Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEmpirical regression quantile process with possible application to risk analysis
105
0
0.0
(
0
)
Added by
Martin Schindler
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The processes of the averaged regression quantiles and of their modifications provide useful tools in the regression models when the covariates are not fully under our control. As an application we mention the probabilistic risk assessment in the situation when the return depends on some exogenous variables. The processes enable to evaluate the expected $alpha$-shortfall ($0leqalphaleq 1$) and other measures of the risk, recently generally accepted in the financial literature, but also help to measure the risk in environment analysis and elsewhere.
rate research
Read More
Various events in the nature, economics and in other areas force us to combine the study of extremes with regression and other methods. A useful tool for reducing the role of nuisance regression, while we are interested in the shape or tails of the basic distribution, is provided by the averaged regression quantile and namely by the average extreme regression quantile. Both are weighted means of regression quantile components, with weights depending on the regressors. Our primary interest is the averaged extreme regression quantile (AERQ), its structure, qualities and its applications, e.g. in investigation of a conditional loss given a value exogenous economic and market variables. AERQ has several interesting equivalent forms: While it is originally defined as an optimal solution of a specific linear programming problem, hence is a weighted mean of responses corresponding to the optimal base of the pertaining linear program, we give another equivalent form as a maximum residual of responses from a specific R-estimator of the slope components of regression parameter. The latter form shows that while AERQ equals to the maximum of some residuals of the responses, it has minimal possible perturbation by the regressors. Notice that these finite-sample results are true even for non-identically distributed model errors, e.g. under heteroscedasticity. Moreover, the representations are formally true even when the errors are dependent - this all provokes a question of the right interpretation and of other possible applications.
We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.
Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.
We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requires the calculation of moments of Nadaraya-Watson quantile regression estimators. This calculation is done by application of higher order Edgeworth expansions.
In this work, we investigate Gaussian process regression used to recover a function based on noisy observations. We derive upper and lower error bounds for Gaussian process regression with possibly misspecified correlation functions. The optimal convergence rate can be attained even if the smoothness of the imposed correlation function exceeds that of the true correlation function and the sampling scheme is quasi-uniform. As byproducts, we also obtain convergence rates of kernel ridge regression with misspecified kernel function, where the underlying truth is a deterministic function. The convergence rates of Gaussian process regression and kernel ridge regression are closely connected, which is aligned with the relationship between sample paths of Gaussian process and the corresponding reproducing kernel Hilbert space.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
