Statistical Inference for the Extreme Value Distribution under Adaptive Type-II Progressive Censoring Schemes Z.S. Ye1; P.S. Chan2†; M. Xie1,3 and H.K.T. Ng4 1
Department of Industrial & Systems Engineering, National University of Singapore 2
3
Department of Statistics, Chinese University of Hong Kong
Department of Systems Engineering and Engineering Management, City University of Hong Kong 4
Department of Statistical Science, Southern Methodist University
[E-mail address:
[email protected];
[email protected];
[email protected];
[email protected].]
ABSTRACT Adaptive Type-II progressive censoring schemes have been shown to be able to strike a balance between the estimation efficiency and the total testing duration. In this paper, some general statistical properties of an adaptive Type-II progressive censoring scheme are first investigated. A bias correction procedure is proposed to reduce the bias of the maximum likelihood estimators. We then focus on the extreme value distributed lifetimes and derive the Fisher information matrix for the maximum likelihood estimators based on these properties. Four different approaches are proposed to construct confidence intervals for the parameters of the extreme value distribution. Performance of these methods is compared via an extensive Monte Carlo simulation.
Key words: Adaptive Type-II progressive censoring, confidence interval, extreme value distribution, information matrix, the bootstrap.
†
Corresponding author.
1
1 INTRODUCTION Before a manufacturer launches a newly-developed product to the market, a life-testing experiment is often required to assess its reliability. During the test, censoring technique is usually adopted to obtain the lifetime information within a reasonable timeframe. Two common censoring schemes are the Type-I and the Type-II censoring, where the experiment terminates, respectively, at a pre-determined time T and upon certain number of failures. To allow for more flexibility in removing un-failed units from the tests, more general censoring approaches are called for. The progressive Type-II right censoring scheme is an appealing one and has attracted much attention in the literature, e.g., Wang (2008), Ismail and Aly (2010), Maturi et al. (2010), Wu (2010), Balakrishnan et al. (2011) Burkschat (2008) and Lin et al. (2011), to name a few. A book length treatment on this topic is found in Balakrishnan and Aggarwala (2000). See also Balakrishnan (2007) for a recent comprehensive review on progressive censoring. A typical progressive Type-II censoring scheme works as follows. Suppose that n identical units are placed on a lifetime test and upon the ith failure, Ri functioning units are randomly withdrawn from the remaining n i R1 Ri 1 surviving units. The experiment continues until the mth failure is observed. Similar to the traditional Type-II censoring approach, a problem associated with the progressive scheme is that with positive probability, the total testing duration might be unacceptably long. To address this issue, Kundu and Joarder (2006) proposed a hybrid variant of the progressive censoring scheme by imposing a time limit T on the test. Under this scheme, the test stops at Xm:m:n if the mth progressively censored observed failure occurs before time T. Otherwise, the experiment terminates at time T with all remaining items censored at T. However, it is possible that the effective sample size is very small or even zero, indicating possible loss of efficiency under this scheme. To strike a balance between the total testing time and the estimation efficiency, Ng et al. (2009) proposed an adaptive Type-II progressive censoring scheme, which works as follows. If Xm:m:n < T, the experiment proceeds with the pre-specified progressive censoring scheme R1 , , Rm . Otherwise the censoring scheme switches to R1 , , RJ , 0,...0, Rm , where J < m is the number of failures observed before T and Rm n m k 1 Rk . The basic idea of this J
scheme is to speed up the test as much as possible when the test duration exceeds a pre2
determined threshold T. When T = ∞, the adaptive variant reduces to a regular progressive TypeII censoring scheme. On the other hand if T = 0, this variant leads to a traditional Type-II censoring scheme. It is noted that Cramer and Iliopoulos (2010) studied the general adaptive progressive Type-II censoring schemes that allow experimenters to choose the next censoring number. After proposing the adaptive Type-II progressive censoring scheme, Ng et al. (2009) applied it to the exponential distributions. Nevertheless, compared with the exponential distribution, the Weibull model is more general and receives more applications. The Weibull distribution is one of the most commonly-used models in the parametric analysis of lifetime data. For example, Ye et al. (2011) used it to fit the interface fatigue data of an electronic device, Hong et al. (2009) used it to fit the transformer failure data. More applications of this distribution can be found in the book by Murthy et al. (2004). Logarithm of a Weibull random variable follows an extreme value distribution, which belongs to the location-scale family. Due to the good properties and wellestablished theorems for this family (e.g. Lawless 2003 Chapters 5 & 6), it is often more convenient to work with the equivalent extreme value distribution instead of directly dealing with the Weibull distribution. In view of the advantages of the adaptive Type-II progressive censoring scheme and the popularities of the Weibull model, it is of particular interest to study the performance of this scheme when applying to the extreme value distribution. Maximum likelihood estimation and the expected total time on test have been investigated by Lin et al. (2009). However, the from the tables given by Lin et al. (2009), it is easily seen that the biases of the MLE is significant when the sample size is not large enough. This motivates us to use the bootstrap bias correction technique to calibrate the point estimation. In addition, this study will also focus on the investigation of general properties of this scheme and on the construction of confidence intervals for the scale and location parameters of the extreme value distribution. We consider four approaches to construct confidence intervals of the unknown parameters. Specifically, we consider the confidence interval constructions based on (i) the Fisher (expected) information matrix, (ii) the observed information matrix, (iii) the parametric percentile bootstrap and (iv) the parametric bootstrap-t method. Compared with Ng et al. (2009) and Lin et al. (2009), we also introduce a notation system that is able to greatly simplify the expressions. This paper is organized as follows. In Section 2, marginal distribution of the order statistics 3
under the adaptive censoring scheme is derived. Section 3 discusses how to use the bootstrap method to correct the estimation biases. Section 4 confines to the extreme value distribution and discusses four approaches in constructing confidence intervals for the parameters. In Section 5, an extensive simulation study is carried out to examine the performance of these four approaches. Section 6 concludes the paper.
2 NOTATIONS AND GENERAL RESULTS Let the random variable X be the lifetime of a product with PDF f x , CDF F x and survival function F x . To assess the lifetime distribution, n units are chosen from the product population with corresponding lifetimes X1, X2, …, Xn, which are n independent copies of X. Before the life test, the effective sample size m ≤ n, the progressive censoring scheme
R1 , , Rm and the scheme switching time T are specified. When t < T, the test is carried on as a regular progressive censoring scheme R1 , , Rm , where Ri functioning units are randomly removed upon the ith failure. Denote the total number of failures observed before time T as J, where J is a random variable with support on 0,1,..., m . If J < m, then after T, the test is switched to the scheme R1 , , RJ , 0,...0, Rm where Rm n m k 1 Rk . After the experiment, J
m ordered lifetimes, denoted by X X1:m:n ,..., X m:m:n , are observed. In addition, the number of observed failures before T, i.e. J, is also recorded. The joint distribution of X and J can be expressed as f X, J x1:m:n ,..., xm:m:n , j cm I x
m
F r xk :m:n f xk :m:n , T k
j:m:n , xk 1:m:n
(1)
k 1
where
R , k j k m k rk 0, j k m , ck s and k rs 1 . s k s 1 j n R k 1 k The distribution of J is of special interest because it reflects the effectiveness of this adaptive 4
scheme. When J equals 0 with a high probability, the adaptive scheme would proceed with a conventional Type-II censoring scheme most of the time. On the other hand if J assigns a large probability to m, this scheme would proceed with a regular progressive Type-II censoring scheme for most of the time. The probability mass function of J can be computed based on (1) as follows. The event J 0 is equal to the case where lifetimes of all units are greater than T. therefore, Pr J 0 F n T
(2)
It can also be shown (in the appendix) that when j > 0, bk , j F j1 T F k T
j
Pr J j c j
k j 1
k 1
,
(3)
j 1 b where k , j and by convention, m1 0 . s 1 s k sk
Next, we shall look at the marginal distribution of Xi:m:n given J. When i j , the marginal distribution can be derived by direct integration as
dx j:m:n
xi:m:n
dx j 1:m:n
T
dx1:m:n
0
x j 1:m:n
T
x j:m:n
x2:m:n
1 f X i:m:n , J xi:m:n | J j Pr J j
dx
m:m:n
x j 2:m:n
dxi 1:m:n
0
xi:m:n
dxi 1:m:n .
f X, J x1:m:n ,..., xm:m:n , j
T
After some routine deduction, it can be shown that
f X i:m:n , J xi:m:n | J j
c j I 0,T xi:m:n
j 1
i
a Pr J j h 1 k i 1
k , h ,i , j
F k T F h k 1 xi:m:n f xi:m:n ,
where i
ak ,h,i , j s 1 sh
j 1 1 1 s h s i 1 s k . s k
Similarly when i > j, direct calculation shows that 5
(4)
f X i:m:n , J xi:m:n | J j
ci I T , xi:m:n
j 1
i
d Pr J j k 1 h j 1
k , h ,i , j
F k h T F h 1 xi:m:n f xi:m:n ,
(5)
where j 1
d k , h ,i , j s 1 sk
i 1 1 s k s j 1 s h . s h
Based on the above marginal distribution, the expected test duration can be computed as m 1
T
j 0
T
0
E X m:m:n Pr J j xf X m:m:n , J x | J j dx Pr J m xf X m:m:n , J x | J m dx .
(6)
To carry out the statistical inference, the log-likelihood function based on the observed data is required. Based on (1), The log-likelihood function can be expressed as m
l | X x, J j log cm rk log F xk:m:n log f xk:m:n ,
(7)
k 1
where Θ is the unknown parameter to estimate. From this equation, it is readily seen that point estimation of Θ is the same as that under the progressive Type-II right censoring scheme
R , , R , 0,...0, R . It is observed that when the sample size is small, the bias of the estimates 1
j
m
is large. This motivates the next section on bias correction.
3 THE BOOTSTRAP BIAS CORRECTED ESTIMATOR ˆ the MLE. Generally speaking, ˆ is a biased Let Θ be the true value of the parameters, and estimator. When the sample size is small, the bias is often very significant. The bias can be corrected based on the bootstrap principle. Let u be an additive adjustment such that ˆ u 0 . Here u can be estimated via the bootstrap method (Hall 1997). The procedure E
is as follows. [1]. Collect the adaptive progressive Type-II censoring data (X, J) and compute the MLE of Θ,
ˆ , by maximizing the log-likelihood function. denoted as
6
[2]. Based on the adaptive censoring scheme R1 , , Rm and the switching time T, generate an adaptive Type-II progressive censoring dataset from the extreme value distribution
ˆ. with distribution parameter ˆ*. [3]. Obtain the MLE based on this simulated dataset. Denote this bootstrap estimate by ˆ *, ˆ * ,..., ˆ* . [4]. Repeat Steps [2] – [3] B times and obtain 1 2 B ˆ [5]. Then u can be estimated as uˆ
1 B ˆ* k . B k 1
The bias-corrected estimate of Θ can then be given by ˆ uˆ 2 ˆ
1 B ˆ* k B k 1
Performance of this bias-corrected estimator will be demonstrated via simulation in Section 5.
4 INTERVAL ESTIMATION FOR THE EXTREME VALUE DISTRIBUTION This section is confined to the special case of the extreme value distribution. The results presented in Section 2 will be utilized here. Note that the extreme value distribution has CDF and PDF as
x F x 1 exp exp , f x
(8)
x x exp exp , 1
(9)
respectively. Therefore, the log-likelihood function can be specified as m x x l l | X x, J j log cm m ln k:m:n rk 1 exp k:m:n , (10) k 1
where Θ = (μ, σ) is the parameter vector. To simplify the notation, define Z
7
X
and
zk:m:n
xk:m:n
. The CDF and PDF of Z are linked to (8) and (9) as FZ z 1 exp exp z and f Z x exp z exp z .
(11)
The partial derivative of l with respect to μ and σ are given by m m 1 l rk 1 exp zk:m:n , k 1
(12)
m m z l k:m:n 1 rk 1 exp zk:m:n . k 1
(13)
MLE of μ and σ can be obtained by simultaneously solving the equations
l 0 and
l 0 . The second derivatives of l with respect to μ and σ are given by m 2 1 2 l 2 rk 1 exp zk:m:n , k 1
m zk2:m:n 2 m m 2 zk:m:n l 1 r 1 exp z r 1 exp zk:m:n , k k :m:n 2 k 2 2 k 1 2 k 1 m z 2 m m 1 l 2 2 rk 1 exp zk:m:n k:m2:n rk 1 exp zk:m:n . k 1 k 1
(14)
(15)
(16)
Because the MLEs for μ and σ do not have explicit expressions, a basic idea for constructing confidence intervals is to make use of the asymptotic properties of the MLE and consider asymptotically pivotal quantities. To construct confidence intervals based on the asymptotically pivotal quantities, the information matrix is required. Two scenarios are considered in the following.
4.1 Confidence Interval Based on Observed Information 8
Matrix The asymptotic variances and covariance of the maximum likelihood estimators can be obtained by inverting the observed information matrix I, which is given by
2 2 l l 2 I 2 2 l l 2 ˆ Because
l 0 and l 0 ，By combining the results of Equations (12) ˆ ˆ
(16), the elements in the observed information matrix can be simplified as
2 m l 2, 2 ˆ ˆ
(17)
2 m m rk 1 2 l zˆ exp zˆk:m:n , 2 ˆ 2 k 1 ˆ 2 k:m:n ˆ
(18)
2 m m 1 l 2 2 zˆk:m:n . ˆ k 1 ˆ ˆ
(19)
Based on the asymptotic theory of the MLE for the extreme value distribution (Meeker and
ˆ is bivariate normal with mean Escobar 1998, pp. 622), the asymptotic distribution of the MLE vector Θ and variance-covariance matrix I 1 . The approximate confidence intervals for μ and σ are then respectively given by
1 1 ˆ z /2 I111 and ˆ exp z /2 I 22 / ˆ , ˆ / exp z /2 I 22 / ˆ , 1 where z /2 is the 100α/2 percentile of the standard normal distribution, while I ij is the (i, j)th
element of I 1 .Here we apply normal approximation to the log-transformed ˆ , construct confidence interval based on this normal approximation, and then transform back to the confidence interval for σ. Efficiency of the log transformation has been verified by Chan et al. (2008) and Ng et al. (2009). Our simulation results also suggest that the log transformation does 9
improve the coverage probability, especially when the sample size is small.
4.2 Confidence
Interval
Based
on
Fisher
Information
Matrix The asymptotic variances and covariance of the maximum likelihood estimators can also be obtained by inverting the expected Fisher information matrix I, which is given by
2 2 l l 2 I E 2 2 l l 2
(20)
l 0 , Equations (14) - (16) can be simplified as Because E l 0 and E 2 m E 2 l 2 ,
(21)
2 m m 1 E 2 l 2 2 E rk 1 Z k2:m:n exp Z k:m:n , k 1
(22)
2 m m 1 E l 2 2 E Z k:m:n . k 1
(23)
Equations (22) and (23) can be computed by first conditioning on J and then taking expectation with respect to J. The detailed expressions are as follows. 2 2 m m m r 1 E 2 l E E 2 l | J 2 k 2 E Z k2:m:n exp Z k:m:n | J j (24) j 0 k 1
2 m m m 1 E l 2 2 E Z k:m:n | J j j 0 k 1
(25)
In order to compute the above expectations, the marginal distribution of Zi:m:n given J should be obtained first. When i j , the marginal distribution of Zi:m:n can be readily obtained based on (4) 10
and (11). Therefore, by direct integration, it can be shown that the moment generating function of Zi:m:n given J is given by
cj
i
j
E exp tZi:m:n | J j Pr J j h1 k i
ak ,h,i , j FZk ZT
h k
t 1
L t 1, h k exp ZT , (26)
s
x 1 t where ZT T / , and L x, s t e dt is the lower incomplete gamma function. 0
The first and second derivatives of Equation (26) with respect to t are given by j i cj ak ,h ,i , j FZk ZT d E exp tZi:m:n | J j L t 1, h k exp ZT dt Pr J j h 1 k i h k t 1
cj
j
i
Pr J j
h 1 k i
ak ,h ,i , j FZk ZT
h k
t 1
log h k L t 1, h k exp ZT
j i cj ak ,h ,i , j FZk ZT d2 E exp tZ | J j t 1, h k exp ZT i:m:n Pr J j dt 2 h1 k i h k t 1 L
2c j
Pr J j cj
Pr J j
i
j
h 1 k i i
j
h 1 k i
ak ,h ,i , j FZk ZT
h k
t 1
ak ,h,i , j FZk ZT
h k
t 1
log h k L t 1, h k exp ZT log 2 h k L t 1, h k exp ZT
where L x, s and L x, s are respectively given by L x, s
s
s
d d2 L x, s log t t x 1et dt and L x, s 2 L x, s log 2 t t x 1et dt . dx dx 0 0
Note that these two derivatives can be expressed as a function of the Meijer G-function (Mathai and Haubold 2008), and thus can be easily computed. 2 Based on these two derivatives, the expectations of Zi:m:n and ri 1 Zi:m:n exp Zi:m:n can be
specified as
11
cj
E Z i:m:n | J j
cj
i
Pr J j
i
Pr J j
j
j
h k
h 1 k i
ak ,h ,i , j FZk ZT
h k
h 1 k i
ak ,h ,i , j FZk ZT
L 1, h k exp ZT
,
log h k L 1, h k exp ZT
(27)
c j Ri 1 i j ak ,h ,i , j FZk ZT E ri 1 Z i2:m:n exp Z i:m:n | J j L 2, h k exp ZT Pr J j h 1 k i h k 2 2c j Ri 1
i
j
ak ,h ,i , j FZk ZT
log h k L 2, h k exp ZT 2 h k c j Ri 1 i j ak ,h ,i , j FZ ZT log 2 h k L 2, h k exp ZT Pr J j h 1 k i h k 2
Pr J j
.(28)
h 1 k i
k
On the other hand, when i > j, the marginal distribution of Zi:m:n can be readily obtained based on (5) and (11). Parallel to the above deduction, it can be shown that j 1 i FZk h ZT ci E Z i:m:n | J j H 1, h exp ZT d k , h ,i , j Pr J j k 1 h j 1 h j 1 i FZk h ZT ci log h H 1, h exp ZT d k , h ,i , j Pr J j k 1 h j 1 h
,
(29)
j 1 i FZk h ZT ci E ri 1 Z i2:m:n exp Z i:m:n | J j d k ,h,i, j 2 H 2, h exp ZT Pr J j k 1 h j 1 h j 1 i FZk h ZT 2ci dk ,h,i, j 2 log h H 2, h exp ZT Pr J j k 1 h j 1 h
.(30)
j 1 i FZk h ZT ci d k , h ,i , j log 2 h H 2, h exp ZT 2 Pr J j k 1 h j 1 h
x 1 t In the above expressions, U x, s t e dt is the upper incomplete gamma function, while s
U x, s and U x, s are respectively given by U x, s
d d2 U x, s log t t x 1et dt and U x, s 2 U x, s log 2 t t x 1et dt . dx dx s s
Similarly, these two derivatives can also be expressed in terms of the Meijer G-function. 12
Based on the results of (27) - (30), Equations (24) and (25) can be computed, after which the Fisher information matrix (20) can be specified. Similarly, the approximate confidence intervals for μ and σ are then respectively given by
1 1 ˆ z /2 I111 and ˆ exp z /2 I22 / ˆ , ˆ / exp z /2 I22 / ˆ .
4.3 Parametric Percentile Bootstrap Approach The normal approximations based on the asymptotic pivots are adequate when the effective sample size is large enough. However, when the number of failures is not sufficient, the normal approximations may not work well. Simulation techniques such as the bootstrap approaches may be able to provide more accurate approximate intervals. A widely used bootstrap method is the parametric percentile bootstrap described by Efron and Tibshirani (1993). To construct the confidence intervals for Θ, a typical parametric percentile bootstrap algorithm works as follows.
[1]. Collect the adaptive progressive Type-II censoring data (X, J) and compute the MLE of Θ,
ˆ , by maximizing (10). denoted as [2]. Based on the adaptive censoring scheme R1 , , Rm and the switching time T, generate an adaptive Type-II progressive censoring dataset from the extreme value distribution ˆ ˆ , ˆ . with distribution parameter
[3]. Obtain the MLE based on this simulated dataset. Denote this bootstrap estimate by ˆ * ˆ * , ˆ * .
ˆ * ˆ * , ˆ * . ˆ *, ˆ * ,..., ˆ * , where [4]. Repeat Steps [2] – [3] B times and obtain i i i 1 2 B * * * [5]. Arrange ˆ1* , ˆ 2* ,..., ˆ B* and ˆ1* , ˆ 2*..., ˆ B* in ascending orders and obtain ˆ1 , ˆ 2 ,..., ˆ B
and ˆ 1 , ˆ 2 ..., ˆ B . *
*
*
A pair of two-sided 100(1 – α)% percentile bootstrap confidence intervals for μ and σ are then * * * * given by ˆ B /2 , ˆ B1 /2 and ˆ B /2 , ˆ B1 /2 . It is noted that the bootstrap estimates
13
ˆ *, ˆ * ,..., ˆ * can also be used for bias correction. We can also construct the confidence interval 1 2 B
for the bias-corrected estimator. But this requires the double bootstrap which is quite timeconsuming, and thus is not discussed here.
4.4 Studentized Bootstrap Approach An advantage of the percentile bootstrap method is its simplicity. However, when the sample size is small, the percentile approach is generally not as accurate as the studentized bootstrap approach, abbreviated as bootstrap-t, with an appropriate transformation (Meeker and Escobar 1998). To obtain the bootstrap-t confidence intervals, the following algorithm can be applied.
[1] – [3]. The same as the steps in the percentile bootstrap above. [4]. Compute the variance-covariance matrix I 1* based on the observed information matrix (alternatively, the Fisher information matrix can be used). Obtain the t-statistics
log log ˆ * log ˆ / T ˆ * ˆ / I111* and T
1* I 22 / ˆ *
[5]. Repeat Steps [2] – [4] B times and obtain T1 , T2 ,..., TB and T1log , T2log ,..., TBlog . [6]. Sort T1 , T2 ,..., TB and T1log , T2log ,..., TBlog in ascending orders and obtain the ordered log log log sequences T1 , T 2 ,..., T B and T1 , T 2 ,..., T B .
A pair of two-sided 100(1 – α)% bootstrap-t confidence intervals for μ and σ are then given by
1 log 1 1 1 ˆ / exp T log ˆ T ˆ ˆ ˆ ˆ B /2 I11 , T B1 /2 I11 and B1 /2 I 22 / , / exp T B /2 I 22 / .
5 SIMULATION RESULTS To evaluate the performance of the approaches based on the observed information matrix (OIM), the Fisher information matrix (FIM), the percentile bootstrap (PB) and the bootstrap-t (TB), a Monte Carlo simulation study is conducted in this section. Because of the lack of invariance properties of the estimated parameters, some representative settings in the simulation study 14
should be chosen rather than exhausting all the possible settings. For illustrative purpose, we consider the standard extreme value distribution with μ = 0 and σ = 1. The following censoring schemes are considered. Scheme 1: R1 R2 ... Rm1 1 ; Scheme 2: R1 R2 ... Rm/2 2 , Rm/2 1 ... Rm1 0 ; Scheme 3: R1 R2 ... Rm/2 0 , Rm/2 1 Rm/2 2 ... Rm1 2 . Here denotes the integer part of the number in the bracket. The biases, the coverage probabilities and the expected widths of the confidence intervals for these different censoring schemes, different sample sizes and T = – 1, 0 and 1 are presented in Tables 1 – 6.
15
Table 1. Coverage probabilities and expected width of the 95% confidence intervals for µ based on different methods for Scheme 1: The number outside the bracket is the coverage probability and the number in the bracket is the expected width.
(n, m)
T
(25, 10)
(50, 10)
(50, 20)
(50, 25)
bias
OIM
FIM
PB
TB
before corr.
after corr.
-1
-0.129
-0.0181
87.3 (1.23) 87.1 (1.23) 78.1 (1.32)
75.3 (1.79)
0
-0.108
0.0039
87.1 (1.23) 87.0 (1.22) 80.0 (1.31)
77.9 (1.78)
1
-0.110
-0.0005
87.1 (1.23) 87.0 (1.22) 80.6 (1.32)
78.1 (1.79)
-1
-0.196
-0.0159
83.9 (1.88) 83.1 (1.82) 78.1 (1.91)
66.8 (2.61)
0
-0.206
-0.0283
84.0 (1.88) 83.0 (1.81) 80.6 (1.92)
68.9 (2.67)
1
-0.247
-0.0710
83.9 (1.88) 83.0 (1.81) 79.7 (1.92)
68.1 (2.64)
-1
-0.056
-0.0027
91.1 (.907) 91.0 (.906) 88.1 (.973)
85.0 (1.12)
0
-0.060
-0.0013
90.9 (.904) 90.9 (.903) 87.4 (.967)
85.4 (1.12)
1
-0.066
-0.0053
90.9 (.916) 90.9 (.904) 88.4 (.967)
85.6 (1.12)
-1
-0.036
0.0011
93.1 (.766) 93.0 (.764) 92.3 (.799)
92.1 (.879)
0
-0.045
-0.0076
92.9 (.765) 92.9 (.764) 89.9 (.784)
89.9 (.858)
1
-0.044
-0.0050
92.9 (.764) 92.9 (.764) 91.1 (.786)
91.3 (.863)
-0.293
-0.0510
83.1 (2.69) 81.8 (2.57) 80.1 (2.63)
68.4 (3.83)
0
-0.265
-0.0193
83.9 (2.72) 82.5 (2.59) 78.5 (2.59)
66.7 (3.80)
1
-0.257
-0.0127
83.2 (2.71) 81.7 (2.57) 78.4 (2.58)
67.5 (3.77)
-0.060
-0.0120
91.4 (.947) 91.2 (.938) 88.5 (.947)
84.1 (1.05)
0
-0.058
-0.0093
91.4 (.946) 91.2 (.937) 90.2 (.946)
85.9 (1.05)
1
-0.048
0.0085
91.5 (.946) 91.3 (.937) 90.4 (.943)
85.6 (1.05)
-0.024
-0.0032
93.8 (.569) 93.8 (.568) 90.8 (.567)
90.1 (.594)
0
-0.025
-0.0057
93.5 (.559) 93.4 (.559) 93.2 (.564)
93.6 (.589)
1
-0.020
0.0009
93.6 (.558) 93.6 (.558) 92.8 (.561)
92.8 (.588)
(100, 10) -1
(100, 30) -1
(100, 50) -1
16
Table 2. Coverage probabilities and expected width of the 95% confidence intervals for σ based on different methods for Scheme 1: The number outside the bracket is the coverage probability and the number in the bracket is the expected width.
(n, m)
T
(25, 10)
(50, 10)
(50, 20)
(50, 25)
Bias
OIM
FIM
PB
TB
before corr.
after corr.
-1
-0.085
-0.0082
90.3 (1.03) 90.0 (1.05) 82.3 (.947)
94.7 (1.36)
0
-0.081
-0.0040
90.3 (1.03) 90.8 (1.05) 81.7 (.931)
95.0 (1.34)
1
-0.082
-0.0060
90.4 (1.03) 91.0 (1.05) 82.7 (.940)
95.4 (1.35)
-1
-0.082
0.0044
90.2 (1.12) 89.6 (1.10) 81.5 (1.00)
92.8 (1.41)
0
-0.094
-0.0096
90.2 (1.12) 89.6 (1.10) 81.8 (1.01)
93.7 (1.44)
1
-0.105
-0.0224
90.0 (1.12) 89.4 (1.10) 81.2 (1.01)
93.2 (1.43)
-1
-0.041
0.0050
92.9 (.742) 93.8 (.770) 87.9 (.718)
94.1 (.854)
0
-0.041
0.0053
92.9 (.742) 93.8 (.770) 88.5 (.711)
95.3 (.847)
1
-0.036
0.0049
92.5 (.741) 93.5 (.769) 88.1 (.710)
95.3 (.845)
-1
-0.027
-0.0017
93.0 (.613) 94.4 (.649) 90.4 (.630)
95.5 (.717)
0
-0.031
-0.0026
93.0 (.613) 94.4 (.649) 89.7 (.593)
95.2 (.675)
1
-0.032
-0.0021
93.3 (.608) 94.6 (.645) 90.0 (.591)
95.5 (.671)
-0.104
-0.0075
90.0 (1.16) 89.1 (1.11) 81.4 (1.04)
93.9 (1.53)
0
-0.093
-0.0053
90.0 (1.16) 89.1 (1.11) 80.7 (1.02)
94.9 (1.52)
1
-0.096
-0.0092
90.2 (1.16) 89.1 (1.12) 79.5 (1.02)
93.8 (1.51)
-0.033
-0.0047
93.5 (.642) 94.0 (.656) 90.6 (.622)
94.6 (.700)
0
-0.031
-0.0021
93.5 (.642) 94.0 (.656) 89.8 (.620)
95.3 (.700)
1
-0.029
-0.0005
93.5 (.641) 94.1 (.657) 90.0 (.619)
95.5 (.698)
-0.015
0.0009
94.2 (.436) 95.8 (.470) 91.1 (.452)
94.7 (.482)
0
-0.016
-0.0010
94.2 (.436) 95.8 (.470) 92.2 (.432)
94.8 (.460)
1
-0.018
-0.0024
94.0 (.431) 95.7 (.467) 91.6 (.426)
94.6 (.454)
(100, 10) -1
(100, 30) -1
(100, 50) -1
17
Table 3. Coverage probabilities and expected width of the 95% confidence intervals for µ based on different methods for Scheme 2: The number outside the bracket is the coverage probability and the number in the bracket is the expected width.
(n, m)
T
(25, 10)
(50, 10)
(50, 20)
(50, 25)
bias
OIM
FIM
PB
TB
before corr.
after corr.
-1
-0.124
-0.0209
87.3 (1.23) 87.1 (1.23) 83.3 (1.26)
83.8 (.908)
0
-0.129
-0.0297
87.1 (1.23) 87.0 (1.22) 81.8 (1.26)
81.3 (1.65)
1
-0.120
-0.0218
87.1 (1.23) 87.0 (1.22) 83.1 (1.27)
82.5 (1.66)
-1
-0.187
-0.0115
83.9( 1.88) 83.1 (1.82) 80.5 (1.87)
82.9 (.995)
0
-0.186
0.0024
84.0 (1.88) 83.0 (1.81) 79.7 (1.89)
71.8 (2.65)
1
-0.187
-0.0172
83.9 (1.88) 83.0 (1.81) 78.8 (1.87)
69.4 (2.64)
-1
-0.056
-0.0013
91.1 (.907) 91.0 (.906) 87.2 (.917)
89.2 (.674)
0
-0.056
-0.0007
90.9 (.904) 90.9 (.903) 87.9 (.914)
87.0 (1.04)
1
-0.055
0.0038
90.9 (.906) 90.9 (.904) 88.1 (.918)
87.5 (1.04)
-1
-0.037
-0.0010
93.1 (.766) 93.0 (.764) 90.4 (.769)
90.8 (.562)
0
-0.031
0.0022
92.9 (.765) 92.9 (.764) 92.1 (.777)
92.8 (.832)
1
-0.040
-0.0101
92.9 (.764) 92.9 (.764) 90.7 (.778)
92.3 (.833)
-0.266
-0.0526
83.0 (2.66) 81.7 (2.53) 78.6 (2.56)
80.9 (1.02)
0
-0.264
-0.0462
83.1 (2.66) 81.9 (2.54) 79.8 (2.57)
67.7 (3.74)
1
-0.266
-0.0090
83.1 (2.66) 81.8 (2.53) 80.3 (2.58)
68.1 (3.73)
-0.046
-0.0002
91.6 (.895) 91.5 (.889) 88.9 (.899)
90.5 (.560)
0
-0.043
0.0017
91.6 (.895) 91.5 (.889) 90.4 (.896)
86.6 (.990)
1
-0.045
0.0052
91.6 (.894) 91.5 (.888) 90.2 (.903)
87.2 (.999)
-0.019
0.0066
94.1 (.548) 94.0 (.547) 93.9 (.550)
92.9 (.398)
0
-0.015
0.0029
93.9 (.548) 93.9 (.547) 92.7 (.551)
93.6 (.568)
1
-0.015
-0.0039
94.0 (.548) 94.0 (.548) 92.8 (.551)
93.3 (.569)
(100, 10) -1
(100, 30) -1
(100, 50) -1
18
Table 4. Coverage probabilities and expected width of the 95% confidence intervals for σ based on different methods for Scheme 2: The number outside the bracket is the coverage probability and the number in the bracket is the expected width.
(n, m)
T
(25, 10)
(50, 10)
(50, 20)
(50, 25)
bias
OIM
FIM
PB
TB
before corr.
after corr.
-1
-0.076
-0.0058
90.7 (.970) 93.0 (1.08) 82.4 (1.64)
95.2 (1.24)
0
-0.077
-0.0081
90.7 (.963) 92.8 (1.06) 83.6 (.895)
95.3 (1.24)
1
-0.080
-0.0054
91.1 (.967) 93.4 (1.07) 83.4 (.903)
94.6 (1.25)
-1
-0.089
-0.0044
90.3 (1.10) 90.4 (1.10) 70.3 (2.59)
94.3 (1.41)
0
-0.088
-0.0003
90.3(1.10)
90.4 (1.10) 82.9 (1.00)
93.8 (1.45)
1
-0.089
-0.0043
90.1 (1.10) 90.1 (1.10) 81.6 (.992)
94.4 (1.44)
-1
-0.037
-0.0041
92.7 (.697) 95.3 (.783) 86.3 (1.04)
96.0 (.781)
0
-0.038
-0.0037
92.7(.693)
95.0 (.773) 88.1 (.668)
94.7 (.780)
1
-0.036
-0.0028
92.7 (.693) 95.0 (.773) 89.5 (.669)
95.0 (.781)
-1
-0.026
0.0012
93.6 (.569) 96.6 (.693) 91.5 (.820)
94.8 (.621)
0
-0.024
0.0012
93.4 (.547) 96.6 (.653) 91.4 (.539)
95.6 (.597)
1
-0.025
0.0022
93.4 (.547) 96.8 (.649) 90.8 (.540)
95.2 (.600)
-0.110
-0.0064
89.8 (1.15) 89.1 (1.13) 66.4 (3.70)
94.2 (1.49)
0
-0.094
-0.0082
89.9 (1.15) 89.3 (1.12) 81.0 (1.02)
93.4 (1.51)
1
-0.093
-0.0075
89.8 (1.15) 89.0 (1.12) 82.3 (1.02)
94.1 (1.50)
-0.027
0.0045
93.7 (.617) 95.1 (.658) 86.4 (.993)
94.8 (.672)
0
-0.027
-0.0058
93.6 (.617) 95.1 (.658) 91.2 (.598)
95.1 (.670)
1
-0.028
0.0077
93.6 (.617) 95.0 (.657) 91.3 (.602)
95.1 (.674)
-0.015
-0.0014
94.3 (.401) 97.7 (.497) 94.4 (.565)
95.8 (.418)
0
-0.013
0.0024
94.1 (.374) 98.0 (.470) 93.4 (.372)
95.2 (.391)
1
-0.012
0.0056
94.1 (.374) 98.0 (.468) 92.0 (.372)
93.5 (.391)
(100, 10) -1
(100, 30) -1
(100, 50) -1
19
Table 5. Coverage probabilities and expected width of the 95% confidence intervals for µ based on different methods for Scheme 3: The number outside the bracket is the coverage probability and the number in the bracket is the expected width.
(n, m)
T
(25, 10)
(50, 10)
(50, 20)
(50, 25)
bias
OIM
FIM
PB
TB
before corr.
after corr.
-1
-0.131
-0.0073
85.2 (1.38) 84.7 (1.35) 80.9 (1.37)
76.9 (1.84)
0
-0.145
-0.0191
85.2 (1.37) 84.7 (1.34) 79.3(1.35)
76.1 (1.86)
1
-0.142
-0.0138
85.2 (1.37) 84.7 (1.34) 80.2 (1.36)
77.3 (1.89)
-1
-0.202
0.0103
83.4 (2.01) 82.2 (1.92) 78.5 (1.94)
64.2 (2.48)
0
-0.203
-0.0594
83.4 (2.01) 82.3 (1.92) 79.4 (1.95)
66.5 (2.57)
1
-0.204
-0.0095
83.3 (2.00) 82.2 (1.92) 78.6 (1.96)
64.7 (2.54)
-1
-0.054
0.0097
90.1 (1.03) 89.8 (1.01) 85.7 (1.03)
83.2 (1.20)
0
-0.067
-0.0058
90.0 (1.02) 89.7 (1.00) 85.9 (1.01)
83.4 (1.19)
1
-0.071
-0.0091
89.9 (1.02) 89.6 (1.00) 87.3 (1.02)
84.5 (1.20)
-1
-0.045
-0.0004
91.5 (.831) 91.4 (.827) 90.1 (.839)
87.9 (.939)
0
-0.043
0.0007
91.6 (.810) 91.5 (.806) 88.7 (.813)
88.0 (.904)
1
-0.035
0.0088
91.6 (.808) 91.5 (.804) 89.5 (.811)
88.9 (.906)
-0.259
-0.0084
82.8 (2.73) 81.5 (2.60) 77.4 (2.60)
65.8 (3.83)
0
-0.304
-0.0568
82.7 (2.73) 81.2 (2.59) 79.5 (2.62)
66.3(3.83)
1
-0.308
-0.0530
82.7 (2.73) 81.4 (2.60) 79.9 (2.62)
67.5 (3.84)
-0.059
-0.0087
91.2 (.995) 90.9 (.983) 89.6 (.991)
84.0 (1.11)
0
-0.069
-0.0182
91.2 (.995) 90.9 (.983) 89.3(.985)
85.1 (1.10)
1
-0.058
-0.0062
91.2 (.994) 91.0 (.982) 88.0 (.986)
84.4 (1.10)
-0.023
0.0041
93.3 (.601) 93.3 (.599) 93.2 (.600)
92.6 (.634)
0
-0.022
0.0004
93.4 (.585) 93.3 (.584) 92.3 (.586)
91.9 (.618)
1
-0.023
0.0002
93.4 (.584) 93.3 (.583) 92.8 (.582)
92.6 (.614)
(100, 10) -1
(100, 30) -1
(100, 50) -1
20
Table 6. Coverage probabilities and expected width of the 95% confidence intervals for σ based on different methods for Scheme 3: The number outside the bracket is the coverage probability and the number in the bracket is the expected width.
(n, m)
T
(25, 10)
(50, 10)
(50, 20)
(50, 25)
bias
OIM
FIM
PB
TB
before corr.
after corr.
-1
-0.095
-0.0128
89.8 (1.09) 88.8 (1.05) 81.5 (.969)
95.0 (1.43)
0
-0.104
-0.0219
90.0 (1.08) 89.2 (1.04) 81.0 (.948)
95.4 (1.40)
1
-0.094
-0.0117
89.8 (1.08) 88.9 (1.04) 81.5 (.956)
95.1 (1.41)
-1
-0.090
-0.0024
89.9 (1.14) 88.6 (1.10) 80.7 (1.00)
91.4 (1.33)
0
-0.109
-0.0039
90.0 (1.15) 88.8 (1.09) 81.4 (1.01)
92.3 (1.38)
1
-0.091
-0.0035
90.0 (1.15) 88.9 (1.10) 81.4 (1.02)
91.6 (1.36)
-1
-0.041
0.0060
92.5 (.789) 92.0 (.776) 87.8 (.748)
95.4 (.903)
0
-0.051
-0.0077
92.4 (.780) 92.0 (.776) 87.0 (.736)
95.2 (.891)
1
-0.047
-0.0035
92.6 (.780) 92.3 (.767) 88.4 (.741)
94.7 (.896)
-1
-0.035
-0.0004
93.2 (.693) 92.8 (.686) 90.3 (.670)
95.0 (.777)
0
-0.032
0.0009
93.0 (.653) 92.6 (.646) 88.6 (.630)
95.1 (.728)
1
-0.038
-0.0032
93.2 (.653) 93.0 (.645) 88.4 (.625)
95.7 (.725)
-0.091
-0.0017
90.1 (1.16) 89.0 (1.12) 79.8 (1.02)
94.0 (1.53)
0
-0.103
-0.0148
89.8 (1.17) 88.4 (1.11) 81.1 (1.03)
93.8 (1.53)
1
-0.109
-0.0219
90.0 (1.17) 88.7 (1.12) 80.9 (1.03)
93.9 (1.53)
-0.036
-0.0060
93.6 (.664) 93.3 (.655) 90.1 (.637)
94.4 (.723)
0
-0.040
-0.0107
93.2 (.663) 93.0 (.654) 89.5 (.635)
95.0 (.723)
1
-0.032
-0.0014
93.4 (.664) 93.1 (.655) 88.8 (.635)
94.4 (.722)
-0.018
0.0072
94.4 (.498) 94.2 (.496) 92.5 (.487)
95.0 (.524)
0
-0.018
-0.0042
94.2 (.471) 94.0 (.469) 93.5 (.460)
95.4 (.494)
1
-0.018
0.0050
94.0 (.468) 93.9 (.465) 91.9 (.451)
94.6 (.486)
(100, 10) -1
(100, 30) -1
(100, 50) -1
21
A simple comparison between columns 3 and 4 in Tables 1-6 reveals that the bias is significantly reduced after the bootstrap bias correction, especially when the sample size is small. From these tables, we can also see that the coverage probabilities based on the observed information matrix are almost the same to those based on the expected information matrix. Therefore, the approach based on observed information matrix is recommended because of its simplicity. It is also interesting to observe that performance of the parametric bootstrap approach is much lower than performance of the approaches based on the information matrices. The bootstrap-t confidence intervals for µ are also not satisfactory. This might be due to the fact that the MLE ˆ and ˆ are negatively biased. When bootstrap estimates are obtained based on these biased estimators, they are again negatively biased. This makes the bootstrap estimates highly biased, leading to low coverage probabilities. Based on this argument, we conduct another simulation (not shown) that simply moves the bootstrap-p confidence interval by uˆ . The results indicate that after the simple shifting, the coverage probabilities have been improved, and are almost the same as those based on the information matrixes. On the other hand, the bootstrap-t confidence intervals for σ are quite close to 95% even when the effective sample size is small. This indicates the effectiveness of the log-transformation. But when the sample size is large enough, say (100, 50), the confidence intervals for σ based on observed/expected information matrices are also satisfactory. Based on this simulation, the bootstrap bias correction is strongly recommended when the sample size is small. We also recommend the approach based on the observed information matrix for constructing confidence intervals for µ regardless of the sample size, the bootstrap-t approach for σ when the sample size is small, and the approach based on observed information matrix for σ when the sample size is large. A fringe benefit of using the bootstrap approach is that the bias correction can be fulfilled using the same bootstrap samples.
6 CONCLUSION This study investigated some general properties of the adaptive Type-II censoring schemes and proposed the bootstrap bias correction method to cut down the bias of the point estimators. It was found that the bias-corrected estimators perform much better than the MLE when the sample size 22
is not large enough. This paper also studied the construction of confidence intervals for the two parameters of the extreme value distribution. Both the observed and the expected information matrices were derived. The performance of different methods in constructing confidence intervals was compared via a Monte Carlo simulation. Some recommendations about construction of the confidence intervals were made.
Acknowledgement: Mr. Ye’s work is partially supported by the Global Scholarship Programme for Research Excellence 2011 provided by the Chinese University of Hong Kong. Prof. Chan and Prof Ng’s work is supported by the GRF grant CUHK 410410 of HKRGC. Prof. Xie’s work is supported by a grant from City University of Hong Kong (Project No.9380058).
APPENDIX
Distribution of J For the lifetime distribution F x , consider the regular progressive Type-II censoring scheme
R1 ,..., Rm with samples Y1 ,..., Ym . When 0
