Conditional Rank Statistics

• Conditional Rank Statistics (CRS) are inference methods that transform data via conditional ranks, allowing analysis based on order and group structure.
• They enable nonparametric estimation, conditional regression, and robust variable selection through techniques like ranked set sampling and maximally selected rank statistics.
• CRS methods provide solid asymptotic theory and computational efficiency, making them valuable in economic, biomedical, and high-dimensional survival analyses.

Conditional Rank Statistics (CRS) refer generically to inferential or modeling approaches that utilize order, ranking, or permutation structure within or across strata defined by covariates or sample design, typically by conditioning either on ranks or covariates. CRS formalizes distributional and inferential properties (e.g., of statistics, estimators, or predictions) that arise when the usual parameters of interest are replaced by or connected to their conditional ranks, or when rank-based procedures are adapted to handle conditioning on observed group structure or covariate values. Applications span nonparametric estimation of distribution functions from ranked samples, conditional rank regression models, random forest split criteria, and modern high-dimensional coverage criteria indexed by rank.

1. Definitions and Conceptual Foundations

Several canonical constructions exemplify Conditional Rank Statistics:

• Conditional Sample Ranks: For a continuous outcome $Y$ and covariates $X$, the conditional rank is defined as $U = F_{Y|X}(Y|X)$, where $F_{Y|X}(y|x)$ denotes the conditional cumulative distribution function (CDF) of $Y$ given $X$ (Chernozhukov et al., 8 Jul 2024). The mapping $Y \mapsto U$ yields $U \sim \mathrm{Uniform}(0,1)$ conditionally on $X$.
• Ranked Set Sampling (RSS) and Judgment Post-Stratification (JPS): Observations $(X_i, R_i)$ are recorded, with $R_i \in \{1,\ldots,k\}$ assigned as the (possibly error-prone) rank of $X_i$ within a hypothetical sample of size $k$. Conditionally, $X_i$ is distributed as the $R_i$th order statistic in a sample from an unknown $F$ (Duembgen et al., 2013).
• Conditional Linear Rank Statistics in Random Forests: For structure discovery in survival forests, linear or maximally selected rank statistics are calculated conditional on covariate-induced splits, adapting classical rank-based hypothesis testing to recursive partitioning (Wright et al., 2016).
• Rank-Conditional Coverage: In high-dimensional multiple testing, confidence coverage is evaluated as a function of estimator rank within a vector of parameters, adjusting classical inference by conditioning on empirical ranking (Morrison et al., 2017).

These divergent settings share the core principle: inference or modeling is not on untransformed data, but on objects whose properties are determined or modulated by ranks, conditioning events, or permutation-invariant statistics.

2. Methodological Frameworks and Estimators

Ranked Set Sampling (RSS) and Distribution Function Estimation

In RSS or JPS, inference on an unknown continuous $F$ is based on samples indexed by observed ranks. Let $N_r = \sum_{i=1}^n 1_{R_i=r}$ and $$\hat F_r(x) = N_r^{-1} \sum_{i:R_i=r} 1_{X_i \le x}$$ the stratum-$r$ empirical CDF. Three key estimators are studied (Duembgen et al., 2013):

Estimator	Definition
Stratified ($\hat F_S$)	$(1/k)\sum_{r=1}^{k} \hat F_r(x)$
Nonparametric MLE ($\hat F_L$)	Maximizes: $$L_n(x,p) = \sum_{r=1}^k N_r\left[\hat F_r(x)\log B_r(p) + (1 - \hat F_r(x))\log(1-B_r(p))\right]$$; $B_r$ is Beta CDF
Moment-based ($\hat F_M$)	Solves: $n\hat F_M(x) = \sum_{r=1}^k N_r B_r(t)$ for $t\in [0,1]$

Where $B_r(p)$ is the cumulative distribution of the $r$th order statistic from a sample of size $k$ (i.e., Beta$(r,k+1-r)$).

Conditional Rank–Rank Regression (CRRR)

Let $Y, Z$ be continuous random variables, $X$ covariates. Conditional ranks are estimated for each $i$ as $\hat U_i := \Lambda(b(X_i)'\hat\beta_Y(Y_i))$ (and analogously $\hat V_i$ for $Z$), via distribution regression using a link function $\Lambda$ (logit or probit) and basis functions $b(x)$. CRRR proceeds as an OLS regression or correlation of $\hat U_i$ on $\hat V_i$:

$$\hat U_i = \alpha + \beta \hat V_i + \epsilon_i, \qquad \beta = \operatorname{Corr}(\hat U_i, \hat V_i)$$

This $\beta$ estimates the average within-$X$ Spearman correlation between $Y$ and $Z$ (Chernozhukov et al., 8 Jul 2024).

Conditional and Maximally Selected Rank Statistics in Random Forests

In split variable selection for random survival forests, linear rank statistics $S_j(t)$ and their standardized analogues are computed for each covariate $j$ and possible split $t$, using weights conditional on sample configuration. The maximally selected rank statistic $M_j = \max_t |T_j(t)|$ across split points $t$ guards against bias toward high-cardinality covariates (Wright et al., 2016).

3. Asymptotic Theory and Large-Sample Properties

Rank-Based Distribution Estimation

Under mild regularity, as $n \to \infty$ (with $k$ fixed), the process $\hat B_n^Z(t) = \hat F_Z(F^{-1}(t))$ for each estimator $Z$ exhibits a uniform linear expansion:

$$\sqrt{n}[\hat B_n^Z(t) - t] \approx \sum_{r=1}^k \gamma_{n,r}^Z(t) V_{n,r}(B_r(t))$$

with $V_{n,r}$ independent Brownian bridges (one per rank stratum), and explicit $\gamma_{n,r}^Z(t)$ weights. In balanced sampling, $\hat F_S$ and $\hat F_M$ are asymptotically equivalent, while $\hat F_L$ attains the smallest asymptotic variance (Duembgen et al., 2013).

Conditional Rank–Rank Regression

The CRRR estimator $$\hat\beta = \operatorname{Corr}(\hat U_i, \hat V_i)$$ is root-$n$ consistent and asymptotically normal:

$$\sqrt{n}(\hat\beta - \beta) \Rightarrow N(0,\sigma_\beta^2)$$

where $\sigma_\beta^2$ is characterized via influence function representation involving the distribution regression estimates (Chernozhukov et al., 8 Jul 2024). Standard errors can be estimated by exchangeable (weighted) bootstrap over the two-stage procedure.

Variable Selection in Random Survival Forests

The maximally selected rank statistic under null is asymptotically Gaussian, permitting analytic or permutation-based $p$-value approximations. Careful control via Brownian-bridge, Bonferroni, or multivariate Gaussian approximations reduces selection bias toward variables with many levels without sacrificing consistency (Wright et al., 2016).

4. Confidence Intervals, Bands, and Coverage Properties

Exact and Simultaneous Confidence Intervals

In ranked set data, conditional on the observed rank allocation, the sum $n\hat F_{naive}(x) = \sum_r Y_r$ ($Y_r \sim \mathrm{Bin}(N_r, B_r(F(x)))$ independently) permits construction of non-asymptotic, exact Clopper–Pearson–type confidence intervals for $F(x)$. Simultaneous Kolmogorov–Smirnov-type bands are built by simulating the null process under $F$ uniform, conditional on ranks (Duembgen et al., 2013).

Rank-Conditional and Coverage-Adjusted Intervals

Rank conditional coverage (RCC) at rank $i$ is defined as $P(\theta_{s(i)} \in CI_{s(i)})$, where $s$ indexes estimator ordering. Bootstrapped (parametric or nonparametric) intervals constructed to target RCC achieve nominal coverage uniformly across all ranks, outperforming both marginal and false coverage-statement rate (FCR) controlling methods in addressing the "winner's curse" common in high-dimensional settings (Morrison et al., 2017).

5. Robustness and Sensitivity to Imperfect Ranking

Simulation studies in the context of RSS indicate that estimators react differently to violation of perfect ranking (signal-plus-noise with less than perfect correlation):

• $\hat F_L$ (nonparametric MLE) exhibits increasing bias and mean-squared error under even mild misranking, particularly in the distribution tails.
• $\hat F_M$ (moment) remains nearly unbiased and highly efficient under both perfect and modestly imperfect ranking; it outperforms $\hat F_L$ in robustness.
• $\hat F_S$ (stratified) remains unbiased but is less efficient (higher variance).

A plausible implication is that the moment estimator $\hat F_M$ can be recommended as a compromise between efficiency under perfect conditions and resilience to rank errors (Duembgen et al., 2013).

6. Computational Algorithms and Practical Implementation

Procedures for CRS-based inference are computationally tractable. For RSS distribution estimation, $\hat F_S$ requires only stratum CDFs, $\hat F_M$ involves root-finding in a monotone sum over Beta CDFs per $x$, and $\hat F_L$ requires maximizing a strictly concave likelihood, both easily solved by bisection or Newton–Raphson. All estimators are piecewise constant on observed order statistics, allowing precomputation and vectorized algorithms in $R$ or similar environments (Duembgen et al., 2013).

Distribution regression for CRRR uses binary-response GLMs on fine grids, tail-restricted extrapolations, and standard correlation computation. Bootstrap inference accommodates arbitrary exchangeable weights and is scalable (Chernozhukov et al., 8 Jul 2024).

Maximally selected rank statistics, when deployed in random forests, leverage analytic or fast permutation-based $p$-value approximations. The "minLau" procedure (minimum of Brownian-bridge and Bonferroni approximations) provides near-unbiased, efficient variable selection even with many candidate splits and large-scale data (Wright et al., 2016).

Bootstrap computation for RCC-controlling intervals is implemented in the R package rcc, providing both parametric (Gaussian) and nonparametric variants. Usage involves resampling, re-ranking, quantile calculation for the error, and interval formation per rank (Morrison et al., 2017).

7. Empirical and Applied Perspectives

CRS and CRRR have demonstrated interpretative value, especially in economic and biomedical science. In intergenerational mobility, CRRR decomposes overall persistence into within-group and between-group components: within-group (conditional rank correlation) and between-group (difference with unconditional rank correlation). An application to Swiss administrative income data demonstrated that within-group persistence explained 62% of total persistence for sons and 52% for daughters (Chernozhukov et al., 8 Jul 2024).

Survival analysis with maximally selected rank statistics yields unbiased variable selection across covariate types and improved predictive performance, as evidenced in simulated and diverse real datasets, including gene expression and GWAS (Wright et al., 2016).

In high-dimensional inference, RCC-based intervals rectify selective inference coverage failures and are computationally practical, with code support in R for a variety of applied contexts (Morrison et al., 2017).

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References

• Dümbgen & Zamanzade (2018): "Inference on a Distribution Function from Ranked Set Samples" (Duembgen et al., 2013)
• Chernozhukov et al.: "Conditional Rank-Rank Regression" (Chernozhukov et al., 8 Jul 2024)
• Hothorn & Lausen, Genz, et al.: "Unbiased split variable selection for random survival forests using maximally selected rank statistics" (Wright et al., 2016)
• Benjamini & Yekutieli, Weinstein & Reid, et al.: "Rank conditional coverage and confidence intervals in high dimensional problems" (Morrison et al., 2017)