Biomedical Engineering Reference
In-Depth Information
2.3. Other Smooth Estimation Methods
Paralleling the approach to density estimation proposed by Olkin and
Spiegelman
105
, Kouassi and Singh
85
proposed the model
h
w
t
(t;
^
) = w
t
h(t;
^
) + (1w
t
)h(t)
(14)
for h(t). Here the weight parameter w
t
2[0; 1] is unknown and depends on
the time point t. The w
t
is estimated by minimizing the mean squared error
(MSE) of h
w
t
(t;
^
), and its estimator w
t
is then used in (14) to obtain the
estimator h
w
t
(t;
^
) for h(t).
^
is the ML estimator of the unknown parameter
or parameter vector in the parametric hazard model h(t; ) that can be,
e.g., Weibull with unknown shape and scale parameters, and h(t;
^
) is the
parametric estimator of h(t). The h(t) is any smooth nonparametric hazard
estimator of h(t). For technical simplicity, Kouassi and Singh took h(t) to be
the kernel estimator h
b
(t)
116;136;157
. The w
t
provides some insight into which
of the parametric or nonparametric estimators is more commensurate with
the data; it is expected to be close to 1 when the parametric model is valid
or close to 0 otherwise. The h
w
t
(t;
^
) is a semiparametric estimator, because
it is a combination of a parametric and nonparametric estimators. When
the parametric model holds, the semiparametric hazard estimator h
w
t
(t;
^
)
converges to the true model at the same rate as the parametric hazard
estimator; otherwise, it converges at the same rate as the nonparametric
hazard estimator. The proposed method leads to a more precise hazard
estimator in the sense that the MSE of the semiparametric estimator is
smaller than those of its parametric and nonparametric competitors.
In addition, Patil
110
and Antoniadis, Gregoire, and Nason
10
estimated
the hazard function h(t) with wavelet methods.
3. Smooth Estimation of Hazard Function with Covariates
The linear PH model (2) is a popular regression tool for the analysis of
censored failure time data, but the linearity assumption of the covariate
eects may not be valid in practice. One can remedy the violation by means
of various nonparametric regression techniques. Therefore, in this section we
will rst introduce some existing nonparametric regression techniques for
modeling covariate eects in the PH model and then some semiparametric
and nonparametric regression models for the conditional hazard functions
as alternatives to the linear PH model. First, we introduce some notations to
be used in the following sections. Let (Y
i
;X
i
;
i
), i = 1;:::;n, be a sample
of i.i.d. random variables, each having the same distribution as (Y;X; ),
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