Quantile Identification With Atoms and Ties

Why Atoms and Ties Change Quantile Identification

Quantile regression is often introduced through a score equation: at a fitted quantile, the fraction of observations below it should balance the target quantile level. That intuition is exact when the outcome distribution is continuous at the quantile. With discrete outcomes, rounded measurements, or repeated values, however, probability can accumulate at a single point. The score then has a jump rather than passing smoothly through zero. A different issue arises when the distribution function is flat across a gap in the support: several candidate values can minimize the same quantile loss.

The distinction matters because a quantile can be defined by a precise convention even when the minimizer of check loss is not unique. The generalized inverse selects one value; the score condition describes the full set of values that minimize the population loss. These are related objects, but they need not be identical as sets.

Definition: Let $F(y\mid x)=P(Y\leq y\mid X=x)$ be a conditional CDF and let $0<\tau<1$. The lower generalized-inverse conditional quantile is $$ Q_\tau(x)=\inf\{y:F(y\mid x)\geq \tau\}. $$ The infimum convention selects the leftmost value at which the CDF reaches or exceeds $\tau$. The set of check-loss minimizers is a separate object and can contain more than this selected value.

The Score Condition With a Possible Tie

For a scalar candidate $q$, the check loss is $\rho_\tau(Y-q)$, where $\rho_\tau(u)=u(\tau-I(u<0))$. Consider its population expectation as a function of $q$. When $Y\ne q$, the derivative is $-\tau$ if $Y>q$ and $1-\tau$ if $Y<q$. At $Y=q$, the loss has a corner, so its derivative is an interval rather than a single number.

The subgradient of the expected loss is therefore the interval from $P(Y<q)-\tau$ to $P(Y\leq q)-\tau$. A convex function is minimized at $q$ precisely when zero belongs to its subgradient. This gives the quantile score condition below.

Theorem: A value $q$ minimizes $E[\rho_\tau(Y-q)\mid X=x]$ if and only if $$ F(q-\mid x)\leq \tau\leq F(q\mid x), $$ where $F(q-\mid x)=P(Y<q\mid X=x)$. Equivalently, there is some $\gamma\in[0,1]$ such that $$ P(Y<q\mid X=x)+\gamma P(Y=q\mid X=x)=\tau. $$ Here $\gamma$ represents a permissible subgradient contribution from probability tied at $q$; it is not a claim that a tied observation has a fractional outcome.

The condition follows directly from the subgradient. At $Y=q$, the derivative with respect to $q$ can be any value in $[-\tau,1-\tau]$. Averaging the lower and upper subgradient endpoints gives $F(q-\mid x)-\tau$ and $F(q\mid x)-\tau$. Zero belongs to this interval exactly when the displayed inequalities hold. Equivalently, choose a tie contribution $\gamma P(Y=q\mid X=x)$ to raise the probability strictly below $q$ to $\tau$. This argument also explains why replacing the inequality with $F(q\mid x)=\tau$ is generally wrong when there is an atom.

What an Atom Does—and Does Not—Imply

An atom at $q$ means $P(Y=q\mid X=x)>0$, so the conditional CDF jumps at $q$. If $\tau$ falls strictly inside that jump, then $F(q-\mid x)<\tau<F(q\mid x)$. The score condition holds at $q$, and values just below or above it do not satisfy the condition. The quantile minimizer is unique in this case, even though the CDF is discontinuous.

If $\tau$ equals one endpoint of the jump, a flat region adjoining the atom can make the minimizer set larger. More generally, the generalized inverse remains a well-defined selected quantile, but the score condition can hold throughout an interval. Thus “there is an atom” does not by itself mean “the quantile is nonunique.” Nonuniqueness occurs when there is a range of candidate values over which the CDF stays at the target level, including possible boundary points.

Worked Example: A Quantile Inside an Atom

Suppose a hypothetical conditional distribution has $P(Y=0\mid X=x)=0.20$, $P(Y=1\mid X=x)=0.50$, and $P(Y=3\mid X=x)=0.30$. Find the generalized-inverse quantile and check the score condition at $\tau=0.60$.

The CDF reaches $0.20$ at 0 and $0.70$ at 1. Therefore $$ Q_{0.60}(x)=\inf\{y:F(y\mid x)\geq 0.60\}=1. $$ At $q=1$, $F(1-\mid x)=0.20$ and $F(1\mid x)=0.70$, so $$ 0.20\leq 0.60\leq 0.70. $$ The score condition holds. In the tie-weight form, $\gamma$ must satisfy $$ 0.20+\gamma(0.50)=0.60, \qquad \gamma=\frac{0.40}{0.50}=0.80. $$ Thus the probability strictly below 1 falls short of the target, and the atom supplies the remaining $0.40$ of probability needed by the condition. Since the target lies strictly between the two CDF values, the minimizer is uniquely 1.

Flat Regions and Set-Valued Minimizers

A CDF is flat on an interval containing no probability mass. If it is flat at exactly $\tau$, every value in that region satisfies the score condition: the probability below the candidate and the probability at or below it both equal $\tau$. Check loss has zero slope throughout the region, so it has the same expected value at all of those candidates. The lower generalized inverse chooses the left endpoint under its infimum convention, while the full set of minimizers can extend to the right.

Worked Example: A Flat CDF at the Target Level

Suppose $P(Y=0)=0.40$ and $P(Y=2)=0.60$, with no other possible values. At $\tau=0.40$, the generalized inverse is $Q_{0.40}=0$, since $F(0)=0.40$. Yet every $q\in[0,2]$ satisfies the score condition. At $q=0$, the CDF bounds are $F(0-)=0$ and $F(0)=0.40$. For $0<q<2$, both bounds equal $0.40$; at $q=2$, they are $0.40$ and 1. Hence zero belongs to the loss subgradient at every such $q$.

The equality of losses can also be checked at the endpoints. At $q=0$, only the outcome 2 contributes, giving $$ E[\rho_{0.40}(Y-0)]=0.60(0.40)(2)=0.48. $$ At $q=2$, only the outcome 0 contributes, giving $$ E[\rho_{0.40}(Y-2)]=0.40(0.60)(2)=0.48. $$ Between 0 and 2, the expected loss has zero slope, so it remains $0.48$. The selected generalized-inverse quantile is 0, but the set of loss minimizers is the whole interval $[0,2]$.

Empirical Ties and the Regression Score

The same distinction appears in a sample. The empirical CDF is a step function, and repeated observed outcomes create jumps. Its generalized-inverse quantile selects the smallest observed-scale value where the empirical CDF reaches $\tau$. The empirical check-loss minimizers, however, are exactly those $q$ for which the empirical CDF immediately below $q$ is at most $\tau$ and the empirical CDF at $q$ is at least $\tau$. If the empirical CDF equals $\tau$ across a gap, all points in the gap minimize the empirical loss.

Worked Example: Tied Observations and a Nonunique Empirical Minimizer

Consider the hypothetical sample $1,1,1,4,9$ and $\tau=0.60$. The empirical CDF at 1 is $3/5=0.60$, so the lower generalized-inverse empirical quantile is 1. For any $q$ strictly between 1 and 4, exactly three of the five observations are below $q$, giving empirical CDF $0.60$. The score condition therefore holds throughout that gap, and the empirical loss is minimized on $[1,4]$.

For a numerical check, at $q=1$, the three observations equal to 1 have zero loss. The other two yield $$ \sum_{i=1}^{5}\rho_{0.60}(Y_i-1) =(0.60)(4-1)+(0.60)(9-1) =1.80+4.80=6.60. $$ At $q=4$, the three observations equal to 1 contribute $3(0.40)(4-1)=3.60$, and the observation equal to 9 contributes $(0.60)(9-4)=3.00$. The total is again $6.60$. This confirms that the endpoints, and hence the entire interval between them, minimize the empirical loss.

For a conditional quantile, apply the scalar condition at each covariate value $x$. In a linear quantile regression with candidate index $X'\beta$, the corresponding score has covariate weights: the subgradient contribution takes the form $X\{I(Y<X'\beta)+\gamma I(Y=X'\beta)-\tau\}$, with an admissible tie contribution $\gamma\in[0,1]$. A minimizing coefficient vector must make zero belong to the aggregate subgradient. This is a vector condition, not a separate scalar equality for every coefficient. Ties can make the score set-valued, and a flat loss direction can leave multiple coefficient vectors minimizing the objective. The pointwise conditional quantile convention alone does not guarantee that a chosen regression specification has a unique coefficient vector.

Common Mistakes and Practical Interpretation

Requiring $F(q\mid x)=\tau$ in every case. At an atom, the correct condition is $F(q-\mid x)\leq\tau\leq F(q\mid x)$. The CDF can jump over $\tau$.
Calling every atom a source of nonuniqueness. If $\tau$ lies strictly inside the jump, the atom can identify a unique minimizing value. Check whether the CDF is flat at the target level before concluding that the minimizer is an interval.
Equating the generalized inverse with the entire minimizer set. The generalized inverse selects one value by convention. The check-loss condition can hold at additional values, especially across a flat region.
Treating a tie contribution as a fractional observation. The $\gamma$ term represents the range of valid subgradients at a loss corner; it does not alter the observed outcome or assign it a fractional value.
Interpreting a zero regression score as unique identification. A zero-subgradient condition establishes minimization under the relevant convexity conditions. It does not alone establish that no other value or coefficient vector also minimizes the objective.

These distinctions are useful in discrete-outcome applications, duration data recorded on a coarse time scale, and finite samples with repeated values. They also clarify what an estimated quantile reports: under the lower generalized-inverse convention it is the first value reaching the requested probability, while the underlying check-loss objective may not distinguish it from nearby values in a flat region.

Key takeaway: At a conditional quantile $q$, the score condition is an inequality across the probability jump, not necessarily an equality of the CDF to $\tau$. An atom can bracket the target and yield a unique minimizer; a flat CDF region at the target can yield many minimizers. State whether you mean the generalized-inverse selection or the full set of check-loss minimizers.

Check Your Understanding

Use the CDF bounds and the check-loss score condition in each answer.

If $F(q-\mid x)=0.35$, $F(q\mid x)=0.65$, and $\tau=0.50$, does $q$ satisfy the quantile score condition? What does the jump imply about uniqueness if it strictly brackets $\tau$?
A CDF remains equal to $0.25$ from $a$ up to, but not including, $b$. For $\tau=0.25$, why can the check-loss minimizer set include values throughout $[a,b]$?
What value does the lower generalized inverse select when $F$ first reaches $\tau$ at the left endpoint of a flat region?
In the tie-weight condition, what does $\gamma$ represent, and why should it not be interpreted as changing an observed outcome?
Why does a zero aggregate subgradient in quantile regression not, by itself, prove that the coefficient vector is unique?