The estimation of conditional quantiles of a response variable is of prime interest in many statistical applications. Quantile regression (QR) is a powerful and popular method usedextensively to estimate conditional quantiles of a response Y in the presence of a covariate X. Moreover, QR can quantify the effect of covariates at different quantile levels. Whilemodeling a rare event, quantiles at high or low tails are of particular interest. In such cases, QR has inevitable shortfalls. Since we have fewer data at tails, QR estimates suffer fromhigh variability. The performance deteriorates further when the underlying distribution is heavy-tailed. Estimation of extreme quantiles is therefore challenging, especially when thedata comes from a heavy-tailed distribution.Extreme value theory (EVT) provides mathematical tools to analyze rare events. In practice, EVT can be used to assess the probability of more extreme events than any previouslyobserved occurrences. The scope of application of EVT includes financial risk assessment, extreme climate modeling, network anomaly detection, etc. Statistical modeling using EVTcan be carried out in two approaches: Block maxima and Peak over Threshold (POT). In the block maxima approach, Generalized extreme value (GEV) distribution is fitted to a seriesof maximums derived from the observations. In the POT approach, one fits Generalized Pareto (GP) distribution to observations exceeding a certain threshold.An important problem while using the POT approach is the choice of the threshold. Models based on EVT use asymptotic arguments to approximate the tail behavior. Hence,the choice of the threshold is crucial in order to fit the GP distribution to the data. POT approach is a popular tool for the estimation of extreme quantiles in heavy-tailed datawherein the excesses over a threshold are modeled as a function of the covariate X. However, the efficiency of POT is severely compromised when the threshold itself varies as a function of the covariate.This dissertation proposes an integrated approach for estimating extreme conditional quantiles from a heavy-tailed distribution. We begin with the case where the threshold doesnot vary as a function of the covariate X. Using the POT approach to model the scaled conditional excess, we propose an estimator for high conditional quantile. We establishlarge sample properties of our estimator in the context of the GP distribution. Through numerical investigations, we demonstrate the superiority of our method over QR at higherquantiles. However, this method is not adequate when the threshold itself varies as a function of the covariate X. In order to circumvent this issue, we propose an extension of ourapproach, which uses standard QR to extract information on the threshold and then model the residuals as a GP distribution with covariate dependent parameters. We establish theasymptotic properties of our method in the context of the GP distribution. Extending further, we thoroughly study the asymptotic performance of this approach for a wide class ofheavy-tailed distributions using numerical simulations. Using simulation studies, we compare our approach with existing methods used in estimation of high conditional quantiles forheavy tailed distributions. As an application, we have implemented our method for the task of precipitation downscaling with data obtained from the Vancouver International Airportweather station. We have also demonstrated how our method of covariate adaptive threshold selection can be implemented in practice.
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