‚X(y) = {x | x ∈ X(y), θx ∈ X(y), 0 < θ < 1}

/

and for Y (x) = {0},

‚Y (x) = {y | y ∈ Y (x), »y ∈ X(y), » > 1}.

/

For y = 0 we have ‚X(y) = {0}, and for Y (x) = {0} ‚Y (x) = {0}.

Returns to scale is a characteristic of the surface of the production set. The

production set exhibits constant returns to scale (CRS) if, for ± ≥ 0 and P ∈ Ψ,

±P ∈ Ψ; it exhibits non-increasing returns to scale (NIRS) if, for 0 ¤ ± ¤ 1

and P ∈ Ψ, ±P ∈ Ψ; it exhibits non-decreasing returns to scale (NDRS) if,

for ± ≥ 1 and P ∈ Ψ, ±P ∈ Ψ. Especially, a convex production set exhibits

non-increasing returns to scale, but note that the converse is not true.

Recall that, given a production set Ψ with the scalar output y, the production

function g can also be de¬ned for x ∈ Rp :

+

g(x) = sup{y | (x, y) ∈ Ψ}.

It may be de¬ned via the input set and the output set as well:

g(x) = sup{y | x ∈ X(y)} = sup{y | y ∈ Y (x)}.

We can interpret returns to scale in terms of the production function g:

• Constant returns to scale: For all t ≥ 0, g(tx) = tg(x).

• Non-decreasing returns to scale: For all t ≥ 1, g(tx) ≥ tg(x)

• Non-increasing returns to scale: For all t ≥ 1, g(tx) ¤ tg(x).

How can one evaluate the performance of a given pair of input and output?

When the outputs are scalars, one may do it with input x and output y by

computing g(x)’y or y/g(x). This is usually inadequate though when multiple

inputs or outputs are incorporated. In order to tackle this problem radial

5.2 Nonparametric Hull Methods 93

e¬ciency measures have been proposed. For a given input-output point (x0 , y0 ),

its input e¬ciency is de¬ned as

θIN (x0 , y0 ) = inf{θ | θx0 ∈ X(y0 )}.

The e¬cient level of input corresponding to the output level y0 is then given

by

x‚ (y0 ) = θIN (x0 , y0 )x0 .

Note that x‚ (y0 ) is the intersection of ‚X(y0 ) and the ray θx0 , θ > 0. The

output e¬ciency score θOUT (x0 , y0 ) can be de¬ned similarly:

θOUT (x0 , y0 ) = sup{θ | θy0 ∈ Y (x0 )}.

The e¬cient level of output corresponding to the input level x0 is given by

y ‚ (x0 ) = θOUT (x0 , y0 )y0 .

5.2 Nonparametric Hull Methods

5.2.1 An Overview

The production set Ψ and the production function g is usually unknown, but

typically a sample of production units or decision making units (DMU™s) is

available instead:

X = {(xi , yi ), i = 1, . . . , n}.

The problem of productivity analysis is to estimate Ψ or g from the data X .

The most popular nonparametric method is based on data envelopment anal-

ysis (DEA), which is an extension of Farrel (1957)™s idea and was popularized

in a modern operational research fashion by Charnes, Cooper, and Rhodes

(1978). Deprins, Simar, and Tulkens (1984), who extended the DEA to the

more ¬‚exible class of production sets, introduced the free disposal hull (FDH)

model dropping convexity assumption for the production set.

Statistical properties of these hull methods such as consistency and minimax

rate of convergence have been studied in the literature. Park (2001) and Simar

and Wilson (1999) provide insightful reviews on the statistical inference of

existing nonparametric frontier models.

94 5 Nonparametric Productivity Analysis

5.2.2 Data Envelopment Analysis

The data envelopment analysis (DEA) of the observed sample X is de¬ned as

the smallest free disposable and convex set containing X :

n n

Rp+q

= {(x, y) ∈ |x ≥ γi xi , y ¤

ΨDEA γi yi

+

i=1 i=1

n

γi = 1, γi ≥ 0 ∀i},

for some (γ1 , . . . , γn ) such that

i=1

where the inequalities between vectors are understood componentwise. A set

Ψ is free disposable, if for given (x, y) ∈ Ψ all (x , y ) with x ≥ x and y ¤ y

belong to Ψ. Free disposability is generally assumed in economic modeling.

The e¬ciency scores for a given input-output level (x0 , y0 ) are estimated by :

θIN (x0 , y0 ) min{θ > 0 | (θx0 , y0 ) ∈ ΨDEA };

=

θOUT (x0 , y0 ) = max{θ > 0 | (x0 , θy0 ) ∈ ΨDEA }.

The e¬cient levels for a given level (x0 , y0 ) are estimated by

x‚ (y0 ) = θIN (x0 , y0 )x0 ; y ‚ (x0 ) = θOUT (x0 , y0 )y0 .

The consistency and the convergence rate of this radial e¬ciency score with

multidimensional inputs and outputs were established analytically by Kneip,

Park, and Simar (1998). For p = 1 and q = 1, Gijbels, Mammen, Park, and

Simar (1999) obtained its limit distribution depending on the curvature of the

frontier and the density at boundary. Jeong and Park (2002) extended this

result to higher dimension.

5.2.3 Free Disposal Hull

The free disposal hull (FDH) of the observed sample X is de¬ned as the smallest

free disposable set containing X :

p+q

ΨFDH = {(x, y) ∈ R+ | x ≥ xi , y ¤ yi , i = 1, . . . , n}.

One de¬ne the estimators of e¬ciency scores by FDH in a similar way to DEA.

By a straightforward calculation we obtain their closed forms for a given input-

5.3 DEA in Practice : Insurance Agencies 95

output level (x0 , y0 ) :

xj

IN i

θ (x0 , y0 ) = min max ;

xj

i:y¤yi 1¤j¤p

0

k

yi

OUT

θ (x0 , y0 ) = max min ,

k

y0

i:x≥xi 1¤k¤q

where v j is the jth component of a vector v. The e¬cient levels for a given

level (x0 , y0 ) are estimated by the same way as those for DEA. Park, Simar,

and Weiner (1999) showed that the limit distribution of the FDH estimator

with multivariate setup is a Weibull distribution depending upon the slope of

the frontier and the density at boundary.