Conditional Quantiles and Other Response Functionals

From a Conditional Distribution to a Target

A conditional response distribution describes the possible values of \(Y\) at a given feature value \(x\). Often, however, a prediction or scientific summary is a single number rather than a whole distribution. The choice of number matters: a conditional mean describes the center under squared-error loss, while a conditional median describes a center under absolute-error loss. A conditional quantile describes a specified point in the distribution, such as its 90th percentile.

Write \(F_x(y)=P(Y\leq y\mid X=x)\) for the conditional distribution function of the response. When a conditional density or mass function exists, write it as \(f(y\mid x)\). The targets below are calculated from this conditional law, not from the marginal distribution of \(Y\). For feature values \(x\) at which a conditional law is specified, each target is a functional of that law.

The generalized-inverse convention is important when the conditional law has point masses. A quantile need not be a value where the distribution function equals \(\tau\): it is the first response value at which the distribution function reaches or exceeds \(\tau\). The defining inequalities are \(F_x(q_\tau(x)-)\leq\tau\leq F_x(q_\tau(x))\).

Other Conditional Functionals

The conditional mean and variance summarize location and spread, respectively. When the relevant moments exist, define

The mean requires a finite first absolute moment, \(E[|Y|\mid X=x]<\infty\); the variance requires a finite second moment. A conditional median and quantile require no moments. This distinction matters for heavy-tailed laws, where a median may be well-defined even if a mean is not.

A mode is another possible summary: for a discrete law, it is a response value with greatest conditional probability; for a density, it is a point where the density attains its maximum. Unlike quantiles, a density mode can depend on the chosen density representation and need not be unique. It is not generally a measure of central location. In practice, the appropriate functional is determined by the question and the consequences of prediction errors.

Quantiles and Their Loss Function

Quantiles are not just descriptive cutoffs. They are optimal predictions for an asymmetric absolute-error criterion. Define the pinball loss at level \(\tau\) by \(\rho_\tau(u)=u(\tau-\mathbf{1}\{u<0\})\), where \(\mathbf{1}\{\cdot\}\) is one when its condition holds and zero otherwise. For an overprediction \(a-Y<0\), this loss charges at rate \(1-\tau\); for an underprediction \(a-Y>0\), it charges at rate \(\tau\).

Proof. For fixed \(y\), as a function of \(a\), \(\rho_\tau(y-a)\) is continuous and piecewise linear. Its slope, except at \(a=y\), is \(\mathbf{1}\{y<a\}-\tau\). Consequently, for \(a>b\),

The integrand is bounded, so expectation and integration can be interchanged on the finite interval \([b,a]\). Taking expectations gives

The quantile inequalities are \(F(q_\tau-)\leq\tau\leq F(q_\tau)\). If \(t>q_\tau\), monotonicity gives \(F(t-)\geq F(q_\tau)\geq\tau\). Therefore, for \(a>q_\tau\), the integral formula yields \(R(a)-R(q_\tau)\geq0\). If \(t<q_\tau\), then \(F(t)\leq F(q_\tau-)\leq\tau\), so \(F(t-)\leq\tau\). Applying the formula with \(b=a\) and upper endpoint \(q_\tau\) shows \(R(q_\tau)-R(a)\leq0\) whenever \(a<q_\tau\). Thus \(R(q_\tau)\leq R(a)\) for every \(a\), proving the result. \(\square\)

At \(\tau=1/2\), pinball loss is one-half of absolute-error loss: \(\rho_{1/2}(u)=|u|/2\). Hence the median minimizes expected absolute error. In contrast, the mean minimizes expected squared error.

Derivation. Let \(\mu=E[Y]\). Expanding \(Y-a=(Y-\mu)+(\mu-a)\), then using \(E[Y-\mu]=0\), gives

The final term is nonnegative and equals zero precisely when \(a=\mu\), so the mean minimizes squared-error risk. Applying these results to the conditional distribution at \(x\) gives the corresponding conditional targets: the conditional mean minimizes conditional squared-error risk, and a conditional median minimizes conditional absolute-error risk, when the required moments exist. \(\square\)

Continuous Conditional Laws

For a continuous conditional law with strictly increasing distribution function, \(q_\tau(x)\) is the unique solution to \(F_x(q_\tau(x))=\tau\). If a conditional density is available, the distribution function is obtained by integrating that density. In familiar parametric families, quantiles can often be expressed using a standard quantile function.

Interpreting the Target Carefully

A response distribution can support several legitimate targets, and they answer different questions. The mean is sensitive to the magnitude of extreme values; the median is more resistant. A high quantile can describe an upper-tail outcome without claiming that the response is usually near that value. Neither the mean nor a quantile recovers an unobserved “true” response for an individual: each summarizes the conditional law.

For discrete or mixed laws, do not assume that a quantile solves \(F_x(y)=\tau\). The correct condition allows a jump across \(\tau\). Also, do not assume the median is always a unique point. For example, if a conditional response equals \(-2\) or \(4\), each with probability \(1/2\), then every \(m\in[-2,4]\) is a median: for such \(m\), both \(P(Y\leq m\mid X=x)\) and \(P(Y\geq m\mid X=x)\) are at least \(1/2\). Under the lower generalized-inverse convention, the median quantile is \(-2\).

Finally, a conditional distribution is generally specified only up to changes on feature values that occur with probability zero. Its functionals inherit that qualification: targets are determined for feature values where the conditional law is specified, and versions that agree almost surely give the same targets almost surely. This is why a conditional target should be understood as a feature-dependent summary of a conditional response law, rather than an automatically unique value at every conceivable \(x\).