , and are the parame-ters. Note that the Weibull distribution, and type I extreme-valuedistributions are special cases of the MWD. The shape of thehazard-rate function of a MWD depends only on the parameter. When , the hazard-rate function is increasing in time;

when , the hazard-rate function is a bathtub shaped curve(decreases in the beginning, and then increases in time).

Lai, Xie, & Murthy [1] discussed the properties & parameterestimation for the MWD based on a complete sample. However,in many life-testing & reliability studies, the experimenter maynot always obtain complete information on failure times for allexperimental units. For example, units may break accidentallyin an industrial experiment. There are a lot of situations in whichthe removal of units prior to failure is pre-planned. Data ob-tained from such experiments are called censored data. Savingthe total time on test, and the cost associated with it are someof the major reasons for censoring. Therefore, we consider esti-mation procedures based on censored samples.

The most common censoring schemes are Type-I & Type-IIcensoring, but the conventional Type-I & Type-II censoringschemes do not have the flexibility of allowing removal of unitsat points other than the terminal point of the experiment. For thisreason, we consider a more general censoring scheme calledProgressive Type-II Right censoring, which is as follows. Con-sider an experiment in which units are placed on a life-test.At the time of the first failure, units are randomly removedfrom the remaining surviving units. At the second failure,

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NG: PARAMETER ESTIMATION FOR A MODIFIED WEIBULL DISTRIBUTION 375

units from the remaining units are randomlyremoved. The test continues until the failure at whichtime, all remainingunits are removed. The are fixed prior to the study. Clearly,the complete sample, and Type-II right censored samples arespecial cases of this scheme. Some early works on progres-sive censoring can be found in Cohen [2], [3], Mann [4], andThomas & Wilson [5]. A practical application of progressiveType-II censoring on aging tests on solid insulating materialshas been illustrated by Montanari & Cacciari [6]. Viveros &Balakrishnan [7] proposed a conditional method of inference toderive exact confidence intervals. A book dedicated completelyto progressive censoring has been published by Balakrishnan& Aggarwala [8]. Recently, the estimation of parameters fromdifferent lifetime distributions based on progressive Type-IIcensored data are studied by many authors including Childs &Balakrishnan [9], Balakrishnan & Kannan [10], Ng, Chan, &Balakrishnan [11], and Balakrishnan et al. [12].

In this paper, we consider progressively Type-II censored datafrom a MWD. In Section II, we first discuss the point estimationof model parameters based on least-squares regression on WPPplot (LSRE). Then we derive the MLE for the parameters, andthe expressions for the observed Fisher information matrix. Sec-tion III provides the interval estimation of parameters based ona -normal approximation of the distribution of the MLE, andthe log-transformed MLE. The results of a Monte Carlo simu-lation study are presented in Section IV, which provides a com-parison of LSRE & MLE, as well as coverage probabilities ofconfidence intervals based on them. Suggestions & commentsare made based on these simulation results. Finally, an illustra-tive example is presented in Section V.

II. POINT ESTIMATION OF MODEL PARAMETERS

Letdenote the progressively Type-II right censored sample, with

being the progressive censoring scheme. Forconvenience, we will suppress the censoring scheme in the no-tation of the . We also denote the observed values ofsuch a progressively Type-II right censored sample by

.

A. Based on Multiple Linear Regression on WPP Plot

In the complete sample case, Lai, Xie, & Murthy [1] sug-gested the use of multiple linear regression on a WPP plot toestimate the parameters , , and . This procedure can be mod-ified for progressively Type-II censored data by computing theempirical cdf as (see Meeker & Escobar [13])

(2.1)

where

The estimates of the parameters can be obtained by least-squares fit of multiple linear regression

(2.2)

where , and ,for

.

B. Maximum Likelihood Estimation

1) Estimation Procedure: The likelihood function based ona progressively Type-II censored sample is given by

(2.3)

where

The log-likelihood function may then be written as

(2.4)

where is a constant. This corrects a mistake in Equation (17)in Lai, Xie, & Murthy [1]. From (2.4), we derive the likelihoodequations for parameters , , and as

(2.5)

(2.6)

(2.7)

(2.8)

Equations (2.4)–(2.8) correct the errors, and generalize Equa-tion (17)–(21) in [1]. Equation (2.5) yields an explicit solutionfor in terms of & as presented in (2.6). Therefore, to obtain


, , & , we can solve the two likelihood (2.7) & (2.8) numer-ically to obtain & , and substitute them into (2.6) to obtain

. Numerical methods can be employed here to solve the equa-tions, and the LSRE can be used as initial values for aniterative procedure.

2) Observed Fisher Information: In this subsection, wecompute the observed Fisher information for the MLE. Thesewill enable us to construct confidence intervals for the param-eters based on the limiting -normal distribution, and examinethe probability coverage through simulation.

We now derive the observed Fisher information for the like-lihood using (2.5), (27), and (2.8). We have

(2.9)where

The observed Fisher information matrix can be inverted toobtain a local estimate of the asymptotic variance-covariancematrix of the MLE as

(2.10)

III. CONFIDENCE INTERVALS FOR THE MODEL PARAMETERS

Following the general asymptotic theory of MLE, the sam-pling distribution of can be approximatingby a standard -normal distribution. A two-sided-normal approximation confidence interval for can then be

constructed as

(3.1)

Similarly, the distributions of

and

can be approximated by a standard -normal distribution; andtwo-sided -normal-approximation confidence in-tervals for , and can be constructed as

(3.2)

and

(3.3)

respectively.Because , , and are positive parameters, it is possible to

use log transformations to obtain approximate confidence inter-vals for these parameters (see Meeker & Escobar [13]) by ap-proximating the distribution of bya standard -normal distribution, where is the esti-mated variance of .

A two-sided -normal-approximation confi-dence interval for obtained in this manner is then given by

(3.4)Following the same procedure, a two-sided-normal-approximation confidence interval for , and can

be constructed as

(3.5)and

(3.6)respectively.

IV. SIMULATION RESULTS

A Monte Carlo simulation study is conducted to compare theperformance of the LSRE & MLE of the model parameters. Thecoverage probabilities of the approximate confidence intervalsbased on MLE & log-transformed MLE are also examined viasimulations. Progressively censored samples from MWD withthe following sets of parameters were generated using the algo-rithm in [14]:

(1) , , ;(2) , , ;(3) , , ;(4) , , .


TABLE IPROGRESSIVE CENSORING SCHEMES USED IN THE

MONTE CARLO SIMULATION STUDY

For different choices of sample sizes & progressive censoringschemes, we generated 1,000 sets of data. For notational conve-nience, Table I lists the different censoring schemes (c.s.) usedin the simulation study. We also simulated the complete samplesituations with sample sizes , 40 & 60. For each set ofsimulated data, we computed the LSRE from (2.2), and MLEfrom (2.5), (2.7), & (2.8). The Newton-Raphson method is em-ployed to solve the nonlinear (2.5), (2.7), & (2.8) with startingvalues . We also computed the local estimate of the vari-ance-covariance matrix by inverting the observed Fisher infor-mation matrix (2.9), and use that to construct two-sided approx-imate 90% & 95% confidence intervals for the parameters from(3.1)–(3.6).

A. Comparison of Estimators

Tables II–V present the simulated biases, and mean squareerrors (MSE) of LSRE, and MLE of the parameters , , &for the four MWD families. From these results, we observe thefollowing:

• For estimation of , MLE is better than LSRE, unless theeffective sample sizes are small (say, less than 20).

• For estimation of , MLE is always better than LSRE.• For estimation of , LSRE is better than MLE only when

the effective sample sizes are small (say, less than 20),and the censoring occurs in the early stage of the exper-iment.

• When the conventional Type-II censoring is used, the bi-ases & MSE of the estimates are relatively higher thanthe other two types of progressive censoring schemesbeing considered in the simulation study, which are cen-soring schemes with censoring occurs only at the firstobserved failure, and censoring occurs evenly at differentobserved failures.

TABLE IICOMPARISON OF (~a, ~b, ~�), AND (a; b; �) FOR a = 0:5, b = 1:0, � = 0:1:

TABLE IIICOMPARISON OF (~a;~b; ~�), AND (a; b; �) FOR a = 0:5, b = 1:4, � = 0:1

Based on the simulation results, we suggest the use of MLEinstead of LSRE for point estimation based on considerationof bias & MSE. However, we should keep in mind that nu-merical methods are required to solve the likelihood equations,which may be computationally involved, while the LSRE canbe obtained readily from any statistical packages (for example,


TABLE IVCOMPARISON OF (~a;~b; ~�), AND (a; b; �) FOR a = 2:0, b = 0:8, � = 0:1

TABLE VCOMPARISON OF (~a;~b; ~�), AND (a; b; �) FOR a = 1:4, b = 0:5, � = 0:1

Minitab, S-PLUS, SAS) with a built-in function to solve a mul-tiple linear regression.

B. Coverage Probabilities

Tables VI–X present for the four MWD families the sim-ulated coverage probabilities for two-sided approximate 90%& 95% confidence intervals for the parameters based on the-normal-approximate distributions of MLE & log-transformed

MLE. For parameters & , the simulation results show that the

TABLE VI90% AND 95% COVERAGE PROBABILITIES FOR APPROXIMATE CONFIDENCE

INTERVALS BASED ON MLE, AND LOG-TRANSFORMED MLE(a = 0:5, b = 1:0, � = 0:1)

TABLE VII90% AND 95% COVERAGE PROBABILITIES FOR APPROXIMATE CONFIDENCE


coverage probabilities of confidence intervals based on MLE donot reach the desired confidence levels, while the coverage prob-abilities of confidence intervals based on log-transformed MLEgive more satisfactory results. On the other hand, for param-eter , the coverage probabilities of confidence intervals based


TABLE VIII90% AND 95% COVERAGE PROBABILITIES FOR APPROXIMATE CONFIDENCE


TABLE IX90% AND 95% COVERAGE PROBABILITIES FOR APPROXIMATE CONFIDENCE


on log-transformed MLE are considerably lower than the de-sired confidence levels, while the coverage probabilities of con-fidence intervals based on MLE are close to the desired confi-dence levels.

TABLE XSIMULATED PROGRESSIVELY CENSORED SAMPLE BASED ON FAILURE-TIME

DATA FROM Aarset [15]

TABLE XITWO-SIDED 90% AND 95% s-NORMAL-APPROXIMATE CONFIDENCE

INTERVALS FOR a, b, AND �

From these results, a -normal approximation to the distribu-tion of log-transformed MLE of & , and a -normal approx-imation to the distribution of MLE of provide more accurateresults. In other words, we suggest the use of (3.4) & (3.5) forconstruction of approximate confidence intervals for & , and(3.3) for construction of approximate confidence intervals for

. We can also observe that the coverage probabilities corre-sponding to conventional Type-II censoring are lower than theother two censoring schemes in most cases.

On the basis of the simulation results, it appears that theconventional Type-II censoring may not be a good censoringscheme from the estimation point of view.

V. ILLUSTRATIVE EXAMPLE

In this section, we present an example to illustrate theuse of the inference procedures discussed in this paper. Weconsider the data from Aarset [15], which were used as anillustrative example by Lai, Xie, & Murthy [1]. Note thatthe LSRE, and MLE of parameters ( , , ) based on thecomplete sample are ,and , respectively.In this case, we assume a progressively Type-II rightcensored sample with , and censoring scheme

, , , 11, 18,25, 32. We simulated the progressively Type-II right censoredsample based on the data from Aarset [15], and the data soobtained are presented in Table X.

The LSRE, and MLE of ( , , ), based on the pro-gressively censored sample presented in Table X, arecomputed as , and

, respectively.From (2.10), the local estimate of the asymptotic variance-

covariance matrix of MLE is

Table XI presents the two-sided 90%, and 95% -normal-ap-proximate confidence intervals for , , and from (3.4), (3.5),and (3.3), respectively.


ACKNOWLEDGMENT

The author’s sincere thanks go to the Associate Editor, Dr.J. H. Lambert, and the Managing Editor, Dr. J. W. Rupe. Theauthor would also like to thank Prof. N. Balakrishnan and Prof.W. R. Schucany for their useful suggestions which resulted in amuch improved version of the manuscript.

REFERENCES

[1] C. D. Lai, M. Xie, and D. N. P. Murthy, “A modified Weibull distribu-tion,” IEEE Trans. Reliab., vol. 52, pp. 33–37, 2003.

[2] A. C. Cohen, “Progressively censored samples in life testing,” Techno-metrics, vol. 5, pp. 327–329, 1963.

[3] , “Life testing and early failure,” Technometrics, vol. 8, pp.539–549, 1966.

[4] N. R. Mann, “Exact three-order-statistic confidence bounds on reliablelife for a Weibull model with progressive censoring,” J. Amer. StatisticalAssoc., vol. 64, pp. 306–315, 1969.

[5] D. R. Thomas and W. M. Wilson, “Linear order statistic estimation forthe two-parameter Weibull and extreme value distributions from Type-IIprogressively censored samples,” Technometrics, vol. 14, pp. 679–691,1972.

[6] G. C. Montanari and M. M. Cacciari, “Progressively-censored agingtests on xlpe-insulated cable models,” IEEE Trans. Electrial Insulation,vol. 23, pp. 365–372, 1988.

[7] R. Viveros and N. Balakrishnan, “Interval estimation of parametersof life from progressively censored data,” Technometrics, vol. 36, pp.84–91, 1994.

[8] N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory,Methods, and Applications. Boston: Birkhäuser, 2000.

[9] A. Childs and N. Balakrishnan, “Conditional inference procedures forthe Laplace distribution when the observed samples are progressivelycensored,” Metrika, vol. 52, pp. 253–265, 2000.

[10] N. Balakrishnan and N. Kannan, “Point and interval estimation for pa-rameters of the logistic distribution based on progressively Type-II cen-sored samples,” in Handbook of Statistics, N. Balakrishnan and C. R.Rao, Eds. Amsterdam: North-Holland, 2001, vol. 20, pp. 431–456.

[11] H. K. T. Ng, P. S. Chan, and N. Balakrishnan, “Estimation of parametersfrom progressively censored data using em algorithm,” ComputationalStatistics and Data Analysis, vol. 39, pp. 371–386, 2002.

[12] N. Balakrishnan, N. Kannan, C. T. Lin, and H. K. T. Ng, “Point andinterval estimation for gaussian distribution, based on progressivelytype-II censored samples,” IEEE Trans. Reliab., vol. 52, pp. 90–95,2003.

[13] W. Q. Meeker and L. A. Escobar, Statistical Methods for ReliabilityData. New York: John Wiley & Sons, 1998.

[14] N. Balakrishnan and R. A. Sandhu, “A simple simulational algorithm forgenerating progressive type-II censored samples,” The American Statis-tician, vol. 49, pp. 229–230, 1995.

[15] M. V. Aarset, “How to identify bathtub hazard rate,” IEEE Trans. Reliab.,vol. 36, pp. 106–108, 1987.

H. K. T. Ng is an Assistant Professor in the Department of Statistical Scienceat Southern Methodist University. He received his Ph.D. degree in Mathematics(2002) from McMaster University, Hamilton, Canada. He has been at SouthernMethodist University since August 2002. His research interests include relia-bility, nonparametric methods, censored data analysis, and statistical inference.He is a member of the American Statistical Association.

parameter estimation for a modified weibull distribution, for progressively type-ii censored samples

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