12.2 Nonparametric Hull Methods
The production set
and the production function
is
usually unknown, but a sample of production units or decision
making units (DMU's) is available instead:
The aim of productivity analysis is to estimate
or
from the data
. Here we consider only the deterministic
frontier model, i.e. no noise in the observations and
hence
with probability
.
For example, when
the structure of
can
be expressed as:
or
where
is the frontier function, and
and
are
the random terms for inefficiency of the observed pair
for
.
The most popular nonparametric method is Data Envelopment Analysis (DEA),
which assumes that the production set is convex and free
disposable. This model is an extension of Farrel (1957)'s idea and was popularized by
Charnes, Cooper, and Rhodes (1978). Deprins, Simar, and Tulkens (1984),
assuming only free disposability on the production set, proposed a
more flexible model, say, Free Disposal Hull (FDH) model.
Statistical properties of these hull methods have been studied in
the literature. Park (2001), Simar and Wilson (2000)
provide reviews on the statistical inference of existing
nonparametric frontier models. For the nonparametric frontier
models in the presence of noise, so called nonparametric stochastic
frontier models, we refer to Simar (2003),
Kumbhakar, Park, Simar and Tsionas (2004) and references therein.
12.2.1 Data Envelopment Analysis
The Data Envelopment Analysis (DEA) of the observed sample
is defined as the smallest free disposable and convex set
containing
:
The DEA efficiency scores for a given input-output level
are obtained via (12.3):
The DEA efficient levels for a given level
are given
by (12.1) and (12.2) as:
Figure 12.4 depicts 50 simulated production units and the
frontier built by DEA efficient input levels. The simulated model is
as follows:
for
, where
denotes the exponential
distribution with mean
. Note that
.
The scenario with an exponential distribution for the logarithm of
inefficiency term and 0.75 as an average of inefficiency are reasonable
in the productivity analysis literature (Gijbels, Mammen, Park, and Simar; 1999).
Figure 12.4:
50 simulated production units (circles),
the frontier of the DEA estimate (solid line),
and the true frontier function
(dotted line).
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The DEA estimate is always downward biased in the sense that
.
So the asymptotic analysis quantifying the discrepancy
between the true frontier and the DEA estimate would be appreciated.
The consistency and the convergence rate of DEA efficiency scores
with multidimensional inputs and outputs were established
analytically by Kneip, Park, and Simar (1998). For
and
,
Gijbels, Mammen, Park, and Simar (1999) obtained its limit distribution
depending on the curvature of the frontier and the density at
the boundary. Jeong and Park (2004) and Kneip, Simar, and Wilson (2003)
extended this result to higher dimensions.
12.2.2 Free Disposal Hull
The Free Disposal Hull (FDH) of the observed sample
is
defined as the smallest free disposable set containing
:
We can obtain the FDH estimates of efficiency scores for a given
input-output level
by substituting
with
in
the definition of DEA efficiency scores. Note that, unlike DEA
estimates, their closed forms can be derived by a straightforward
calculation:
where
is the
th component of a vector
. The efficient
levels for a given level
are obtained by the same way
as those for DEA. See Figure 12.5 for an illustration by
a simulated example:
for
. Park, Simar, and Weiner (1999) showed that the
limit distribution of the FDH estimator in a multivariate setup is
a Weibull distribution depending on the slope of the frontier
and the density at the boundary.
Figure 12.5:
50 simulated production units (circles)
the frontier of the FDH estimate (solid line),
and the true frontier function
(dotted line).
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