A Simulation Comparison of Estimators of Conditional Extreme Value Index under Right Random Censoring
In extreme value analysis, the extreme value index plays a vital role as it determines the tail heaviness of the underlying distribution and is the primary parameter required for the estimation of other extreme events. In this paper, we review the estimation of the extreme value index when observations are subject to right random censoring and the presence of covariate information. In addition, we propose some estimators of the extreme value index, including a maximum likelihood estimator from a perturbed Pareto distribution. The existing estimators and the proposed ones are compared through a simulation study under identical conditions. The results show that the performance of the estimators depend on the percentage of censoring, the underlying distribution, the size of extreme value index and the number of top order statistics. Overall, we found the proposed estimator from the perturbed Pareto distribution to be robust to censoring, size of the extreme value index and the number of top order statistics.
Keywords: Extreme value index; random censoring; covariate information; moving window; simulations.
The study of extreme events has received much attention in many fields of application due to the nature of their impact. For instance, extreme earth quakes cause many deaths and destruction to properties; large price movements in equities result in huge losses, profits or collapse of financial markets; large insurance claims lead to solvency problems.
Unlike traditional statistical methods that focus on the central part of distributions, statistics of extremes focuses on the tail of the underlying distribution. Interest is then on parameters associated with the tail of the underlying distribution, such as high quantiles and exceedance probabilities. For inference on such parameters, distributional results are needed on the extreme observations. The first such result in extreme value theory was obtained by Fisher and Tippett (1928) and further developed by e.g. Gnedenko (1943) and de Haan (1970). These asymptotic distributions form the basis for carrying out inference in extreme value analysis.
Let be an independent and identically distributed sample on some random variable and let be the corresponding order statistics. The mentioned results state that, if there exists normalising constants and and some nondegenerate function such that
then the constants can be redefined such that for
Here, (2) is the so-called Generalised Extreme Value distribution and is the extreme value index (or tail index). The parameter is the primary parameter needed in extreme value analysis and determines the tail heaviness of the underlying distribution. If belongs to the Pareto domain (heavy tailed); if belongs to the Weibull domain (short-tailed); and if belongs to the Gumbel domain (light-tailed). An underlying distribution function, for which (1) and (2) hold, is said to be in the domain of attraction of denoted
The estimation of has been addressed in many papers, including Hill (1975); de Haan and Peng (1998); Tsourti and Panaretos (2003); Beirlant et al. (2004); Gomes et al. (2008); Gomes and Guillou (2014). When covariate information is available, the focus is to include it in the estimation by modelling the parameters of the extreme value distribution as a function of the covariate(s). For example, Davison and Smith (1990) fitted a Generalised Pareto (GP) distribution with parameters taken as an exponential function of the covariates; Gardes and Girard (2008) used moving-window methodology; Beirlant and Goegebeur (2003) and Wang and Tsai (2009) used a conditional exponential regression model; and Beirlant and Goegebeur (2004) employed repeated fitting of local polynomial maximum likelihood estimation.
In the case of censoring, Beirlant et al. (2007) and Einmahl et al. (2008) proposed an inverse probability-of-censoring weighted method to adapt classical extreme value index estimators to censoring. Similarly, Gomes and Neves (2011) and Brahimi et al. (2013) used this idea to adapt various estimators to censoring. In addition, Beirlant et al. (2010) addressed the issue of censoring, obtaining maximum likelihood estimators by adapting the likelihood function of the generalised Pareto distribution to censoring. Also, Worms and Worms (2014) considered estimators based on Kaplan-Meier integration and censored regression. Furthermore, Ameraoui et al. (2016) estimated the extreme value index from a Bayesian perspectives and Beirlant et al. (2017) proposed a reduced-bias estimator based on an extended Pareto distribution.
In the case of the presence of both covariate information and censoring, Ndao et al. (2014) proposed three estimators for the estimation of the conditional extreme value index and extreme quantiles for heavy-tailed distributions. In particular, the Hill, generalised Hill and moment type estimators were proposed using the moving window method (Gardes and Girard, 2008) and adapting the estimators to censoring by utilising the inverse probability-of-censoring weighted method (Beirlant et al., 2007; Einmahl et al., 2008). Unlike Ndao et al. (2014), Stupfler (2016) proposed a moment estimator valid for all domains of attraction. In addition, Ndao et al. (2016) addressed the estimation of the extreme value index under censoring and the presence of random covariates.
Although quite a number of papers have compared the available estimators of the extreme value index, no paper has compared the estimators of the conditional extreme value index when observations are subject to random censoring. The aim of this paper is two-fold. Firstly, to adapt some classical estimators to the current context, including a reduced-bias maximum likelihood estimator based on a perturbed Pareto distribution in Beirlant et al. (2004). Secondly, we review the available estimators that were proposed in the literature for estimating conditional extreme value index under right random censoring and compare them together with the proposed ones in a simulation study.
The remainder of the paper is organised as follows. In Section 2, we set out the framework for the estimation of the parameter of interest i.e. extreme value index. In addition, the existing estimators are reviewed and we present the proposed estimators. In section 3, we conduct a simulation study to assess the performance of these estimators. Lastly, general conclusions from the simulation results are presented in section 4.
2 Estimation of the Extreme Value Index
Consider as independent copies of a positive random variable, and let be the values of an associated -dimensional covariate vector, where Also, in order to incorporate the presence of censoring, let be independent copies of another positive random variable also associated with the covariate vector We assume that for all the random variables, and are independent. Furthermore, for every we assume that the random variables and have respective conditional distribution functions, and where and are real functions. We consider the case where and are in the Pareto domain of attraction i.e. and For this domain of attraction, the distribution functions can be represented as
or equivalently in terms of the tail quantile function,
Here, and are slowly varying functions associated with and respectively and defined as
In this context, we observe the triplets where and By the independent assumption of and the conditional distribution function of the random variable is related to and as
Therefore, is also in the Pareto domain with conditional extreme value index given in Einmahl et al. (2008) as
Before presenting the estimators, we define a ball in where and are the center and radius respectively. Thus,
In addition, let be a positive integer that approaches as . The estimators of the EVI are based on observations of for which the corresponding values of fall within the ball The proportion of the design points falling within the ball is defined as
where is the indicator variable. Relation (9) plays an important role in this procedure as it describes how the points are concentrated around the neighbourhood of when approaches (Gardes and Girard, 2008). The number of nonrandom observations in is given by .
Let denote the pair, that have their corresponding -values falling within the ball as defined in (8). Also, let be the corresponding order statistics of ’s and be the values of ’s associated with The values of form the basis for adapting the classical estimators of the conditional extreme value index presented below to censoring.
2.1 The Existing Estimators
The existing estimators result from the application of the moving window technique (Gardes and Girard, 2008) and the inverse probability-of-censoring weighted method (Beirlant et al., 2007; Einmahl et al., 2008) to adapt classical estimators to censoring. Ndao et al. (2014) used this approach to adapt the Hill, generalised Hill and moment estimators to censoring. In this section, we review these estimators and follow a similar approach to propose other estimators of the extreme value index in the next section.
The estimators introduced by Ndao et al. (2014) are presented are the following:
The Hill-type estimator: The Hill estimator (Hill, 1975) is arguably the most common estimator of in the Pareto case i.e. To take into account the available covariate information, the Hill estimator is defined for the -largest order statistics as
The Moment-type estimator: Dekkers et al. (1989) introduced the moment estimator as an adaptation of the Hill estimator valid for all domains of attraction. It is defined to take into account the covariate information as,
The generalised Hill Estimator: Beirlant et al. (1996) proposed the generalised Hill (GH) estimator as an attempt to extend the Hill estimator to the case where The GH estimator is obtained as the slope of the ultimately linear part of the generalised Pareto quantile plot of the observations within the defined window as,
The estimators (10), (11) and (13) were adapted to censoring by dividing each estimator by the proportion of noncensored observations in the largest order statistics of ’s. Thus, an adapted estimator of the conditional extreme value index is given by
2.2 The Proposed Estimators
In this section, we propose some estimators for the estimation of conditional extreme value index when observations are subject to right random censoring. We follow closely the methodology in Ndao et al. (2014).
Firstly, the estimators are presented to take into account the covariate and are subsequently adapted to censoring.
The Zipf estimator: Kratz and Resnick (1996) derived the Zipf estimator as a smoother version of the Hill estimator through unconstrained least squares fit to the largest observations on the generalised Pareto quantile plot method of Beirlant et al. (1996). The estimator is valid for and in the case of a covariate, is given by,
The Moment Ratio: The Moment Ratio estimator was introduced by Danielsson et al. (1996) as a moment based estimator to reduce bias in the Hill estimator. The moment ratio estimator is valid for the Pareto domain of attraction only. In the case of a covariate, it is given by
where is obtained from (12).
The Peng Moment Estimator: Deheuvels et al. (1997) reports on a variant of the moment estimator for the no covariate case suggested by Liam Peng. This estimator is designed to reduce bias in the moment estimator and it is adapted to the covariate case as
where is as before obtained from (12). This estimator is valid for all domains of attraction.
The Perturbed Pareto Estimator
Beirlant et al. (2004) derived the perturbed Pareto estimator as a reduced-biased Hill type estimator by making use of the second-order properties on the underlying distribution function, Here, we consider adapting this estimator to censoring and the presence of covariate information. Considering (3) and (5), we can write
Therefore, (18) can be interpreted as
for large and Now, consider the relative excesses for a large threshold where is the index of the th exceedance. Then, it seems natural to consider the strict Pareto distribution as the approximate distribution of the relative excesses, The maximum likelihood estimator of the parameter results in the Hill estimator in (10). However, if the strict Pareto approximation is poor, the Hill estimator has large bias, and hence, a second-order refinement is needed to address the departure from the strict Pareto distribution (see Beirlant et al., 2004). Assume that the function, satisfies the second-order assumption:
Assumption I: There exists a real constant and a rate function satisfying as such that for all ,
where (Beirlant et al., 1999, page 602).
where is regularly varying with index - Ignoring the error term, (21) becomes a mixture of two Pareto distributions. The survival function for such a distribution is given by
where and In practice, the perturbed Pareto distribution is fitted to the relative excesses, and the parameters of the distribution can be estimated through the maximum likelihood method. The resulting estimator is denoted by Similar to Ndao et al. (2014), the estimators of the the extreme value index from (i) through to (iv) are adapted to censoring using (14).
Furthermore, we extend the two estimators of the extreme value index introduced in Worms and Worms (2014) when observations are subject to random censoring to the case where covariate information is available.
The first estimator is given by
where and are respectively the Kaplan-Meier estimators for and given by
for Here, is defined as a function of the form
The second alternative estimator is a weighted version of the Hill-type estimator (23),
In the next section, the performance of the existing and proposed estimators will be compared via a simulation study.
3 Simulation Study
In this section, we present a simulation study to assess the performance of the estimators discussed in the previous section as the asymptotic distribution of most of the estimators are unknown.
Several sample sizes were considered in the simulation and for simplicity, we report on the simulation for samples of size generated from three distributions i.e. Burr, Fréchet and Pareto only. For each distribution, our interest is in the estimation of the conditional extreme value index (EVI) function, Here, and were chosen as and such that the values of for are within (0,1). This range of values of the extreme value index is the most common in extreme value theory literature for simulation studies and practical applications (see for e.g. Gilli and Këllezi, 2006; Gomes and Neves, 2011; Einmahl et al., 2008; Stupfler, 2016). In particular, we selected values of equal to, and corresponding respectively to values, 0.63 (large), 0.31 (medium) and 0.10 (small). The choice of parameter functions for each distribution are presented in Table 1.
In addition, the distribution of is chosen such that the percentage of censoring in the right tail is 10%, 35% and 55%. The performance measures used for examining the estimators of are Mean Square Error (MSE) and median bias (hereafter referred to as bias).
The following algorithm was implemented to obtain the performance measures:
Generate random observations from,
Generate random samples from the distributions of and with parameters, and respectively. To maintain an approximately equal percentage of censoring in each sample, is chosen as where is the percentage of noncensored observations.
Let and to obtain the triplets
Choose a covariate value of interest, window size, and obtain the observations with its values falling within the window where is the number of observations within the window.
Compute an estimate of using at each number of top order statistics for the sample in A4.
At each value, compute the median bias
and the MSE,
3.2 Results and Discussion
The results of the simulation study are presented in this section. For brevity and ease of presentation, we present the results for Burr distribution and leave that of Pareto and Fréchet to the appendix. Figures 1, 2 and 3 show the results for estimators of and respectively. From these figures, it can easily be seen that most of the estimators’ performance diminish as increases. This is expected as more intermediate observations are included in the estimation leading to bias. In addition, we observed that the bias and to a larger extent MSE, increases with decreasing value of Furthermore, the performance of the estimators of decreases as the censoring percentage increases. This is in conformity to the findings in Ndao et al. (2014).
We now turn attention to the performance of the individual estimators. Firstly, we found that the Hill estimator has large MSE and bias as increases. This is in contrast to the simulation results and Corollary 4.2 in Ndao et al. (2014). Thus, we may conclude that the performance of the Hill estimator depends on the choice of parameter function, However, this result is consistent with the performance of the Hill estimator in the case where there is no covariate information nor censoring (see e.g. Beirlant et al., 2004, and references therein).
For samples generated from the Burr distribution, the PPD, GH and MOM estimators are the most robust to censoring in most cases. These estimators have smaller bias and MSE compared to the other estimators of In particular, the proposed PPD estimator is seen to have the smallest bias and MSE as the percentage of censoring increases.
With regard to the Pareto distribution, the performance of the estimators are similar to that of the Burr distribution. However, some differences occur which are mentioned. Firstly, the PMom estimator competes with the best three estimators i.e. PPD, MOM and GH in terms of bias and MSE. Secondly, in the case, of large censoring, the PPD estimator has the smallest MSE and relatively good bias.
In the case of samples generated from the Fréchet distribution, all the estimators have good MSE values with the exception of small and large values where some of the estimators’ performance deteriorate. In terms of bias, the performance does not differ significantly from that of MSE. Interestingly, the Hill estimator is seen to compete with the PPD, Zipf and WW.KM in terms of bias and MSE as the percentage of censoring and increase.
Overall, we found that the performance of estimators depend on the distribution, the number of top order statistics and the percentage of censoring. Contrary to what was reported in Ndao et al. (2014), we found that the Hill estimator has large bias and MSE except for samples generated from the Fréchet distribution. The proposed PPD estimator is universally competitive in estimating regardless of its size, percentage of censoring and number of top order statistics.
In this paper, the central issues were a review and proposals of estimators of conditional extreme value index when observations are subject to right random censoring. In the latter, we proposed adapting some classical extreme value index to censoring and presence of covariate information. The existing and the proposed estimators were compared in a simulation study. Some interesting results were obtained and are outlined as follows:
The performance of the estimators depend on the underlying distribution of the sampled data. Thus, no estimator was universally the best under all the simulation conditions considered. However, a closer look at the results reveal that some general conclusions can be reached on some estimators that can be considered as appropriate for estimating
The performance of the estimators depend on the size of Bias and MSE values were generally larger for small values and smaller for larger values of Therefore, we recommend that in practice before proceeding to use any estimator for one should assess the potential range of the true value of In such cases, several estimators can be considered and a median or average value used as an estimate to help in the selection of the preferred estimator.
The performance of the estimators generally deteriorates as the percentage of censoring increases. However, the estimators that are robust to censoring, to a larger extent, maintain their performance under increased percentage of censoring. These estimators include PPD, MOM and PMom and GH. Conspicuously missing is the Hill estimator which was shown to have large bias and MSE except for samples from the Fréchet distribution.
Overall, we found that the proposed PPD estimator mostly has the best MSE and bias behaviour under small and heavy censoring. In addition, it has less bias as the number of top order statistics, increases.
Some additional research is needed to establish the consistency and asymptotic normality of the PPD estimator. This will enhance statistical inference and will be considered in future research.
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Appendix A Pareto Distribution
For each figure, the following description of the panels apply. Leftmost column: Middlemost column : Rightmost column:
Appendix B Fréchet Distribution
For each figure, the following description of the panels apply. Leftmost column: Middlemost column : Rightmost column: