1. INTRODUCTION
An important problem in productivity and efficiency analysis is to
characterize and to estimate the production frontier, i.e., the set of the
most efficient production process. The idea is to analyze how firms
combine their inputs to produce in an efficient way the output. We are
then interested in the production frontier because it represents a
reasonable benchmark value or reference frontier. Let us introduce the
basic concepts and notation.
According to economic theory (Koopmans,
1951; Debreu, 1951; Shephard, 1970), the production set, where the
activity is described through a set of p inputs
used to produce an univariate output
,
is defined as the set of physically attainable points
(x,y):
This set can be described mathematically by its sections
where, for any level of inputs x, the requirement set
Y(x) represents the set of all outputs that a firm can
produce using x as inputs. Assuming that Ψ is compact, the
maximal achievable level of output for a given level of inputs x
defines the output-efficient function ∂Y(x) = max
Y(x). From an economic point of view, this function is
supposed monotone nondecreasing, and it is then called the production
function and its graph, which represents the efficient boundary of Ψ,
is called the production frontier. Other different assumptions can be
assumed on Ψ, such as free disposability, i.e., if
(x,y) ∈ Ψ then
(x′,y′) ∈ Ψ for any
x′ ≥ x and y′ ≤ y;
or convexity, i.e., every convex combination of feasible production plans
is also feasible; or no free lunch, i.e., for all y > 0 we
have y ∉ Y(0) (see, e.g., Shephard, 1970).
The production process, which generates observations
χn =
{(Xi,Yi)|
i = 1,…,n} is defined, e.g., through the joint
distribution of a random vector (X,Y) on
,
where X represents the inputs and Y is the output. In
the case where Ψ is equal to the support of the distribution of
(X,Y), another way to define the production frontier is
given as follows. The production function, which we denote from now on by
φ, is characterized for a given level of inputs x by the
upper boundary of the support of the conditional distribution of
Y given X ≤ x, i.e.,
where F(·/x) =
F(x,·)/FX(x)
is the conditional distribution function of Y given X
≤ x, with F being the joint distribution function of
(X,Y) and FX the marginal
distribution function of X. It is supposed here that
FX(x) > 0 or that x is
an interior point of the support of the distribution of X. The
inequality X ≤ x has to be understood componentwise.
As a matter of fact, the function φ is the smallest monotone
nondecreasing function that is larger than or equal to the
output-efficient function ∂Y(.). Its graph defines the
production frontier. If the efficient boundary of Ψ is monotone
nondecreasing (a quite reasonable assumption in practice), it coincides
with the production frontier. So, we have, in some sense, just
reparametrized the definition of the efficient frontier of Ψ. This
new formulation of the production frontier is due to Cazals, Florens, and
Simar (2002).
A large amount of literature is devoted to the estimation of the
production frontier from a random sample of production units
χn. Two different approaches have been mainly
developed: the deterministic frontier models, which suppose that with
probability one, all the observations in χn
belong to Ψ, and the stochastic frontier models, where random noise
allows some observations to be outside of Ψ.
In deterministic frontier models, there are mainly two nonparametric
methods based on envelopment techniques: the free disposal hull (FDH) and
the data envelopment analysis (DEA). The FDH estimator was introduced by
Deprins, Simar, and Tulkens (1984) and relies
only on the free disposability assumption on Ψ. The DEA estimator,
which was initiated by Farrell (1957) and
popularized as a linear programming estimator by Charnes, Cooper, and
Rhodes (1978), requires stronger assumptions: it
relies on the free disposability assumption and the convexity of Ψ.
Note that the convexity assumption is widely used in economics but it is
not always valid. The production set might admit increasing returns to
scale, i.e., the output increases faster than the inputs, or there might
be lumpy goods, i.e., fractional values of inputs or outputs do not exist.
Hence, the FDH is a more general estimator than the DEA. The asymptotic
distribution of the FDH estimator was derived by Park, Simar, and Weiner
(2000) in the case of multivariate input and
output, and the asymptotic distribution of the DEA estimator was derived
by Gijbels, Mammen, Park, and Simar (1999) in
the univariate case. The statistical theory of these estimators is now
mostly available. See Simar and Wilson (2000)
for a recent survey of the available results.
In stochastic frontier models, where noise is allowed, only parametric
restrictions on the shape of the frontier and on the data generating
process allow identification of the noise from the efficiency frontier and
estimation of this frontier. Aigner, Lovell, and Schmidt (1977), Meeusen and van den Broek (1977), Olsen, Schmidt, and Waldman (1980), Stevenson (1980), and
Battese and Coelli (1988) specified a model for
the production function and a specific distributional form for the error
and then used maximum likelihood methods to estimate the parameters of the
production function. These methods may lack robustness if the assumed
distributional form does not hold. In particular, outliers in the data may
unduly affect the estimate of the frontier function, or, it may be biased
if the error structure is not correctly specified. Furthermore, as
illustrated by Caudill, Ford, and Groper (1995),
heteroskedasticity in the error term, if not properly accounted for, can
lead to significant biases when estimating the production frontier.
Nonparametric deterministic frontier models are very appealing because
they rely on very few assumptions, but, by construction, they are very
sensitive to extreme values and to outliers. Recently, a robust
nonparametric envelopment estimator of the production frontier has been
suggested by Cazals et al. (2002). They
introduce the concept of expected maximal output frontier of order
,
where
denotes the set of all integers m ≥ 1. It is defined as the
expected maximum achievable level of output among m firms drawn
in the population of firms using less than a given level of inputs.
Formally, for a fixed integer
and a given level of inputs x, the frontier function of order
m is defined as
where
(Y1,…,Ym) are
m independent identically distributed random variables generated
by the distribution of Y given X ≤ x. Its
nonparametric estimator is defined by
where
is the empirical version of F(y/x),
with
As pointed out in Cazals et al. (2002), the
FDH estimator of the production function can be viewed as a plug-in
estimator of φ(x), where the unknown
F(y/x) in the formulas (1) has been
replaced by its empirical analogue
.
It is given by
Because of the trimming nature of the order-m frontier, the
estimator
does not envelop all the data points, and so it is more robust to extreme
values than the FDH estimator
.
By choosing m appropriately as a function of the sample size
estimates the production function φ(x) itself while keeping
the asymptotic properties of the FDH estimator.
Hendricks and Koenker (1992, p. 58) stated,
“In the econometric literature on the estimation of production
technologies, there has been considerable interest in estimating so called
frontier production models that correspond closely to models for extreme
quantiles of a stochastic production surface.” The present paper can
be viewed as the first work to actually implement the idea of Hendricks
and Koenker: we construct a new nonparametric estimator of the production
frontier that is more robust to extreme values than the standard
DEA/FDH estimators and than the nonparametric estimator of Cazals et
al. It is based on extreme quantiles of the conditional distribution of
Y given X ≤ x. These nonstandard conditional
quantiles define a natural concept of a partial production frontier in
place of the m-trimmed frontier. The idea is nice and attractive,
because here the “trimming” is continuous in terms of the
order-α quantile where α ∈ [0,1]. Quantile methods
are known for their robustness. More precisely, conditional quantiles are
not very sensitive to large observations in the output direction. We show
that our new partial frontier and its resulting estimator share most of
the properties of the order-m frontier and its estimator.
The paper follows the structure of Cazals et al. (2002) initially very closely, adapting their technique
to the output oriented case and extending their basic ideas, thus sharing
similar comments. It is organized as follows. Section 2 motivates our
concept of quantile-frontier of order α and investigates its
properties and its relation to the order-m frontier and to the
true production frontier. In Section 3, we define a nonparametric
estimator of our order-α frontier, which is very easy to derive, very
fast to compute, and does not envelop all the observed data points. In
Section 4, we show that this estimator converges at the rate
and is asymptotically normally distributed. We also derive a
nonparametric estimator of the efficient production frontier and analyze
its asymptotic distribution. In Section 5, a numerical illustration is
proposed with some simulated examples and a data set on labor (as input)
and mail volumes (as output) about 10,000 French post offices. We show how
resistant to outliers our estimators are compared with the estimators of
the expected maximal output frontiers of order m. Section 6
concludes the paper. The proofs appear in the Appendix.
2. A NEW CONCEPT OF PRODUCTION
FRONTIER
Let
be the probability space on which the vector of inputs X and the
output variable Y are defined. In this approach, we define the
attainable set Ψ to be the support of the joint distribution of
(X,Y), and we will concentrate on the set Ψ* =
{(x,y) ∈
Ψ|FX(x) > 0}, which
contains the interior of Ψ.
From its definition, φ(x), the value of the production
function coincides with the order one quantile of the law of Y
given X ≤ x,
This suggests introducing a concept of production function of
continuous order α ∈ [0,1], as the quantile function of
order α of the law of Y given that X does not exceed
a given level of inputs. This function takes, for a given level of inputs
x, the value
This conditional quantile is the production threshold exceeded by
100(1 − α)% of firms that use less than the level x as
inputs. The function F−1(./x) is
the so-called generalized inverse of F(·/x).
If the distribution function F(·/x) is
strictly increasing, its inverse coincides with the generalized inverse
F−1(./x). Using this property, we
easily obtain the following result.
PROPOSITION 2.1. Assume that for every x such that
FX(x) > 0, the conditional
distribution function F(·/x) is strictly
increasing on the support [0,φ(x)].
Then,
From property (2), we see that any production unit
(x,y) in Ψ* belongs to some α-order quantile
curve. Then unit (x,y) produces more than 100α% of
all production units using inputs smaller than or equal to x and
produces less than the 100(1 − α)% remaining units. Thus the
quantile function qα(x) quantifies the
production efficiency of unit (x,y) by comparing it with
all units that use the same level of inputs x and also with those
that use strictly less than x. This motivates our interest in the
distribution of Y given X ≤ x.
But the most attractive property of this quantile function is that it
can be easily nonparametrically estimated without the drawbacks of the
methods trying to estimate the frontier function itself: it will be less
sensitive to noise, extreme values, or outliers. This is developed in the
next section.
As it is shown by property (2), the quantile curves
{(x,qα(x))|FX(x)
> 0} cover the whole production set Ψ*. As can be seen in the next
proposition, this does not hold for expected order-m frontiers of
Cazals et al. (2002)
{(x,φm(x))|FX(x)
> 0}.
PROPOSITION 2.2. Under the assumption of Proposition 2.1 and if we
assume furthermore the free disposability of outputs, i.e.,
then the functions φm do not
satisfy the following property:
Let us compare how the expected maximal production function and the
quantile function can be useful in terms of practical efficiency analysis.
Suppose a production unit uses a quantity of inputs x0
and produces an output y0;
φm(x0) gives the expected
maximum production among a fixed number of m firms using less
than x0 as inputs. This value indicates how efficient
the unit (x0,y0) is, compared with
these m units. This is achieved by comparing its level
y0 with the value of
φm(x0). For this particular
unit, we know that it belongs to a quantile frontier. The order of this
frontier, which is known, gives the proportion of units that produce less
than y0 among all firms using less than
x0. Hence the quantile function gives a clearer
indication on the production performance, and it can be viewed as a
reasonable benchmark value.
We can, however, establish an asymptotic relationship between the two
families of production functions φm and
qα. Namely, we can state the following
proposition.
PROPOSITION 2.3. For every x such that the conditional
distribution function F(·/x) is twice
differentiable with first derivative f (·/x)
strictly positive on the support
[0,φ(x)], we have as m → ∞
and α → 1,
where ψx′(α) =
−F′′(qα(x)/x)/f3(qα(x)/x).
From its definition, it is clear that for any fixed x such
that FX(x) > 0,
qα(x) is a monotone nondecreasing
function of α. The limiting case when α → 1 is of particular
interest. It converges to the efficient frontier: by letting m
tend to infinity in (3) and using limm→∞
φm(x) = φ(x), we obtain
φ(x) − qα(x) =
o(α) when α → 1. We can prove this property directly
by using the monotonicity of quantiles
qα(x) with respect to α as indicated
by the next proposition. Even more strongly it is shown, under some
regularity conditions, that the order-α production function
qα converges uniformly to the true production
function φ.
PROPOSITION 2.4.
(i) For any fixed value of x such that
FX(x) > 0, we have
limα→1 [searr] qα(x)
= φ(x).
(ii) Assume that for every α ∈ [0,1],
the quantile function qα(.) is continuous on the
interior of the support of X. Then for any compact K interior to the
support of X,
The function qα converges to a monotone
nondecreasing function φ as α → 1, but it is not monotone
nondecreasing itself unless we add the following assumption:
This assumption is not needed for all the results of this paper except
for the next proposition, but it appears to be quite reasonable: it says
that the chance of producing less than a value y decreases if a
firm uses more inputs. This assumption is necessary and also
sufficient.
PROPOSITION 2.5. The quantile function x [map ]
qα(x) is monotone nondecreasing on
the set
for every order α ∈ [0,1] if and only if
the function x [map ] F(y/x) is
monotone nonincreasing on the set
for any output
.
Note that the results established in Proposition 2.4 are very similar
to those obtained for the order-m frontier. Indeed
φm(x) converges simply and uniformly to
φ(x) as m → ∞. However for Proposition
2.5, Cazals et al. (2002, Theorem A.3) only
prove that if assumption (4) holds then
φm(x) is monotone nondecreasing in
x.
3. NONPARAMETRIC ESTIMATION
To estimate the conditional quantile
qα(x), it is natural to use the
conditional empirical quantile obtained by inverting the conditional
empirical distribution function
,
This estimator may be computed explicitly as follows. Let
Nx be the number of observations
Xi smaller than or equal to x,
i.e.,
,
and, for j = 1,…,Nx, denote
by Y(ij) the jth
order statistic of the observations Yi such
that Xi ≤ x:
Y(i1) ≤
Y(i2) ≤
··· ≤
Y(iNx).
We have, for x such that Nx ≠
0,
Hence,
Therefore, we obtain for every α > 0,
where [αNx] denotes the
integral part of αNx: the largest integer
less than or equal to αNx. The
conditional empirical quantile
is thus computed very easily as being the simple empirical quantile of
observations yi such that
xi ≤ x.
For comparison, note that an exact formula is available to compute
.
It is as simple as the formula (5) but is restricted to the case of no
ties among the inputs. The nonparametric estimator
can also be approximated in practice by using a Monte-Carlo algorithm,
even in the full multivariate case (several inputs and several outputs),
which we do not treat in our paper. For instance, in the univariate output
case, the Monte-Carlo method can be described as follows. For a given
x, draw a random sample of size m with replacement among
these yi such that
xi ≤ x and denote this sample by
(yb1,…,ybm).
Then compute
.
Redo this for b = 1,…,B where B is
large. Finally, we have
where the quality of the approximation can be tuned by the choice of
B.
Note also that the relation between the order-m frontier and
the true frontier remains valid with their estimators
,
i.e.,
.
Similarly it is easily seen, for any fixed value of inputs x for
which the estimator
is well defined for every order α ∈ [0,1], that
is a monotone nondecreasing function of α, and thus
Note that even for large values of α < 1, the estimator
is less sensitive to extreme values than the FDH estimator
,
which by construction envelops all the observations. The asymptotic
theory is discussed in Section 4. Note also that
is not necessarily monotone nondecreasing with respect to x.
Indeed, even if assumption (4) is assumed for the true conditional
distribution function, it could happen that its empirical counterpart does
not satisfy it. Of course we know that for large sample size n,
it will mostly be the case.
Another property that
shares with
lies in the fact that both the nonparametric partial frontiers
underestimate the full frontier φ(x), for every order. In
our case, for any value of inputs x for which
are well defined for any order α ∈ [0,1], we have
Indeed, because the production function φ(·) is monotone
nondecreasing and greater than or equal to the efficient-output function
∂Y(.), for each i such that
Xi ≤ x we have almost surely
Yi ≤
∂Y(Xi) ≤
φ(Xi) ≤ φ(x).
Therefore
.
On the other hand we have
for every α ∈ [0,1].
4. ASYMPTOTIC PROPERTIES
For the unconditional case where ξα denotes the
order-α quantile of a distribution function
FZ of a random variable Z, and
denotes the empirical quantile of a sample
(Z1,…,Zn) of
Z, if FZ is differentiable in
ξα and such that
FZ′(ξα) > 0, the
Bahadur representation theorem gives
The direct application of this result to the distribution function
FZ(·) =
F(·/x) does not serve our purpose because our
data do not yield a sample from this distribution. However, as for
unconditional quantiles ξα, we focus here on pairs
(x,α) that satisfy the following property:
As a consequence of this property, F(·/x)
is a bijective transformation from a neighborhood of
qα(x) onto a neighborhood of α. In
particular the generalized inverse
F−1(·/x) is equal to the
inverse of F(·/x) in the neighborhood of
α. This property will be used in the proof of the following theorem,
which summarizes the asymptotic properties of our estimator
.
THEOREM 4.1. Let α ∈ (0,1) be a fixed order and
let x be a fixed value such that FX(x)
> 0. Assume that the conditional distribution function
F(·/x) is differentiable at
qα(x) with derivative f
(qα(x)/x) > 0.
Then,
where
It is important to note that here, also, the equivalent properties
hold with the nonparametric estimator of the order-m frontier.
Indeed it is easy to see that
converges at the rate
and is asymptotically unbiased and normally distributed:
,
where σ2(x,m) = E
[Γm2(x,X,Y)],
with
Moreover for a vector
,
the asymptotic r-variate normal distribution is obtained with
asymptotic covariances given by
Σm(xk,xl)
= E
[Γm(xk,X,Y)Γm(xl,X,Y)].
Similarly we have the following more general result for the estimator of
the conditional quantile frontier function.
THEOREM 4.2. Let
x1,…,xr be r
levels of the input X that satisfy the assumption of Theorem 4.1 for a
given order α ∈ (0,1). Then,
where
with
In applied work, the variance factors
σ2(x,α) and
Σα(xk,xl)
must be estimated. For instance, consistent estimators for these factors
can be obtained by plugging in nonparametric estimators for the
conditional density f (·/x) and the marginal
distribution function FX(x) and
taking the empirical mean for the expectation. Note that, as for
unconditional quantiles, quantiles in the tail of the conditional
distribution where the conditional density is low are inherently more
difficult to estimate.
Note also that Cazals et al. (2002) obtained
an asymptotic representation for their nonparametric estimator
where the error term is uniform in x, whereas the error term
involved in our approach depends on x (see the proof of Theorem
4.1). This can be explained as follows. Both
are representable as functionals of the empirical distribution function
.
The corresponding functional for
is differentiable in the Frechet sense w.r.t. the sup-norm, whereas that
corresponding to
is only differentiable in the Gâteaux sense. The uniformity of the
error term allowed Cazals et al. (2002, Appendix
B) to improve the convergence results of
by a functional limit theorem, which is not the case in our approach.
It is also interesting to compare
with the estimator of the standard conditional quantile of the
distribution of Y given X = x. First note that
this latter estimate requires a smoothing procedure, which is not the case
when the distribution of Y is conditioned by X ≤
x. To compare their asymptotic variance, let us recall that the
smooth estimators of the quantiles ξα(x) of
the distribution function Fx of Y
given X = x, are obtained by inverting a kernel
estimator of Fx and satisfy the following
result:
where μ2(x,α) = α(1 −
α)R(K)/fx2(ξα(x)),
with fx(y) =
(∂/∂y)Fx(y),
R(K) = ∫K2(u)
du, and K and hn are,
respectively, a kernel and a bandwidth satisfying some specific
constraints (see, e.g., Berlinet, Cadre, and Gannoun,
2001; Ducharme, Gannoun, Guertin, and Jequier,
1995).
Let us now turn to the convergence to the full frontier function
φ(x). We know that the estimator
converges to the FDH estimator
as α → 1. We also know from Park et al. (2000) that under regularity conditions, as n
→ ∞, the FDH estimator
converges to the true unknown frontier φ(x). The idea is
then to define α as a function of n such that
as n → ∞. We thus derive an estimator of the true
production frontier φ(x) and show in the next theorem that
it converges to the same asymptotic distribution as the FDH estimator and
as the nonparametric envelopment estimator of Cazals et al. (2002). The rate of convergence of the order
α(n) to 1 is provided.
THEOREM 4.3. Assume that the joint probability measure of
(X,Y) on the compact support Ψ provides a
strictly positive density on the frontier
{(x,φ(x))|FX(x)
> 0} and that the function φ is continuously
differentiable. Then for any x interior to the support of X we have, as
n → ∞,
where μx is a constant and the
order α(n) is such that
The constant μx appearing in the limiting
Weibull depends on the slope of the frontier and the value of the density
near the frontier point (x,φ(x)). A consistent
nonparametric estimator of this unknown constant has been proposed in Park
et al. (2000).
Like the approach of Cazals et al. (2002),
here also we lose the
-consistency because we use
to estimate the full frontier φ(x) and not the partial
frontier qα(x).
5. NUMERICAL ILLUSTRATIONS
In this section, we illustrate our procedure through some numerical
examples with simulated and real data. In the simulation study, the
observations are simulated according to the same data generating process
used in Simar (2003).
5.1. Example 1
We first consider a situation where the attainable set is convex. We
simulate a sample of n = 500 data points
(xi,yi)
according to the Cobb–Douglas log-linear frontier model given by
Y = X0.5 × exp(−U),
where X is uniform on (0,1) and U is exponential with
mean 1/3. The true frontier function is φ(x) =
x0.5.
Figure 1 illustrates the simulated data and
the quantile curves
and the expected maximal frontiers
(B = 1,000) for several different values of α and
m. In the solid lines, the estimates
(Figure 1a) with α = 0.7, 0.97, 0.98, 0.99,
1 are compared with the estimates
(Figure 1b) with m = 2, 25, 50, 75,
∞. The true frontier φ is in dash-dotted lines. The frontiers
are monotone nondecreasing with respect to the order. For Figure 2, we add in the data set three outliers, and
we plot the same frontiers
.
n = 500. Comparison between
(a) and
(b), output vs. input.
n = 503. Comparison between
(a) and
(b) with three outliers included, output vs. input.
From Figures 1 and 2, it is clear that the frontiers
are more resistant to the three outliers than the FDH frontier, but they
are less resistant to the outliers than the quantile frontiers of orders
α < 1. Indeed, the quantile frontier
is influenced by only one outlier, and it comes back down immediately,
whereas the frontiers
with m = 25, 50, 75 are attracted by all the outliers and
moreover continue to grow after each jump. So in this particular example
the frontier
is more robust to the outliers than the three frontiers
,
whereas it envelops all these frontiers in absence of the three
outliers.
5.2. Example 2
We now simulate a sample of n = 500 data points
(xi,yi) with a
nonconvex production set. We choose here the model Y =
exp(−5 + 10X)/(1 + exp(−5 +
10X))exp(−U), where X is uniform on (0,1)
and U is exponential with mean 1/3.
Figure 3 plots the simulated data and, in
the solid lines, the frontiers
(B = 1,000) with the same orders as in the preceding example
and, in the dash-dotted lines, the true frontier φ. Note that, here
also, the frontier
is above all the frontiers
.
We again add in the data set three outliers, as shown in Figure 4, and we plot the frontiers
for the same orders. It is clear that the quantile curves of orders α
< 1 are more resistant to the three outliers than the expected maximal
output frontiers
and the FDH frontier
.
n = 500. Comparison between
(a) and
(b), output vs. input.
n = 503. Comparison between
(a) and
(b) with three outliers included, output vs. input.
5.3. Example 3
We now test the robustness of both estimators
for a small sample size n = 100. In Figures
5a and 5b we plot, in the dotted lines, the quantile frontier of
order α = 0.93 and, in the solid lines, the frontiers
(B = 1,000) of orders m = 5, 7, 50, 75. In Figure 5a, the data points are simulated according to
the same model used in Example 1, and in Figure
5b, they are simulated according to the same model used in Example
2. Observe that the quantile frontier
is below the frontiers
of orders m = 50, 75 and is above those of orders m =
5, 7.
n = 100. In the solid lines, the order-m frontiers
and in the dotted lines, the quantile frontier
.
In Figure 6 we add to the preceding two data
sets the same three outliers used in Examples 1 and 2, and we plot the
same frontiers. We remark that the frontiers
of orders m = 50, 75 are highly influenced; even those of very
low orders m = 5, 7 are attracted by the three outliers, whereas
the quantile frontier is slightly perturbed.
n = 103. In the solid lines, the order-m frontiers
and in the dotted lines, the quantile frontier
with three outliers included.
We repeated the same exercise with many other simulated data sets,
leading to the same kind of results.
5.4. Frontier Analysis of French Post Offices
We examine here a real data set in a univariate situation: this data
set about the cost of the delivery activity of the postal services in
France is analyzed by Cazals et al. (2002).
There are n = 9,521 post offices observed in 1994. For each post
office i, the input xi is the labor
cost measured by the quantity of labor, which represents more than 80% of
the total cost of the delivery activity. The output
yi is defined as the volume of delivered mail
(in number of objects).
The 4,000 observed post offices with the smallest input levels are
plotted in Figure 7, along with the estimates of
quantile frontiers (Figure 7a) and of expected
maximal output frontiers (Figure 7b), for
several different orders α and m. Here we obtain the
frontiers
with B = 2,000 bootstrap loops.
The 4,000 observations with the smallest input levels. (a) The
frontiers
of orders α = 0.3, 0.5, 0.7, 0.9, 0.97, 0.98, 0.99, 0.995, 0.999 and
the FDH frontier (α = 1). (b) The frontiers
of orders m = 1, 2, 25, 50, 100, 200, 300, 400, 600 and the FDH
frontier (m = ∞). Output vs. input.
By using (5), it is very easy to check that every post office
i belongs to the quantile curve of order
.
On the other hand, the frontiers
do not cover the observations below the first frontier
(12% of the observed data) and the observations between the frontiers of
successive orders
.
This disadvantage of frontiers
with respect to frontiers
is due to the fact that the order m is discrete.
Note that the order αi of the quantile
frontier that passes through the post office
(xi,yi) is equal
to the percentage of post offices that produce less than
yi among all the post offices using inputs
smaller than or equal to xi. In other words,
this order indicates that the ith post office produces more than
100α% of all post offices using inputs smaller or equal to
xi and produces less than the 100(1 −
α)% remaining post offices. This is why one sees in Figure 7a that, if αi is close
to one, then the post office
(xi,yi) can be
seen to be performing relatively efficiently, and likewise, if
αi is close to zero, then the post office would be
performing relatively inefficiently. Thus the order of the empirical
quantile frontier
defines a reasonable benchmark value. Note also that the nonparametric
estimation of the expected frontier
can be viewed as a mark of good practice for post offices when studying
their performance. However, this benchmark is less clear than the
empirical quantile frontier because it is less easy to interpret and does
not cover the whole data set.
We also remark that the frontiers
(Figure 7b) are perturbed by the extreme
observations from the order m = 25, whereas the frontiers
are not influenced except for those having orders almost equal to one
(α ≥ 0.999).
Figure 8a (resp. Figure
8b) indicates how the percentage p(α) (resp.
p(m)) of observations above the quantile estimates
(resp. the expected maximum cost estimates
) decreases with α (resp. m). We remark that the
percentage p(α) decreases very slowly until the order α =
0.8 of approximately 24% of observations. It means that the quantile
frontiers of orders 0 ≤ α ≤ 0.8 are very tight. The 24%
observations below the frontier
have an intermediate production performance and can be relatively
inefficient. However, the percentage p(α) falls dramatically
from the order α = 0.8, which means that the quantile frontiers of
extreme orders 0.8 ≤ α ≤ 1 are very spaced and are spread out
over 76% of the observations. In particular, 10% of the observations are
above the frontier
and 3% of the observations are above the frontier
.
It is what explains notably the fact that only quantile frontiers of
orders very close to one are influenced by superefficient units.
Evolution of the percentage of observations above the frontiers (a)
and (b)
.
In Figure 8b, we observe an opposite
phenomenon: first the percentage p(m) falls severely
until the order m = 50 of approximately 80% of observations, and
then it continues to decrease but very slowly. Consequently the frontiers
of orders m ≥ 50 are very tight. In particular we just have
9% of observations between the two frontiers
,
and only 3% of observations are above the frontier
.
The 20% observations above the frontier
are extreme and could be outliers or noisy observations. In summary the
frontiers
are very tight from the order m = 50 and are spread out over
extreme observations; it is then natural that these frontiers would be
more sensitive to extreme values than the quantile frontiers.
We can illustrate this result more clearly by considering the
following inverse problem: for a given percentage p0,
denote by α(p0) (resp.
m(p0)) the order of the frontier
(resp.
) above which the percentage of observations is equal to
p0; we have p(α(p0))
= p(m(p0)) = p0.
Inverting the relationship between α and p(α) and between
m and p(m) in Figure
8, we get the evolution of α and m as functions of
p. When the percentage p varies between 0 and 10%, we
remark that the order α(p) is almost constant
(α(p) ≈ 1), whereas the order m(p) falls
rapidly from m = 600 to m ≈ 100. This means that the
10% extreme observations influence all the frontiers
with orders 100 ≤ m ≤ ∞, whereas only the frontiers
with orders almost equal to 1 are influenced by these extreme
observations. This can be understood because the FDH frontier
envelops all the observed data.
This result is also illustrated in Figure 9,
where the curve of evolution of α(p) with respect to
m(p) is nearly flat from the point (100, 0.995) which
corresponds to the percentage p ≈ 10%. This plot establishes
an empirical relationship between the two families of frontiers
.
Given a frontier
,
we can determine the frontier
above which we have the same percentage of observations and vice
versa.
An empirical relationship between
.
6. CONCLUSIONS
In this paper, we propose a new statistical concept of a production
frontier that allows a more subtle tuning than the expected maximal output
frontier of order
(Cazals et al., 2002). We define a frontier of
continuous order α ∈ [0,1] of the production set
Ψ, for a given level of inputs x, by the conditional
α-quantile of the distribution of Y given X ≤
x.
Our quantile frontiers satisfy at least the same statistical
properties as the expected maximal output frontiers of order m.
Moreover they have the advantage, from an economic point of view, of
covering the interior of the attainable set entirely, thus giving a
clearer indication of the production efficiency. This benefit is due to
the continuity of the index α of our conditional quantiles.
A nonparametric estimator of the quantile function of order α <
1 is very easy to derive by inverting the empirical version of the
conditional distribution function. It does not envelop all the observed
data points, and so it is more robust to extreme values than the standard
DEA/FDH nonparametric envelopment estimators. Also it is easier to
interpret than the nonparametric estimator of the expected function of
order m. Moreover our estimator achieves the
-consistency and is asymptotically unbiased and normally
distributed, which is reasonable because the conditioning set X
≤ x has a positive probability measure. By choosing α as
an appropriate function of n, it estimates the true frontier
function and satisfies the asymptotic properties of the FDH estimator.
The method is illustrated using simulated and real data. It shows that
the nonparametric quantile frontiers are more resistant to large
observations in the output direction than the nonparametric estimates of
expected maximal output frontiers and that the continuous order α
represents a good benchmark value. The robustness revealed by the
numerical illustrations needs to be confirmed by some theoretical
properties. This question is currently being investigated.
It should be clear that, unlike the approach of Cazals et al. (2002), the conditional quantile approach is not
extended here to multivariate Y. Serfling (2002) stated, “Despite the absence of a natural
ordering of Euclidean space for dimension greater than one, effort to
define vector-valued quantile functions for multivariate distributions has
generated several approaches.” The methods based on depth functions
recommended by Serfling might be adapted to generalize in a reasonable way
our univariate conditional quantiles. This problem is worth
investigating.
APPENDIX: PROOFS
Proof of Proposition 2.1. Let (x,y) ∈
Ψ* and set α = F(y/x). It is an
immediate consequence of the strict monotonicity of
F(·/x) that
qα(x) =
F−1(α/x) = y.
█
Proof of Proposition 2.2. Let us assume the contrary. Then we
obtain
where Supp(X) is the support of the distribution of
denotes its interior. Let
be fixed such that ∂Y(x) > 0. Because the
production function φ is greater than or equal to the
output-efficient function ∂Y(.), we have φ(x)
= ∂Y(x) or φ(x) >
∂Y(x).
If φ(x) = ∂Y(x), we know from
Cazals et al. (2002, Appendix A) that
φ(x) = lim [searr]m→∞
φm(x), so there exists an integer
such that φmx(x)
< φmx+1(x) ≤
∂Y(x) (else, we would have
φm(x) =
φm+1(x) for every
,
so that φ(x) = φ1(x); consequently
we would obtain
,
which is impossible because the function
F(·/x) is strictly increasing on
[0,φ(x)]). Let y be a real number such
that φmx(x) <
y <
φmx+1(x). Using the
free disposability assumption of outputs, it is easily seen that
Y(x) ≡ [0,∂Y(x)],
so that y ∈ Y(x). Then by (A.1), there
exists an integer
such that y =
φmx,y(x).
It follows that
φmx(x) <
φmx,y(x)
< φmx+1(x), and
thus mx <
mx,y <
mx + 1 because
φm(x) is a monotone nondecreasing
function of m. This contradicts the fact that
mx,y is an integer.
Now if φ(x) > ∂Y(x), first
note that ∂Y(x) ∈ Y(x)
yields by (A.1) that ∂Y(x) =
φmx,∂Y(x)(x)
where
.
Because of lim [searr]m→∞
φm(x) = φ(x), there exists
an order mx >
mx,∂Y(x) such that
φ(x) ≥ φm(x) >
∂Y(x) when m ≥
mx, and
φm(x) ≤ ∂Y(x)
when m < mx. For any y
∈ Y(x) we have y ≤
∂Y(x) so that y <
φmx(x). We also have
by (A.1) y =
φmx,y(x)
where
.
Hence
φmx,y(x)
< φmx(x).
Therefore, again using the monotonicity of
φm(x) with respect to m, we get
mx,y <
mx. In summary,
Because φm(x) ∈
[0,∂Y(x)] = Y(x) for
every m < mx, the map m
[map ] φm(x) is well defined and is onto
from {1,…,mx − 1} to
Y(x). As a consequence, the finite set
{φ1(x),…,φmx−1(x)}
coincides with the interval [0,∂Y(x)],
which implies the contradiction. █
Proof of Proposition 2.3. Let x be an input that
satisfies the condition of Proposition 2.3. We have by definition
φm(x) = E
[max(Y1,…,Ym)],
where Y1,…,Ym are
m independent identically distributed random variables generated
by the distribution function F(·/x). Let
be the empirical distribution function of
(Y1,…,Ym). The
empirical quantile of order α ∈ (0,1] of this sample is then
defined by
We know that
qαm(x) is equal to
Y(αm) if αm is an integer
and to Y([αm]+1) otherwise.
Then q1m(x) =
Y(m). Because the family
(qαm(x))0<α<1
increases to q1m(x) when
α → 1, the dominated convergence theorem yields
φm(x) = limα→1
E
[qαm(x)],
and thus we can write φm(x) = E
[qαm(x)] +
ε1(α), where ε1(α) =
o(α) when α → 1. On the other hand, according to the
representation theorem of Bahadur (see, e.g., Serfling, 1980, Theorem 2.5.1, p. 91), we have for every α
∈ (0,1),
where, with probability 1,
Rm,x(α) =
O(m−3/4(log
m)3/4) as m → ∞. By using
Kiefer's theorem (see, e.g., Serfling, 1980, Theorem D, p. 101), it can easily be seen that a
more precise expression of the remainder is given by
where, almost surely and uniformly in α, we have
It follows that
Now consider the function ψx(p) =
1/f
(qp(x)/x), p
∈ [0,1]. We have as m → ∞,
Because F(·/x) has a positive continuous
density f (./x) in the neighborhood
(0,φ(x)) of qα(x), for any
α ∈ (0,1), we obtain according to Shorack and Wellner (1986, Proposition 6, p. 9) that the partial derivative
(∂/∂α)qα(x) exists and
equals 1/f
(qα(x)/x). Then, for every
α ∈ (0,1), the derivative of ψx(α)
with respect to α is given by
Using the fact that ψx(1) −
ψx(α) = (1 −
α)ψx′(α) + (1 −
α)ε2(α), where ε2(α) =
o(α) when α → 1, we obtain as m →
∞ and α → 1,
which proves (3). █
Proof Proposition 2.4.
1. Because the family
{qα(x)}0≤α≤1 is
monotone nondecreasing and bounded by q1(x),
qα(x) converges pointwise to
q1(x) when α → 1.
2. Let K be a compact subset of
.
The term {qα(.)}0<α<1 is a
nondecreasing sequence of real valued functions that are continuous on
K. Moreover it converges pointwise to the continuous function
q1(.) as α tends to one. Then by Dini's
theorem (Schwartz, 1991, p. 325) the convergence
is uniform on K. █
Proof of Proposition 2.5. Suppose that for every y
≥ 0, the function x [map ]
F(y/x) is monotone nonincreasing on
.
Let α ∈ [0,1] and x1 ≤
x2 such that
FX(x1) > 0. Then
F(qα(x2)/x1)
≥
F(qα(x2)/x2)
≥ α. It follows that
qα(x2) ≥
inf{y| F(y/x1)
≥ α} = qα(x1).
Conversely, suppose that the quantile function is monotone
nondecreasing for every order on
.
Let
such that FX(x1) > 0.
Set α = F(y/x2). We have
qα(x2) =
inf{u|F(u/x2)
≥ α}, so that y ≥
qα(x2). Because
qα(x1) ≤
qα(x2), y ≥
qα(x1), and thus
F(y/x1) ≥
F(qα(x1)/x1)
≥ α = F(y/x2).
█
The following lemma will be useful in the proof of Theorem 4.1.
LEMMA 6.1. Let {Vn},
{Wn} be two sequences of random variables
satisfying the following conditions.
(i) For all δ > 0, there exists a
λ (depending on δ) s.t.
P(|Wn| > λ) <
δ.
(ii) For all k and all ε > 0
Then
.
The proof of this lemma can be found in Ghosh (1971, Lemma 1, p. 1958). Now let us demonstrate
Theorem 4.1.
Proof of Theorem 4.1. Consider the statistical functional
Tα,x that associates to a distribution
function G on
the real value
The conditional quantile qα(x) and
its estimator
are then given by
.
Let
where
is the first Gâteaux differential of
Tα,x at F in the direction of
1(Xi ≤ .,Yi
≤ .). Using property (6), we obtain by a straightforward
computation
Hence,
Therefore,
where
We have
so that, by the central limit theorem,
and by the law of large numbers,
Let
.
Then we obtain via (A.2)
Using Lemma 6.1, we will show that
converges in probability to zero. We have for every real number
t,
where
Because F(·/x) is differentiable at
qα(x) with derivative
,
which implies
.
We know from the law of large numbers that
,
and thus,
On the other hand we have
By a simple computation we find that
Using the continuity of F(·/x) in
qα(x), we get E
[(Zt,n −
Wnα,x)2]
→ 0 as n → ∞, and thus,
Now, using the results (A.3) and (A.6)–(A.8), we will show that
Vnα,x and
Wnα,x satisfy the two
conditions of Lemma 6.1. As E
[(Wnα,x)2]
= σ2(x,α), the first condition follows from a
trivial application of the Markov inequality. For any k and any
ε > 0, setting t = k, we have by (A.6),
Hence,
Therefore, combined with (A.7) and (A.8),
limn→∞
P(Vnα,x ≤
k,Wnα,x ≥
k + ε) = 0. Now applying (A.6) to t = k +
ε, we get
Then (A.7) and (A.8) yield limn→∞
P(Vnα,x ≥
k +
ε,Wnα,x ≤
k) = 0. Hence, the second condition of Lemma 6.1 is also
satisfied. Therefore
,
as n → ∞, i.e.,
In particular
Rnα,x converges in
probability to zero as n → ∞. Thus,
The consistency follows from results (A.5) and (A.9), and the
asymptotic normality is obtained by (A.4) and (A.10). █
Proof of Theorem 4.2. It follows from (A.10), as n
→ ∞,
Hence, the multivariate central limit theorem yields
where
This ends the proof. █
Proof of Theorem 4.3. From Park et al. (2000) and Cazals et al. (2002) we know that
So by using the following decomposition:
we want to find a function α of n such that
From (5) we have for any α > 0,
Set for every k ∈
{1,…,Nx − 1}
and let Cx(n) =
max{Cx,k(n) | 1 ≤
k ≤ Nx − 1}. Then we
have
because
.
Now, using that
,
we get
It follows from (A.11)–(A.13) that
so that
Because the support Ψ of (X,Y) is compact, the
support of Y is bounded. Let M > 0 be its upper
bound. Then for any k =
1,…,Nx − 1,
Hence,
Therefore,
We deduce from (A.14)
We know from the strong law of large numbers that
.
So to achieve our goal, it is sufficient to choose α(n) such
that
Indeed we find,
This completes the proof. █
