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CPS1810 Jin-Jian Hsieh et al.
generalized solution of ( |0 ) can be rewritten as the minimizer of the

following function,
log( − ) −
0
( |0 ) = ∑ | ( ) |0 | + | − ∑ |0 |

̂
̂
( )

=1 =1

+ | + 2(1 − ) ∑ { ≥ } |0 |,
0

=1
̂ ( )

̂

where M is an extremely large positive value larger then ∑ | 0 / ( )

=1

̂

and ∑ 2 | 0 (1 − ){ ≥ } / ( ). Because the variance of ̂ |0 is

0
=1

difficult to estimate, we use the bootstrap resampling method to estimate the
variance of ̂ |0 . Firstly, we obtain a resampling data from the original data as
∗
∗
∗
∗
∗
{( , , , , ), = 1, … , }. Secondly, estimate the parameter |0 based

̂ ∗
on the bootstrapping sample, denoted as . Then, repeat this process
̂ ∗
times. We obtain the estimators { , = 1, … , }. Therefore, we can estimate

2
1
̅ ∗
̂ ∗
̅ ∗
̂ ∗
the variance of |0 by ̂ ( ̂ |0 ) = −1 ∑ ( − ) , where = ∑ /

=1
=1
. Hence, we can construct the 100(1 − )% confidence interval for |0 as
̂
̂
̂
|0 ± (/2), where √( ̂ ), (/2) = Φ −1 (1 − /2) and Φ(·) is
| 0
the cumulative distribution function of (0,1).

3. Result
In this section, we conduct simulation studies to examine the finite sample
performance of the proposed approach. We consider the log-linear quantile
residual life model as:

− quantile{log( − )| ≥ 0, } = + , (5)
0
1

0

where we take the true values of = −1.5 and = −0.5. The covariate
1

0
is generated from bernoulli distribution with mean 0.5. The terminal event time
is generated from exponential distribution with mean , and the non-
terminal event time is generated from exponential distribution with mean
1/{−(1 − )(−( + ))} which is associated with model (5).
0
1
Further, (, ) follow the Clayton copula, where () = ( − − 1)/

and (, ) = ( − + − − 1) −1/ . The right censoring time follows

a uniform distribution on [0, ]. We set Kendall’s = 0.3, 0.5, 0.7, quantile
= 0.5, sample size = 200 and = 0, 0.07, 0.17, 0.35 which is the
0
0, 25%, 50% and 75% quantile of . We consider the settings (, , ) =
(0.5,0.57,3). For each case, we replicate 400 simulation runs with 100
bootstrapping times.
We compare our proposed method with the method by Jung et al. (2009),
which didn’t consider the association between and . We present the bias
of the proposed estimator (Bias), the empirical standard deviation (EmpSd),
the average of estimated standard deviation (AveSd) based on the bootstrap
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