Conditional Coverage Probability Research Articles

Conditional predictive inference for stable algorithms

We investigate generically applicable and intuitively appealing prediction intervals based on k-fold cross-validation. We focus on the conditional coverage probability of the proposed intervals, given the observations in the training sample (hence, training conditional validity), and show that it is close to the nominal level, in an appropriate sense, provided that the underlying algorithm used for computing point predictions is sufficiently stable when feature-response pairs are omitted. Our results are based on a finite sample analysis of the empirical distribution function of k-fold cross-validation residuals and hold in nonparametric settings with only minimal assumptions on the error distribution. To illustrate our results, we also apply them to high-dimensional linear predictors, where we obtain uniform asymptotic training conditional validity as both sample size and dimension tend to infinity at the same rate and consistent parameter estimation typically fails. These results show that despite the serious problems of resampling procedures for inference on the unknown parameters (cf. in A Festschrift for Erich L. Lehmann (1983) 28–48 Wadsworth; Ann. Statist. 24 (1996) 307–335; J. Mach. Learn. Res. 19 (2018) 5), cross-validation methods can be successfully applied to obtain reliable predictive inference even in high dimensions and conditionally on the training data.

Conditional Contact Angle Distribution in LEO Satellite-Relayed Transmission

This letter characterizes the contact angle distribution based on the condition that the relay low earth orbit (LEO) satellite is in the communication range of both the ground transmitter and the ground receiver. As one of the core distributions in stochastic geometry-based routing analysis, the analytical expression of the Cumulative Distribution Function (CDF) of the conditional contact angle is derived. Furthermore, the conditional contact angle is applied to analyze the inaccessibility of common satellites between the ground transmitter and receiver. Finally, with the help of the conditional contact angle, coverage probability and achievable data rate in LEO satellite-relayed transmission are studied.

A User Centric Blockage Model for Wireless Networks

This paper proposes a cascade blockage model for analyzing the vision that a user has of a wireless network. This model, inspired by the classical multiplicative cascade models, has a radial structure meant to analyze blockages seen by the receiver at the origin in different angular sectors. The main novelty is that it is based on the geometry of obstacles and takes the joint blockage phenomenon into account. We show on a couple of simple instances that the Laplace transforms of total interference satisfies a functional equation that can be solved efficiently by an iterative scheme. This is used to analyze the coverage probability of the receiver and the effect of blockage correlation and penetration loss in both dense and sparse blockage environments. Furthermore, this model is used to investigate the effect of blockage correlation on user beamforming techniques. Another functional equation and its associated iterative algorithm are proposed to derive the coverage performance of the best beam selection in this context. In addition, the conditional coverage probability is also derived to evaluate the effect of beam switching. The results not only show that beam selection is quite efficient for multi-beam terminals, but also show how the correlation brought by blockages can be leveraged to accelerate beam sweeping and pairing.

Bootstrap prediction intervals with asymptotic conditional validity and unconditional guarantees

Abstract It can be argued that optimal prediction should take into account all available data. Therefore, to evaluate a prediction interval’s performance one should employ conditional coverage probability, conditioning on all available observations. Focusing on a linear model, we derive the asymptotic distribution of the difference between the conditional coverage probability of a nominal prediction interval and the conditional coverage probability of a prediction interval obtained via a residual-based bootstrap. Applying this result, we show that a prediction interval generated by the residual-based bootstrap has approximately $50\%$ probability to yield conditional under-coverage. We then develop a new bootstrap algorithm that generates a prediction interval that asymptotically controls both the conditional coverage probability as well as the possibility of conditional under-coverage. We complement the asymptotic results with several finite-sample simulations.

On the Uplink SIR Meta Distribution for AF Relay Networks

A fine-grained performance analysis of uplink transmission is provided in dual-hop amplify-and-forward (AF) relay networks. In order to improve the cell-edge coverage, multiple relay nodes are deployed around the base station and AF relay transmission is established via the aid of the scheduled relay node. The locations of the cell-edge users and relay nodes are modeled by correlated stochastic geometry models. In contrast to the conventional performance analysis based on the coverage probability, the distribution of the conditional coverage probability (CCP), which is called signal-to-interference ratio (SIR) meta distribution, is investigated by considering the random location of the nodes as well as the underlying channel. The moments of the CCP are analytically derived and the approximation of uplink SIR meta distribution is presented in semi-closed-form expression. Numerical results verify the accuracy of the analytical expressions and show the impact of the system parameters on the network performance.

The Improved Value-at-Risk for Heteroscedastic Processes and Their Coverage Probability

A risk measure commonly used in financial risk management, namely, Value-at-Risk (VaR), is studied. In particular, we find a VaR forecast for heteroscedastic processes such that its (conditional) coverage probability is close to the nominal. To do so, we pay attention to the effect of estimator variability such as asymptotic bias and mean square error. Numerical analysis is carried out to illustrate this calculation for the Autoregressive Conditional Heteroscedastic (ARCH) model, an observable volatility type model. In comparison, we find VaR for the latent volatility model i.e., the Stochastic Volatility Autoregressive (SVAR) model. It is found that the effect of estimator variability is significant to obtain VaR forecast with better coverage. In addition, we may only be able to assess unconditional coverage probability for VaR forecast of the SVAR model. This is due to the fact that the volatility process of the model is unobservable.

Conditional dual‐connectivity decoupling design framework for OMA and NOMA heterogeneous networks and performance comparison

In order to enhance simultaneously the association robustness of mobile users and spectrum efficiency (SE) of systems, by integrating the decoupled uplink (UL)/downlink (DL) association (DUDA) and cross-tier dual connectivity (DC), the orthogonal multiple access (OMA)-mode and non-OMA (NOMA)-mode design schemes are investigated over a three-tier heterogeneous network (HetNet) accompanying with the proposition of a conditional association analysis method. The scheme exploits effectively the DC requirement of NOMA. Additionally, in the proposed conditional association design and analysis method, the decoupling UL association designs are conducted only under a given DL association condition. Owing to low complexity, this method shows that it is very easy to achieve the possible decoupling UL DC designs and the corresponding condition association probabilities. Then, the conditional coverage probabilities and SEs of the OMA-mode and NOMA-mode HetNets are derived by specially considering DUDA and coupled UL/DL association strategies. The proposed conditional design framework can be easily extended to multi-tier HetNets and is helpful to select feasible DL candidate from all possible DL associations so that the resulting UL associations are optimal. Besides this, the cross-tier DC design ensures the effective exploitation of the whole network resources because different tiers have different serving levels.

On the meta distribution in spatially correlated non-Poisson cellular networks

In this paper, we consider a cellular network in which the locations of the base stations are spatially correlated. We introduce an analytical framework for computing the distribution of the conditional coverage probability given the point process, which is referred to as the meta distribution and provides one with fine-grained information on the performance of cellular networks beyond spatial averages. To this end, we approximate, from the typical user standpoint, the spatially correlated (non-Poisson) cellular network with an inhomogeneous Poisson point process. In addition, we employ a new and recently proposed definition of the coverage probability and introduce an efficient numerical method for computing the meta distribution. The accuracy of the proposed approach is validated with the aid of numerical simulations.

Sense-and-Predict: Harnessing Spatial Interference Correlation for Cognitive Radio Networks

Cognitive radio (CR) is a key enabler realizing future networks to achieve higher spectral efficiency by allowing spectrum sharing between different wireless networks. It is important to explore whether spectrum access opportunities are available, but conventional CR based on transmitter (TX) sensing cannot be used to this end because the paired receiver (RX) may experience different levels of interference, according to the extent of their separation, blockages, and beam directions. To address this problem, this paper proposes a novel form of medium access control (MAC) termed sense-and-predict (SaP), whereby each secondary TX predicts the interference level at the RX based on the sensed interference at the TX; this can be quantified in terms of a spatial interference correlation between the two locations. Using stochastic geometry, the spatial interference correlation can be expressed in the form of a conditional coverage probability, such that the signal-to-interference ratio (SIR) at the RX is no less than a predetermined threshold given the sensed interference at the TX, defined as an opportunistic probability (OP). The secondary TX randomly accesses the spectrum depending on OP. We optimize the SaP framework to maximize the area spectral efficiencies (ASEs) of secondary networks while guaranteeing the service quality of the primary networks. Testbed experiments using USRP and MATLAB simulations show that SaP affords higher ASEs compared with CR without prediction.

Approximation of Meta Distribution and its Moments for Poisson Cellular Networks

The notion of meta distribution as the distribution of the conditional coverage probability was introduced in the literature before. In this letter, we show how we can reconstruct the entire meta distribution only from its moments using Fourier–Jacobi expansion. As an example, we specifically consider Poisson cellular networks. We also provide a simple closed-form approximation for its moments, along with its error analysis. Lastly, we apply the approximation to obtain a power scaling law for downlink Poisson cellular networks.

Equi-Coverage Contours in Cellular Networks

In this letter, we introduce a general cellular network model, where: 1) users and base stations (BSs) are distributed as two general point processes that may be coupled; 2) pathloss is assumed to follow a multi-slope power-law pathloss model; and 3) fading (power) is assumed to be independent across all wireless links. For this setup, we first obtain a set of contours representing the same meta distribution of signal-to-interference ratio , which is the distribution of the conditional coverage probability given the point process, for different values of the parameters of the pathloss function and BS and user point processes. This general result is then specialized to 3GPP-inspired user and BS configurations obtained by combining Poisson point process and Poisson cluster process.

A simple procedure for computing improved prediction intervals for autoregressive models

Abstract. This article concerns the construction of prediction intervals for time series models. The estimative or plug-in solution is usually not entirely adequate, since the (conditional) coverage probability may differ substantially from the nominal value. Prediction intervals with improved (conditional) coverage probability can be defined by adjusting the estimative ones, using rather complicated asymptotic procedures or suitable simulation techniques. This article extends to Markov process models a recent result by Vidoni, which defines a relatively simple predictive distribution function, giving improved prediction limits as quantiles. This new solution is fruitfully considered in the challenging context of prediction for time-series models, with particular regard to AR and ARCH processes.

Conditional and Unconditional Confidence Intervals Following a Group Sequential Test

ABSTRACT After a group sequential test, the naïive confidence interval (CI) is usually biased in the sense that it does not cover the true parameter at the correct nominal level. Furthermore, when the stopping time is taken into account, the actual conditional confidence coverage probability can be much less accurate. In this article, we study the conditional coverage probability and other related properties of the naïve CI and different versions of exact CI's. It is demonstrated that only correcting the overall confidence level does not necessarily improve the confidence level at any given stopping stage. Conditional inference can be applied to construct an exact conditional CI but it is not without serious undesirable properties. We propose a two-step restricted conditional confidence interval (RCCI) which considerably improves the conditional confidence level while minimizing the undesirable properties. Numerical comparisons are made between the proposed method and existing methods. The results show that the RCCI not only improves the conditional coverage probability considerably from the exact CI's but also is free of the major undesirable properties displayed by the pure conditional CI. Differences between the conditional and unconditional CI's and their respective strengths are also discussed.

Response prediction in mixed effects models

Although prediction in mixed effects models usually concerns the random effects, in this paper we deal with the problem of prediction of a future, or yet unobserved, response random variable, belonging to a given cluster. In particular, the aim is to define computationally tractable prediction intervals, with conditional and unconditional coverage probability close to the target nominal value. This solution involves the conditional density of the future response random variable given the observed data, or a suitable high-order approximation based on the Laplace method. We prove that, unless the amount of data is very limited, the estimative or naive predictive procedure gives a relatively simple, feasible solution for response prediction. An application to generalized linear mixed models is presented.

The adjustment of prediction intervals to account for errors in parameter estimation

Abstract. Standard approximate 1 − α prediction intervals (PIs) need to be adjusted to take account of the error in estimating the parameters. This adjustment may be aimed at setting the (unconditional) probability that the PI includes the value being predicted equal to 1 − α. Alternatively, this adjustment may be aimed at setting the probability that the PI includes the value being predicted equal to 1 − α, conditional on an appropriate statistic T. For an autoregressive process of order p, it has been suggested that T consist of the last p observations. We provide a new criterion by which both forms of adjustment can be compared on an equal footing. This new criterion of performance is the closeness of the coverage probability, conditional on all of the data, of the adjusted PI and 1 − α. In this paper, we measure this closeness by the mean square of the difference between this conditional coverage probability and 1 − α. We illustrate the application of this new criterion to a Gaussian zero‐mean autoregressive process of order 1 and one‐step‐ahead prediction. For this example, this comparison shows that the adjustment which is aimed at setting the coverage probability equal to 1 − α conditional on the last observation is the better of the two adjustments.

Improved prediction intervals for stochastic process models

Abstract. This paper reviews some recent results on the construction of improved prediction limits for time series models and presents a simple solution based on a fully conditional approach. A prediction limit, expressed as a modification of the estimative one, is obtained so that its conditional and unconditional coverage probability equals the target value to third‐order accuracy. Although the specification of the ancillary statistic is not required, it respects the conditionality principle, to the relevant order of approximation. Moreover, the corresponding predictive density is specified in a relatively simple closed form. Simple examples show the usefulness of this conditional approach to prediction.

Conditional Inference Following Group Sequential Testing

AbstractThe work of Fisher (1959) and Buehler (1959) discuss the importance of conditioning on recognizable subsets of the sample space. The stopping time yields an easily identifiable partition of the sample space when considering group sequential testing. We first present confidence intervals that are correct when conditioning on the subset of data for which a trial stopped at a particular analysis. These intervals have very desirable properties for observations that are highly unusual (given any value of the mean). In addition, they provide insight into how information about the mean is distributed between the two sufficient statistics. We then use conditional coverage probabilities to compare the sample mean, stagewise, and repeated confidence intervals. We find that none of these intervals outperforms the others when conditioning on stopping time, and no interval is a uniformly acceptable performer.

On an Asymptotic Theory of Conditional and Unconditional Coverage Probabilities of Empirical Bayes Confidence Intervals

Empirical Bayes (EB) methodology is now widely used in statistics. However, construction of EB confidence intervals is still very limited. Following Cox (1975), Hill (1990) and Carlin & Gelfand (1990, 1991), we consider EB confidence intervals, which are adjusted so that the actual coverage probabilities asymptotically meet the target coverage probabilities up to the second order. We consider both unconditional and conditional coverage, conditioning being done with respect to an ancillary statistic.

Sequential analysis of matched dichotomous data from prospective case-control studies.

Sequential analysis of randomized controlled clinical trials and epidemiological prospective (matched) case-control studies can have ethical or economical advantages above a fixed sample size approach. It offers the possibility to stop early when enough evidence for an apparent effect of the risk factor or lack of the expected effect is achieved. In clinical trials it is well accepted to stop the trial early in favour of the alternative hypothesis. In epidemiological studies, in general, the need is not felt to stop early in case of a clear exposure effect. Little attention has been paid, however, to early stopping and accepting the null hypothesis. In metabolic epidemiological studies, where analysis destroys the biological material, the question of efficient use of samples, for example, those stored in a biobank, becomes crucial. Also a slow accrual of cases or the costs of follow-up of a cohort nested study can make it desirable to stop a study early once it becomes clear that no relevant exposure effect will be found. Matching can further reduce the amount of information necessary to reach a conclusion. We derived test statistics Z (efficient score) and V (Fisher's information) for the sequential analysis of studies with dichotomous data where each case can be matched to one or more controls. A variable matching ratio is allowed. These test statistics can be entered into the software PEST to monitor the course of the study. The double sequential probability ratio test and the double triangular test were evaluated with simulated data for odds ratios equal to 1.5, 2.0 and 2.5 and various type I and type II error probabilities both under H(0) and under H(1). Our simulations resulted in average and median values for the amount of information (V) that are far less than those for a fixed sample size study. Efficiency gain ranges from 32 per cent to 60 per cent. The proposed sequential analysis was applied in an investigation on the possible relationship between the polymorphism of the MTHFR-gene and rectal cancer in a cohort of women with cases matched by age to one and to three controls. A sequential analysis of matched data can lead to early stopping in favour of H(0) or H(1), thus conserving valuable resources for future testing. A sequentially designed study can be more economical and less arbitrary than a study that makes use of conditional power or conditional coverage probability calculations to decide early stopping.

On the Construction of Bayes–Confidence Regions

Summary We obtain approximate Bayes–confidence intervals for a scalar parameter based on directed likelihood. The posterior probabilities of these intervals agree with their unconditional coverage probabilities to fourth order, and with their conditional coverage probabilities to third order. These intervals are constructed for arbitrary smooth prior distributions. A key feature of the construction is that log-likelihood derivatives beyond second order are not required, unlike the asymptotic expansions of Severini.