Conical-Hull Estimators in DEA

• Conical-hull estimators in DEA are nonparametric methods that construct the production frontier under constant returns to scale using a conical extension of input-output data.
• They achieve faster convergence rates by reducing the effective dimensionality of the estimation problem compared to convex-hull and FDH approaches.
• Simulation-based bias correction facilitates robust efficiency analysis by providing precise confidence intervals and reducing finite-sample bias.

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 (Park et al., 2010, Daouia et al., 2010).

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

$$\hat{\lambda}(\mathbf{x}_0, \mathbf{y}_0) = \sup \{ \lambda > 0 : (\mathbf{x}_0, \lambda \mathbf{y}_0) \in \widehat{\mathcal{T}_n} \}$$

(Park et al., 2010).

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 $\hat{g}(\mathbf{x}_0)$ (Park et al., 2010).

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 (Park et al., 2010).

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 $b_1, b_2$ (Park et al., 2010).

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 (Daouia et al., 2010).

Key comparison points:

Estimator Class	Rate	Limiting Distribution
DEA–CRS (conical-hull)	$O_p(n^{-2/(p+1)})$	Gaussian (support function of random hull) (Park et al., 2010)
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 (Park et al., 2010).

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 (Daouia et al., 2010).

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 (Park et al., 2010, Daouia et al., 2010).