Bias Reduction in Kernel Tail Index Estimation for Randomly Truncated Pareto-Type Data

Abstract

A bias reduction to a kernel estimator of the tail index of randomly right-truncated Pareto-type distributions is made. The asymptotic normality of the derived estimator is established by assuming the second-order condition of regular variation. A simulation study is carried out to evaluate the finite sample behavior of the proposed estimator and compare it to those with non-reduced bias. An application to a real dataset of lifetimes of automobile brake pads is done.

Appendices

Appendix: A

A.1: Instrumental Result

Proposition A.1.

Assume that $\overline {\mathbf {F}}\in \mathcal {R}\mathcal {V}_{2}\left (-1/\gamma _{1};\tau _{1},A_{\mathbf {F}}\right )$ and K satisfies assumptions $\left [ A1\right ] -\left [ A4\right ] ,$ then: $\left (i\right )$ $E_{t}\left (\beta \right ) =\eta _{2}+\eta _{3}A_{\mathbf {F}}\left (t\right ) \left (1+o\left (1\right ) \right ) ,$ for β > 0, and for a given twice-differentiable function f : $\left (ii\right )$ $f\left (L_{t,K}\right ) =f\left (\gamma _{1}\right ) +f^{\prime }\left (\gamma _{1}\right ) \eta _{1}A_{\mathbf {F}}\left (t\right ) \left (1+o\left (1\right ) \right ) ,$ where η_i = η_i,K, i = 1, 2, 3 are stated in (1.7) and Section 1.1 respectively. Moreover

$$ \left( iii\right) \text{ }A_{\mathbf{F}}\left( t\right) =\frac {E_{t}\left( \beta\right) -f_{\ast}\left( L_{t}\right) }{\eta_{3}-f_{\ast }^{\prime}\left( \gamma_{1}\right) \eta_{1}}\left( 1+o\left( 1\right) \right) ,\text{ as }t\rightarrow\infty, $$

where f_∗ is as in (1.12), $E_{t}\left (\beta \right ) :=E_{t,K}\left (\beta \right )$ and L_t := L_t,K.

Proof 1.

Recall that (1.10), and let us decompose $E_{t}\left (\beta \right )$ into the sum of

$$ E_{t}^{\left( 1\right) }\left( \beta\right) :=1-\beta{\int}_{1}^{\infty }x^{-\beta-1}K^{\ast}\left( x^{-1/\gamma_{1}}\right) dx $$

and

$$ E_{t}^{\left( 2\right) }\left( \beta\right) :=-\beta{\int}_{1}^{\infty }x^{-\beta-1}\left( K^{\ast}\left( \frac{\overline{\mathbf{F}}\left( tx\right) }{\overline{\mathbf{F}}\left( t\right) }\right) -K^{\ast}\left( x^{-1/\gamma_{1}}\right) \right) dx. $$

Using the change of variables $s=x^{-1/\gamma _{1}}$ we readily show that $E_{t}^{\left (1\right ) }\left (\beta \right ) =\eta _{2},$ for β > 0. Applying Taylor’s expansion to K^∗, we may rewrite $E_{t}^{\left (2\right ) }\left (\beta \right )$ into the sum of

$$ E_{t}^{\left( 2,1\right) }\left( \beta\right) :=-\beta{\int}_{1}^{\infty }x^{-\beta-1}\left( \frac{\overline{\mathbf{F}}\left( tx\right) } {\overline{\mathbf{F}}\left( t\right) }-x^{-1/\gamma_{1}}\right) K\left( x^{-1/\gamma_{1}}\right) dx $$

and

$$ E_{t}^{\left( 2,2\right) }\left( \beta\right) :=-\frac{\beta}{2}{\int}_{1}^{\infty}x^{-\beta-1}\left( \frac{\overline{\mathbf{F}}\left( tx\right) }{\overline{\mathbf{F}}\left( t\right) }-x^{-1/\gamma_{1}}\right)^{2}K^{\prime}\left( \xi_{t}\left( x\right) \right) dx, $$

where $\xi _{t}\left (x\right )$ is between $\overline {\mathbf {F}}\left (tx\right ) /\overline {\mathbf {F}}\left (t\right )$ and $x^{-1/\gamma _{1}}.$ Since HCode $\overline {\mathbf {F}}\in \mathcal {R}\mathcal {V}_{2}(-1/\gamma _{1};\tau _{1},$ A_F), then making use of Potters inequalities corresponding to the second-order condition of df F, we write

$$ \left\vert \frac{\frac{\overline{\mathbf{F}}\left( tx\right) } {\overline{\mathbf{F}}\left( t\right) }-x^{-1/\gamma_{1}}}{A_{\mathbf{F} }\left( t\right) }-x^{-1/\gamma_{1}}\frac{x^{\tau_{1}/\gamma_{1}}-1} {\tau_{1}\gamma_{1}}\right\vert \leq\epsilon x^{-1/\gamma_{1}+\tau_{1} /\gamma_{1}+\epsilon}, $$

(A.1)

for any 𝜖 > 0, for all x > 1 and for all large t, see for instance Theorem 2.3.9 in de Haan and Ferreira (2006). Using this inequality, we end up with

$$ E_{t}^{\left( 2,1\right) }\left( \beta\right) =-\left( 1+o\left( 1\right) \right) A_{\mathbf{F}}\left( t\right) \beta{\int}_{1}^{\infty }x^{-\beta-1/\gamma_{1}-1}\frac{x^{\tau_{1}/\gamma_{1}}-1}{\tau_{1}\gamma_{1} }K\left( x^{-1/\gamma_{1}}\right) dx. $$

Once again, using change of variables $s=x^{-1/\gamma _{1}},$ we show easily that the previous integral equals to η₃. In view of assumption $\left [ A4\right ] ,$ the function $K^{\prime }$ is bounded, it follows that

$$ E_{t}^{\left( 2,2\right) }\left( \beta\right) =O\left( 1\right) {\int}_{1}^{\infty}x^{-\beta-1}\left\vert \frac{\overline{\mathbf{F}}\left( tx\right) }{\overline{\mathbf{F}}\left( t\right) }-x^{-1/\gamma_{1} }\right\vert^{2}dx. $$

Let 𝜖 > 0 so small such that − 1/γ₁ + 𝜖 < 0. The Potters inequalities corresponding to the first order condition of regularly varying functions says:

$$ \left\vert \frac{\overline{\mathbf{F}}\left( tx\right) }{\overline {\mathbf{F}}\left( t\right) }-x^{-1/\gamma_{1}}\right\vert \leq\epsilon x^{-1/\gamma_{1}+\epsilon}<\epsilon, $$

for all x ≥ 1 and for all large t, see for instance Proposition B.1.9, assertion 5 in de Haan and Ferreira (2006). It follows that

$$ E_{t}^{\left( 2,2\right) }\left( \beta\right) =o\left( 1\right) {\int}_{1}^{\infty}x^{-\beta-1}\left\vert \frac{\overline{\mathbf{F}}\left( tx\right) }{\overline{\mathbf{F}}\left( t\right) }-x^{-1/\gamma_{1} }\right\vert dx. $$

Let us write

$$ E_{t}^{\left( 2,2\right) }\left( \beta\right) =o\left( A_{\mathbf{F} }\left( t\right) \right) {\int}_{1}^{\infty}x^{-\beta-1}\left\vert \frac{\frac{\overline{\mathbf{F}}\left( tx\right) }{\overline{\mathbf{F} }\left( t\right) }-x^{-1/\gamma_{1}}}{A_{\mathbf{F}}\left( t\right) }\right\vert dx. $$

Subtracting $x^{-1/\gamma _{1}}\frac {x^{\tau _{1}/\gamma _{1}}-1}{\tau _{1} \gamma _{1}},$ inside the sign of the previous absolute value, and adding the same quantity, then applying inequality (A.1), we may readily show that $E_{t}^{\left (2,2\right ) }\left (\beta \right ) =o\left (A_{\mathbf {F}}\left (t\right ) \right ) ,$ as $t\rightarrow \infty ,$ that we omit further details. Thereby $E_{t}^{\left (2,1\right ) }\left (\beta \right ) =\left (1+o\left (1\right ) \right ) \eta _{3}$ leading to the result of assertion $\left (ii\right ) .$ To show the second assertion $\left (ii\right ) ,$ is suffices to use Taylor’s expansion to function f and similar arguments as used to first assertion $\left (i\right ) ,$ that we omit details. Let us now focus on the third one $\left (iii\right ) .$ Multiplying, the equation of assertion $\left (i\right )$ by $f\left (\gamma _{1}\right ) ,$ we write

$$ f\left( \gamma_{1}\right) E_{t}\left( \beta\right) =f\left( \gamma_{1}\right) \eta_{2}+f\left( \gamma_{1}\right) \eta_{3}A_{\mathbf{F} }\left( t\right) \left( 1+o\left( 1\right) \right) , $$

(A.2)

and from assertion $\left (ii\right )$ we have $f\left (\gamma _{1}\right ) =f\left (L_{t}\right ) -\eta _{1}f^{\prime }\left (\gamma _{1}\right ) A_{\mathbf {F}}\left (t\right ) \left (1+o\left (1\right ) \right ) .$ Substituting the previous expression into the first term of the right-side of equation (A.2), yields

$$ f\left( \gamma_{1}\right) E_{t}\left( \beta\right) =\left( f\left( L_{t}\right) -\eta_{1}f^{\prime}\left( \gamma_{1}\right) A_{\mathbf{F} }\left( t\right) \right) \eta_{2}\left( 1+o\left( 1\right) \right) +f\left( \gamma_{1}\right) \eta_{3}A_{\mathbf{F}}\left( t\right) \left( 1+o\left( 1\right) \right) , $$

which gives

$$ A_{\mathbf{F}}\left( t\right) =\frac{E_{t}\left( \beta\right) -\frac{f\left( L_{t}\right) }{f\left( \gamma_{1}\right) }\eta_{2}} {\eta_{3}-\frac{f^{\prime}\left( \gamma_{1}\right) }{f\left( \gamma_{1}\right) }\eta_{2}\eta_{1}}\left( 1+o\left( 1\right) \right) . $$

In particular, for f = f_∗, we have $f_{\ast }\left (\gamma _{1}\right ) =\eta _{2},$ it follows that

$$ A_{\mathbf{F}}\left( t\right) =\frac{E_{t}\left( \beta\right) -f_{\ast }\left( L_{t}\right) }{\eta_{3}-f_{\ast}^{\prime}\left( \gamma_{1}\right) \eta_{1}}\left( 1+o\left( 1\right) \right) , $$

which gives the third assertion $\left (iii\right ) .$ The proof of Proposition A.1 is now completed. □

A.2: High Quantile Estimation

An extreme quantile of df F is a value q_ν defined in terms of the generalized inverse by $q_{\nu }:=U_{\mathbf {F}}\left (1/\nu \right )$ for v ↓ 0. In other words, it is an X-value which is sufficiently large so that the probability of exceeding it is very small. Also known as value-at-risk (VaR), this quantity is largely used, as a risk measure, in several fields such as in finance, insurance, hydrology and reliability. For asymptotic needs, we suppose that v is a function of the observed sample size n, denoted by v = v_n, and assumed to be much smaller than 1/n. The estimation of high quantiles of heavy-tailed distributions, in the case of complete data, has been extensively studied in the literature (de Haan and Ferreira, 2006, see, for instance,). The well-known Weissman estimator (Weissman, 1978) of high quantile q_ν adapted to our new tail index estimator $\widehat {\gamma }_{1,K}^{\ast }$ is given by

$$ \widehat{q}_{\nu}:=X_{n-k:n}\left( \frac{\nu}{\overline{\mathbf{F}}_{n}\left( X_{n-k:n}\right) }\right)^{-\widehat{\gamma}_{1,K}^{\ast}}, $$

where F_n is Woodroofe’s nonparametric estimator of df F, given in Section 1.

Appendix: B

Figure 2

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Burr distribution truncated by another Burr distribution, with β = 1 and γ₁ = 0.6 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 3

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Burr distribution truncated by another Burr distribution, with β = 0.5 and γ₁ = 0.6 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 4

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Burr distribution truncated by another Burr distribution, with β = 2 and γ₁ = 0.6 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 5

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Burr distribution truncated by another Burr distribution, with β = 1 and γ₁ = 0.8 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 6

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Burr distribution truncated by another Burr distribution, with β = 0.5 and γ₁ = 0.8 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 7

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Burr distribution truncated by another Burr distribution, with β = 2 and γ₁ = 0.8 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 8

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Fréchet distribution truncated by another Fréchet distribution, with β = 1 and γ₁ = 0.6 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 9

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Fréchet distribution truncated by another Fréchet distribution, with β = 0.5 and γ₁ = 0.6 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 10

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Fréchet distribution truncated by another Fréchet distribution, with β = 2 and γ₁ = 0.6 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 11

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Fréchet distribution truncated by another Fréchet distribution, with β = 1 and γ₁ = 0.8 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 12

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Fréchet distribution truncated by another Fréchet distribution, with β = 0.5 and γ₁ = 0.8 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 13

Absolute biases (left panel) and RMSE’s (right panel) of $\widehat {\gamma }_{1,K}^{\ast },\widehat {\gamma }_{1,K},\widehat {\gamma }_{1},\widetilde {\gamma }_{1}^{\ast }$ and $\overline {\gamma }_{1,K}^{\ast }$ for a Fréchet distribution truncated by a nother Fréchet distribution, with β = 2 and γ₁ = 0.8 under the following cases: p = 0.55 (top), p = 0.7 (middle) and p = 0.9 (bottom). The simulation is based on 2000 replicates of size 500

Figure 14

Box-plots corresponding to estimators $\widehat {\gamma }_{1,K}^{\ast },$ $\overline {\gamma }_{1,K}^{\ast },$ $\widetilde {\gamma }_{1}^{\ast },$ $\widehat {\gamma }_{1,K}$ and $\widehat {\gamma }_{1}$ for a Fréchet distribution truncated by another Fréchet distribution, with β = 1, γ₁ = 0.6, p = 0.55 and p = 0.9 based on 2000 replicates of size N = 500

Figure 15

Box-plots corresponding to estimators $\widehat {\gamma }_{1,K}^{\ast },$ $\overline {\gamma }_{1,K}^{\ast },$ $\widetilde {\gamma }_{1}^{\ast },$ $\widehat {\gamma }_{1,K}$ and $\widehat {\gamma }_{1}$ for a Fréchet distribution truncated by another Fréchet distribution, with β = 1, γ₁ = 0.8, p = 0.55 and p = 0.9 based on 2000 replicates of size N = 500

Figure 16

Box-plots corresponding to estimators $\widehat {\gamma }_{1,K}^{\ast },$ $\overline {\gamma }_{1,K}^{\ast },$ $\widetilde {\gamma }_{1}^{\ast },$ $\widehat {\gamma }_{1,K}$ and $\widehat {\gamma }_{1}$ for a Burr distribution truncated by another Burr distribution, with β = 1, γ₁ = 0.6, p = 0.55 and p = 0.9 based on 2000 replicates of size N = 500

Figure 17

Box-plots corresponding to estimators $\widehat {\gamma }_{1,K}^{\ast },$ $\overline {\gamma }_{1,K}^{\ast },$ $\widetilde {\gamma }_{1}^{\ast },$ $\widehat {\gamma }_{1,K}$ and $\widehat {\gamma }_{1}$ for a Burr distribution truncated by another Burr distribution, with β = 1, γ₁ = 0.8, p = 0.55 and p = 0.9 based on 2000 replicates of size N = 500