Divergence based robust estimation of the tail index through an exponential regression model

Abstract

The extreme value theory is very popular in applied sciences including finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction characterized by its tail index; there are three broad classes of tails—the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the assumed model. There have been some recent attempts to produce asymptotically unbiased or robust alternative to the Hill estimator; however all the robust alternatives work for any one type of tail. This paper proposes a new general estimator of the tail index that is both robust and has smaller bias under all the three tail types compared to the existing robust estimators. This essentially produces a robust generalization of the estimator proposed by Matthys and Beirlant (Stat Sin 13:853–880, 2003) under the same model approximation through a suitable exponential regression framework using the density power divergence. The robustness properties of the estimator are derived in the paper along with an extensive simulation study. A method for bias correction is also proposed with application to some real data examples.

Appendices

Appendix 1: Assumptions (A1)–(A7) of Ghosh and Basu (2013) under the Assumed Exponential Regression Model

We assume the set-up of an exponential regression model, where the random variables $W_1, \ldots , W_n, \ldots$ are independent, but for each j, $W_j$ follows an exponential distribution with mean $\theta _j = \frac{\gamma }{1 - \left( \frac{j}{k+1}\right) ^\gamma }$. In this paper we have approximated the distribution of the transformed variable $Y_j$, as defined in Sect.  2, by the distribution of $W_j$ for each $j=1, \ldots , k-1$. Now we will present a brief argument to show that Assumptions (A1)–(A7) of Ghosh and Basu (2013), required for asymptotic consistency and normality of the MDPDE, hold under the present set-up of exponential regression model.

First note that Assumption (A1)–(A3) and (A5) hold directly from the form of an exponential distribution function. Next, as shown in the proof of Theorem 1, the matrix $J^{(i)}$, as per the notation of Ghosh and Basu (2013), is a positive scalar given by

$$\begin{aligned} J^{(i)} = \frac{(1+\alpha ^2)}{(1+\alpha )^3} \widetilde{J}\left( \frac{i}{k+1}\right) \theta _i^{\alpha -2}. \end{aligned}$$

So, the matrix $\varPsi _n$ is in fact a positive scalar with $\lambda _0 = \lim \limits _{k\rightarrow \infty }\varPsi _n = a_\gamma >0$; this implies that (A4) also holds. Finally we need to prove three limiting statements of Assumptions (A6) and (A7) of Ghosh and Basu (2013). We only present the proof of first one, namely (noting that we are dealing with scalar parameter $\gamma$ here)

$$\begin{aligned} \lim _{N\rightarrow \infty } \sup _{k>1} \left\{ \frac{1}{k-1} \sum _{i=1}^{k-1} E \left[ |\nabla _{g}V_i(W_i;\theta )| I(|\nabla _{g}V_i(W_i;\theta )| > N)\right] \right\} = 0. \end{aligned}$$

(17)

Here $\nabla _g$ represents the derivative with respect to our parameter of interest $\gamma$. The proof of the others are similar and hence omitted.

To prove (17), note that under this present model, we have for each i,

$$\begin{aligned} \nabla _{g}V_i(W_i;\theta ) = C_i \left[ \frac{\alpha }{(1+\alpha )^2} + \left( \frac{W_i}{\theta _i} - 1\right) e^{-\frac{\alpha W_i}{\theta _i}}\right] = C_i \psi \left( \frac{W_i}{\theta _i}\right) , \end{aligned}$$

where $C_i= (1+\alpha )\widetilde{J}_\alpha \left( \frac{i}{k+1}\right) \theta _i$ and $\psi (w) = \frac{\alpha }{(1+\alpha )^2} + (w-1) e^{-\alpha w}$. However, letting $W_i^* = \frac{W_i}{\theta _i}$, we get that $W_1^*, \ldots , W_{k-1}^*$ are independent and identically distributed observations from a standard exponential distribution with mean 1. So, we have

$$\begin{aligned}&\frac{1}{k-1} \sum _{i=1}^{k-1} E \left[ |\nabla _{g}V_i(W_i;\theta )| I(|\nabla _{g}V_i(W_i;\theta )|> N)\right] \\&\quad = \frac{1}{k-1} \sum _{i=1}^{k-1} E \left[ |C_i| \left| \psi \left( \frac{W_i}{\theta _i}\right) \right| I(|C_i| \left| \psi \left( \frac{W_i}{\theta _i}\right) \right|> N)\right] \\&\quad = \frac{1}{k-1} \sum _{i=1}^{k-1} |C_i| E\left[ \left| \psi (W_i^*)\right| I\left( \left| \psi (W_i^*)\right|> \frac{N}{\max _{1\le i \le k-1}|C_i|}\right) \right] \\&\quad = E\left[ \left| \psi (W_1^*)\right| I( \left| \psi (W_1^*)\right| >\frac{N}{\max _{1\le i \le k-1}|C_i|})\right] \left( \frac{1}{k-1} \sum _{i=1}^{k-1}|C_i|\right) . \end{aligned}$$

However, it is easy to check that both the terms $\left( \frac{1}{k-1} \sum _{i=1}^{k-1}|C_i|\right)$ and $\left( \max _{1\le i \le k-1}|C_i|\right)$ are bounded as $k\rightarrow \infty$. Thus, by Dominated Convergence Theorem, we have

$$\begin{aligned} \lim \limits _{N\rightarrow \infty } ~ E\left[ \left| \psi _1(W_1^*)\right| I\left( \left| \psi _1(W_1^*)\right| >\frac{N}{\max _{1\le i \le k-1}|C_i|}\right) \right] =0, \end{aligned}$$

and hence (17) holds.

Appendix 2: Some comments on the estimator proposed by Vandewalle et al. (2004)

Vandewalle et al. (2004) presented an interesting and practically important footstep in statistics by justifying the necessity of combining two apparently contradictory theory of extreme value statistics and robust statistics, primarily for the Pareto-Type tails ($\gamma >0$). They have used the robust regression method proposed by Marazzi and Yohai (2004) and the exponential regression model developed in Beirlant et al. (1999) given by

$$\begin{aligned} Y_j \sim _d \left( \gamma + b_{n,k}\left( \frac{\gamma }{k+1}\right) ^{-\rho }\right) g_j, \quad j=1, \ldots , k, \end{aligned}$$

(18)

where $g_j$ are independent and identically distributed standard exponential random variables. The proposed estimator was examined through an interesting real data example where its robustness was illustrated clearly.

While developing the robust estimator, Vandewalle et al. (2004) transformed the above model into a liner form given by Eq. (3.1) of their paper, which reads

$$\begin{aligned} Y_j \sim _d \gamma + b_{n,k}\left( \frac{\gamma }{k+1}\right) ^{-\rho } + \gamma e_j, \quad j=1, \ldots , k, \end{aligned}$$

(19)

where $e_j = g_j -1$. Here comes our first little doubt by noting that the RHS of the Equations (18) and (19) are not equal; the closest form to the second that equals the first is

$$\begin{aligned} \gamma + b_{n,k}\left( \frac{\gamma }{k+1}\right) ^{-\rho }g_j + \gamma e_j. \end{aligned}$$

So, it needs to be clarified the reason of dropping $g_j$ from the second term. After assuming the linearized form (19), they have re-parametrize it as

$$\begin{aligned} Y_j = \theta _i + \theta _2t_j + \sigma e_j, \quad j=1, \ldots , k, \end{aligned}$$

(20)

where $t_j = \left( \frac{\gamma }{k+1}\right) ^{-\rho }$, $\theta _1 = \gamma$, $\theta _2= b_{n,k}$ and $\sigma =\gamma$. Then, for the case $\gamma > 0$, they have used the robust regression method proposed by Marazzi and Yohai (2004) to estimate the parameters $(\theta _1, ~\theta _2, ~\sigma )$. This regression method has high breakdown and efficiency for usual regression set-up that they have noted for proposing the robust estimator of $\gamma$; However, the approach is computationally complicated. Moreover, under the transformed set-up (20) it is to be noted that $\theta _1 = \sigma$; this constraint needs to be taken care of while solving for the estimator numerically and may have potential effect on the properties of the resulting estimator. This needs to be examined extensively through simulation or theoretical results, that was missing in the work of Vandewalle et al. (2004). They have also noted similar limitation of the work and made a comment in the “conclusion” that they would consider this issues in their future work. Considering all this doubts, we have decided not to consider this proposal in our simulation studies.