---
title: Conical-Hull Estimators in DEA
url: https://www.emergentmind.com/topics/conical-hull-estimators-in-dea
type: topic
---

# Conical-Hull Estimators in DEA

Conical-hull estimators in Data Envelopment Analysis (DEA) are nonparametric estimators of the attainable technology set under assumptions such as constant or variable returns to scale. In contrast to classical convex-hull methods appropriate for variable returns-to-scale (VRS) technologies, conical-hull approaches are requisite under the more restrictive and prevalent constant returns-to-scale (CRS) hypothesis. The mathematical underpinning of conical-hull estimators, their asymptotic behavior, statistical performance, and their relationship to free disposal hull (FDH) estimators and extreme value theory (EVT) are central to quantitative efficiency analysis in productive systems [1010.0312], [1011.5722].

## 1. Definition and Structure of Conical-Hull Estimators

Let $\mathbf{x}\in\mathbb{R}_+^p$ denote a vector of nonnegative inputs and $y\in\mathbb{R}_+$ an output. The CRS technology set is defined as
\[
\mathcal{T} = \left\{ (\mathbf{x}, y) \in \mathbb{R}_+^{p+1} : 0 \le y \le g(\mathbf{x}) \right\},
\]
where the boundary function $g: \mathbb{R}_+^p \to \mathbb{R}_+$ is convex and homogeneous of degree one ($g(a\mathbf{x}) = a g(\mathbf{x})$ for all $a>0$). Geometrically, $\mathcal{T}$ is the conical extension (with vertex at the origin) of the set $A = \{ (\mathbf{x}, y) : 0 \le y \le g(\mathbf{x}), \|\mathbf{x}\| = 1 \}$.

Given i.i.d. data $\{(\mathbf{X}_i, Y_i)\}_{i=1}^n$, the conical-hull DEA estimator under CRS is constructed as the smallest convex cone through the origin containing the rays
\[
R_i = \left\{ (\gamma \mathbf{X}_i, \gamma Y_i) : \gamma \ge 0 \right\}.
\]
The sample estimator of $\mathcal{T}$ is
\[
\widehat{\mathcal{T}_n} = \mathrm{conv} \left( \{ R_1, \ldots, R_n \} \cup \{ (\mathbf{x}, 0) : \mathbf{x} \in \mathbb{R}_+^p \} \right),
\]
and the estimated boundary at $\mathbf{x}_0$ is
\[
\hat{g}(\mathbf{x}_0) = \sup \{ y>0 : (\mathbf{x}_0, y) \in \widehat{\mathcal{T}_n} \}.
\]
For multiple outputs ($q>1$), the estimator targets the directional edge
\[
\lambda(\mathbf{x}, \mathbf{y}) = \sup \{ \lambda > 0: (\mathbf{x}, \lambda \mathbf{y}) \in \mathcal{T} \},
\]
with the analogous estimator
\[
\hat{\lambda}(\mathbf{x}_0, \mathbf{y}_0) = \sup \{ \lambda > 0 : (\mathbf{x}_0, \lambda \mathbf{y}_0) \in \widehat{\mathcal{T}_n} \}
\]
[1010.0312].

## 2. Asymptotic Properties and Rates of Convergence

Under standard regularity (twice continuous differentiability of $g$ near $\mathbf{x}_0$ and positivity of the sampling density at the boundary), the conical-hull estimator under CRS satisfies
\[
\hat{g}(\mathbf{x}_0) - g(\mathbf{x}_0) = O_p\left(n^{-2/(p+1)}\right),
\]
whereas the convex-hull estimator for the VRS case achieves only $O_p\left(n^{-2/(p+2)}\right)$. This improvement arises because homogeneity reduces the effective dimension of the estimation problem. The convergence rate for the edge $\hat{\lambda}(\mathbf{x}_0, y_0)$ is identical to that of $\hat{g}(\mathbf{x}_0)$ [1010.0312].

## 3. Limiting Distributions and Practical Simulation

The limiting distribution of $n^{2/(p+1)} \{ \hat{g}(\mathbf{x}_0) - g(\mathbf{x}_0) \}$ is characterized as the support function of a random convex hull of points $(\mathbf{V}_{2i}, W_i)$ uniformly distributed on a deterministic region $R_n(\kappa)$ in a transformed coordinate system. The key constant $\kappa$ incorporates the local geometry of the boundary (via the Hessian of $g$ at $\mathbf{x}_0$) and the boundary density. The limit law can be simulated in practice by:

- Plug-in estimation of interface density and Hessian,
- Generating i.i.d. samples in $R_n(\hat{\kappa})$,
- Linear programming to compute sample realizations of the support function,
- Using the empirical distribution of these values for inference and bias correction.

The procedure is directly extendable to directions in the output space for $q>1$ by appropriate orthogonal transformation [1010.0312].

## 4. Bias Correction and Confidence Interval Construction

Simulation of the limit law enables estimation of the mean asymptotic bias. The bias-corrected estimator is
\[
\tilde{g}(\mathbf{x}_0) = \hat{g}(\mathbf{x}_0) - n^{-2/(p+1)} \bar{Z}_n,
\]
where $\bar{Z}_n$ is the average simulated support-function value. For directional edges, the adjustment is $\tilde{\lambda}(\mathbf{x}_0, \mathbf{y}_0) = \tilde{g}(\mathbf{x}_0) / y_0$. Confidence intervals for the boundary function at a point are constructed using the empirical quantiles of the simulated distribution:
\[
\left[ \hat{g}(\mathbf{x}_0) - n^{-2/(p+1)} Z^{(b_1)}_n(0),\; \hat{g}(\mathbf{x}_0) - n^{-2/(p+1)} Z^{(b_2)}_n(0) \right]
\]
for appropriate order statistics $b_1, b_2$ [1010.0312].

## 5. Comparison with FDH and EVT-Based Estimators

The FDH estimator defines the attainable set via the union of orthants anchored at each observation, generating a conical hull under free disposability. The FDH boundary at $x$ is
\[
\hat{g}_n(x) = \max_{i : X_i \le x} Y_i.
\]
Extreme-value theory (EVT) provides an exact asymptotic characterization for FDH: under regular variation, the rate and limit law are Weibull-type, and the rate is driven by the right tail parameter $\rho_x$. However, the non-Gaussian limit and high sensitivity to outliers motivate robust, quantile-based, and asymptotically Gaussian estimators $\tilde{g}^*_n(x)$, defined through order statistics and EVT tail index estimates [1011.5722].

Key comparison points:

| Estimator Class           | Rate                       | Limiting Distribution            |
|--------------------------|----------------------------|----------------------------------|
| DEA–CRS (conical-hull)   | $O_p(n^{-2/(p+1)})$        | Gaussian (support function of random hull) [1010.0312] |
| DEA–VRS (convex-hull)    | $O_p(n^{-2/(p+2)})$        | Gaussian (but slower rate)       |
| FDH                      | $(n\ell_x)^{-1/\rho_x}$    | Weibull-type (non-Gaussian)      |
| EVT–robust frontier      | $O_p\left(\left(k_n\right)^{-1/2}\right)$ | Gaussian                        |

## 6. Implementation and Simulation Evidence

Practical simulation for bias correction and frequency interpretation of quantiles is recommended due to otherwise slow convergence and persistent finite-sample bias. Monte Carlo evidence (for $p=q=2$, Cobb–Douglas frontier, $n=100,400$) demonstrates that the bias-corrected estimator $\tilde{\lambda}_0$ outperforms the naive DEA–CRS estimator: median squared error is reduced to approximately 64–82% of its uncorrected counterpart. This improvement persists across moderate and large sample sizes, confirming both the substantial finite-sample bias and the effectiveness of simulation-based correction [1010.0312].

For FDH- and EVT-based smooth conical-hull estimators, robust versions display superior finite-sample bias and mean squared error properties, achieving nominal confidence coverage and strong resistance to outlier contamination in both simulation and empirical socioeconomic data [1011.5722].

## 7. Extensions and Contextual Discussion

The conical-hull estimator's framework readily generalizes from the scalar-output case ($q=1$) to higher dimensions, requiring only coordinate transformations. Its strict reliance on CRS is both a strength (enhanced rate, reduced effective dimension) and a limitation (restricting applicability in non-CRS environments). By contrast, FDH and monotone hull estimators are applicable under the more general free-disposability assumption, but incur slower asymptotics and less regular limit theory.

A plausible implication is that, in efficiency analysis where homogeneity of the technology set can be defensibly imposed, conical-hull estimators should be preferred for both statistical efficiency and interpretability. Where robustness to outliers or coverage accuracy is critical, quantile-based or EVT-corrected frontiers are demonstrably advantageous [1010.0312], [1011.5722].

Source: https://www.emergentmind.com/topics/conical-hull-estimators-in-dea