Recovering Conditional Quantiles From Distribution Regression

From a Conditional Distribution to a Quantile

A conditional CDF describes the whole distribution of an outcome at a given covariate value. A conditional quantile answers a more focused question: what outcome value marks a specified point in that distribution? Recovering quantiles from a fitted conditional CDF makes it possible to move between these two summaries while retaining the distributional information encoded by the CDF.

For example, the conditional median is the outcome level at which the conditional distribution has accumulated one half of its probability. With discrete outcomes, the CDF may jump over one half; with continuous outcomes, it may increase gradually; and in either case it may remain flat over a range. The definition of a quantile must handle all three possibilities.

Definition: For outcome $Y$, covariates $X=x$, and quantile level $0<\tau<1$, the conditional CDF is $F(y\mid x)=P(Y\le y\mid X=x)$. The conditional $\tau$-quantile is its generalized inverse: $$ Q_\tau(x)=\inf\{y:F(y\mid x)\ge \tau\}. $$ The infimum convention selects the leftmost outcome value at which the CDF reaches or exceeds $\tau$.

At a quantile value $q=Q_\tau(x)$, the CDF need not equal $\tau$. For a distribution with an atom at $q$, it can jump from below $\tau$ to above it. The corresponding characterization is $F(q-\mid x)\le \tau\le F(q\mid x)$, where $F(q-\mid x)=P(Y<q\mid X=x)$. A flat segment also does not make the generalized inverse ambiguous: the infimum selects its left endpoint when the CDF has already reached $\tau$ there.

Distribution regression estimates $F(y\mid x)$ by fitting a binary-response model at outcome thresholds $y$, using the indicator $1\{Y\le y\}$ as the response. In practice, an analyst may fit models at a finite set of thresholds, then evaluate their fitted probabilities at a covariate value $x$. The quantile estimate is obtained by finding the first threshold at which the fitted probability reaches the target $\tau$.

Estimation condition: To interpret an inversion as a quantile estimate, the fitted values should form a valid conditional CDF as a function of $y$: they should be nondecreasing, lie between zero and one, and have the appropriate limiting behavior at the extremes. The outcome thresholds must also cover the range relevant to the desired quantile. Separate threshold regressions do not automatically guarantee monotonicity.

Inverting Fitted Values on a Threshold Grid

Suppose the fitted CDF is available at ordered thresholds $y_1<\cdots<y_J$. The grid-based inverse at level $\tau$ is the first threshold whose fitted CDF value is at least $\tau$. If fitted values at the covariate value of interest are $\widehat F(y_j\mid x)$, this is

\widehat Q_\tau(x)=y_{j^*},\qquad j^*=\min\{j:\widehat F(y_j\mid x)\ge \tau\}.

This rule returns a grid value, not necessarily a precise quantile on the outcome scale. If $\widehat F(y_{j-1}\mid x)<\tau\le\widehat F(y_j\mid x)$, the grid brackets the crossing: under a nondecreasing fitted CDF, the generalized inverse is greater than $y_{j-1}$ and no greater than $y_j$. A finer grid can narrow that bracket. Interpolation can provide a numerical approximation for a continuous outcome, but it adds an assumption about how the CDF changes between thresholds. It is not generally appropriate for a discrete outcome, whose CDF is a step function.

Worked Example: Inverting a Conditional Median on a Grid

Suppose a fitted distribution regression gives the following hypothetical conditional CDF values for a continuous outcome measured in minutes, at a particular covariate value $x$.

Threshold $y$ (minutes)	$\widehat F(y\mid x)$
10	0.18
20	0.44
30	0.72
40	0.91

For the median, $\tau=0.50$. The first fitted CDF value at least 0.50 is 0.72, at 30 minutes. Thus the grid-based estimate is $\widehat Q_{0.50}(x)=30$ minutes, and the grid brackets the crossing between 20 and 30 minutes because $0.44<0.50\le0.72$.

If, solely as a local approximation, we assume the fitted CDF increases linearly between 20 and 30 minutes, the interpolated estimate is

20+\frac{0.50-0.44}{0.72-0.44}(30-20) =20+\frac{0.06}{0.28}(10) \approx 22.14\text{ minutes}.

The grid inverse and interpolated value are not contradictory: 30 minutes is the first crossing on the specified grid, while 22.14 minutes estimates a crossing between grid points under the stated linearity approximation. Without that approximation or a fitted CDF that can be evaluated between thresholds, the grid values alone do not identify the more precise number.

Atoms and Flat CDF Regions

When $Y$ is discrete, its CDF is constant between possible outcome values and jumps at those values. The size of a jump at $y$ is the conditional probability mass $P(Y=y\mid X=x)=F(y\mid x)-F(y-\mid x)$. A target probability can fall anywhere within that jump; the generalized inverse still returns the outcome at which the jump occurs. Thus an estimated quantile can remain at the same outcome for a range of $\tau$ values.

A plateau requires careful interpretation. If $F$ reaches $\tau$ at the left endpoint of a flat region, the generalized inverse is that left endpoint—not the right endpoint or the midpoint. If the CDF remains below $\tau$ throughout a flat region, the quantile lies beyond it, at the first later point where the CDF reaches $\tau$. A plateau is not evidence that every value in the interval is an equally valid generalized-inverse quantile.

Worked Example: A Discrete Outcome With a Plateau

Consider a hypothetical outcome $Y$ that takes the values 0, 1, and 3 hours at a fixed covariate value, with conditional probabilities 0.20, 0.50, and 0.30, respectively. The CDF is 0.20 at 0 hours, 0.70 at 1 hour, and 1.00 at 3 hours. It is constant at 0.70 for values from 1 hour up to, but not including, 3 hours.

For $\tau=0.60$, the CDF at 0 hours is $0.20<0.60$, while at 1 hour it is $0.70\ge0.60$. Therefore $Q_{0.60}(x)=1$ hour. The jump at 1 hour spans the target probability: $F(1-\mid x)=0.20\le0.60\le0.70=F(1\mid x)$.

For $\tau=0.70$, the CDF first reaches the target at 1 hour, so $Q_{0.70}(x)=1$ hour as well. The plateau does not shift the inverse to 3 hours: the infimum of the set of outcomes with CDF at least 0.70 is 1 hour. By contrast, for $\tau=0.80$, the CDF remains below the target until the jump at 3 hours, so $Q_{0.80}(x)=3$ hours.

Why the Generalized Inverse Works

The inversion follows directly from the meaning of a conditional CDF. For any candidate outcome $y$, $F(y\mid x)\ge\tau$ means that at least a fraction $\tau$ of the conditional distribution lies at or below $y$. The smallest $y$ satisfying that condition is precisely the boundary outcome that defines the $\tau$-quantile. This argument does not require a continuous outcome or a strictly increasing CDF.

Q_\tau(x)=\inf\{y:P(Y\le y\mid X=x)\ge\tau\}.

For a discrete distribution, the probability may accumulate in a jump, so $F(Q_\tau(x)\mid x)$ can exceed $\tau$. For a continuous distribution with a strictly increasing CDF, the crossing is typically exact: $F(Q_\tau(x)\mid x)=\tau$. In both cases, the definition uses the first crossing and remains well-defined.

Monotonicity Before Inversion

A true CDF is nondecreasing in its outcome argument. But distribution regression estimates separate binary-response models across thresholds, and their fitted probabilities can violate this ordering in a finite sample. For example, the estimate at a higher threshold might be slightly lower than the estimate at a lower threshold. Applying the first-crossing rule to such a sequence can produce an unstable or misleading result.

Before inversion, inspect the fitted values across thresholds. If they are not nondecreasing, use a documented monotonicity correction or a model that enforces the CDF restriction, then invert the resulting curve. A correction changes the fitted values, so its choice and its potential effect on the quantile should be reported. This is distinct from interpolating between ordered values: interpolation addresses grid resolution, not violations of the CDF condition.

Worked Example: A Nonmonotone Fitted CDF

At a fixed covariate value, suppose the hypothetical fitted probabilities at thresholds 0, 10, 20, and 30 units are 0.15, 0.62, 0.58, and 0.90. The estimate at 20 is below the estimate at 10, so these values do not form a valid CDF. For $\tau=0.61$, the raw first-crossing rule returns 10 units, but the apparent decrease from 0.62 to 0.58 shows that the fitted sequence is not coherent as a distribution.

For illustration, apply equal-weight isotonic regression to these four values. The violating adjacent pair 0.62 and 0.58 is pooled at their average, $(0.62+0.58)/2=0.60$, giving the nondecreasing sequence 0.15, 0.60, 0.60, 0.90. The first corrected value at least 0.61 is 0.90, at 30 units, so the corrected grid inverse is 30 units.

This large change illustrates why the raw inverse should not be treated as reliable when monotonicity fails. The numerical correction here is an explicit illustrative choice, not a universal prescription; different weighting or correction methods can yield different fitted curves and therefore different quantiles.

Comparing With Direct Quantile Regression

Direct quantile regression estimates a conditional quantile at a chosen level $\tau$, often by specifying a model such as $Q_\tau(x)=z(x)'\beta_\tau$ and estimating its coefficients using check loss. In contrast, distribution regression estimates conditional probabilities at outcome thresholds and recovers quantiles by inversion. The approaches target the same conditional quantile when their models correctly represent the relevant conditional distribution or quantile, but their finite-sample estimates need not coincide.

The distribution-regression route uses information across thresholds, and can recover several quantile levels from the same fitted conditional CDF. Its practical accuracy depends on threshold coverage, grid resolution, and coherence of the estimated CDF. Direct quantile regression targets selected quantiles directly and depends on the chosen quantile-regression specification. A linear quantile-regression curve and an inverted distribution-regression curve can differ because of sampling variability, functional-form restrictions, or grid approximation. Neither estimate should automatically be interpreted as causal.

Worked Example: Comparing Two Estimated Quantile Curves

Suppose a hypothetical analysis estimates the conditional median of a continuous outcome $Y$, measured in thousands of dollars, using both methods. At covariate value $x=0$, the distribution-regression CDF predictions are 0.10, 0.45, and 0.80 at $y=0$, 10, and 20, respectively. Assume for illustration that the CDF is linearly interpolated between 10 and 20. At $x=1$, the corresponding predictions are 0.05, 0.25, and 0.60.

At $x=0$, the median crossing lies between 10 and 20. The interpolated distribution-regression estimate is

10+\frac{0.50-0.45}{0.80-0.45}(20-10) =10+\frac{0.05}{0.35}(10) \approx 11.43

thousand dollars. At $x=1$, it is

10+\frac{0.50-0.25}{0.60-0.25}(20-10) =10+\frac{0.25}{0.35}(10) \approx 17.14

thousand dollars. Now suppose direct median regression gives $\widehat Q_{0.50}(x)=15+6x$, also in thousands of dollars. Its fitted medians are 15 thousand dollars at $x=0$ and 21 thousand dollars at $x=1$. These differ from the interpolated distribution-regression values by about 3.57 and 3.86 thousand dollars, respectively.

The comparison is between estimates of the same conditional median at these covariate values, not evidence by itself that either method is wrong. The distribution-regression values depend on the fitted CDF and the stated interpolation; the direct estimates depend on the linear median specification. Assessing which specification is more suitable requires examining the data and model assumptions.

Common Pitfalls and Practical Checks

Using equality instead of a first crossing. Requiring $\widehat F(y\mid x)=\tau$ can fail when the CDF jumps over $\tau$. Find the smallest $y$ with $\widehat F(y\mid x)\ge\tau$.
Treating a discrete plateau as a continuous interval of quantiles. The generalized inverse selects the leftmost value where the CDF reaches the target; it does not select the midpoint of a flat segment.
Reporting interpolation as though it were identified by the grid. State the interpolation rule and its assumptions. For discrete outcomes, preserve the support and invert the step CDF rather than interpolating across impossible outcome values.
Ignoring nonmonotone fitted probabilities. Check ordering across thresholds and document any correction before interpreting the inverse.
Comparing estimates without checking their specifications. Differences from direct quantile regression may reflect different functional forms, sampling variation, or grid approximation, not a difference in the definition of a quantile.

Key takeaway: A conditional quantile is the generalized inverse of the conditional CDF: the first outcome value at which the CDF reaches the target probability. For estimation, preserve the distinction between a grid crossing, an interpolation-based approximation, and a direct quantile-regression estimate.

Check Your Understanding

Use the generalized-inverse definition and distinguish the true CDF conditions from estimation choices.

A fitted CDF is 0.32 at $y=5$ and 0.68 at $y=10$. What does the grid imply about the location of the estimated median?
For a discrete outcome, the CDF jumps from 0.40 at values below 2 to 0.75 at 2. What is the generalized-inverse quantile for $\tau=0.60$, and why need it not satisfy $F(q\mid x)=\tau$?
A CDF reaches 0.50 at 4 and stays at 0.50 up to 8. What is the generalized-inverse quantile for $\tau=0.50$: 4, 6, or 8? Explain.
Why can separate threshold regressions produce fitted values that should not be inverted without checking them?
Give one reason an interpolated distribution-regression quantile may differ from a direct quantile-regression estimate, even when both target the same conditional quantile.