Conditional CDFs and Quantile Functions as Estimands

Three Representations of a Conditional Distribution

A conditional distribution can be described by its probabilities, its cumulative probabilities, or its quantiles. These representations answer related but different questions: the conditional law specifies probabilities for sets, the conditional CDF summarizes probabilities below thresholds, and the conditional quantile gives thresholds associated with specified probability levels. Treating these objects as estimands makes clear both what a distributional model is trying to recover and what can be learned from its different summaries.

Let \(X\) be a predictor and \(Y\) a real-valued response. In general, a conditional distribution at a predictor value \(x\) is not described by a density: it may have discrete masses, a continuous component, or both. Conditional CDFs and quantiles remain well-defined in all these cases.

Because \(X\) and \(Y\) are real-valued, such a regular conditional distribution exists. Its values are determined only up to sets of predictor values having \(P_X\)-probability zero. Thus, \(F_x\) and \(Q_x\) are generally identified as functions of \(x\) only \(P_X\)-almost surely. A pointwise interpretation at an individual \(x\) requires a specified version, and claims at predictor values outside the distribution of \(X\) may depend on modeling assumptions.

For each fixed \(x\), \(F_x\) is nondecreasing, right-continuous, and has limits 0 and 1 at negative and positive infinity. The definition of \(Q_x\) uses an infimum rather than an ordinary inverse so it applies when \(F_x\) has jumps or flat regions. In particular, no conditional density and no strict increase of the CDF are required.

Generalized Inversion, Including Atoms

For a real-valued distribution with CDF \(F\), define \(Q(\tau)=\inf\{y:F(y)\geq\tau\}\). The following properties explain exactly what this definition guarantees. They apply to \(F_x\) for each \(x\) at which a version of the conditional law is specified.

Proof. Since \(F(y)\) tends to 1 as \(y\) tends to positive infinity, the set \(S=\{y:F(y)\geq\tau\}\) is nonempty. Since \(F(y)\) tends to 0 as \(y\) tends to negative infinity, \(S\) is bounded below. Therefore its infimum \(q\) is finite. Choose \(z_n\in S\) with \(q\leq z_n<q+1/n\). Right-continuity and monotonicity give \(F(q)\geq\lim_{n\to\infty}F(z_n)\geq\tau\). For every \(z<q\), \(F(z)<\tau\), so taking the limit as \(z\) increases to \(q\) gives \(F(q-)\leq\tau\). If \(q\leq y\), monotonicity gives \(F(y)\geq F(q)\geq\tau\). Conversely, if \(F(y)\geq\tau\), then \(y\in S\), so \(q=\inf S\leq y\). Finally, \(F(q)-F(q-)=P(Y=q)\) under the distribution with CDF \(F\), by continuity of probability for increasing sets. This proves the claims. \(\square\)

In conditional form, the theorem gives \(F_x(Q_x(\tau)-)\leq\tau\leq F_x(Q_x(\tau))\). If there is an atom at the quantile, probability levels throughout the jump correspond to the same threshold. By contrast, if \(F_x\) is continuous and strictly increasing at the relevant range, the quantile is the ordinary inverse there and \(F_x(Q_x(\tau))=\tau\). That equality is a special case, not the general definition.

The Quantile Function Determines the Law

A conditional CDF and its quantile function are not competing targets that retain unrelated information. The generalized inverse preserves the whole distribution, including its jumps. The next identity makes that precise.

Proof. By the generalized-inverse equivalence, \(Q(\tau)\leq y\) holds exactly when \(\tau\leq F(y)\). The set of levels \(\tau\in(0,1)\) satisfying this condition is \((0,F(y)]\cap(0,1)\), whose length is \(F(y)\). This holds for every real \(y\), so the quantile function recovers the CDF everywhere. \(\square\)

Applying the identity to a conditional CDF shows that \(Q_x\) determines \(F_x\) for each specified \(x\). Equivalently, if \(U\) is uniform on \((0,1)\), then \(Q_x(U)\) has CDF \(F_x\): for any real \(y\),

This is a distributional representation, not a claim that a model must generate outcomes by first drawing an observed uniform variable. It shows why quantiles at all levels together encode the complete conditional law, while a small selection of quantiles provides only selected features of it.

Location, Spread, and Tail Behavior

A conditional quantile at level \(\tau\) is a threshold on the response scale. For example, \(Q_x(0.5)\) is a conditional median under the stated generalized-inverse convention, and the conditional interquartile range \(Q_x(0.75)-Q_x(0.25)\) summarizes the spread of the central half of the distribution. These summaries describe different aspects of the law: a median is a location measure, while a difference between quantiles is a scale measure. Neither requires a conditional mean or variance to exist.

Upper-tail quantiles also have a direct probability interpretation. If \(q=Q_x(\tau)\), the theorem gives \(P(Y<q\mid X=x)\leq\tau\leq P(Y\leq q\mid X=x)\), for the chosen conditional law. When there is no atom at \(q\), these probabilities coincide at \(\tau\). When there is an atom, a threshold can include more than the requested probability level because the response cannot be divided at that point. The 90th percentile, for instance, is an upper-tail threshold associated with a nominal 10% exceedance level, but with an atom at the threshold the actual exceedance probability may be smaller than 10%.

The full curve \(\tau\mapsto Q_x(\tau)\) adds information beyond a location and a spread summary. Its lower levels describe the lower tail, its upper levels describe the upper tail, and changes in its slope across levels reflect how rapidly response values expand across probability ranges. A conditional CDF gives the corresponding threshold probabilities on the response scale. Since either complete function determines the conditional law, the choice between them is often guided by the scientific question and the form in which an answer will be used.

Why These Estimands Matter

A distributional analysis can target \(F_x\), \(Q_x\), or the conditional law \(K(x,\cdot)\), depending on the intended use. If a decision needs the probability that a response falls below a clinical threshold, \(F_x\) gives that quantity directly. If a decision needs a threshold associated with a specified risk level, \(Q_x\) is the natural summary. If many probability queries are needed, the whole conditional law provides the common object underlying both.

These are estimands: population-level quantities defined by the joint distribution of \((X,Y)\), rather than particular estimates produced by a fitted model. Their conditional nature matters. A marginal quantile of \(Y\) need not equal the conditional quantile at any particular \(x\), and averaging conditional quantiles across predictor values generally does not produce a marginal quantile. Each \(x\) indexes its own conditional distribution.