Least Absolute Deviation

=

estimating function:

( ) _∑ ( )

In presence of censoring censored data, it follows that

( )

Ying, Jung and Wei (1995) proposed the estimating equation resembling

( )

4.8

( )

Notice that the above modification utilizes the weighting technique to correct the censoring bias.

4.3 M-estimator

The M-estimator which minimizes

_∑ ( )

= solution to the following estimating function

(

^Zi ^Z

) (

^ZTi^β

)

⁰ estimator of its conditional expected value given the data. Recall that

β Z β = _i − ^T_i

i X

e^*( ) . It follows that

))

The resulting estimating function becomes

β 0 affects the quantitle calculation. It is easy to see that

C}

X , the quantile criteria is the same. That is, the objective function can be written as

However the above expression is still not directly applicable since C is subject to censoring by T . The imputation principle is also applies to replace the unknown

C by an estimator of its conditional expected value. Notice that when δ =0, C is observed but when δ =1 , C is censored. In summary to minimize

( )

(

ρτ ^{X −min(ZTβ,C)}

)

, Hornoré, Khan and Powell (1992) proposed to use

( ) ( )

Pr(C >c . Note that the same idea of imputation has been used by Hornoré, Khan and Powell (2002) in M-estimation.

4.5 Rank-based estimators

If we do not consider the previous way to change the structure of data, we can attempt to use the primary information of data – rank. We have discuss that we can explain R-estimator in different meanings but the equivalent result. The following methods are based on modify the solving equation

( )

^{3.18 .}

In presence of censoring, direct ranking is impossible. Therefore the alternative expression in terms of pairwise comparison in

(

3.17

)

becomes useful. Fygenson and Ritov (1994) suggested to select “comparable pair” I

(

e^*j

( )

β ^>e^*i

( )

β ^,δi ⁼¹

)

. Note that

as long as δ_i =1, we know the value I

(

ej

( )

^β >ei

( )

^β

)

despite of censoring. They proposed the following estimating function

( ) ( ( ) ( ) )

which has a nice monotonic property that guarantees unit-root. Note that the resulting estimator is a U-statistic useful for large-sample analysis.

Tsiatis (1990) proposed a log-rank type estimator of the form

( )

^{β ⋅ ⋅}

[

^Z_i ^{− β}

]

⁼⁰

where w_i

()

^. is some nonnegative weight function and

( ) ( )

( )

( ) ( )

( )

∑

∑

=

=

≥

⋅

≥

= _n

j

i j

n j

i j

i

e e

I

e e

I Z

1

*

* 1

*

*

) (

β β

Z β β

β

j

(

4.16

)

can be interpreted as the average value of covariate for subjects in the risk set:

( )

^}

: { )

|

(t j e t

R β = _j β ≥ ,

for t=e_i

( )

β with δ_i =1. When w_i

( )

β =1, the estimator is the log-rank estimator.

Chapter 5 Numerical Analysis

In this chapter, w evaluate finite-sample performances of several estimators by Monte-Carlo simulations. We consider the following model with h

( )

t =log

( )

t such that

( )

T_i =β₀ +β₁Z_i +ε_i

log ,

where Z_i ^{~ N}

( )

^0,1 and ε_i follows different types of distributions. Under censoring, the observable variable becomes

(

Z

)

C

X = β₀ +β₁ +ε ∧ ,

where the censoring variable C is generated from a uniform variable distributed in the interval [-1.5, 1.5]. The overall censoring rates vary between 25% and 30%. Sample size set to be n= 100 and 200 with 1000 replications.

5.1 Error with the Standard Normal Distribution

Consider that ε has the standard normal distribution with the density depicted in Figure 5.1:

Figure 5.1: ε ~ N(0,1)

The results are given in Table 5.1. Under this case, LAD and the quantile estimator

(

π =0.5

)

yield the same result. The two rank-based methods also have nice

results. Since ε is symmetric around zero and the chance of observing extreme observations is pretty low, we can expect that the LS has the best performance and all the methods should be valid.

5.2 Error with the Student’s T Distribution

Consider that ε has the student’s t distribution T_(v) with the density

( ) (

R

)

v v v

v f

v

⎟⎟ ∈

⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎠

⎜ ⎞

⎝ Γ⎛

⎟⎠

⎜ ⎞

⎝ Γ⎛ +

=

⎟⎠

⎜ ⎞

⎝

−⎛ +

ε ε π

ε 1

2 2

1

2 1 2

,

where v is the degree of freedom. The density for T₍₂₎ is depicted in Figure 5.2.

Figure 5.2:

ε

~T₍₂₎ vs .

ε

~N(0,1)

For comparison, we also plot the density of the standard normal distribution. We see that T₍₂₎ tends to produce more extreme observations. Under the adaptive choice of k, Huber’s estimator performs the best. The LAD and quantile methods with π =0.5 are also the same since ε is symmetric around zero. They become superior to the LS method, which is vulnerable to outliers, under the T distribution with heavy tails. The two log-rank methods still have nice results without being affected by extreme

observations.

5.3 Error with the Gumbel Distribution (right-skewed)

Consider that ε has the Gumbel distribution with the density

( )

e ^{( )}e ^{( )}

(

R

)

f = ^e ∈

− −

−

− −

γ ε

ε ^γ

α ε

γ α ε

1

,

where α is the location parameter and γ is the scale parameter. The density with

( ) ( )

α,γ = 0,5 is depicted in Figure 5.3:

Figure 5.3: ε ~Gumbel⁺(0,5)

The distribution is asymmetric such that it yields positive extreme values but with low frequency. Based on the results in Table 5.3, we see that most of the methods cannot accurately estimate β₀, the intercept term, except for the quantile method. Notice that for Z =0, log T

( )

=β₀ +ε. It is easy to see that rank-based procedure cannot detect β₀. The first three methods fail too since ε is asymmetric around zero. The quantile can flexibly adjust for this situation. For the slope estimation, all the methods are valid.

5.4 Error with the Gumbel Distribution (left-skewed)

Consider that ε has the Gumbel distribution (left-skewed) with the density

( )

e^{( )}e ^{( )} R

f = ^e ∈

−

−

−

γ ε

ε ^γ

α ε

γ α ε

1 ,

,

where α is the location parameter and γ is the scale parameter. The density with

( ) ( )

α,γ = 0,5 is depicted in Figure 5.4:

Figure 5.4: ε ~Gumbel⁻(0,5),

The distribution is asymmetric such that yields negative extreme values. Table 5.4 indicates that most of the methods significantly underestimate β₀, except for the quantile method. Similarly the rank-based methods cannot detect β₀ either. The asymmetry of ε also violates the assumption of the first three methods still. Only the quantile estimator can handle this problem. For the slope estimation, all the methods are valid.

5.5 Performances under Censoring

Here we focus on the estimation of β , the slope parameter under censoring. ₁ From Table 5.5 to Table 5.8, we see the same pattern again that the LS method has better performance under the normal error distribution. Here the LAD and qunatile method withτ =0.5 are no longer the same since they use different methods to adjust for censoring. The imputation approach discussed in §4.4 seems to perform better

than the weighting method. Besides, the quantile method still can handle asymmetric error data as well. Most methods, which use the Kaplan-Meier estimator in estimation is much affected by the censoring rate. The linear rank estimator without using Kaplan-Meier estimator is more robust to the censoring condition.

Chapter 6 Conclusion

In this thesis, we consider a class of general linear model and the major objective is to estimate the regression parameter. The underlying error distribution is unknown but usually practitioners may have some prior knowledge about its shape. Such information is useful for choosing an appropriate inference method for parameter estimation. For example, the least squares estimator is a suitable choice if the error is symmetric around zero with light tails like the normal distribution. On the other hand, if the chance of extreme observations is not low, the LAD estimator or rank-based procedures are more appropriate. If the error is skewed, most of the methods can not capture the intercept information except the quantile regression estimator. Generally speaking, the rank-based estimators have nice performances in most situations.

There are two different ways of handling missing data. One is imputation and the other is weighting. We have seen that these two approaches are also adopted for analyzing censored data as well. In the previous discussions, the LS estimator is more sensitive to the tail behavior. It is also more vulnerable to censoring since the Kaplan-Meier estimator does not estimate the tail well. The of quantile estimator which uses the imputation approach to handle censoring is valid in most situations.

The rank-based estimators perform pretty well even under censoring. In fact, they are frequently adopted in related problems.

As mentioned at the beginning, another area of research considers the situation that the distribution of ε is known but the form of transformation is unspecified.

Such a model is called the transformation model. It seems that most research in this area has not addressed on the relationship between ε and the objective function. We think that this may deserve further investigation.

References

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[3] Cheng, S. C.; Wei, L. J.; Ying, Z. (1995). Analysis of Transformation Models with Censored Data. Biometrika, 82, 835-845.

[4] Fine, J. P.; Ying, Z.;Wei, L. J.(1998). On the linear transformation model for censored data. Biometrika, 85, 980-986.

[5] Fygenson, M.; Ritov, Y. (1994). Monotone Estimating Equations for Censored Data. The Annals of Statistics, 22, 732-746.

[6] Harrington, D. P.; Fleming, T. R. (1982). A Class of Rank Test Procedures for Censored Survival Data. Biometrika, 69, 553-566.

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[8] Huang, J.; Ma, S.; Xie, H. (2005). Least Absolute Deviations Estimation for the Accelerated Failure Time Model. The University of Iowa, Department of Statistics and Actuarial Science, Technical Report No. 350.

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The Annals of Statistics, 18, 303-328.

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Biostatistics, 0, 1–15.

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Regression Models. Journal of the American Statistical Association, 90, 178-184.

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Appendix

Table 5.1. A: Performances of Different Estimators )

Table 5.1. B: Performances of Different Estimators )

Table 5.2. A: Performances of Different Estimators

Table 5.2. B: Performances of Different Estimators )

Table 5.3. A: Performances of Different Estimators

Table 5.3. B: Performances of Different Estimators

~

Table 5.4. A: Performances of Different Estimators

Table 5.4. B: Performances of Different Estimators censoring rate=0, ε ~ Gumbel^-(0, 5), n = 200

Table 5.5. A: Performances of Different Estimators

LS LAD Quantile Linear

rank

Table 5.5. B: Performances of Different Estimators )

LS LAD Quantile Linear

rank

Table 5.6. A: Performances of Different Estimators

LS LAD Quantile Linear

rank

Table 5.6. B: Performances of Different Estimators

( )

2

LS LAD Quantile Linear

rank

Table 5.7. A: Performances of Different Estimators ^{censoring rate=26.2%, ε ~Gumbel+}

( )

^{0,5 , n = 100}

LS LAD Quantile Linear

rank

LS LAD Quantile Linear

rank

Table 5.8.A: Performances of Different Estimators

^{censoring rate=26.2%, ε ~Gumbel−}

( )

^{0,5 , n = 100}

LS LAD Quantile Linear

rank

Table 5.8.B: Performances of Different Estimators

^{censoring rate=25.8%, ε ~Gumbel−}

( )

^{0,5 , n = 200}

LS LAD Quantile Linear

rank

在文檔中 線性模型於不同誤差假設下之統計推論 (頁 26-0)