Modern Stochastics: Theory and Applications

Let ${X_{1}},\dots ,{X_{n}}$ be independent and identically distributed (i.i.d.) random variables with a common distribution function F. Suppose that the right tail $1-F$ is a regularly varying function with index $-1/\gamma $ (written $1-F\in {\mathrm{RV}_{-1/\gamma }}$), that is, for $x\gt 0$, where $\gamma \gt 0$ is the positive extreme value index (EVI). Put Condition (1) is equivalent to see, e.g. pg. 73 in [5]. For a large class of quantile type functions U satisfying (2) there exists a function $A(t)\to 0$ of constant sign for large values of t, such that for every $x\gt 0$, where $\rho \le 0$ is a second order parameter and

Numerous of the existing estimators of a positive extreme value index (see, e.g. [18, 9, 8]) are constructed by using parameterized statistics where ${X_{1,n}}\le {X_{2,n}}\le \cdots \le {X_{n,n}}$ denote the ascending order statistics of ${X_{1}},\dots ,{X_{n}}$. ${M_{n}}(k,1)$ is the classical Hill estimator ([18]), while ${M_{n}}(k,2)$ was introduced in [8]. The asymptotic properties (including weak consistency and asymptotic normality) of ${M_{n}}(k,r)$ when r is real positive were considered in [13]. To learn more about estimators for a positive extreme value index, we refer to the review [10], where more than one hundred estimators are collected.

Several papers are devoted to the investigation of the asymptotic properties of estimators (with two tuning parameters), defined by where $\Gamma (1+r)$, $r\ge 0$ denotes the gamma function defined by the integral $\Gamma (1+r)={\textstyle\int _{0}^{\infty }}{t^{r}}\exp \{-t\}\mathrm{d}t$. The estimators (4) were presented in [3]. It is important to note that the family of estimators (4) generalizes several classical estimators. The estimator ${\hat{\gamma }_{n}}(k,1,1)$ coincides with the Hill estimator, while the estimator ${\hat{\gamma }_{n}}(k,1,2)=\sqrt{{M_{n}}(k,{r_{1}})/2}$ was proposed in [9] as an alternative to the Hill estimator. Also, the estimator ${\hat{\gamma }_{n}}(k,2,1)={M_{n}}(k,2)/\left(2{M_{n}}(k,1)\right)$ is a moment ratio estimator, introduced by de Vries; see [6].

For completeness, we recall the result regarding the asymptotic normality of estimators (4).

The subfamily ${\hat{\gamma }_{n}}(k,1,r)$, $r\gt 0$ of the family of estimators (4) was investigated in [12]. The subfamilies ${\hat{\gamma }_{n}}(k,r,2)$, $r\ge 1$ and ${\hat{\gamma }_{n}}(k,r,1)$, $r\ge 1$ of (4) were considered in [4] and [22], respectively.

Recall from Prop. 1 in [4] that if (2) and (5) hold, then for $r\gt 0$, Motivated by the construction of the moment ratio estimator, we consider the ratio ${M_{n}}(k,{r_{2}})/{M_{n}}(k,{r_{1}})$, $0\le {r_{1}}\lt {r_{2}}$ and by combining (8) with Slutsky’s theorem (see e.g. [24]) we find that ${M_{n}}(k,{r_{2}})/{M_{n}}(k,{r_{1}})$ converges in probability to ${\gamma ^{{r_{2}}-{r_{1}}}}\Gamma ({r_{2}}+1)/\Gamma ({r_{1}}+1)$ as $n\to \infty $. This leads to the new family of semi–parametric estimators of $\gamma \gt 0$: It should be noted that the family of estimators (9) generalizes the Hill estimator (${r_{1}}=0$, ${r_{2}}=1$), the alternative Hill estimator (${r_{1}}=0$, ${r_{2}}=2$) and the moment ratio estimator (${r_{1}}=1$, ${r_{2}}=2$).

Our main result states that the estimators in (9) are asymptotically normal for $\gamma \gt 0$.

Obviously, that for any $\rho \lt 0$ and $0\lt {r_{1}}\lt {r_{2}}$. Recall that $A(t)\to 0$ as $t\to \infty $. Thus, under assumption (5), the asymptotic bias ${\nu _{1}}(\rho ,{r_{1}},{r_{2}})$ $A(n/k)$ tends to zero as $n\to \infty $, but estimators satisfying (11) are referred to as asymptotically biased; see, e.g. [19].

Next, consider three families of asymptotically biased estimators for $\gamma \gt 0$. Namely, by taking ${r_{2}}=2{r_{1}}$ in (9) we obtain the estimators Using a slight modification of the comparison scheme for biased estimators (proposed in [6]), we will compare the estimator ${\hat{\gamma }_{n}^{(1)}}(k,r)$ with the following estimators: In [4] it is noted that ${\hat{\gamma }_{n}}(k,r,2)$, $r\ge 1$ are asymptotically unbiased estimators, and thus we eliminated these estimators from our comparison. We refer to [21] for a discussion of difficulties related to the comparison of an asymptotically unbiased estimator with an asymptotically biased estimator. Taking into account that ${\hat{\gamma }_{n}^{(2)}}(k,r)={\hat{\gamma }_{n}^{(1)}}(k,0,r)$, $r\gt 0$ and ${\hat{\gamma }_{n}^{(3)}}(k,r)={\hat{\gamma }_{n}^{(1)}}(k,r-1,r)$, $r\ge 1$, the asymptotic normality of the estimators ${\hat{\gamma }_{n}^{(\ell )}}(k,r)$, $\ell =1,2,3$ follows directly from Theorem 2.

The paper is organized as follows. In the next Section, we compare estimators ${\hat{\gamma }_{n}^{(\ell )}}(k,r)$, $\ell =1,2,3$ theoretically. In Section 3, a small-scale simulation study is undertaken. In Section 4, an application to the exchange rate data is presented to illustrate the behavior of the estimator ${\hat{\gamma }_{n}^{(1)}}(k,r)$. Section 5 contains conclusions, while all proofs are collected in Section 6.

The asymptotic second moment of ${\hat{\gamma }_{n}^{(\ell )}}(k,r)$ is ${A^{2}}\left(n/k\right){\lambda _{\ell }^{2}}(\rho ,r)+{\gamma ^{2}}{\varsigma _{\ell }^{2}}(r)/k$, see [6] for the definition of the asymptotic second moment. Firstly, following [20] (see also [1]), we fix r and find the asymptotic behavior (as $n\to \infty $) of the so-called minimal mean squared error where ${k_{n,\ell }^{\ast }}$ is the minimizing sequence.

Further, we will assume the classical restriction $\rho \lt 0$. It eliminates the case where $|A|$ is a slowly varying function at infinity. In addition, under assumption $\rho \lt 0$ in (3), there exists a positive decreasing function $a\in {\mathrm{RV}_{2\rho -1}}$ such that see [7].

Note that ${\lambda _{\ell }}(\rho ,r)\ne 0$ in (12) is an essential assumption. It allows us to balance the rate of decay of squared asymptotic bias and asymptotic variance.

By applying Lemma 2.8 in [7], we get that the minimizing sequence ${k_{n,\ell }^{\ast }}={k_{n,\ell }^{\ast }}(r)$ satisfies the relation where ${a^{\gets }}$ is the inverse function of a. Following the lines in [6] it can be proven that Substituting (14)–(15) into (12) we get as $n\to \infty $. The next step is to minimize the right-hand side of (16) with respect to r, or equivalently, to minimize the product with respect to r. Using Wolfram Mathematica 10.4 the minimization of (17) is performed numerically. If product (17) attains its minimum at a point $r={r_{\ell }^{\ast }}$, then ${r_{\ell }^{\ast }}$ is called the optimal choice of the tuning parameter r. The graphs of the optimal choices ${r_{\ell }^{\ast }}={r_{\ell }^{\ast }}(\rho )$, $\ell =1,2,3$ are provided in Figures 1 and 2.

The graphs of ${r_{\ell }^{\ast }}(\rho )$, $\ell =1,2,3$ are plotted using a set $\{-i/100,\hspace{3.33333pt}1\le i\le 500\}$ of values of ρ. Consider ${r_{\ell }^{\ast }}(\rho )$, $\ell =1,2,3$ as functions of ρ on $[-i/100,-1/100]$, $1\lt i\le 500$. Then the difference ${r_{\ell }^{\ast }}(-1/100)-{r_{\ell }^{\ast }}(-i/100)$ is the width of the range of ${r_{\ell }^{\ast }}(\rho )$ on $[-i/100,-1/100]$. For example, taking $i=100$ we find that the widths of the range of ${r_{1}^{\ast }}(\rho )$, ${r_{2}^{\ast }}(\rho )$ and ${r_{3}^{\ast }}(\rho )$ are 0.38, 0.73 and 0.58, respectively. Calculating the widths of the range of ${r_{1}^{\ast }}(\rho )$, ${r_{2}^{\ast }}(\rho )$ and ${r_{3}^{\ast }}(\rho )$ for values $i\gt 100$ allows us to conclude that the width of the range of ${r_{1}^{\ast }}(\rho )$ is smaller than the widths of the range of ${r_{2}^{\ast }}(\rho )$ and ${r_{3}^{\ast }}(\rho )$. Thus, ${r_{1}^{\ast }}(\rho )$ is less sensitive to the estimation of the parameter ρ than the other two optimal choices.

Now we can compare the estimator ${\hat{\gamma }_{n}^{(1)}}({k_{n,1}^{\ast }}({r_{1}^{\ast }}),{r_{1}^{\ast }})$ with the estimators Following [6], we write for the limit of the ratio of the minimal mean squared errors of ${\hat{\gamma }_{n}^{(\ell )}}({k_{n,\ell }^{\ast }}({r_{\ell }^{\ast }}),{r_{\ell }^{\ast }})$ and ${\hat{\gamma }_{n}^{(1)}}({k_{n,1}^{\ast }}({r_{1}^{\ast }}),{r_{1}^{\ast }})$. Using (16) we get that where We say that the estimator ${\hat{\gamma }_{n}^{(1)}}({k_{n,1}^{\ast }}({r_{1}^{\ast }}),{r_{1}^{\ast }})$ outperforms ${\hat{\gamma }_{n}^{(\ell )}}({k_{n,\ell }^{\ast }}({r_{\ell }^{\ast }}),{r_{\ell }^{\ast }})$ on a half-line $\{(\rho ,\gamma ):\rho ={\rho _{0}},\gamma \gt 0\}$ if ${\phi _{\ell ,1}}({\rho _{0}})\gt 1$.

Graphs of the functions ${\phi _{2,1}}(\rho )$, $-5\le \rho \lt 0$ and ${\phi _{3,1}}(\rho )$, $-5\le \rho \lt 0$ are presented in Fig. 3. Whence one can deduce that the estimator ${\hat{\gamma }_{n}^{(1)}}({k_{n,1}^{\ast }}({r_{1}^{\ast }}),{r_{1}^{\ast }})$ outperforms the estimators ${\hat{\gamma }_{n}^{(\ell )}}({k_{n,\ell }^{\ast }}({r_{\ell }^{\ast }}),{r_{\ell }^{\ast }})$, $\ell =2,3$ in the area $\{(\gamma ,\rho ):\hspace{3.33333pt}\gamma \gt 0,-5\le \rho \lt 0\}$.

We end this Section with a comparison of the estimators Theorem 1 in [3] states that for any $\rho \lt 0$ and ${r_{2}}\gt 1$ there is a value ${\bar{r}_{1}}={\bar{r}_{1}}(\rho )$ such that $\nu (\rho ,{\bar{r}_{1}}(\rho ),{r_{2}})=0$, where $\nu (\rho ,{r_{1}},{r_{2}})$ is the same as in Theorem 1. As for the case $0\lt {r_{2}}\le 1$, we add the following statement: $\nu (\rho ,{r_{1}},{r_{2}})\ne 0$ for any $\rho \lt 0$ and $({r_{1}},{r_{2}})\in [1,\infty )\times (0,1]$, see Lemma 1 in Section 6. With the goal of eliminating unbiased estimators from our consideration, we will compare estimators (9) with ${\hat{\gamma }_{n}}(k,{r_{1}},{r_{2}})$, ${r_{1}}\ge 1$, $0\lt {r_{2}}\le 1$. In fact, this comparison reduces to the comparison of estimators ${\hat{\gamma }_{n}^{(1)}}({\tilde{K}_{n}}({\tilde{R}_{1}}(\rho ),{\tilde{R}_{2}}(\rho )),{\tilde{R}_{1}}(\rho ),{\tilde{R}_{2}}(\rho ))$ and ${\hat{\gamma }_{n}}({\tilde{k}_{n}}({\tilde{r}_{1}}(\rho ),{\tilde{r}_{2}}(\rho )),{\tilde{r}_{1}}(\rho ),{\tilde{r}_{2}}(\rho ))$, where are optimal choices for estimators ${\hat{\gamma }_{n}^{(1)}}(k,{r_{1}},{r_{2}})$ and ${\hat{\gamma }_{n}}(k,{r_{1}},{r_{2}})$, respectively.

One can verify that the limit of the ratio of the minimal mean squared errors of equals Numerically we obtain that the estimator ${\hat{\gamma }_{n}^{(1)}}({\tilde{K}_{n}}({\tilde{R}_{1}}(\rho ),{\tilde{R}_{2}}(\rho )),{\tilde{R}_{1}}(\rho ),{\tilde{R}_{2}}(\rho ))$ dominates the estimator ${\hat{\gamma }_{n}}({\tilde{k}_{n}}({\tilde{r}_{1}}(\rho ),{\tilde{r}_{2}}(\rho )),{\tilde{r}_{1}}(\rho ),{\tilde{r}_{2}}(\rho ))$ in the area $\{(\gamma ,\rho ):\gamma \gt 0,-5\le \rho \lt 0\}$, see Fig. 4.

Let ${\hat{\gamma }_{n}^{(\ell )}}({k_{n,\ell }^{\ast }}({r_{\ell }^{\ast }}),{r_{\ell }^{\ast }})$, $\ell =1,2,3$ be the estimators discussed in Section 2. In this section, we use numerical simulations to verify the theoretical result provided in Fig. 3. For this purpose, the i.i.d. samples ${X_{1}},\dots ,{X_{n}}$ of sizes $n=1000$ were simulated $N=500$ times from Burr distribution with several values of positive extreme value index γ and second order parameter $\rho \lt 0$.

We recall that the Burr (type XII) distribution has a distribution function $F(x)=1-{\left(1+{x^{-\rho /\gamma }}\right)^{1/\rho }}$, $x\ge 0$, while the appropriate quantile type function has the form $U(t)={t^{\gamma }}{\left(1-{t^{\rho }}\right)^{-\gamma /\rho }}$, $t\gt 1$.

The Burr distribution belongs to the Hall’s class of Pareto-type distributions ([16, 17]), i.e. its quantile type function $U(t)$ satisfies the relation with $(C,\beta )=(1,1)$.

It is well-known that under assumption (18) a condition (3) is satisfied with $A(t)=\beta \gamma {t^{\rho }}$. Now, using (13) one can find that By substituting (19) into (14) we get Put To estimate the second order parameter ρ we apply the estimator This estimator was introduced in [11]. To decide which value (0 or 1) of the parameter τ to take in the above mentioned estimator, we implemented the algorithm provided in [15]. To estimate the parameter β we use the estimator where ${V_{i}}=i\left(\ln ({X_{n-i+1,n}})-\ln ({X_{n-i}})\right)$, $1\le i\le \kappa $, which was introduced in [14]. Following the recommendations in [2], for both estimators we used $\kappa =[{n^{0.995}}]$, where $[\cdot ]$ denotes the integer part.

Replacing β and ρ by estimators ${\hat{\beta }_{n}}(\kappa )$ and ${\hat{\rho }_{n}}(\kappa ,\tau )$ in (20) we obtain empirical values of ${k_{n,\ell }^{\ast }}(r)$.

To calculate the estimators ${\hat{\gamma }_{n}^{(\ell )}}({k_{n,\ell }^{\ast }}({r_{\ell }^{\ast }}),{r_{\ell }^{\ast }})$, $\ell =1,2,3$ we use the following algorithm:

1. Estimate the parameters β and ρ using the estimators ${\hat{\beta }_{n}}(\kappa )$ and ${\hat{\rho }_{n}}(\kappa ,\tau )$, respectively.

2. Find numerically ${r_{\ell }^{\ast }}=\mathrm{argmin}\left\{r\gt 0:\hspace{3.33333pt}{\lambda _{\ell }^{2}}({\hat{\rho }_{n}}(\kappa ,\tau ),r){\left({\varsigma _{\ell }^{2}}(r)\right)^{-2{\hat{\rho }_{n}}(\kappa ,\tau )}}\right\}$ for $\ell =1,2$ and ${r_{3}^{\ast }}=\mathrm{argmin}\left\{r\gt 1:\hspace{3.33333pt}{\lambda _{3}^{2}}({\hat{\rho }_{n}}(\kappa ,\tau ),r){\left({\varsigma _{3}^{2}}(r)\right)^{-2{\hat{\rho }_{n}}(\kappa ,\tau )}}\right\}$.

3. Compute ${\hat{k}_{n,\ell }^{\ast }}({r_{\ell }^{\ast }})$ and then find estimate ${\hat{\gamma }_{n}^{(\ell )}}({k_{n,\ell }^{\ast }}({r_{\ell }^{\ast }}),{r_{\ell }^{\ast }})$.

The results of simulations are summarized in Fig. 5. For the Burr distribution, we took parameters ρ and γ from intervals $[-4,0)$ and $(0,2]$, respectively. We divide the rectangle $[-4,0)\times (0,2]$ into squares Taking the true values of the parameters ρ and γ as coordinates of the center of each square ${V_{i,j}}$, we performed Monte-Carlo simulations. Let $EMS{E_{\ell }}(\rho ,\gamma )$ denote the empirical MSE of the estimator ${\hat{\gamma }_{n}^{(\ell )}}({k_{n,\ell }^{\ast }}({r_{\ell }^{\ast }}),{r_{\ell }^{\ast }})$ when observations are simulated from the Burr distribution with true parameters ρ and γ. The square ${V_{i,j}}$ is colored in black, gray and white if respectively.

The graphical result in Fig. 5 indicates that additional simulation is needed for the case $\rho =-1$. We consider a model of i.i.d. observations from the Fréchet distribution with d.f. $F(x)=\exp \left\{-{x^{-1/\gamma }}\right\}$, $x\gt 0$. Also, we consider a stable model, i.e., when observations are absolute values of i.i.d. stable random variables with characteristic function $\varphi (t)=\exp \left\{-|t{|^{1/\gamma }}\right\}$. The results of these simulations are presented in Table 1 and Table 2, respectively. The best result in each column is presented in bold.

The findings of our Monte-Carlo simulations are the following:

Here we deal with daily exchange rates of U.S. Dollar expressed in Chinese Yuans (CHY) and South Korea Wons (KRW). We choose the samples of width $n=1248$ for a period of almost five years, from August 24, 2020, to August 22, 2025. These daily data are taken from Federal Reserve Bank of St. Louis (https://fred.stlouisfed.org). We analyze the absolute values of the logarithmic returns where $E{R_{t}^{(1)}}$ denotes the USD/CHY rate, while $E{R_{t}^{(2)}}$ – USD/KRW rate at a time t. Zero and small log-returns are deleted by using a rule ${R_{t}^{(i)}}\lt 0.0001$. After deleting time series ${R_{t}^{(1)}}$ and ${R_{t}^{(2)}}$ contain ${n_{1}}=1175$ and ${n_{2}}=1234$ observations, see Fig. 6.

We applied the algorithm (with $\ell =1,2,3$) provided in Section 3 to calculate the estimate of the parameter γ. The estimates of $(\rho ,\beta )$ are $(-0.712,1.040)$ and $(-0.697,1.045)$ for time series ${R_{t}^{(1)}}$, $1\le t\le 1175$ and ${R_{t}^{(2)}}$, $1\le t\le 1234$, respectively. Regarding the estimates of the parameter γ, they are collected in Table 3.

It is worth noting that one of the stylized econometric facts states that in most cases the absolute values of log-returns of economic and financial real data exhibit heavy tail phenomena and their distribution satisfies (1) with extreme value index $1/4\le \gamma \le 1/2$. The estimates of γ presented in Table 3 allow to conclude that the effect of heavy tails is observed in time series ${R_{t}^{(1)}}$, $1\le t\le 1175$ and ${R_{t}^{(2)}}$, $1\le t\le 1234$. In addition, all estimates do not contradict the mentioned stylized fact.

We introduced the new family of estimators (parameterized by two tuning parameters) for a positive EVI. This family is quite rich: it generalizes several classical estimators and parameterized families of estimators, previously presented in the statistical literature. We proved the asymptotic normality of the newly introduced estimators. This allows us to compare estimators proposed in this paper with other estimators for a positive EVI theoretically, in the sense of asymptotic MMSE. It is shown that the family of newly introduced estimators ${\hat{\gamma }_{n}^{(1)}}(k,r)$ (at the optimal levels of parameters) dominates the families of estimators proposed in [13] and [22].

For practical purposes, we discussed a subfamily of newly introduced estimators. This subfamily is parameterized in one tuning parameter, which, as a function of a second order parameter ρ, has quite small width of range. Nowadays, estimation of the parameter ρ is still at a poor level. Thus noting that the tuning parameter is less sensitive to the estimation of parameter ρ is a quite significant advantage against other estimators for a positive EVI. The performance of the considered subfamily of estimators has been exhibited in a small-scale Monte-Carlo study and for two real data sets.

To prove Theorem 2 we apply a result from [13], adopted for our purposes. A generalization of the following Theorem can be found in [23].

We are now ready to prove Theorem 2.