194,222 research outputs found
Bootstrap Prediction Bands for Functional Time Series
A bootstrap procedure for constructing pointwise or simultaneous prediction
intervals for a stationary functional time series is proposed. The procedure
exploits a general vector autoregressive representation of the time-reversed
series of Fourier coefficients appearing in the Karhunen-Lo\`{e}ve
representation of the functional process. It generates backwards-in-time,
functional replicates that adequately mimic the dependence structure of the
underlying process and have the same conditionally fixed curves at the end of
each functional pseudo-time series. The bootstrap prediction error distribution
is then calculated as the difference between the model-free,
bootstrap-generated future functional observations and the functional forecasts
obtained from the model used for prediction. This allows the estimated
prediction error distribution to account for not only the innovation and
estimation errors associated with prediction but also the possible errors from
model misspecification. We show the asymptotic validity of the bootstrap in
estimating the prediction error distribution of interest. Furthermore, the
bootstrap procedure allows for the construction of prediction bands that
achieve (asymptotically) the desired coverage. These prediction bands are based
on a consistent estimation of the distribution of the studentized prediction
error process. Through a simulation study and the analysis of two data sets, we
demonstrate the capabilities and the good finite-sample performance of the
proposed method
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Quantile-based methods for prediction, risk measurement and inference
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The focus of this thesis is on the employment of theoretical and practical quantile methods in addressing prediction, risk measurement and inference problems. From a prediction perspective, a problem of creating model-free prediction intervals for a future unobserved value of a random variable drawn from a sample distribution is considered. With the objective of reducing prediction coverage error, two common distribution transformation methods based on the normal and exponential distributions are presented and they are theoretically demonstrated to attain exact and error-free prediction intervals respectively.
The second problem studied is that of estimation of expected shortfall via kernel smoothing. The goal here is to introduce methods that will reduce the estimation bias of expected shortfall. To this end, several one-step bias correction expected shortfall estimators are presented and investigated via simulation studies and compared with one-step estimators.
The third problem is that of constructing simultaneous confidence bands for quantile regression functions when the predictor variables are constrained within a region is considered. In this context, a method is introduced that makes use of the asymmetric Laplace errors in conjunction with a simulation based algorithm to create confidence bands for quantile and interquantile regression functions. Furthermore, the simulation approach is extended to an ordinary least square framework to build simultaneous bands for quantiles functions of the classical regression model when the model errors are normally distributed and when this assumption is not fulfilled.
Finally, attention is directed towards the construction of prediction intervals for realised volatility exploiting an alternative volatility estimator based on the difference of two extreme quantiles. The proposed approach makes use of AR-GARCH procedure in order to model time series of intraday quantiles and forecast intraday returns predictive distribution. Moreover, two simple adaptations of an existing model are also presented
Post-selection Inference for Conformal Prediction: Trading off Coverage for Precision
Conformal inference has played a pivotal role in providing uncertainty
quantification for black-box ML prediction algorithms with finite sample
guarantees. Traditionally, conformal prediction inference requires a
data-independent specification of miscoverage level. In practical applications,
one might want to update the miscoverage level after computing the prediction
set. For example, in the context of binary classification, the analyst might
start with a prediction sets and see that most prediction sets contain
all outcome classes. Prediction sets with both classes being undesirable, the
analyst might desire to consider, say prediction set. Construction of
prediction sets that guarantee coverage with data-dependent miscoverage level
can be considered as a post-selection inference problem. In this work, we
develop uniform conformal inference with finite sample prediction guarantee
with arbitrary data-dependent miscoverage levels using distribution-free
confidence bands for distribution functions. This allows practitioners to trade
freely coverage probability for the quality of the prediction set by any
criterion of their choice (say size of prediction set) while maintaining the
finite sample guarantees similar to traditional conformal inference
Representing uncertainty by possibility distributions encoding confidence bands, tolerance and prediction intervals
For a given sample set, there are already different methods for building possibility distributions encoding the family of probability distributions that may have generated the sample set. Almost all the existing methods are based on parametric and distribution free confidence bands. In this work, we introduce some new possibility distributions which encode different kinds of uncertainties not treated before. Our possibility distributions encode statistical tolerance and prediction intervals (regions). We also propose a possibility distribution encoding the confidence band of the normal distribution which improves the existing one for all sample sizes. In this work we keep the idea of building possibility distributions based on intervals which are among the smallest intervals for small sample sizes. We also discuss the properties of the mentioned possibility distributions
Quantile-based methods for prediction, risk measurement and inference
The focus of this thesis is on the employment of theoretical and practical quantile methods in addressing prediction, risk measurement and inference problems. From a prediction perspective, a problem of creating model-free prediction intervals for a future unobserved value of a random variable drawn from a sample distribution is considered. With the objective of reducing prediction coverage error, two common distribution transformation methods based on the normal and exponential distributions are presented and they are theoretically demonstrated to attain exact and error-free prediction intervals respectively. The second problem studied is that of estimation of expected shortfall via kernel smoothing. The goal here is to introduce methods that will reduce the estimation bias of expected shortfall. To this end, several one-step bias correction expected shortfall estimators are presented and investigated via simulation studies and compared with one-step estimators. The third problem is that of constructing simultaneous confidence bands for quantile regression functions when the predictor variables are constrained within a region is considered. In this context, a method is introduced that makes use of the asymmetric Laplace errors in conjunction with a simulation based algorithm to create confidence bands for quantile and interquantile regression functions. Furthermore, the simulation approach is extended to an ordinary least square framework to build simultaneous bands for quantiles functions of the classical regression model when the model errors are normally distributed and when this assumption is not fulfilled. Finally, attention is directed towards the construction of prediction intervals for realised volatility exploiting an alternative volatility estimator based on the difference of two extreme quantiles. The proposed approach makes use of AR-GARCH procedure in order to model time series of intraday quantiles and forecast intraday returns predictive distribution. Moreover, two simple adaptations of an existing model are also presented.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Quantile-based methods for prediction, risk measurement and inference
The focus of this thesis is on the employment of theoretical and practical quantile methods in addressing prediction, risk measurement and inference problems. From a prediction perspective, a problem of creating model-free prediction intervals for a future unobserved value of a random variable drawn from a sample distribution is considered. With the objective of reducing prediction coverage error, two common distribution transformation methods based on the normal and exponential distributions are presented and they are theoretically demonstrated to attain exact and error-free prediction intervals respectively. The second problem studied is that of estimation of expected shortfall via kernel smoothing. The goal here is to introduce methods that will reduce the estimation bias of expected shortfall. To this end, several one-step bias correction expected shortfall estimators are presented and investigated via simulation studies and compared with one-step estimators. The third problem is that of constructing simultaneous confidence bands for quantile regression functions when the predictor variables are constrained within a region is considered. In this context, a method is introduced that makes use of the asymmetric Laplace errors in conjunction with a simulation based algorithm to create confidence bands for quantile and interquantile regression functions. Furthermore, the simulation approach is extended to an ordinary least square framework to build simultaneous bands for quantiles functions of the classical regression model when the model errors are normally distributed and when this assumption is not fulfilled. Finally, attention is directed towards the construction of prediction intervals for realised volatility exploiting an alternative volatility estimator based on the difference of two extreme quantiles. The proposed approach makes use of AR-GARCH procedure in order to model time series of intraday quantiles and forecast intraday returns predictive distribution. Moreover, two simple adaptations of an existing model are also presented.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
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