Non-parametric quantile estimation

Transformed kernel estimation - R programming

Ramon Alemany, Catalina Bolancé & Montserrat Guillén

• Alemany, R., Bolancé, C. and Guillén, M. (2013) A nonparametric approach to calculating value-at-risk, Insurance: Mathematics and Economics, 52(2), 255-262.
• Alemany, R., Bolancé, C. and Guillén, M. (2014) Accounting for severity of risk when pricing insurance products. UB Riskcenter Working Papers Series 2014-05

DATA DESCRIPTION

Name	Content of operational risk loss data
Internal data set	75 observed loss amounts
External data set	700 observed loss amounts
Public data risk no. 1	1000 observed loss amounts for category no. 1
Public data risk no. 2	400 observed loss amounts for category no. 2

Download Internal data set here.	Download Public data risk no.1 here.
Download External data set here.	Download Public data risk no.2 here.

VALUE AT RISK WITH KERNEL AND TRANSFORMED KERNEL ESTIMATION

For a random sample of n independent and identically distributed observations x₁, x₂,..., x_n of a random variable X with pdf f_X, the value at risk can be calculated using classifcal kernel density estimator and transformed kernel estimation:

- Classical kernel cdf estimation

\begin{equation} \begin{array}{c} \ \ \ \ \widehat{F}_{X}(x)=\int_{-\infty }^{x} \widehat{f}_{X}(u)du =\int_{-\infty }^{x}\frac{1}{nb}\sum_{i=1}^{n}k\left( \frac{u-X_{i}}{b}\right)du \\ =\frac{1}{n}\sum_{i=1}^{n}\int_{-\infty }^{\frac{x-X_{i}}{b}}k\left( t\right) dt=\frac{1}{n}\sum_{i=1}^{n}K\left( \frac{x-X_{i}}{b} \right). \label{kdist} \end{array} \end{equation}
K is the kernel function and b is the bandwidth.
To estimate value at risk, the Newton-Raphson method is applied

- Transformed kernel cdf estimation

\begin{equation} \widehat{F}_{X}\left( x\right) =\widehat{F}_{T\left( X\right) }(T\left( x\right) )=\frac{1}{n}\sum_{i=1}^{n}K\left( \frac{T\left( x\right) -T\left( X_{i}\right) }{b}\right) \label{tkdist} \end{equation}
T(·) is the transformation, K is the kernel function and b is the bandwidth.
To estimate value at risk, the Newton-Raphson method is applied

• Download the R script here

• Download the SAS program here

• Download the results here.

REFERENCES

[1] Bolancé, C. (2010) Optimal Inverse Beta(3,3) Transformation in kernel density estimation, SORT Statistics and Operations Research Transaction, 34, 223-238.

[2] Bolancé, C., Guillén, M., Gustafsson J. and Nielsen, J.P. (2012) Quantitative Operational Risk Models Chapman & Hall/CRC.

[3] Bolancé, C., Guillén, M. and Nielsen, J.P. (2009) Transformation kernel estimation of insurance claim cost distribution, in Corazza, M. and Pizzi, C. (Eds), Mathematical and Statistical Methods for Actuarial Sciences and Finance, Springer, Roma, 223-231.

[4] Bolancé, C., Guillén, M. and Nielsen, J.P. (2008) Inverse Beta transformation in kernel density estimation. Statistics & Probability Letters, 78, 1757-1764.

[5] Bolancé, C., Guillén, M. and Nielsen, J.P. (2003) Kernel density estimation of actuarial loss functions, Insurance: Mathematics and Economics, 32, 19-36.

[6] Bolancé, C., Guillén, M. and Pitt, D. (2014) Non-parametric models for univariate claim severity distributions - an approach using R, UB Riskcenter Working Papers Series 2014-01.

[7] Buch-Larsen, T., Guillen, M., Nielsen, J.P. and Bolancé, C. (2005) Kernel density estimation for heavy-tailed distributions using the Champernowne transformation. Statistics, 39, 503-518.

[8] Clements, A.E., Hurn, A.S. and Lindsay, K.A. (2003) Möebius-like mappings and their use in kernel density estimation, Journal of the American Statistical Association, 98, 993-1000.