Semi-nonparametric Estimation of Operational Risk Capital with Extreme Loss Events

Bank operational risk capital modeling using the Basel II advanced measurement approach (AMA) often lead to a counter-intuitive capital estimate of value at risk at 99.9% due to extreme loss events. To address this issue, a flexible semi-nonparametric (SNP) model is introduced using the change of variables technique to enrich the family of distributions to handle extreme loss events. The SNP models are proved to have the same maximum domain of attraction (MDA) as the parametric kernels, and it follows that the SNP models are consistent with the extreme value theory peaks over threshold method but with different shape and scale parameters from the kernels. By using the simulation dataset generated from a mixture of distributions with both light and heavy tails, the SNP models in the Frechet and Gumbel MDAs are shown to fit the tail dataset satisfactorily through increasing the number of model parameters. The SNP model quantile estimates at 99.9 percent are not overly sensitive towards the body-tail threshold change, which is in sharp contrast to the parametric models. When applied to a bank operational risk dataset with three Basel event types, the SNP model provides a significant improvement in the goodness of fit to the two event types with heavy tails, yielding an intuitive capital estimate that is in the same magnitude as the event type total loss. Since the third event type does not have a heavy tail, the parametric model yields an intuitive capital estimate, and the SNP model cannot provide additional improvement. This research suggests that the SNP model may enable banks to continue with the AMA or its partial use to obtain an intuitive operational risk capital estimate when the simple non-model based Basic Indicator Approach or Standardized Approach are not suitable per Basel Committee Banking Supervision OPE10 (2019).Comment: There are 32 pages, including tables, figures, appendix and reference. The research was presented at the MATLAB Annual Computational Finance Conference, September 27-30, 202

Efficient Semiparametric Estimation of Censored and Truncated Regressions via a Smoothed Self-Consistency Equation

An asymptotically efficient likelihood-based semiparametric estimator is derived for the censored regression (tobit) model, based on a new approach for estimating the density function of the residuals in a partially observed regression. Smoothing the self-consistency equation for the nonparametric maximum likelihood estimator of the distribution of the residuals yields an integral equation, which in some cases can be solved explicitly. The resulting estimated density is smooth enough to be used in a practical implementation of the profile likelihood estimator, but is sufficiently close to the nonparametric maximum likelihood estimator to allow estimation of the semiparametric efficient score. The parameter estimates obtained by solving the estimated score equations are then asymptotically efficient. A summary of analogous results for truncated regression is also given. Copyright The Econometric Society 2004.

Efficiency Bounds for Distribution-free Estimators of the Binary.

Lower bounds are derived for the asymptotic variances of regular distribution-free (or semiparametric) estimators of the parameters of the binary-choice model and the censored-regression (Tobit) model. A semiparametric estimator is one that does not require any assumption about the distribution of the stochastic error term in the model, apart from regularity conditions. Comparison of the bounds with the corresponding asymptotic Cramer-Rao bounds for the classical parametric problem shows the loss of information due to lack of a priori knowledge of the functional form of the error distribution. Copyright 1987 by The Econometric Society.

Environmental Quality Preference and Benefit Estimation in Multinomial Probit Models: A Simulation Approach

Simulated maximum likelihood is used to estimate a random parameter multinomial probit model of destination choice for recreational fishing trips, formulated to accommodate varying tastes and varying perceptions of environmental quality across individuals. The restricted likelihood ratio test strongly rejects the independent probit model, which is similar to the independent logit model in both the parameter and benefit estimates. Furthermore, both the Krinsky-Robb and bootstrapping procedures suggest that the benefit (standard deviation) of an environmental policy is found to be markedly lower (higher) when heterogeneous preferences are taken into account. Copyright 1998, Oxford University Press.

Estimation of AUC or Partial AUC Under Test-Result-Dependent Sampling

The area under the ROC curve (AUC) and partial area under the ROC curve (pAUC) are summary measures used to assess the accuracy of a biomarker in discriminating true disease status. The standard sampling approach used in biomarker validation studies is often inefficient and costly, especially when ascertaining the true disease status is costly and invasive. To improve efficiency and reduce the cost of biomarker validation studies, we consider a test-result-dependent sampling (TDS) scheme, in which subject selection for determining the disease state is dependent on the result of a biomarker assay. We first estimate the test-result distribution using data arising from the TDS design. With the estimated empirical test-result distribution, we propose consistent nonparametric estimators for AUC and pAUC and establish the asymptotic properties of the proposed estimators. Simulation studies show that the proposed estimators have good finite sample properties and that the TDS design yields more efficient AUC and pAUC estimates than a simple random sampling (SRS) design. A data example based on an ongoing cancer clinical trial is provided to illustrate the TDS design and the proposed estimators. This work can find broad applications in design and analysis of biomarker validation studies