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On the maximum likelihood estimators for parameters of a Weibull distribution under random censoring

Namhyun Kim

aDepartment of Science, Hongik University, Korea
Correspondence to: 1Department of Science, Hongik University, 94 Wausan-Ro, Mapo-Gu, Seoul 04066, Korea. E-mail: nhkim@hongik.ac.kr
Received March 30, 2016; Revised May 18, 2016; Accepted May 19, 2016.
Abstract

In this paper, we consider statistical inferences on the estimation of the parameters of a Weibull distribution when data are randomly censored. Maximum likelihood estimators (MLEs) and approximate MLEs are derived to estimate the parameters. We consider two cases for the censoring model: the assumption that the censoring distribution does not involve any parameters of interest and a censoring distribution that follows a Weibull distribution. A simulation study is conducted to compare the performances of the estimators. The result shows that the MLEs and the approximate MLEs are similar in terms of biases and mean square errors; in addition, the assumption of the censoring model has a strong influence on the estimation of scale parameter.

Keywords : approximate maximum likelihood estimators, Koziol-Green model, maximum likelihood estimators, power comparison, random censoring, Weibull distribution
1. Introduction

In survival studies, data may be subject to random censoring. Let T1, . . . , Tn denote lifetimes with distribution function F and probability density function (pdf) f . The censoring times C1, . . . , Cn drawn independently of the Ti are from distribution function G and pdf g. The Ti’s are censored on the right by Ci, so we observe n i.i.d. random pairs (Xj, δj), j = 1, . . . , n, where

$Xj=min?(Tj,Cj)?????????and?????????δj={1,if?Xj=Tj,0,if?Xj=Cj.$

For a model of random censorship, Koziol and Green (1976) introduced a special model with

$1-G=(1-F)β,?????????for?some?β>0.$

Under this model, we have

$P(Ti>Ci)=∫-∞∞(1-F(x))?dG(x)=∫01β(1-x)βdx=ββ+1.$

Hence, β/(β + 1) is the expected proportion of the censored observations and β is called the censoring parameter. The case β = 0 corresponds to no censoring. Cs?rg? and Horv?th (1981) called (1.2) the Koziol-Green model of random censorship. The motivation and characterization of this model is discussed in Chen et al. (1982) and Kim (2014).

Suppose the lifetime distribution follows a Weibull distribution Weibull(λ, α) with the distribution function

$F(x)=1-e-λxα,?????????x>0,$

and the pdf

$f(x)=αλxα-1e-λxα,?????????x>0,$

where λ and α are the scale and shape parameters, respectively, and are positive.

This paper derives the maximum likelihood estimators (MLEs) and the approximate MLEs of the parameters of a Weibull distribution under random censoring. The analysis of censored data usually assumes that the censoring distribution does not involve any parameters of interest. We find the MLEs and the approximate MLEs in this case and those under the Koziol-Green model in (1.2) to compare them. In Section 2, we obtain the MLEs of the parameters of a Weibull distribution. We also derive the approximate MLEs using the type I extreme value distribution of the minimum since the MLEs cannot be expressed in explicit form. In Section 3, we conduct a simulation study to compare the MLEs and the approximate MLEs. We generate censored random samples under the Koziol-Green model and compute the estimators in two different ways. One is the estimators ignoring the Koziol-Green model (assuming the censoring distribution does not involve any parameters) and the other is the estimators under the true model. We compare the two of them to see what happens when we ignore the random censorship model. Section 4 ends the paper with some concluding remarks.

2. MLEs and the approximate MLEs

### 2.1. MLEs and the approximate MLEs under the assumption that the censoring distribution does not involve any parameters

For the observed random pairs (Xj, δj) in (1.1), we assume that Xj’s are the ordered observations without loss of generality. The likelihood function based on (Xj, δj) is given by (Tableman and Kim, 2003)

$L=n!∏i=1n(f(xi)G?(xi))δi(g(xi)F?(xi))1-δi=n!∏i=1nf(xi)δiF?(xi)1-δi∏i=1ng(xi)1-δiG?(xi)δi,$

where F? = 1 ? F, G? = 1 ? G. If we assume that the distribution of the censoring time Ci’s does not involve any parameters of interest, the part $∏i=1ng(xi)1-δiG?(xi)δi$ in (2.1) plays no role in the estimation process. Hence the likelihood function can be taken to be

$L∝∏i=1nf(xi)δiF?(xi)1-δi.$

If the lifetime distribution follows Weibull(λ, α) given in (1.4) or (1.3), then the log-likelihood function l = ln L becomes

$l∝∑i=1nδi?[ln?α+ln?λ+(α-1)?ln?xi-λxiα]+∑i=1n(1-δi)?(-λxiα).$

The MLEs of α and λ can be derived by solving the likelihood equations

$∂l∂λ=nuλ-∑i=1nxiα=0,$$∂l∂α=nuα+∑i=1nδi?ln?xi-λ∑i=1nxiα?ln?xi=0,$

where $nu=∑i=1nδi$ is the number of uncensored data. From (2.2), we get

$λ(α)=nu∑i=1nxiα.$

Substituting back λ(α) into (2.3), we obtain the equation of α as

$1α=∑i=1nxiα?ln?xi∑i=1nxiα-1nu∑i=1nδi?ln?xi.$

Balakrishnan and Kateri (2008) discussed the existence and uniqueness of the MLEs of a Weibull distribution. The equation (2.5) does not have a solution in a compact form and it should be solved in an iterative way.

If the random variable T follows Weibull(α, λ), then ln T follows the type I extreme value distribution of the minimum EV(μ, σ) with the distribution function and the pdf,

$FY(y)=1-e-ey-μσ,?????????fY(y)=1σey-μσ-ey-μσ,?????????-∞

where μ and σ are location and scale parameters respectively, and μ = ?(1/α) ln λ, σ = 1/α. For the estimation of the extreme value distribution, we refer to Chen (1998) and Engelhardt and Bain (1974).

Pareek et al. (2009) mentioned that Weibull(α, λ) and EV(μ, σ) are equivalent models because the procedure developed under one model can be used for the other. Nonetheless, we could have an easier estimation process by working with the model EV(μ, σ) in (2.6) since μ and σ are the location and scale parameters. A similar technique was used by Kundu (2007) and Kang and Han (2009) for different censoring schemes.

Letting Yj = ln Xj, the log-likelihood function based on (Yj, δj) becomes

$l∝∑i=1nδi?[-ln?σ+yi-μσ-eyi-μσ]+∑i=1n(1-δi)?(-eyi-μσ).$

Remind that Z = (Y ? μ)/σ has a standard extreme value distribution EV(0, 1) with the distribution function and the pdf

$FZ(z)=1-e-ez,?????????fZ(z)=ez-ez$

when Y follows EV(μ, σ). Substituting zi = (yi ? μ)/σ in (2.7), we get

$l(μ,σ)∝nu(-ln?σ)+∑i=1nδizi-∑i=1nezi.$

Since ∂zi/∂μ = ?(1/σ), ∂ezi/∂σ = ?(1/σ)ezi , ∂zi/∂σ = ?(1/σ)zi, and ∂ezi/∂σ = ?(1/σ)ziezi , we have

$∂l∂μ=-nuσ+1σ∑i=1nezi=0,∂l∂σ=-nuσ-1σ∑i=1nδizi+1σ∑i=1nziezi=0,$

by taking derivatives with respect to μ and σ of l(μ, σ) in (2.8). Equivalently

$nu-∑i=1nezi=0,$$nu+∑i=1nδizi-∑i=1nziezi=0.$

Clearly, (2.9) and (2.10) do not have explicit solutions. By expanding h(zj) = ezi in Taylor series keeping only the first 2 terms around

$ξj=FZ-1(pj)=ln(-ln?qj),?????????qj=1-pj,$

for some pj defined later, we approximate

$h(zj)=ezj?h(ξj)+h′(ξj)(zj-ξj)≡aj+bjzj$

with aj = (? ln qj)(1 ? ln(? ln qj)), bj = ? ln qj. Substituting (2.12) and zj = (yj ? μ)/σ into (2.9) and (2.10), we have

$σ?[∑i=1nai-nu]+∑i=1nbiyi=μ∑i=1nbi,$$σ2nu+σ?[∑i=1nδi(yi-μ)-∑i=1nai(yi-μ)]-∑i=1nbi(yi-μ)2=0.$

From (2.13), we obtain the approximate solution μ? as

$μ?=A+Bσ?$

with $A=∑i=1nbiyi/∑i=1nbi$ and $B=(∑i=1nai-nu)/∑i=1nbi$. Substituting (2.15) into (2.14), we get the quadratic equation

$Cσ2-Dσ-E=0?????????or?????????nuσ2-Dσ-E=0,$

where

$C=nu-B?[nu-∑i=1nai+B∑i=1nbi]=nu,D=∑i=1nai(yi-A)-2B∑i=1nbi(yi-A)-∑i=1nδi(yi-A),E=∑i=1nbi(yi-A)2>0.$

Since bj = ? ln qj > 0 and E > 0,

$σ?=D+D2+4nuE2nu$

is the only positive root of the quadratic equation (2.16), and it can be the approximate MLE of σ. The approximate MLEs of α and λ can be α? = 1/σ? and λ? = λ(α?) from (2.4).

Let us think about the plotting position pj in (2.11). For randomly censored data, we can use the Kaplan-Meier estimator

$pjKM=1-∏i≤j(n-in-i+1)δi,$

which has been studied in Kaplan and Meier (1958), Efron (1967), Breslow and Crowley (1974), and Meier (1975). The modified one is

$pj(1)=1-∏i≤j(n-i+1n-i+2)δi?(nn+1)1-δ1.$

We put the part (n/(n + 1))1?δ1 on the similar quantile probabilities proposed by Herd (1960) and Johnson (1964) to avoid 0 values. Another simple choice is the usual Weibull plotting position $pj(2)=j/(n+1)$. Note that $pj(1)$ reduce to $pj(2)=j/(n+1)$ for a complete sample. We call σ?1, α?1, λ?1 for the approximate MLEs when we use $pj(1)$ in (2.17) and σ?2, α?2, λ?2 when $pj(2)=j/(n+1)$ is used.

### 2.2. MLEs and the approximate MLEs under the Koziol-Green model

If we assume the Koziol-Green model (1.2) in the case that the lifetime distribution follows Weibull(λ, α), the censoring distribution also follows a Weibull distribuion with the same shape parameter α. Let us assume G follows Weibull(η, α), then we have the censoring parameter β = η/λ and the expected proportion of the censored data η/(λ + η). The likelihood function in (2.1) still gives the MLE of λ as in (2.4) and the equation of α as

$1α=∑i=1nxiα?ln?xi∑i=1nxiα-1n∑i=1nln?xi,$

which is also the likelihood equation of α for a complete sample. We can obtain the likelihood equation (2.18) from the fact that the distribution of Xj = min(Tj, Cj) follows Weibull(λ + η, α).

As for the approximate MLE of α, we use the extreme value distribution EV(μ, σ) in (2.6) with μ = ?(1/α) ln(λ + η), σ = 1/α. The same approach in Section 2.1 gives

$μ?=A+Bσ?$

and the quadratic equation in σ

$nσ2-Dσ-E=0,$

where

$A=∑i=1nbiyi∑i=1nbi,B=∑i=1nai-n∑i=1nbi,D=∑i=1nai(yi-A)-2B∑i=1nbi(yi-A)-∑i=1n(yi-A),E=∑i=1nbi(yi-A)2>0,$

and aj = (? ln qj)(1 ? ln(? ln qj)), bj = ? ln qj, qj = 1 ? pj, pj = i/(n + 1). Therefore the approximate MLE of σ becomes the positive root of the equation (2.19),

$σ?=D+D2+4nE2n$

and α? = 1/σ?.

3. Simulation results

In this section, a simulation study is conducted to compare the performance of the MLEs with the approximate MLEs in terms of biases and mean squared errors (MSEs) under the different assumptions of the censoring model. Random samples are generated when the true model is the Koziol-Green model in (1.2). We use the sample sizes n = 20, 30, 40, 50 and the parameter values α = 2, 0.5, (λ, η) = (0.4, 0.6), (0.6, 0.4), (0.8, 0.2), which give the proportion of the censoring data 0.6, 0.4, 0.2, respectively. Tables 1?4 provide averages of the MLEs and the approximate MLEs, their MSEs, differences between them, and relative efficiency (RE) for the parameters α and λ, respectively, under the assumption that the censoring distribution does not involve any parameters. Tables 5?8 provide the same statistics under the Koziol-Green model. We replicate each process 2,000 times. The RE, for example, between two estimates α?, α? is defined as

$RE(α?,α^)=MSE(α?)MSE(α^)×100.$

From Tables 1?4, where the estimation process is conducted under the assumption that the censoring distribution does not involve any parameters, we observe the following. First, the approximate MLEs α?1, λ?1 using the modified Kaplan-Meier estimator (2.17) give the closer values to the MLEs α?, λ? on average than the approximate MLEs α?2, λ?2 with $pj(2)=j/(n+1)$ for all censoring ratios and sample sizes considered. This phenomena happen for both parameters α and λ. Second, the estimators α?2, λ?2 have the smallest MSEs, and the MLEs α?, λ? have the largest MSEs for almost all cases. The MSEs are almost the same as for parameter λ, and usually the MSEs of the MLE α? are significantly bigger than the MSEs of the approximate MLEs as for parameter α. We can also see that the REs between the approximate MLEs and the MLEs tend to decrease as the censoring ratio increases (λ decreases) or the sample size decrease.

From Tables 5?8, where the estimation is conducted under the true model (the Koziol-Green model in (1.2)), the average differences between the MLEs and the approximate MLEs are negligible. Apparently the MSEs depend on sample sizes rather than censoring ratios. The assumption of the censoring distribution has a strong influence on the estimation of α rather than that of λ, especially when the censoring ratio is big.

4. Concluding remarks

In this paper, we study randomly censored data from a Weibull distribution. We derive the approximate MLEs in explicit forms using the type I extreme value distribution of the minimum since the MLEs of the parameters do not have explicit solutions. In the simulation study, we generate random samples under the Koziol-Green model, compute the estimators ignoring the censoring model, and compare them with the estimators from the true model. Consequently, we find that the censoring model has a strong influence on the scale parameter α rather than the shape parameter λ. The MSE of the scale parameter becomes larger when we ignore the censoring model. The approximate MLEs are quite similar to the MLEs in terms of biases and MSEs in any situation.

TABLES

### Table 1

Averages of the MLE α? and the approximate MLEs α?1, α?2, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 2, λ = 0.4, 0.6, 0.8 under the assumption that the censoring distribution does not involve any parameters

λnα?α?1α?2MSE(α?)MSE(α?1)MSE(α?2)mean(α? ? α?1)mean(α? ? α?2)RE(α?1, α?)RE(α?2, α?)
0.4202.2852.1292.0340.6660.3750.2380.1550.25156.24335.738
302.1532.1082.0230.3160.2420.1510.0450.13076.70947.707
402.1102.0621.9970.2250.1690.1050.0470.11275.04046.785
502.0922.0742.0120.1520.1450.0860.0180.07995.27656.290

0.6202.1932.1362.0790.3450.2870.2240.0560.11483.33264.858
302.1412.1102.0660.2080.1730.1390.0310.07583.00666.802
402.0952.0782.0410.1330.1230.0990.0170.05492.24774.165
502.0882.0792.0470.1060.0980.0780.0090.04192.16273.708

0.8202.1682.1462.1210.2340.2140.1960.0230.04791.43583.964
302.1022.0912.0730.1360.1300.1210.0100.02995.57988.635
402.0802.0762.0610.1000.0970.0910.0040.01997.16690.474
502.0712.0622.0490.0770.0770.0720.0090.02299.12893.077

MLE = maximum likelihood estimator, RE = relative efficiency, MSE = mean squared error.

### Table 2

Averages of the MLE λ? and the approximate MLEs λ?1, λ?2, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 2, λ = 0.4, 0.6, 0.8 under the assumption that the censoring distribution does not involve any parameters

λnλ?λ?1λ?2MSE(λ?)MSE(λ?1)MSE(λ?2)mean(λ? ? λ?1)mean(λ? ? λ?2)RE(λ?1, λ?)RE(λ?2, λ?)
0.4200.4070.4160.4240.0260.0250.024?0.010?0.01795.31292.540
300.4030.4060.4140.0170.0170.016?0.003?0.01197.56694.282
400.4040.4070.4130.0120.0110.011?0.003?0.00997.58894.681
500.3990.4000.4060.0100.0090.009?0.001?0.00799.59895.153

0.6200.6160.6210.6280.0420.0410.040?0.006?0.01297.85395.309
300.6090.6120.6180.0250.0240.024?0.003?0.00997.63195.252
400.6110.6130.6180.0190.0190.018?0.002?0.00798.32496.440
500.6070.6080.6120.0150.0150.015?0.001?0.00599.37597.387

0.8200.8270.8300.8330.1080.1080.108?0.002?0.00699.854100.396
300.8140.8150.8180.0810.0810.082?0.001?0.00499.926100.799
400.8130.8130.8160.0690.0700.0700.000?0.003100.201101.267
500.8080.8090.8110.0620.0620.063?0.001?0.003100.837101.928

MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.

### Table 3

Averages of the MLE α? and the approximate MLEs α?1, α?2, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 0.5, λ = 0.4, 0.6, 0.8 under the assumption that the censoring distribution does not involve any parameters

λnα?α?1α?2MSE(α?)MSE(α?1)MSE(α?2)mean(α? ? α?1)mean(α? ? α?2)RE(α?1, α?)RE(α?2, α?)
0.4200.5600.5280.5030.0400.0230.0140.0320.05756.97334.912
300.5450.5260.5060.0210.0150.0100.0190.03872.21745.068
400.5270.5200.5020.0130.0120.0070.0070.02486.64352.821
500.5220.5150.5000.0100.0090.0050.0080.02385.35552.461

0.6200.5490.5350.5210.0230.0170.0140.0140.02877.58660.953
300.5290.5210.5100.0120.0100.0080.0080.01984.45367.114
400.5210.5190.5090.0080.0080.0060.0020.01190.33172.481
500.5140.5130.5050.0060.0060.0050.0010.01099.53980.320

0.8200.5390.5350.5290.0150.0150.0130.0040.01098.55390.588
300.5280.5260.5210.0080.0080.0080.0020.007101.96694.120
400.5190.5160.5130.0060.0060.0050.0020.00694.03987.951
500.5130.5120.5090.0050.0040.0040.0010.00497.63891.554

MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.

### Table 4

Averages of the MLE λ? and the approximate MLEs λ?1, λ?2, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 0.5, λ = 0.4, 0.6, 0.8 under the assumption that the censoring distribution does not involve any parameters

λnλ?λ?1λ?2MSE(λ?)MSE(λ?1)MSE(λ?2)mean(λ? ? λ?1)mean(λ? ? λ?2)RE(λ?1, λ?)RE(λ?2, λ?)
0.4200.4040.4110.4190.0270.0260.025?0.007?0.01596.38793.325
300.4020.4070.4140.0170.0160.016?0.006?0.01293.12190.685
400.4040.4050.4120.0130.0120.012?0.002?0.00897.58193.924
500.4050.4080.4140.0100.0090.009?0.003?0.00895.67193.205

0.6200.6160.6220.6280.0890.0890.091?0.005?0.012100.767102.439
300.6100.6140.6200.0700.0710.073?0.004?0.010101.851104.216
400.6080.6090.6140.0620.0620.0640.000?0.00599.876102.249
500.6050.6060.6100.0560.0560.058?0.001?0.005100.045102.423

0.8200.8380.8400.8430.2490.2510.253?0.002?0.005100.629101.450
300.8160.8170.8200.2080.2090.211?0.001?0.004100.506101.394
400.8130.8140.8160.1950.1960.198?0.001?0.004100.410101.269
500.8140.8140.8160.1900.1910.1920.000?0.003100.053100.850

MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.

### Table 5

Averages of the MLE α? and the approximate MLE α?, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 2, λ = 0.4, 0.6, 0.8 under the Koziol-Green model

λnα?α?MSE(α?)MSE(α?)mean(α? ? α?)RE(α?, α?)
0.4202.1522.1430.1960.1920.00897.737
302.1032.1000.1140.1130.00398.783
402.0652.0630.0760.0760.00299.039
502.0642.0630.0590.0590.001100.314

0.6202.1432.1350.1780.1750.00898.278
302.1122.1070.1130.1110.00498.351
402.0752.0730.0760.0760.00299.309
502.0752.0740.0620.0620.001100.179

0.8202.1542.1460.1810.1770.00897.821
302.0962.0920.1080.1070.00499.238
402.0782.0760.0790.0790.00299.509
502.0632.0620.0630.0630.001100.080

MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.

### Table 6

Averages of the MLE λ? and the approximate MLEs λ?, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 2, λ = 0.4, 0.6, 0.8 under the Koziol-Green model

λnλ?λ?MSE(λ?)MSE(λ?)mean(λ? ? λ?)RE(λ?, λ?)
0.4200.4180.4190.0240.024?0.000399.791
300.4100.4100.0160.016?0.000199.830
400.4080.4090.0110.011?0.000199.971
500.4020.4020.0090.0090.0000100.001

0.6200.6230.6240.0410.041?0.000599.776
300.6140.6150.0240.024?0.000399.850
400.6150.6150.0180.018?0.000199.881
500.6090.6090.0150.0150.000099.970

0.8200.8310.8310.1080.108?0.0006100.030
300.8160.8160.0810.081?0.0004100.123
400.8140.8140.0700.070?0.0002100.077
500.8100.8100.0620.0620.0000100.020

MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.

### Table 7

Averages of the MLE α? and the approximate MLE α?, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 0.5, λ = 0.4, 0.6, 0.8 under the Koziol-Green model

λnα?α?MSE(α?)MSE(α?)mean(α? ? α?)RE(α?, α?)
0.4200.5330.5310.0110.0110.00299.037
300.5250.5240.0070.0070.00198.778
400.5180.5180.0050.0050.00099.569
500.5130.5130.0030.0030.00099.698

0.6200.5380.5360.0110.0110.00298.230
300.5210.5200.0060.0060.00199.288
400.5170.5160.0050.0050.00099.520
500.5110.5110.0040.0040.000100.309

0.8200.5360.5340.0120.0120.00298.801
300.5260.5250.0070.0070.00198.819
400.5170.5160.0050.0050.00199.561
500.5120.5120.0040.0040.00099.591

MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.

### Table 8

Averages of the MLE λ? and the approximate MLEs λ?, MSEs, differences between the MLE and the approximate MLE and RE of two estimates when α = 0.5, λ = 0.4, 0.6, 0.8 under the Koziol-Green model

λnλ?λ?MSE(λ?)MSE(λ?)mean(λ? ? λ?)RE(λ?, λ?)
0.4200.4130.4140.0250.025?0.000499.877
300.4100.4100.0160.016?0.000299.951
400.4080.4080.0120.012?0.0001100.029
500.4100.4100.0090.0090.000099.977

0.6200.6230.6240.0900.090?0.0004100.069
300.6160.6160.0720.072?0.0003100.077
400.6110.6110.0630.063?0.0001100.071
500.6080.6080.0570.0570.0000100.006

0.8200.8410.8420.2510.252?0.0007100.168
300.8180.8190.2100.210?0.0004100.114
400.8140.8150.1960.196?0.0002100.078
500.8150.8150.1910.191?0.0001100.028

MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.

References
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