Censored Quantile Regression With A Known Censoring Point

Why a Known Censoring Point Changes Quantile Regression

In some data, values below a known threshold are not recorded separately: they are all reported at the threshold. A measured duration might be recorded as zero whenever it is nonpositive, or a quantity might be reported only down to a known detection limit. The reported distribution then has an artificial pile of observations at the censoring point. Ordinary quantile regression on the reported outcomes describes that observed distribution, but it does not automatically recover quantiles of the latent outcome.

This tutorial considers lower censoring at a known point $c$. The central distinction is between a latent conditional quantile that lies above $c$, which can be visible in the observed conditional distribution, and one at or below $c$, which may not be recoverable from it.

Model and Conditional Quantiles

Definition: Let $Y_i^*$ be an outcome that may not be fully observed, and let $c$ be a known censoring point. Under lower censoring, the observed outcome is $$ Y_i=\max(c,Y_i^*). $$ For a quantile level $0<\tau<1$, define the conditional quantile by $$ Q_\tau(Y^*\mid X=x)=\inf\{y:F_{Y^*\mid X}(y\mid x)\geq \tau\}. $$ A linear latent quantile model specifies $Q_\tau(Y^*\mid X=x)=x'\beta_\tau$, where $x$ is a vector of observed covariates and $\beta_\tau$ is the coefficient vector for quantile level $\tau$.

The censoring rule is deterministic once $Y_i^*$ and $c$ are given. This is a specific known-threshold setting: the threshold is not an unknown parameter, and this model does not cover general censoring whose threshold varies with unobserved information.

Because $y\mapsto\max(c,y)$ is nondecreasing, the observed conditional quantile is

Q_\tau(Y\mid X=x)=\max\{c,Q_\tau(Y^*\mid X=x)\} =\max(c,x'\beta_\tau).

Thus, the observed quantile equals the latent quantile when $x'\beta_\tau>c$. When $x'\beta_\tau\leq c$, the observed quantile is $c$. In that region, the observed quantile does not reveal how far below $c$ the latent quantile lies.

Worked Example: How Censoring Alters a Conditional Quantile

Suppose, hypothetically, $c=0$, $X=x$, and $Y^*\mid X=x$ is uniform on $[x-2,x+2]$. At $\tau=0.75$, the latent quantile is $x-2+0.75(4)=x+1$. The latent quantile model therefore has intercept 1 and slope 1.

At $x=-2$, the latent outcome is uniform on $[-4,0]$, so its 0.75 quantile is $-1$. Censoring maps every outcome to 0, and the observed 0.75 quantile is 0. The formula gives $\max(0,-1)=0$.

At $x=0$, the latent outcome is uniform on $[-2,2]$, with 0.75 quantile 1. Half the outcomes lie below 0 and are censored. The observed distribution has probability 0.5 at 0; its 0.75 quantile is 1, as predicted by $\max(0,1)=1$.

At $x=2$, the latent outcome is uniform on $[0,4]$, with 0.75 quantile 3. The observed 0.75 quantile is also 3. The latent quantile is visible at $x=0$ and $x=2$, but not at $x=-2$.

Identification of the Latent Quantile

A coefficient vector is identified when the observed-data distribution determines it uniquely under the model. In this setting, observations for which the latent conditional quantile exceeds $c$ are especially informative: there, the observed conditional quantile equals $x'\beta_\tau$. Covariate support in that region must be rich enough to distinguish the coefficients.

Identification condition: A useful sufficient condition is that the set of covariate values with positive probability for which $x'\beta_\tau>c$ contains enough variation to identify every component of $\beta_\tau$. A rank formulation is that $$ E\!\left[\mathbf{1}\{X'\beta_\tau>c\}XX'\right] $$ is positive definite. This condition requires informative, full-rank covariate variation in the region where the latent quantile is uncensored. It is not a universal necessary condition, but it makes clear why an entirely censored quantile region cannot identify the latent coefficients.

For regular estimation, one also commonly assumes a correctly specified linear latent conditional quantile, a unique conditional quantile in the informative region, and a conditional density that is positive and well behaved near that quantile. These conditions support stable estimation; they do not make a quantile below $c$ visible. Identification and precision are separate issues.

The restriction $x'\beta_\tau>c$ is a condition on the latent quantile, not a claim that every observed outcome at that covariate value must be uncensored. Some latent outcomes can fall below $c$ even when their $\tau$-quantile is above it.

Worked Example: Checking Where the Latent Quantile Is Visible

Suppose the hypothetical model has $x=(1,z)'$, $\beta_{0.75}=(1,1)'$, and $c=0$. Covariate values $z=-2,0,2$ give latent 0.75 quantiles $-1,1,3$, respectively. The first value is at or below the censoring point; the other two are above it.

If the population has positive probability at both $z=0$ and $z=2$, the design vectors in the uncensored-quantile region are $(1,0)'$ and $(1,2)'$. They are linearly independent: the determinant of the matrix with these rows is $2$, not 0. These two distinct design directions can distinguish an intercept from a slope in this example.

If instead the only covariate value with $x'\beta_{0.75}>0$ were $z=0$, the informative design vector would always be $(1,0)'$. Its outer products cannot provide full rank for the two coefficients. The latent intercept and slope would not both be identified by this uncensored-quantile region alone.

Powell-Style Censored Quantile Estimation

For a residual $u$, the quantile check loss is $\rho_\tau(u)=u\{\tau-\mathbf{1}(u<0)\}$. For a sample with known censoring point $c$, censored quantile regression estimates $\beta_\tau$ by minimizing

\widehat{\beta}_\tau \in \operatorname*{arg\,min}_{b} \sum_{i=1}^{n} \rho_\tau\!\left(Y_i-\max(c,X_i'b)\right).

This criterion compares each observed outcome with the model's predicted observed quantile, $\max(c,X_i'b)$, rather than with the uncensored prediction $X_i'b$. At the median, $\tau=0.5$, check loss is one half of absolute loss. Multiplying the objective by 2 does not change its minimizer, so the median estimator is Powell's censored least-absolute-deviation (CLAD) estimator:

\widehat{\beta}_{0.5} \in \operatorname*{arg\,min}_{b} \sum_{i=1}^{n}\left|Y_i-\max(c,X_i'b)\right|.

The justification starts with the quantile property of check loss. Conditional on $X=x$, if the outcome has a continuous distribution near its $\tau$-quantile $q$, the derivative of expected check loss for a proposed value $a$ is $F(a\mid x)-\tau$. It changes sign at $q$, where $F(q\mid x)=\tau$. More generally, a quantile minimizes expected check loss when $F(q-\mid x)\leq\tau\leq F(q\mid x)$. Under the specified observed-quantile model, the model's fitted value is $\max(c,x'\beta_\tau)$, motivating the sample criterion above.

Censoring makes the objective generally nonconvex in the coefficients: the fitted value is flat at $c$ for $X_i'b\leq c$, then increases with $X_i'b$. The ordinary linear-programming formulation for uncensored quantile regression therefore does not automatically solve this objective. A computation must seek a suitable global minimizer and assess whether the sample contains enough informative observations. Inference also requires appropriate methods; the point-estimation objective alone does not provide standard errors.

Worked Example: Computing a Censored Median Estimate

Suppose a hypothetical sample has $c=0$, an intercept-only model, and observed outcomes $0,0,2,4,5$. The model predicts $\max(0,b)$, so the CLAD objective is $$ L(b)=\sum_{i=1}^{5}|Y_i-\max(0,b)|. $$

When $b\leq0$, the fitted value is 0 and $L(b)=|0|+|0|+|2|+|4|+|5|=11$. When $b=2$, the fitted value is 2 and $L(2)=2+2+0+2+3=9$. When $b=4$, the fitted value is 4 and $L(4)=4+4+2+0+1=11$.

For any $b>0$, the fitted value is $q=b$, so the objective is the sum of absolute deviations from $q$. Such a sum is minimized at a sample median. The ordered outcomes are $0,0,2,4,5$, whose unique median is 2. Since 2 is above the threshold and is attainable as the fitted value, the minimizing coefficient is $\widehat b_{0.5}=2$. This is a sample estimate, not evidence by itself that the population latent median equals 2.

Worked Example: A Group With No Identified Latent Median

Consider a hypothetical intercept for each of two groups, with $c=0$ and $\tau=0.5$. Group A has observed outcomes $0,0,1,3,4$. Its observed median is 1, and its absolute-loss objective at fitted value 1 is $1+1+0+2+3=7$. The median uniquely minimizes the absolute deviations, so the fitted observed median is 1. The corresponding latent group median can be estimated in this sample as 1, above $c$.

Group B has observed outcomes $0,0,0,0,2$. At fitted value $q=0$, its absolute-loss objective is 2. For $0\leq q\leq2$, the objective is $4q+(2-q)=2+3q$, which is minimized at $q=0$. Fitted values below 0 are impossible under the censoring model because $\max(0,b)=0$ for every $b\leq0$.

Consequently, every latent intercept $b\leq0$ yields the same fitted observed median and the same minimum objective. The observed data do not distinguish among those latent medians. Reporting a unique latent median for Group B would claim information the censored distribution does not supply.

Common Mistakes and Practical Interpretation

Fitting ordinary quantile regression to the reported outcomes. This estimates a quantile of $Y$, not directly the latent quantile of $Y^*$. The threshold-induced mass at $c$ can make the two differ.
Treating a reported value at $c$ as an exact latent value. Under the model, it means $Y_i^*\leq c$; it does not reveal the latent value below the threshold.
Assuming the coefficient is identified just because $c$ is known. Knowing the threshold specifies the censoring rule, but identification still depends on informative covariate support and a quantile region above the threshold.
Using the uncensored prediction in the loss. The prediction for the observed outcome is $\max(c,X_i'b)$. Omitting the maximum fits the wrong observed-quantile model.
Assuming the optimization is an ordinary convex quantile-regression problem. The maximum creates flat and changing regions in the objective. Computation and inference must account for that structure.

The practical interpretation is bounded by what the observed distribution identifies. Where the latent conditional quantile is above $c$, the model can relate that quantile to covariates under the stated assumptions. Where it is at or below $c$, the observed quantile gives a threshold value or inequality information, not a precise latent quantile.

Check Your Understanding

Use the known-threshold model and identification conditions to answer the following.

If $c=5$ and $x'\beta_\tau=8$, what is the observed conditional quantile? What if $x'\beta_\tau=3$?
Why can many latent quantiles below $c$ produce the same observed conditional quantile?
For $\tau=0.5$, write the Powell-style censored least-absolute-deviation objective for observations $(Y_i,X_i)$.
Why is positive probability of an uncensored latent quantile not, by itself, enough to identify every coefficient?
In the group B example, what feature of the observed data prevents a unique latent median estimate?