Distributionally Robust Conditional Quantile Prediction with Fixed Design

2021 ◽  
Author(s):  
Meng Qi ◽  
Ying Cao ◽  
Zuo-Jun (Max) Shen
Keyword(s):  
Empirical Distribution ◽  
Random Variable ◽  
Practical Method ◽  
Data Driven ◽  
Finite Sample ◽  
Response Variable ◽  
Conditional Quantile ◽  
Measure Concentration ◽  
Fixed Design ◽  
Distributionally Robust

Conditional quantile prediction involves estimating/predicting the quantile of a response random variable conditioned on observed covariates. The existing literature assumes the availability of independent and identically distributed (i.i.d.) samples of both the covariates and the response variable. However, such an assumption often becomes restrictive in many real-world applications. By contrast, we consider a fixed-design setting of the covariates, under which neither the response variable nor the covariates have i.i.d. samples. The present study provides a new data-driven distributionally robust framework under a fixed-design setting. We propose a regress-then-robustify method by constructing a surrogate empirical distribution of the noise. The solution of our framework coincides with a simple yet practical method that involves only regression and sorting, therefore providing an explanation for its empirical success. Measure concentration results are obtained for the surrogate empirical distribution, which further lead to finite-sample performance guarantees and asymptotic consistency. Numerical experiments are conducted to demonstrate the advantages of our approach. This paper was accepted by Hamid Nazerzadeh, Special Issue on Data-Driven Prescriptive Analytics.

scholarly journals Differentially Private Inference for Binomial Data

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Jordan Alexander Awan ◽  
Aleksandra Slavkovic
Keyword(s):  
Confidence Intervals ◽  
Differential Privacy ◽  
Empirical Distribution ◽  
Random Variable ◽  
Linear Constraints ◽  
Type I ◽  
Finite Sample ◽  
Hypothesis Tests ◽  
Binomial Data ◽  
Uniformly Most Powerful

We derive uniformly most powerful (UMP) tests for simple and one-sided hypotheses for a population proportion within the framework of Differential Privacy (DP), optimizing finite sample performance. We show that in general, DP hypothesis tests can be written in terms of linear constraints, and for exchangeable data can always be expressed as a function of the empirical distribution. Using this structure, we prove a `Neyman-Pearson lemma' for binomial data under DP, where the DP-UMP only depends on the sample sum. Our tests can also be stated as a post-processing of a random variable, whose distribution we coin ``Truncated-Uniform-Laplace'' (Tulap), a generalization of the Staircase and discrete Laplace distributions. Furthermore, we obtain exact p-values, which are easily computed in terms of the Tulap random variable. Using the above techniques, we show that our tests can be applied to give uniformly most accurate one-sided confidence intervals and optimal confidence distributions. We also derive uniformly most powerful unbiased (UMPU) two-sided tests, which lead to uniformly most accurate unbiased (UMAU) two-sided confidence intervals. We show that our results can be applied to distribution-free hypothesis tests for continuous data. Our simulation results demonstrate that all our tests have exact type I error, and are more powerful than current techniques.

scholarly journals Remaining Useful Life Prediction of MOSFETs via the Takagi–Sugeno Framework

Energies ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 2135
Author(s):  
Marcin Witczak ◽  
Marcin Mrugalski ◽  
Bogdan Lipiec
Keyword(s):  
Historical Data ◽  
Field Effect Transistors ◽  
Failure Process ◽  
Practical Method ◽  
Remaining Useful Life ◽  
Data Driven ◽  
Data Repository ◽  
Uncertainty Interval ◽  
Useful Life ◽  
Takagi Sugeno

The paper presents a new method of predicting the remaining useful life of technical devices. The proposed soft computing approach bridges the gap between analytical and data-driven health prognostic approaches. Whilst the former ones are based on the classical exponential shape of degradation, the latter ones learn the degradation behavior from the observed historical data. As a result of the proposed fusion, a practical method for calculating components’ remaining useful life is proposed. Contrarily to the approaches presented in the literature, the proposed ensemble of analytical and data-driven approaches forms the uncertainty interval containing an expected remaining useful life. In particular, a Takagi–Sugeno multiple models-based framework is used as a data-driven approach while an exponential curve fitting on-line approach serves as an analytical one. Unlike conventional data-driven methods, the proposed approach is designed on the basis of the historical data that apart from learning is also applied to support the diagnostic decisions. Finally, the entire scheme is used to predict power Metal Oxide Field Effect Transistors’ (MOSFETs) health status. The status of the currently operating MOSFET is determined taking into consideration the knowledge obtained from the preceding MOSFETs, which went through the run-to-failure process. Finally, the proposed approach is validated with the application of real data obtained from the NASA Ames Prognostics Data Repository.

scholarly journals QUANTILE DOUBLE AUTOREGRESSION

2021 ◽  
pp. 1-47
Author(s):  
Qianqian Zhu ◽  
Guodong Li
Keyword(s):  
Time Series ◽  
Financial Time Series ◽  
Simulation Studies ◽  
Finite Sample ◽  
Conditional Quantile ◽  
New Model ◽  
Conditional Quantiles ◽  
Financial Time ◽  
Portmanteau Tests ◽  
Strict Stationarity

Many financial time series have varying structures at different quantile levels, and also exhibit the phenomenon of conditional heteroskedasticity at the same time. However, there is presently no time series model that accommodates both of these features. This paper fills the gap by proposing a novel conditional heteroskedastic model called “quantile double autoregression”. The strict stationarity of the new model is derived, and self-weighted conditional quantile estimation is suggested. Two promising properties of the original double autoregressive model are shown to be preserved. Based on the quantile autocorrelation function and self-weighting concept, three portmanteau tests are constructed to check the adequacy of the fitted conditional quantiles. The finite sample performance of the proposed inferential tools is examined by simulation studies, and the need for use of the new model is further demonstrated by analyzing the S&P500 Index.

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