--- title: One-Sided Conformal Intervals url: https://www.emergentmind.com/topics/one-sided-conformal-intervals type: topic --- # One-Sided Conformal Intervals Searching arXiv for the cited paper and closely related work on one-sided conformal intervals. First, retrieving the main paper by arXiv id 2606.18199. Now searching for related arXiv papers on one-sided conformal prediction, VaR, bounded outcomes, survival, and computational aspects. One-sided conformal intervals are conformal prediction sets of the form \([L_{n+1},\infty)\) or \((-\infty,U_{n+1}]\) that control left-tail and right-tail miscoverage separately, rather than only the aggregate error of a two-sided interval. In supervised regression, this refines the classical conformal objective from global marginal coverage \(1-\alpha\) to directional guarantees \(\mathbb{P}(Y_{n+1}U_{n+1})\le \alpha^{+}\), with \(\alpha^{-}+\alpha^{+}=\alpha\). Recent work formulates lower and upper one-sided split conformal intervals, proves finite-sample guarantees under exchangeability and asymptotic guarantees under non-exchangeability, and derives a two-sided interval by intersecting the one-sided bounds [2606.18199]. ## 1. Coverage target and formal definition The modern one-sided conformal framework is stated in the standard regression setting with training sample \[ \mathcal{D}_n=\{(X_i,Y_i)\}_{i=1}^n, \] where \(X_i\in\mathbb{R}^p\) and \(Y_i\in\mathcal{Y}\subseteq\mathbb{R}\), and prediction set \(\mathcal{C}_{n+1}=\mathcal{C}(X_{n+1})\subseteq\mathcal{Y}\) for a new response \(Y_{n+1}\). Classical conformal prediction targets only \[ \mathbb{P}(Y_{n+1}\in \mathcal{C}_{n+1})\ge 1-\alpha. \] For a single interval \([L_{n+1},U_{n+1}]\), one-sided conformalization instead specifies lower and upper miscoverage budgets \(\alpha^{-},\alpha^{+}\in(0,1)\) with \(\alpha^{-}+\alpha^{+}=\alpha\), and requires \[ \mathbb{P}(Y_{n+1}\in[L_{n+1},U_{n+1}])\ge 1-\alpha, \] \[ \mathbb{P}(Y_{n+1}U_{n+1})\le \alpha^{+}. \] Tail-specific coverage is strictly stronger than global coverage because it decomposes miscoverage into directional components [2606.18199]. The split conformal construction uses a training/calibration partition \[ \mathcal{D}_n=\mathcal{D}_{n_t}\cup\mathcal{D}_{n_c}, \qquad |\mathcal{D}_{n_t}|=n_t,\;|\mathcal{D}_{n_c}|=n_c, \] a fitted predictor \(\widehat f_{n_t}\) or quantile model \(\widehat q_{\tau,n_t}\), and calibration scores \[ S_i=s_{n_t}((X_i,Y_i)), \qquad i\in\mathcal{D}_{n_c}. \] For level \(\alpha\), the empirical conformal quantile is the empirical \((1-\alpha)(1+1/n_c)\)-quantile of \((S_i)_{i\in\mathcal{D}_{n_c}}\); the \((1+1/n_c)\) factor yields exact finite-sample coverage under exchangeability [2606.18199]. A recurring misconception is that any two-sided conformal interval with valid global coverage automatically achieves directional calibration. It does not: classical split conformal intervals control only the sum of left- and right-tail errors, not the two tails separately [2606.18199]. ## 2. One-sided score constructions The core construction is to calibrate two separate one-sided intervals: \[ \mathcal{C}^L_{n+1,\alpha^-}(X_{n+1})=[L_{n+1},\infty), \qquad \mathcal{C}^U_{n+1,\alpha^+}(X_{n+1})=(-\infty,U_{n+1}]. \] The framework is developed for three score families: residual, standardized residual, and quantile-based scores [2606.18199]. For lower intervals, the residual score is \[ s^{\mathrm{res},L}(x,y)=\widehat f_{n_t}(x)-y, \] with calibration scores \(S_i^{\mathrm{res},L}=\widehat f_{n_t}(X_i)-Y_i\). If \(Q^{\mathrm{res},L}_{\alpha^-,n_c}\) is the conformal quantile, the lower bound is \[ L^{\mathrm{res}}(x)=\widehat f_{n_t}(x)-Q^{\mathrm{res},L}_{\alpha^-,n_c}, \] and the interval is \([\widehat f_{n_t}(x)-Q^{\mathrm{res},L}_{\alpha^-,n_c},\infty)\). With a scale estimate \(\widehat\sigma_{n_t}(x)\), the standardized residual score becomes \[ s^{\mathrm{s\text{-}res},L}(x,y)=\frac{\widehat f_{n_t}(x)-y}{\widehat\sigma_{n_t}(x)}, \] yielding \[ \mathcal{C}^{\mathrm{s\text{-}res},L}_{\alpha^-}(x) = \Bigl[ \widehat f_{n_t}(x)-\widehat\sigma_{n_t}(x)Q^{\mathrm{s\text{-}res},L}_{\alpha^-,n_c}, \infty \Bigr). \] For quantile-based lower bounds, the recommended score is the non-truncated signed version \[ s^{\mathrm{q\text{-}sgn},L}(x,y)=\widehat q_{\alpha^-,n_t}(x)-y, \] which yields \[ \mathcal{C}^{\mathrm{q\text{-}sgn},L}_{\alpha^-}(x) = \Bigl[ \widehat q_{\alpha^-,n_t}(x)-Q^{\mathrm{q\text{-}sgn},L}_{\alpha^-,n_c}, \infty \Bigr). \] A truncated variant, \[ s^{\mathrm{q},L}(x,y)=\max\{\widehat q_{\alpha^-,n_t}(x)-y,0\}, \] is also available, but it can be overly conservative for the opposite tail; the signed version is preferable [2606.18199]. Upper intervals are defined symmetrically. Residual scores use \[ s^{\mathrm{res},U}(x,y)=y-\widehat f_{n_t}(x), \] giving \[ \mathcal{C}^{\mathrm{res},U}_{\alpha^+}(x) = \bigl(-\infty,\widehat f_{n_t}(x)+Q^{\mathrm{res},U}_{\alpha^+,n_c}\bigr]. \] The standardized version is \[ s^{\mathrm{s\text{-}res},U}(x,y)=\frac{y-\widehat f_{n_t}(x)}{\widehat\sigma_{n_t}(x)}, \] and the signed quantile version is \[ s^{\mathrm{q\text{-}sgn},U}(x,y)=y-\widehat q_{1-\alpha^+,n_t}(x), \] which yields \[ \mathcal{C}^{\mathrm{q\text{-}sgn},U}_{\alpha^+}(x) = \bigl(-\infty,\widehat q_{1-\alpha^+,n_t}(x)+Q^{\mathrm{q\text{-}sgn},U}_{\alpha^+,n_c}\bigr]. \] Under exchangeability, the lower and upper one-sided split conformal intervals satisfy \[ \mathbb{P}(Y_{n+1}\in\mathcal{C}^L_{n+1,\alpha^-})\ge 1-\alpha^-, \qquad \mathbb{P}(Y_{n+1}\in\mathcal{C}^U_{n+1,\alpha^+})\ge 1-\alpha^+, \] equivalently, \[ \mathbb{P}(Y_{n+1}U_{n+1})\le \alpha^+. \] If the one-sided scores have no ties almost surely, the corresponding upper bounds are also controlled up to \(1/(n_c+1)\) [2606.18199]. ## 3. Induced two-sided intervals by intersection Once lower and upper one-sided intervals are available, the induced two-sided interval is their intersection: \[ \mathcal{C}^{\cap}_{n+1,\underline{\alpha}} = \mathcal{C}^L_{n+1,\alpha^-}\cap \mathcal{C}^U_{n+1,\alpha^+} = [L_{n+1},U_{n+1}], \] with \(\underline{\alpha}=(\alpha^{-},\alpha^{+})\). The construction assumes the conformity scores are quasi-convex in \(y\), so the level sets are intervals rather than unions of disjoint sets [2606.18199]. The central coverage theorem states that, under exchangeability, \[ 1-(\alpha^-+\alpha^+) \le \mathbb{P}(Y_{n+1}\in\mathcal{C}^{\cap}_{n+1,\underline{\alpha}}). \] If scores are almost surely distinct, then \[ \mathbb{P}(Y_{n+1}\in\mathcal{C}^{\cap}_{n+1,\underline{\alpha}}) \le 1-\max\{\alpha^-,\alpha^+\}+\frac{1}{n_c+1}. \] For the symmetric allocation \(\alpha^-=\alpha^+=\alpha/2\), \[ 1-\alpha \le \mathbb{P}(Y_{n+1}\in\mathcal{C}^{\cap}_{n+1,(\alpha/2,\alpha/2)}) \le 1-\frac{\alpha}{2}+\frac{1}{n_c+1}. \] The lower bound follows from \[ \{Y_{n+1}\notin[L_{n+1},U_{n+1}]\} = \{Y_{n+1}U_{n+1}\}, \] together with a union bound and the one-sided tail bounds [2606.18199]. This intersection construction clarifies the relation between global and directional validity. The induced interval inherits both tail-specific guarantees and a global coverage guarantee at least \(1-\alpha\). At the same time, the probability bounds do not shrink to the same point as \(n_c\to\infty\); with symmetric splitting there remains an \(\alpha/2\) gap, which quantifies the efficiency cost of separate tail control [2606.18199]. A second misconception follows here: asymmetry of interval shape is not sufficient for directional validity. Standard conformalized quantile regression can produce asymmetric intervals, but it still calibrates a single two-sided score and can therefore misallocate error between tails [2606.18199]. ## 4. Exchangeable and non-exchangeable regimes In the exchangeable case, the guarantees are finite-sample. The assumptions are that calibration observations and the test point are exchangeable, and that score functions are symmetric in the calibration sample and quasi-convex in \(y\). Under these conditions, one-sided coverage and the intersection theorem are exact finite-sample bounds of the same kind as standard split conformal prediction [2606.18199]. For non-exchangeable data, such as time series or covariate shift, the framework uses Adaptive Conformal Inference (ACI) and DtACI. Writing the lower- and upper-tail miscoverage indicators at time \(i\) as \[ \mathrm{err}^-_i=\mathbf{1}\{Y_i\notin\mathcal{C}^L_{i,\alpha^-_i}\}, \qquad \mathrm{err}^+_i=\mathbf{1}\{Y_i\notin\mathcal{C}^U_{i,\alpha^+_i}\}, \] ACI updates the tail levels online according to \[ \alpha_{i+1}^\pm=\alpha_i^\pm+\gamma^\pm(\alpha^\pm-\mathrm{err}_i^\pm), \] with step sizes \(\gamma^\pm>0\). The resulting empirical tail error rates satisfy \[ \left| \frac{1}{N}\sum_{i=1}^N \mathrm{err}_i^\pm-\alpha^\pm \right| \le \frac{\max\{\alpha_1^\pm,1-\alpha_1^\pm\}+\gamma^\pm}{N\gamma^\pm}, \] and therefore \[ \frac{1}{N}\sum_{i=1}^N \mathrm{err}_i^\pm \xrightarrow[N\to\infty]{a.s.}\alpha^\pm. \] Defining total error \[ \mathrm{err}^{\mathrm{tot}}_i=\mathrm{err}^-_i+\mathrm{err}^+_i, \qquad \alpha^{\mathrm{tot}}=\alpha^-+\alpha^+, \] the global error also obeys an asymptotic guarantee: \[ \frac{1}{N}\sum_{i=1}^N \mathrm{err}^{\mathrm{tot}}_i \xrightarrow[N\to\infty]{a.s.} \alpha^-+\alpha^+. \] DtACI runs multiple ACI experts with different \(\gamma_j\) in parallel, aggregates them with exponential weights, and yields analogous asymptotic guarantees for both tails and the intersection interval, together with short-term regret bounds describing how quickly the adaptive levels track an oracle [2606.18199]. The practical significance is that one-sided conformal intervals are not restricted to exchangeable regression. They extend to nonstationary sequences, time-varying volatility, and drift scenarios, but the form of validity changes from exact finite-sample coverage to long-run asymptotic tail-specific and global calibration [2606.18199]. ## 5. Directional calibration, skewness, and asymmetric-risk applications Directional calibration means \[ \mathbb{P}(Y_{n+1}U_{n+1})\approx \alpha^+. \] Classical two-sided intervals guarantee only \[ \mathbb{P}(Y_{n+1}U_{n+1})\le \alpha, \] and can therefore be globally valid while being directionally imbalanced. The skewed-data case is the canonical failure mode: symmetric scores may produce too many misses on the left and none on the right [2606.18199]. Simulation studies identify this imbalance explicitly. In skewed Student-\(t\) settings, classical conformal methods based on residuals, standardized residuals, and standard CQR, together with a parametric CLT-based AR(1) benchmark, systematically under-cover the left tail and over-cover the right tail, with upper-tail coverage near \(100\%\). The one-sided intersection intervals calibrate each tail separately, bringing empirical lower and upper coverages close to \(1-\alpha^{-}\) and \(1-\alpha^{+}\), at a mild cost in interval width. The truncated one-sided quantile score can produce extreme upper overcoverage in skewed data because many upper-tail scores are zero; the signed scores avoid this and show better directional calibration [2606.18199]. A principal application is financial return forecasting and Value-at-Risk (VaR). For daily log-returns \[ Y_n=\log(P_n/P_{n-1}), \] the framework is illustrated on SPY, TQQQ, and XLE using a GARCH(1,1) model with Student-\(t\) innovations. If the conformal interval is \[ \mathcal{C}^{\cap}_{n+1,\underline{\alpha}}=[L_{n+1},U_{n+1}], \] the conformal VaR estimator is defined as \[ \widehat{\mathrm{VaR}}_{n+1,\alpha^-}=L_{n+1}. \] Tail-specific validity implies \[ \mathbb{P}(Y_{n+1}<\widehat{\mathrm{VaR}}_{n+1,\alpha^-})\le \alpha^-, \] so the VaR break frequency is controlled by design, not only tested ex post. Empirically, the GARCH-based VaR and classical two-sided conformal intervals systematically under-cover the lower tail and over-cover the upper tail, whereas one-sided conformal intervals with adaptive DtACI align empirical left-tail coverage closely with the nominal \(1-\alpha^{-}\) across changing volatility regimes, including the COVID-19 crisis [2606.18199]. Related work on one-sided VaR recalibration isolates the strength of state dependence through a proxy-reliance parameter \(\rho\in[0,1]\) in the adjusted lower quantile \[ \widehat q_{\alpha,t}^{\,adj}=\widehat q_{\alpha,t}+c_{\rho,t}v_t^\rho. \] That framework treats proxy reliance as a design choice between an approximately constant-shift correction and a fully proxy-scaled correction, and shows that lower or intermediate proxy reliance can outperform fully proxy-scaled recalibration in stressed left-tail VaR control when the volatility proxy underreacts in stress [2603.22569]. ## 6. Related variants and broader context One-sided conformal intervals generalize beyond unbounded regression. For bounded continuous outcomes \(Y\in(0,1)\), transformation regression and beta regression supply one-sided constructions of the form \[ (-\infty,U_{1-\alpha}(x_{n+1})]\cap(0,1) \quad\text{or}\quad [L_{1-\alpha}(x_{n+1}),\infty)\cap(0,1). \] A closely related formulation uses one-sided scores based on signed residuals or on the fitted beta CDF, \[ s_i=F(y_i;\hat\mu_i,\hat\phi_i) \quad\text{or}\quad s_i=\Phi^{-1}(F(y_i;\hat\mu_i,\hat\phi_i)), \] which suggests one-sided bounds that respect support, skewness, and heteroscedasticity. The underlying paper does not explicitly derive one-sided conformal intervals, but all the ingredients are there, and the split and full conformal machinery extends in this direction [2507.14023]. In right-censored survival analysis, one-sided conformal intervals arise naturally as lower bounds \[ [\widehat{\mathrm{LB}}(X_{n+1}),\infty), \] because \(\tilde T=\min(T,C)\le T\). A split conformal lower bound for \(\tilde T\) is automatically a valid, but conservative, lower bound for \(T\). A hybrid survival procedure combines such a one-sided lower bound with a classifier for censoring status: some individuals receive a two-sided interval, while others receive only the lower one-sided interval, with finite-sample coverage under i.i.d. assumptions [2410.24136]. The duality between one-sided intervals and hypothesis-like decision procedures has also been made explicit under the label “conjecture testing.” In distributional conformal prediction based on estimated PIT values, the standard two-sided score \[ \widehat V_i(y)=\left|\widehat U_i'(X_f,y)-\tfrac12\right| \] can be replaced by a one-sided conformity measure such as \[ \widehat V_i(y)=-\widehat U_i'(X_f,y), \] which produces a prediction interval of the form \((c,\infty)\) and thereby supports one-sided conjecture tests through interval inversion [2109.12156]. From a computational perspective, one-sided intervals can be extracted from full conformal prediction by locating a single endpoint of the typicalness level set \[ \Gamma^{(\alpha)}(x_{n+1})=\{z:\pi(z)\ge \alpha\}. \] Root-finding approaches compute interval boundaries by solving \(\pi(z)=\alpha\) via bisection; once the two-sided set is interval-shaped, obtaining a one-sided conformal interval is conceptually straightforward by keeping only one endpoint and extending to \(-\infty\) or \(+\infty\) [2104.06648]. A separate, non-conformal asymptotic literature studies one-sided lower confidence limits through convex tangent cones. There the efficient procedure is based on projection of the influence curve onto the closed convex cone rather than its closed linear span, yielding higher efficiency together with weaker, only one-sided, regularity and stability. This is not conformal prediction, but it provides a useful asymptotic analogue of the broader principle that one-sided inferential goals are not reducible to symmetric two-sided calibration [1412.1701]. One-sided conformal intervals therefore occupy a distinct position within predictive inference. They are not merely truncated versions of two-sided sets. They formalize directional risk control, make \(\alpha^{-}\) and \(\alpha^{+}\) user-settable design parameters, and support applications in skewed regression, time series, finance, survival analysis, and bounded-outcome modeling where the direction of error matters as much as its aggregate frequency [2606.18199]. Source: https://www.emergentmind.com/topics/one-sided-conformal-intervals