^{1}

^{*}

^{2}

In this article, we present a new family of estimators for the regression parameter β in the Additive Hazards Model which represents a gain in robustness not only against outliers but also against unspecific contamination schemes. They are consistent and asymptotically normal and furthermore, and they have a nonzero breakdown point. In Survival Analysis, the Additive Hazards Model proposes a hazard function of the form , where is a common nonparametric baseline hazard function and z is a vector of independent variables. For this model, the seminal work of Lin and Ying (1994) develops an estimator for the regression parameter β which is asymptotically normal and highly efficient. However, a potential drawback of that classical estimator is that it is very sensitive to outliers. In an attempt to gain robustness, álvarez and Ferrarrio (2013) introduced a family of estimators for β which were still highly efficient and asymptotically normal, but they also had bounded influence functions. Those estimators, which are developed using classical Counting Processes methodology, still retain the drawback of having a zero breakdown point.

In Survival Analysis, a main goal is how to model a random variable

A particular type of survival models with great appeal among practitioners focuses on the so-called hazard function

Within the Cox model, the potential harmful effects of outliers were commented by Kalbfleisch and Prentice (1980, ch. 5) [

As for the Additive Hazards Model, the proposal of robust alternatives has received much less atention in the literature. In Álvarez and Ferrario (2013) [

where

where H is the noncontaminated distribution that belongs to the additive hazards family and Q represents a point mass at its argument. For the practitioner, estimators with bounded influence functions are of interest when (s) he seeks a guard against a very small proportion of outliers.

Appart from the fact that the contamination scheme above is very specific, a further drawback of the estimators presented in Álvarez and Ferrario (2013) [

In this article we propose a new family of robust estimators for the additive hazards model in a manner similar to Bednarski (1993) [

The advantage of the estimators we present in this paper over previous proposals arises whenever a dataset contains outliers. When a sample is contaminated by unusual observations, the classical estimator (Ling and Yings) rapidly becomes nonsensical (in that its value drifts away towards zero or infinity). The estimators in Álvarez and Ferrario [

Let

so-called at risk process defined by

where for a column vector v, we denote the matrix

Using classical Counting Process theory, Lin and Ying prove that their

In order to propose a Fréchét differentiable alternative to the classical (Lin and Ying’s) estimator we need to express the estimator as a functional of the joint empirical distribution function and we need to make explicit the structure of the Additive Hazard Family of distributions. We pursue this as follows.

Event times: Let ^{*} has a hazard (conditional) function

Covariates: The covariates

Censoring: Conditional on Z, censoring and event times are independent, i.e.

Observed times: Due to censoring, the observed times are

the joint density of T and Z is

We now develop the joint bivariate distribution function

Censoring indicator: Let

Thus taking the derivative with respect to t we obtain

We define that a cummulative joint distribution function

Now we express the clasical estimator

where we introduce the process

empirical distributions.

In Alvarez and Ferrario (2013) [

In order to define contamination e-neighborhoods, let

that G is in a neighborhood

point mass at some triplet

We propose here an estimating equation by introducing weight functions in the classical formulation, i.e.

where

Naturally, in the the special case where

Let us denote by

where FD is a linear functional called “Fréchét derivative”. Notice that we opt here for a uniform type of differentiability over

In order to avoid excessive notation we will in the sequel develop the proofs without censoring. Let

sponding to the true value of the parameter

and a real function

by

In order that our family of estimators become Fréchét differentiable we will need the following assumptions.

Assumptions

A1) For all

A2) All the functions in

Assumptions A1) and A2) ensure differentiability. The compactness assumption in A2) is needed to allow posibly adaptive choices of W based on some preeliminary estimate of

We seek now a linear approximation of

For the first difference in functionals above, the following Lemma gives a linear approximation:

Lemma 1. Under assumptions A1) and A2),

where

Moreover,

As for the second difference in (11) we have:

Lemma 2. For any

where

Further, the following result gives a bound of

Lemma 3. Under assumptions A1) and A2) there are constants

At this point, for further results we need to add another assumption that guarantees the existence of the inverse of

A3) There is a pair of constants

Thus, the consistency of the estimator in a neighborhood of

Theorem 1. Let the family of functions

for all

Moreover, Fréchét differentiability is asserted as follows:

Theorem 2. Let

This implies that the Fréchét derivative of

In the following theorem we investigate convergence in distribution under contiguous alternatives to some distribution in the additive hazards family

Theorem 3. Let

for some constant

and

where

The result above implies that asymptotic normality holds not only under the true model but also under contiguous alternatives.

In this section, we evaluate the performance of our proposed family of estimators via simulations. Specifically, we carry out three simulation experiments choosing for simplicity a single covariate

In the first simulation we study the behavior or our estimator, denoted “RD” (Robust Differentiable) for increasing sample sizes. We take

In the second simulation, we do a comparison among the classical estimator (LY), the bounded-influence- function (BIF) estimators proposed in Álvarez and Ferrario (2013) [

results of this experiment. For the BIF estimators we take the weight function

Estimator | LY | RD | |||
---|---|---|---|---|---|

n | |||||

50 | coef. | 0.59 | 0.51 | 0.52 | 0.54 |

(s.e.) | (0.32) | (0.50) | (0.43) | (0.34) | |

200 | coef. | 0.51 | 0.52 | 0.52 | 0.51 |

(s.e.) | (0.15) | (0.20) | (0.17) | (0.13) | |

500 | coef. | 0.51 | 0.51 | 0.51 | 0.51 |

(s.e.) | (0.09) | (0.13) | (0.10) | (0.09) | |

1000 | coef. | 0.51 | 0.49 | 0.49 | 0.50 |

(s.e.) | (0.07) | (0.09) | (0.08) | (0.06) | |

10,000 | coef. | 0.50 | 0.51 | 0.51 | 0.50 |

(s.e.) | (0.02) | (0.03) | (0.02) | (0.02) |

Pure Sample | With Outliers | ||||
---|---|---|---|---|---|

% | Estimators | LY | LY | BIF | RD |

0 | coef. | 0.55 | 0.55 | 0.54 | 0.53 |

(s.e.) | (0.19) | (0.19) | (0.16) | (0.22) | |

0.5 | coef. | 0.54 | 0.40 | 0.46 | 0.52 |

(s.e.) | (0.18) | (0.23) | (0.16) | (0.23) | |

5 | coef. | 0.53 | 0.11 | 0.22 | 0.54 |

(s.e.) | (0.14) | (0.56) | (0.41) | (0.18) | |

15 | coef. | 0.50 | 0.040 | 0.08 | 0.52 |

(s.e.) | (0.15) | (0.66) | (0.59) | (0.16) | |

25 | coef. | 0.50 | 0.02 | 0.05 | 0.51 |

(s.e.) | (0.14) | (0.67) | (0.64) | (0.16) |

Lastly, we carry out a third simulation experiment in order to detect what the breakdown points of the RD estimators may be under a different type of model departure. As a model for the contaminating distribution, we chose point masses on the line

Pure Sample | Contaminated Sample | ||||
---|---|---|---|---|---|

% | |||||

0.2 | coef. | 0.55 | 0.52 | 0.55 | 0.52 |

(s.e.) | (0.28) | (0.14) | (0.28) | (0.14) | |

3 | coef. | 0.50 | 0.51 | 0.50 | 0.51 |

(s.e.) | (0.27) | (0.14) | (0.25) | (0.13) | |

10 | coef. | 0.50 | 0.52 | 0.51 | 0.21 |

(s.e.) | (0.29) | (0.14) | (0.23) | (0.42) | |

20 | coef. | 0.50 | 0.50 | 0.50 | -0.13 |

(s.e.) | (0.27) | (0.13) | (0.15) | (0.89) | |

25 | coef. | 0.48 | 0.51 | 0.51 | -0.16 |

(s.e.) | (0.27) | (0.14) | (0.15) | (0.93) | |

30 | coef. | 0.51 | 0.53 | 0.22 | -0.17 |

(s.e.) | (0.27) | (0.14) | (0.40) | (0.94) |

sample a size

Intuitively, the finite sample breakdown point of an estimator is the largest proportion of contaminated observations, and a method can resist before the estimates become nonsensical, which usually means that the estimate drifts away towards zero of infinity, or in general towards the boundaries of a parameter space. Equ- ivalently, its functional version is called the asymptotic breakdown point and it measures the largest proportion of contamination. A statistical functional could tolerate before becoming nonsensical in the same sense (e.g. Maronna, Martin and Yohai 2006 [

We thank the editor and the referee for their comments. This work has been financed in part by UNLP Grants PPID/X003 and PID/X719. Julieta Ferrario further wishes to thank Tadeus Bednarski for generously sharing otherwise electronically unavailable manuscripts.

EnriqueE. Álvarez,JulietaFerrario, (2015) Robust Differentiable Functionals for the Additive Hazards Model. Open Journal of Statistics,05,631-644. doi: 10.4236/ojs.2015.56064

Proof of Lemma 1. Rearranging,

So that substracting

Let

To simplify notation, let

where after distributing the inner brackets

A2), we can choose large enough values

the support of any function in

Take

where

where in order to simplify notation, we introduced the operator

Denote also the set

Since we chose the

In consequence,

Hence for all

which is bounded because of A1) and A2). i.e. for some constant,

Thus

so that

to claim that for some finite constants

focus on the terms

Similar calculations hold for the other

For the last assertion of the Lemma, substitute H by

Proof of Lemma 2. For any fixed

which is independent of

So substracting,

Proof of Lemma 3. Express

where

arguments as in Lemma 1, relying on integration by parts and Assumptions A1)-A2), we see that all the terms

are

Proof of Theorem 1. By Lemmas 1 and 3, for all

Also by Lemma 2,

Take now

for some fixed

So for

Equation (18), if some

which implies that

Proof of Theorem 2. Since

Also by Lemmas 1 and 2 respectively, for all

Adding the above Equations in (19) we get

Note that by Theorem 1, for

Now since by Assumption A3)

Proof of Theorem 3. Decompose

and by Glivenko-Cantelli’s Theorem

Since