Large deviations for method-of-quantiles estimators of
one-dimensional parameters∗

arXiv:1611.04765v2 [] 5 Apr 2018

Valeria Bignozzi†

Claudio Macci‡

Lea Petrella§

Abstract
We consider method-of-quantiles estimators of unknown one-dimensional parameters, namely
the analogue of method-of-moments estimators obtained by matching empirical and theoretical
quantiles at some probability level λ ∈ (0, 1). The aim is to present large deviation results for
these estimators as the sample size tends to infinity. We study in detail several examples; for
specific models we discuss the choice of the optimal value of λ and we compare the convergence
of the method-of-quantiles and method-of-moments estimators.
AMS Subject Classification: 60F10; 62F10; 62F12.
Keywords: location parameter; methods of moments; order statistics; scale parameter; skewness
parameter.

1

Introduction

Estimation of parameters of statistical or econometric models is one of the main concerns in the
parametric inference framework. When the probability law is specified (up to unknown parameters),
the main tool to solve this problem is the Maximum Likelihood (ML) technique; on the other hand,
whenever the assumption of a particular distribution is too restrictive, different solutions may be
considered. For instance the Method of Moments (MM) and the Generalized Method of Moments
(GMM) provide valuable alternative procedures; in fact the application of these methods only
requires the knowledge of some moments.
A different approach is to consider the Method of Quantiles (MQ), that is the analogue of
MM with quantiles; MQ estimators are obtained by matching the empirical percentiles with their
theoretical counterparts at one or more probability levels. Inference via quantiles goes back to [1]
where the authors consider an estimation problem for a three-parameter log-normal distribution;
their approach consists in minimizing a suitable distance between the theoretical and empirical
quantiles, see for instance [12]. Successive papers deal with the estimation of parameters of extreme
value (see [8] and [11]), logistic (see [9]) and Weibull (see [10]) distributions. A more recent reference
is [2] where several other distributions are studied. We also recall [6] where the authors consider an
indirect inference method based on the simulation of theoretical quantiles, or a function of them,
∗
The support of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of
the Istituto Nazionale di Alta Matematica (INdAM) is acknowledged.
†
Dipartimento di Statistica e Metodi Quantitativi, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8,
I-20126 Milano, Italia. e-mail: valeria.bignozzi@unimib.it
‡
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italia.
e-mail: macci@mat.uniroma2.it
§
Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza, Sapienza Università di Roma, Via
del Castro Laurenziano 9, I-00161 Roma, Italia. e-mail: lea.petrella@uniroma1.it

1

when they are not available in a closed form. In [15], an iterative procedure based on ordinary
least-squares estimation is proposed to compute MQ estimators; such estimators can be easily
modified by adding a LASSO penalty term if a sparse representation is desired, or by restricting
the matching within a given range of quantiles to match a part of the target distribution. Quantiles
and empirical quantiles represent a key tool also in quantitative risk management, where they are
studied under the name of Value-at-Risk (see for instance [13]).
In our opinion, MQ estimators deserve a deeper investigation because of several advantages.
They allow to estimate parameters when the moments are not available and they are invariant
with respect to increasing transformations; moreover they have less computational problems, and
behave better when distributions are heavy-tailed or their supports vary with the parameters.
The aim of this paper is to present large deviation results for MQ estimators (as the sample size
tends to infinity) for statistical models with one-dimensional unknown parameter θ ∈ Θ, where the
parameter space Θ is a subset of the real line; thus we match empirical and theoretical quantiles
at one probability level λ ∈ (0, 1). The theory of large deviations is a collection of techniques
which gives an asymptotic computation of small probabilities on an exponential scale (see e.g. [4]
as a reference on this topic). Several examples of statistical models are considered throughout the
paper, and some particular distributions are studied in detail. For most of the examples considered,
we are able to find an explicit expression for the rate function which governs the large deviation
principle of the MQ estimators and, when possible, our investigation provides the optimal λ that
guarantees a faster convergence to the true parameter (see Definition 3.1). Further we compare MQ
and MM estimators in terms of the local behavior of the rate functions around the true value of the
parameter in the spirit of Remark 2.1. Which one of the estimators behaves better strictly depends
on the type of parameter we have to estimate and varies upon distributions. However, we provide
explicit examples (a part from the obvious ones where the MM estimators are not available) where
MQ estimators are preferable.
We conclude with the outline of the paper. In Section 2 we recall some preliminaries. Sections
3 and 4 are devoted to the results for MQ and MM estimators, respectively. In Section 5 we present
examples for different kind of parameters (e.g. scale, location, skewness, etc.), and for each example
specific distributions are discussed in Section 6.

2

Preliminaries

In this section we present some preliminaries on large deviations and we provide a rigorous definition
of the MQ estimators studied in this paper (see Definition 2.1 below).

2.1

Large deviations

We start with the concept of large deviation principle (LDP for short). A sequence of random
variables {Wn : n ≥ 1} taking values on a topological space W satisfies the LDP with rate function
I : W → [0, ∞] if I is a lower semi-continuous function,
lim inf
n→∞

and
lim sup
n→∞

1
log P (Wn ∈ O) ≥ − inf I(w) for all open sets O
w∈O
n
1
log P (Wn ∈ C) ≤ − inf I(w) for all closed sets C.
w∈C
n

We also recall that a rate function I is said to be good if all its level sets {{w ∈ W : I(w) ≤ η} : η ≥ 0}
are compact.
2

Remark 2.1 (Local comparison between rate functions around the unique common zero). It is
known that, if I(w) uniquely vanish at some w0 ∈ W, then the sequence of random variables
converges weakly to w0 . Moreover, if we have two rate functions I1 and I2 which uniquely vanish
at the same point w0 ∈ W, and if I1 (w) > I2 (w) > 0 in a neighborhood of w0 (except w0 ) then any
sequence which satisfies the LDP with rate function I1 converges to w0 faster than any sequence
which satisfies the LDP with rate function I2 .
We also recall a recent large deviation result on order statistics of i.i.d. random variables
(see Proposition 2.1 below) which plays a crucial role in this paper. We start with the following
condition.
Condition 2.1. Let {Xn : n ≥ 1} be a sequence of i.i.d. real valued random variables with
distribution function F , and assume that F is continuous and strictly increasing on (α, ω), where
−∞ ≤ α < ω ≤ ∞. Moreover let {kn : n ≥ 1} be such that kn ∈ {1, . . . , n} for all n ≥ 1 and
limn→∞ knn = λ ∈ (0, 1).
We introduce the following notation: for all k ≥ 1, X1:k ≤ · · · ≤ Xk:k are the order statistics of
the sample X1 , . . . , Xk ; for p, q ∈ (0, 1) we set
H(p|q) := p log

1−p
p
+ (1 − p) log
,
q
1−q

(1)

that is the relative entropy of the Bernoulli distribution with parameter p with respect to the
Bernoulli distribution with parameter q.
Proposition 2.1 (Theorem 3.2 in [7] for λ ∈ (0, 1)). Assume that Condition 2.1 holds. Then
{Xkn :n : n ≥ 1} satisfies the LDP with good rate function Iλ,F defined by

H(λ|F (x)) for x ∈ (α, ω)
Iλ,F (x) :=
∞
otherwise.
00 (F −1 (λ)) as the inverse of an asymptotic variance). Theorem 7.1(c) in [3]
Remark 2.2 (Iλ,F
√
states that, under suitable conditions, { n(Xkn :n − F −1 (λ)) : n ≥ 1} converges weakly to the
λ(1−λ)
centered Normal distribution with variance σ 2 := (F 0 (F
−1 (λ)))2 . Then, if we assume that F is twice
00 (F −1 (λ)) =
differentiable, we can check that Iλ,F

1
σ2

with some computations.

A more general formulation of Proposition 2.1 could be given also for λ ∈ {0, 1} but, in view of
the applications presented in this paper, we prefer to consider a restricted version of the result with
λ ∈ (0, 1) only. This restriction allows to have the goodness of the rate function Iλ,F (see Remark
1 in [7]) which is needed to apply the contraction principle (see e.g. Theorem 4.2.1 in [4]).

2.2

MQ estimators

Here we present a rigorous definition of MQ estimators. In view of this, the next Condition 2.2
plays a crucial role.
Condition 2.2. Let {Fθ : θ ∈ Θ} be a family of distribution functions where Θ ⊂ R and, for all
θ ∈ Θ, Fθ is continuous and strictly increasing on some (αθ , ωθ ), where −∞ ≤ αθ < ωθ ≤ ∞ (as
happens for the distribution function F Sin Condition 2.1). Moreover, for λ ∈ (0, 1), consider the
−1
function F(•)
(λ) : Θ → M, where M := θ∈Θ (αθ , ωθ ), defined by
−1
[F(•)
(λ)](θ) := Fθ−1 (λ).

3

−1
We assume that, for all m ∈ M, the equation [F(•)
(λ)](θ) = m admits a unique solution (with
−1
respect to θ ∈ Θ) which will be denoted by (F(•)
(λ))−1 (m).

Now we are ready to present the definition.
Definition 2.1. Assume that Condition 2.2 holds. Then
quence of MQ estimators (for the level λ ∈ (0, 1)).

n
o
−1
(F(•)
(λ))−1 (X[λn]:n ) : n ≥ 1 is a se-

Proposition 3.1 below provides the LDP for the sequence of estimators in Definition 2.1 (as
the sample size n goes to infinity) when thentrue value of the parameter
o is θ0 ∈ Θ. Actually we
−1
−1
give a more general formulation in terms of (F(•) (λ)) (Xkn :n ) : n ≥ 1 , where {kn : n ≥ 1} is a
sequence as in Condition 2.1.

3

Results for MQ estimators

In this section we prove the LDP for the sequence of estimators in Definition 2.1. Moreover we
discuss some properties of the rate function; in particular Proposition 3.2 (combined with Remark
2.1 above) leads us to define a concept of optimal λ presented in Definition 3.1 below.
We start with our main result and, in view of this, we present the following notation:
hλ,θ0 (θ) := Fθ0 (Fθ−1 (λ)) (for Fθ−1 (λ) ∈ (αθ0 , ωθ0 )).

(2)

Proposition 3.1 (LD for MQ estimators). Assume that {kn : n ≥ 1} is as in Condition 2.1 and
that Condition 2.2 holds. Moreover assume that, for some θ0 ∈ Θ, {Xn : n ≥ 1} are i.i.d. random
−1
variables with distribution function Fθ0 . Then, if the restriction of (F(•)
(λ))−1 on (αθ0 , ωθ0 ) is
n
o
−1
continuous, (F(•)
(λ))−1 (Xkn :n ) : n ≥ 1 satisfies the LDP with good rate function Iλ,θ0 defined by
(
Iλ,θ0 (θ) :=

λ log hλ,θλ (θ) + (1 − λ) log 1−h1−λ
λ,θ (θ)

for θ ∈ Θ such that Fθ−1 (λ) ∈ (αθ0 , ωθ0 )

∞

otherwise,

0

0

where hλ,θ0 (θ) is defined by (2).
−1
(λ))−1 on (αθ0 , ωθ0 ) is continuous, a straightforward application
Proof. Since the restriction of (F(•)
n
o
−1
of the contraction principle yields the LDP of (F(•)
(λ))−1 (Xkn :n ) : n ≥ 1 with good rate function
Iλ,θ0 defined by
n
o
−1
Iλ,θ0 (θ) := inf Iλ,Fθ0 (x) : x ∈ (αθ0 , ωθ0 ), (F(•)
(λ))−1 (x) = θ ,

where Iλ,Fθ0 is the good rate function in Proposition 2.1, namely the good rate function defined by
n
o
−1
Iλ,Fθ0 (x) := H(λ|Fθ0 (x)), for x ∈ (αθ0 , ωθ0 ). Moreover the set x ∈ (αθ0 , ωθ0 ) : (F(•)
(λ))−1 (x) = θ
has at most one element, namely
n
o  {F −1 (λ)} for θ ∈ Θ such that F −1 (λ) ∈ (α , ω )
θ0
θ0
−1
θ
θ
x ∈ (αθ0 , ωθ0 ) : (F(•)
(λ))−1 (x) = θ =
∅
otherwise;
thus we have Iλ,θ0 (θ) = H(λ|Fθ0 (Fθ−1 (λ))) = H(λ|hλ,θ0 (θ)) for θ ∈ Θ such that Fθ−1 (λ) ∈ (αθ0 , ωθ0 ),
and Iλ,θ0 (θ) = ∞ otherwise. The proof is completed by taking into account the definition of the
function H in (1).
4

Remark 3.1 (Rate function invariance with respect to increasing transformations). Let {Fθ : θ ∈
Θ} be a family of distribution functions as in Condition 2.2 and assume that there exists an interval
(α, ω) such that (αθ , ωθ ) = (α, ω) for all θ ∈ Θ. Moreover let ψ : (α, ω) → R be a strictly increasing
function. Then, if we consider the MQ estimators based on the sequence {ψ(Xn ) : n ≥ 1} instead
of {Xn : n ≥ 1}, we can consider an adapted version of Proposition 3.1 with (ψ(α), ψ(ω)) in place
of (α, ω), Fθ ◦ ψ −1 in place of Fθ and, as stated in Property 1.5.16 in [5], ψ ◦ Fθ−1 in place of Fθ−1 .
The LDP provided by this adapted version of Proposition 3.1 is governed by the rate function Iλ,θ0 ;ψ
defined by

H(λ|Fθ0 ◦ ψ −1 (ψ ◦ Fθ−1 (λ))) for θ ∈ Θ such that ψ ◦ Fθ−1 (λ) ∈ (ψ(α), ψ(ω))
Iλ,θ0 ;ψ (θ) :=
∞
otherwise
instead of

Iλ,θ0 (θ) :=

H(λ|Fθ0 (Fθ−1 (λ)))
∞

for θ ∈ Θ such that Fθ−1 (λ) ∈ (α, ω)
otherwise.

One can easily realize that Iλ,θ0 ;ψ and Iλ,θ0 coincide.
By taking into account the rate function in Proposition 3.1, it would be interesting to compare
two rate functions Iλ1 ,θ0 and Iλ2 ,θ0 in the spirit of Remark 2.1 for a given pair λ1 , λ2 ∈ (0, 1); namely
it would be interesting to have a strict inequality between Iλ1 ,θ0 and Iλ2 ,θ0 in a neighborhood of θ0
(except θ0 ).
Thus, if both rate functions are twice differentiable, Iλ1 ,θ0 is locally larger (resp. smaller) than
Iλ2 ,θ0 around θ0 if we have Iλ001 ,θ0 (θ0 ) > Iλ002 ,θ0 (θ0 ) (resp. Iλ001 ,θ0 (θ0 ) < Iλ002 ,θ0 (θ0 )). So it is natural to
00 (θ ) under suitable hypotheses.
give an expression of Iλ,θ
0
0
00 (θ )). Let I
Proposition 3.2 (An expression for Iλ,θ
0
λ,θ0 be the rate function in Proposition 3.1.
0
−1
Assume that Fθ0 (·) and F(·) (λ) are twice differentiable. Then

00
Iλ,θ
(θ0 ) =
0

(h0λ,θ0 (θ0 ))2
λ(1 − λ)

=

{Fθ00 (Fθ−1
(λ))}2
0
λ(1 − λ)

d −1
F (λ)
dθ θ

!2
,
θ=θ0

where hλ,θ0 (θ) is defined by (2).
Proof. One can easily check that
hλ,θ0 (θ0 ) = λ and h0λ,θ0 (θ0 ) = Fθ00 (Fθ−1
(λ)) ·
0

d −1
F (λ)
dθ θ

.
θ=θ0

Moreover after some computations we get
0
Iλ,θ
(θ)
0

=

h0λ,θ0 (θ)



1−λ
λ
−
1 − hλ,θ0 (θ) hλ,θ0 (θ)



and
00
Iλ,θ
(θ)
0

=

h00λ,θ0 (θ)



1−λ
λ
−
1 − hλ,θ0 (θ) hλ,θ0 (θ)

0
00 (θ ) = (h0
2
Thus Iλ,θ
(θ0 ) = 0 and Iλ,θ
0
λ,θ0 (θ0 ))
0
0

taking into account the expression of

h0λ,θ0 (θ0 )





1
λ

+

(h0λ,θ0 (θ))2

+

1
1−λ

above.
5



=

1−λ
+
2
hλ,θ0 (θ) (1 − hλ,θ0 (θ))2
λ

(h0λ,θ (θ))2
0
λ(1−λ) .

!
.

The proof is completed by

Finally, by taking into account Proposition 3.2 (and what we said before it), it is natural to
consider the following
00 (θ ), namely if we
Definition 3.1. A value λmax ∈ (0, 1) is said to be optimal if it maximizes Iλ,θ
0
0
00
00
have Iλmax ,θ0 (θ0 ) = supλ∈(0,1) Iλ,θ0 (θ0 ).

4

Results for MM estimators

The aim of this section is to present a version of the above results for MM estimators; namely
the LDP and an expression of Jθ000 (θ0 ), where Jθ0 is the rate function which governs the LDP
of MM estimators. In particular, when we compare MM and MQ estimators in terms of speed
00 (θ ) in
of convergence by referring to Remark 2.1, the value Jθ000 (θ0 ) will be compared with Iλ,θ
0
0
Proposition 3.2.
We start with the following condition which allows us to define the MM estimators.
Condition 4.1. Let {Fθ : θ ∈ Θ} be a familySof distribution functions as in Condition 2.2, and
consider the function µ : Θ → M, where M := θ∈Θ (αθ , ωθ ), defined by
Z ωθ
µ(θ) :=
xdFθ (x).
αθ

We assume that, for all m ∈ M, the equation µ(θ) = m admits a unique solution (with respect to
θ ∈ Θ) which will be denoted by µ−1 (m).
From now on, in connection with this condition, we introduce the following function:
Z ωθ
∗
eγx dFθ (x).
Λθ (x) := sup {γx − Λθ (γ)} , where Λθ (γ) := log
γ∈R

(3)

αθ

It is well-known that, if {Xn : n ≥ 1} are i.i.d. random variables with distribution function Fθ , and
n
if we set X̄n := X1 +···+X
for all n ≥ 1, then {X̄n : n ≥ 1} satisfies the LDP with rate function Λ∗θ
n
in (3) by Cramér Theorem on R (see e.g. Theorem 2.2.3 in [4]).
Then we have the following result.
Proposition 4.1 (LD for MM estimators). Assume that Condition 4.1 holds. Moreover assume
that, for some θ0 ∈ Θ, {Xn : n ≥ 1} are i.i.d. random variables with distribution function Fθ0 .
(i) If µ−1 (m) := c1 m + c0 for some c1 , c0 ∈ R such that c1 6= 0, then {µ−1 (X̄n ) : n ≥ 1} satisfies
the LDP with rate function Jθ0 defined by
 ∗
Λθ0 (µ(θ)) for θ ∈ Θ such that µ(θ) ∈ (αθ0 , ωθ0 )
Jθ0 (θ) :=
∞
otherwise.
(ii) If the restriction of µ−1 on (αθ0 , ωθ0 ) is continuous and if Λ∗θ0 is a good rate function, the same
LDP holds and Jθ0 is a good rate function.
−1
0
Proof. (i) In this case µ(θ) := θ−c
c1 and {µ (X̄n ) : n ≥ 1} is again a sequence of empirical means of
i.i.d. random variables. Then the LDP still holds by Cramér Theorem on R, and the rate function
Jθ0 is defined by
Jθ0 (θ) := sup {γθ − Λθ0 (c1 γ) − γc0 } ,
γ∈R

6

which yields



θ − c0
Jθ0 (θ) = sup c1 γ
− Λθ0 (c1 γ) = sup {c1 γµ(θ) − Λθ0 (c1 γ)} = Λ∗θ0 (µ(θ)),
c1
γ∈R
γ∈R
as desired.
(ii) Since the restriction of the function µ−1 on (αθ0 , ωθ0 ) is continuous and Λ∗θ0 is a good rate func
tion, a straightforward application of the contraction principle yields the LDP of µ−1 (X̄n ) : n ≥ 1
with good rate function Jθ0 defined by

Jθ0 (θ) := inf Λ∗θ0 (x) : x ∈ (αθ0 , ωθ0 ), µ−1 (x) = θ .

Moreover the set x ∈ (αθ0 , ωθ0 ) : µ−1 (x) = θ has at most one element, namely


{µ(θ)} for θ ∈ Θ such that µ(θ) ∈ (αθ0 , ωθ0 )
−1
x ∈ (αθ0 , ωθ0 ) : µ (x) = θ =
∅
otherwise;
thus we have Jθ0 (θ) = Λ∗θ0 (µ(θ)) for θ ∈ Θ such that µ(θ) ∈ (αθ0 , ωθ0 ), and Jθ0 (θ) = ∞ otherwise.
Now, in the spirit of Remark 2.1, it would be interesting to have a local strict inequality between
the rate function Iλ,θ0 (θ) in Proposition 3.1 for MQ estimators (for some λ ∈ (0, 1)), and the rate
function Jθ0 (θ) in Proposition 4.1 for MM estimators.
Then we can repeat the same arguments which led us to Proposition 3.2. Namely, if both rate
functions Jθ0 and Iλ,θ0 (for some λ ∈ (0, 1)) are twice differentiable, Jθ0 is locally larger (resp.
00 (θ ) (resp. J 00 (θ ) < I 00 (θ )). So it is natural to
smaller) than Iλ,θ0 around θ0 if Jθ000 (θ0 ) > Iλ,θ
0
θ0 0
λ,θ0 0
0
00
give an expression of Jθ0 (θ0 ) under suitable hypotheses.
Proposition 4.2 (An expression for Jθ000 (θ0 )). Let Jθ0 be the rate function in Proposition 4.1.
Assume that, for all θ ∈ Θ, the function Λθ in (3) is finite in a neighborhood of the origin γ = 0
Rω
0
2
and that µ(·) is twice differentiable. Then Jθ000 (θ0 ) = (µσ2(θ(θ00))) , where σ 2 (θ) := αθθ x2 dFθ (x) − µ2 (θ)
is the variance function.
Proof. One can easily check that
Jθ0 0 (θ) = (Λ∗θ0 )0 (µ(θ))µ0 (θ) and Jθ000 (θ) = (Λ∗θ0 )00 (µ(θ))(µ0 (θ))2 + µ00 (θ)(Λ∗θ0 )0 (µ(θ)).
Then, since we have (Λ∗θ0 )00 (µ(θ0 )) =

1
σ 2 (θ0 )

and (Λ∗θ0 )0 (µ(θ0 )) = 0 (this equalities are well-known,

and can be easily checked), we immediately get the desired equality Jθ000 (θ0 ) =

(µ0 (θ0 ))2
.
σ 2 (θ0 )

Remark 4.1 (On the functions Λθ and Λ∗θ in (3)). The function Λθ is finite in a neighborhood of
the origin γ = 0 when we deal with empirical means (of i.i.d. random variables) with light-tailed
distribution. Typically Λ∗θ is a good rate function only in this case.

5

Examples

The aim of this section is to present several examples of statistical models with unknown parameter
θ ∈ Θ, where Θ ⊂ R; in all the examples we always deal with one-dimensional parameters assuming
all the others to be known.
Let us briefly introduce the examples presented below. We investigate distributions with scale
parameter in Example 1, with location parameter in Example 2, and with skewness parameter in
7

Example 3. We remark that in Example 3 we use the epsilon-Skew-Normal distribution defined
in [14]; this choice is motivated by the availability of an explicit expression of the inverse of the
distribution function giving us the possibility of obtaining explicit formulas. Moreover we present
Example 4 with Pareto distributions, which allows to give a concrete illustration of the content
of Remark 3.1. In all these statistical models the intervals {(αθ , ωθ ) : θ ∈ Θ} do not depend on
θ and we simply write (α, ω). Finally we present Example 5 where we have (αθ , ωθ ) = (0, θ) for
θ ∈ Θ := (0, ∞); namely for this example θ is a right-endpoint parameter.
00 (θ ) (as a consequence of PropoIn all examples (except Example 4) we give a formula for Iλ,θ
0
0
sition 3.2) which will be used for the local comparisons between rate functions (in the spirit of
Remark 2.1) analyzed in Section 6.
In what follows we say that a distribution function F on R has the symmetry property if it is
a distribution function of a symmetric random variable, i.e. if F (x) = 1 − F (−x) for all x ∈ R. In
such a case we have F −1 (λ) = −F −1 (1 − λ) for all λ ∈ (0, 1).
Example 1 (Statistical model with a scale parameter θ ∈ Θ := (0, ∞)). Let Fθ be defined by
x
Fθ (x) := G
for x ∈ (α, ω),
θ
where G is a strictly increasing distribution function on (α, ω) = (0, ∞) or (α, ω) = (−∞, ∞).
Then


θ
Fθ−1 (λ) := θG−1 (λ) and hλ,θ0 (θ) = Fθ0 (Fθ−1 (λ)) = G
· G−1 (λ) ;
θ0
it is important to remark that, when (α, ω) = (−∞, ∞), the value λ = G(0) (which yields G−1 (λ) =
0) is not allowed. Now we give a list of some specific examples studied in this paper.
For the case (α, ω) = (0, ∞) we consider the Weibull distribution:
G(x) := 1 − exp(−xρ ) (where ρ > 0) and G−1 (λ) := (− log(1 − λ))1/ρ .

(4)

We also give some specific examples where (α, ω) = (−∞, ∞) and, in each case, η ∈ R is a known
location parameter (and the not-allowed value λ = G(0) depends on η): the Normal distribution
G(x) := Φ(x − η) and G−1 (λ) := η + Φ−1 (λ),
where Φ is the standard Normal distribution function; the Cauchy distribution
 

1
π
1
G(x) :=
π ;
arctan(x − η) +
and G−1 (λ) := η + tan
λ−
π
2
2

(5)

(6)

the logistic distribution

1
−1 ;
λ

(7)

G(x) := exp(−e−(x−η) ) and G−1 (λ) := η − log(− log λ).

(8)

G(x) :=

1
1 + e−(x−η)

and G−1 (λ) := η − log



the Gumbel distribution

If G is twice differentiable we have
00
Iλ,θ
(θ0 ) =
0

{G0 (G−1 (λ))G−1 (λ)}2
λ(1 − λ)θ02
8

(9)

by Proposition 3.2; so, if it is possible to find an optimal λmax , such a value does not depend on θ0
(on the contrary it could depend on the known location parameter η as we shall see in Section 6).
00 (θ ) = 0 if we consider the not-allowed value λ = G(0) ∈ (0, 1)
Moreover one can check that Iλ,θ
0
0
00 (θ ) = I 00
(when (α, ω) = (−∞, ∞)) because G−1 (λ) = 0, and that Iλ,θ
0
1−λ,θ0 (θ0 ) (for all λ ∈ (0, 1))
0
if G is symmetric as it happens, for instance, in (5), (6) and (7) with η = 0.
Example 2 (Statistical model with a location parameter θ ∈ Θ := (−∞, ∞)). Let Fθ be defined
by
Fθ (x) := G(x − θ) for x ∈ (α, ω) = (−∞, ∞),
where G is a strictly increasing distribution function on (α, ω) = (−∞, ∞). Then

Fθ−1 (λ) := θ + G−1 (λ) and hλ,θ0 (θ) = Fθ0 (Fθ−1 (λ)) = G θ + G−1 (λ) − θ0 .
We give some specific examples studied in this paper and, in each case, s > 0 is a known scale
parameter: the Normal distribution
x
G(x) := Φ
and G−1 (λ) := s · Φ−1 (λ);
(10)
s
the Cauchy distribution

 
1
x π
1
−1
G(x) :=
arctan +
and G (λ) := s · tan
λ−
π ;
π
s
2
2

(11)

the logistic distribution

1
−1 ;
λ

(12)

G(x) := exp(−e−x/s ) and G−1 (λ) := −s · log(− log λ).

(13)

1
G(x) :=
and G−1 (λ) := −s · log
−x/s
1+e



the Gumbel distribution

If G is twice differentiable we have
00
Iλ,θ
(θ0 ) =
0

{G0 (G−1 (λ))}2
λ(1 − λ)

(14)

by Proposition 3.2; so, if it is possible to find an optimal λmax , such a value does not depend on θ0
00 (θ ) = I 00
and on the known scale parameter s. Moreover one can check that Iλ,θ
0
1−λ,θ0 (θ0 ) (for all
0
λ ∈ (0, 1)) if G has the symmetry property (as happens for G in (10), (11) and (12), and not for
G in (13)).
Example 3 (Statistical model with a skewness parameter θ ∈ Θ := (−1, 1)). Let Fθ be defined by

x
(1 + θ)G( 1+θ
)
for x ≤ 0
Fθ (x) :=
with x ∈ (α, ω) = (−∞, ∞),
x
θ + (1 − θ)G( 1−θ ) for x > 0,
where G is a strictly increasing distribution function on (α, ω) = (−∞, ∞) with the symmetry
property. Then

λ
) for λ ∈ (0, 1+θ
(1 + θ)G−1 ( 1+θ
2 ]
Fθ−1 (λ) :=
1+θ
(1 − θ)G−1 ( λ−θ
)
for
λ
∈
(
1−θ
2 , 1)
9

and




 (1 + θ0 )G 1+θ G−1 ( λ )
1+θ0
1+θ

hλ,θ0 (θ) = Fθ0 (Fθ−1 (λ)) =
 θ0 + (1 − θ0 )G 1−θ G−1 ( λ−θ )
1−θ0
1−θ

for θ ≥ 2λ − 1
for θ < 2λ − 1.

We can consider the same specific examples presented in Example 2, i.e. the functions G in (10),
(11) and (12) for some known scale parameter s > 0.
If G is twice differentiable and G00 (0) = 0 we have

h

i2
1
0 (G−1 ( λ )) G−1 ( λ ) − λ (G−1 )0 ( λ )
0

G
for λ ∈ (0, 1+θ
1+θ0
1+θ0
1+θ0
1+θ0
2 ]
λ(1−λ)
00

i2
h
Iλ,θ0 (θ0 ) =

1
0 (G−1 ( λ−θ0 )) −G−1 ( λ−θ0 ) + λ−1 (G−1 )0 ( λ−θ0 )
0
for λ ∈ ( 1+θ
G
1−θ0
1−θ0
1−θ0
1−θ0
2 , 1)
λ(1−λ)
by Proposition 3.2, and therefore

h
i2
1
0 (G−1 ( λ ))G−1 ( λ ) − λ

G
1+θ0
1+θ0
1+θ0
λ(1−λ)
00
h
i2
(θ0 ) =
Iλ,θ
0

λ−θ0
λ−θ0
λ−1
1
0
−1
−1
−G
))G
)
+
(G
(
(
1−θ0
1−θ0
1−θ0
λ(1−λ)

0
for λ ∈ (0, 1+θ
2 ]
0
for λ ∈ ( 1+θ
2 , 1);

(15)

so one can expect that, if it is possible to find an optimal λmax , such a value depends on θ0 (this is
00 (θ ) in (15) does not depend on s,
what happens in Section 6). Moreover one can check that Iλ,θ
0
0
00
00
00
I1/2,0 (0) = 1 and Iλ,θ0 (θ0 ) = I1−λ,−θ0 (−θ0 ) (for all λ ∈ (0, 1)).
Example 4 (Statistical model with Pareto distributions with θ ∈ Θ := (0, ∞)). Let Fθ be defined
by
Fθ (x) := 1 − x−1/θ for x ∈ (α, ω) = (1, ∞).
Then
Fθ−1 (λ) := e−θ log(1−λ) = (1 − λ)−θ and hλ,θ0 (θ) = Fθ0 (Fθ−1 (λ)) = 1 − (1 − λ)θ/θ0 .

(16)

We remark that, if we consider Example 1 with G as in (4) with ρ = 1, namely
F̃θ (x) = 1 − e−x for (α̃, ω̃) = (0, ∞),
we can refer to Remark 3.1 with
ψ(x) := ex for x ∈ (α̃, ω̃) := (0, ∞)
(note that (ψ(α̃), ψ(ω̃)) = (1, ∞) = (α, ω)). Then, as pointed out in Remark 3.1, Iλ,θ0 ;ψ and Iλ,θ0
coincide; in fact hλ,θ0 (θ) in (16) meets the analogue expression in Example 1 with G as in (4) with
ρ = 1, namely




θ
θ
−1
−1
F̃θ0 (F̃θ (λ)) = G
· G (λ) = 1 − exp − · (− log(1 − λ)) = 1 − (1 − λ)θ/θ0 .
θ0
θ0
Example 5 (Statistical model with a “right endpoint”parameter θ ∈ Θ := (0, ∞)). Let Fθ be
defined by
G(x)
Fθ (x) :=
for x ∈ (αθ , ωθ ) := (0, θ),
G(θ)
where G : [0, ∞) → [0, ∞) is a strictly increasing function such that G(0) = 0. Then
Fθ−1 (λ) := G−1 (λG(θ)) and hλ,θ0 (θ) = Fθ0 (Fθ−1 (λ)) =
10

λG(θ)
(for λG(θ) ∈ (0, G(θ0 ))).
G(θ0 )

Moreover, after some computations, we get
(
(1−λ)G(θ0 )
0)
λ log G(θ
G(θ) + (1 − λ) log G(θ0 )−λG(θ)
Iλ,θ0 (θ) :=
∞

for 0 < θ < G−1



G(θ0 )
λ



otherwise.

As a specific example we can consider G(x) = x (for all x); in such a case Fθ is the distribution
function concerning the uniform distribution on (0, θ). Finally, if G is twice differentiable, we have
00
(θ0 ) =
Iλ,θ
0

λ(G0 (θ0 ))2
(1 − λ)G2 (θ0 )

(17)

by Proposition 3.2.

6

Local comparisons between rate functions for some examples

In this section we analyze the examples presented in Section 5. We consider local comparisons
between rate functions in the spirit of Remark 2.1 and, more precisely, the following two issues.
• A discussion on the choice of optimal values of λ in order to get the best rate of convergence.
00 (θ ) (varying λ ∈ (0, 1)) in order to find an
More precisely we study the behavior of Iλ,θ
0
0
optimal λmax in the sense of Definition 3.1.
• The comparison of the convergence of the MQ estimators and of the MM estimators. More
00 (θ ) in Proposition 3.2 (for
precisely, when we deal with MM estimators, we compare Iλ,θ
0
0
00
some λ ∈ (0, 1)) and Jθ0 (θ0 ) in Proposition 4.2; obviously, when we have an optimal λmax , we
take λ = λmax . In the single case presented below where MM estimators are not defined, we
compare the convergence of the MQ estimators and of suitable GMM estimators.
We find at least an optimal λmax for all examples except for Example 5 (where we should consider
00 (θ ) = I 00
λ = 1). For several particular examples where (α, ω) = (−∞, ∞) we have Iλ,θ
0
1−λ,θ0 (θ0 )
0
for all λ ∈ (0, 1); then, in those cases, we have the optimal value λmax = 1/2 or, by symmetry, two
distinct optimal values λmax,1 and λmax,2 = 1 − λmax,1 .
In view of what follows it is useful to consider two suitable values λ̃1 , λ̃2 ∈ (0, 1) presented
in the next Lemma 6.1. The value λ̃1 appears in the computations for the Weibull distribution
in Example 1 (and also in the computations for Example 4 as a trivial consequence), while the
λ̃2 appears in the computations for the Gumbel distribution in both Examples 1 and 2; however,
interestingly, Lemma 6.1(iii) shows the close relationship between λ̃1 and λ̃2 .
Lemma 6.1 (The values λ̃1 and λ̃2 ). The following statements hold.
2
(i) Let f1 be the function defined by f1 (λ) := (1−λ)(log(1−λ))
. Then supλ∈(0,1) f1 (λ) = f1 (λ̃1 ), where
λ
λ̃1 ' 0.7968 is the unique value (1/2, 1) such that −2λ̃1 − log(1 − λ̃1 ) = 0.
λ)2
(ii) Let f2 be the function defined by f2 (λ) := λ(log
1−λ . Then supλ∈(0,1) f2 (λ) = f2 (λ̃2 ), where
λ̃2 ' 0.2032 is the unique value (0, 1/2) such that log λ̃2 + 2 − 2λ̃2 = 0.
(iii) We have λ̃1 + λ̃2 = 1.
Proof. (i) The derivative of f1 is
d
log(1 − λ)[−2λ − log(1 − λ)]
f1 (λ) =
dλ
λ2

11

and −2λ − log(1 − λ) < 0 (resp. −2λ − log(1 − λ) > 0) if λ < λ̃1 (resp. λ > λ̃1 ); therefore we have
d
d
dλ f1 (λ) > 0 for λ ∈ (0, λ̃1 ) and dλ f1 (λ) < 0 for λ ∈ (λ̃1 , 1).
(ii) The derivative of f2 is
d
log λ · [log λ + 2 − 2λ]
f2 (λ) =
dλ
(1 − λ)2
and log λ + 2 − 2λ < 0 (resp. log λ + 2 − 2λ > 0) if λ < λ̃2 (resp. λ > λ̃2 ); therefore we have
d
d
dλ f2 (λ) > 0 for λ ∈ (0, λ̃2 ) and dλ f2 (λ) < 0 for λ ∈ (λ̃2 , 1).
(iii) The equality λ̃1 + λ̃2 = 1 can be easily checked by taking into account how the values λ̃1 and
λ̃2 are defined in statements (i) and (ii).

6.1

Example 1

In this section we consider the particular example of the Weibull distribution with (α, ω) = (0, ∞),
and the particular examples with (α, ω) = (−∞, ∞). In each part we analyze the MQ estimators,
and we conclude with the MM estimators.
Analysis of MQ estimators for Weibull distribution. Here we consider G in (4). By (9)
we have
ρ2 (1 − λ)(log(1 − λ))2
00
Iλ,θ
(θ0 ) =
.
0
λθ02
Then we have a unique optimal value λmax = λ̃1 (for every ρ and θ0 ), where λ̃1 is the value in
2
00 (θ ) = ρ f1 (λ) .
Lemma 6.1(i); in fact, if we consider the function f1 in that lemma, we have Iλ,θ
0
0
θ2
0

MM versus MQ estimators for Weibull distribution. We start with the analysis of MM
m
estimators. We have µ(θ) := θΓ(1 + 1/ρ), and therefore µ−1 (m) := Γ(1+1/ρ)
, where Γ is the usual
−1
Gamma function. Thus, by Proposition 4.1(i), {µ (X̄n ) : n ≥ 1} satisfies the LDP with rate
1
function Jθ0 with c1 = Γ(1+1/ρ)
and c0 = 0. In what follows we consider the light-tailed case ρ ≥ 1
and the heavy-tailed case ρ ∈ (0, 1); some more details on the exponential distribution case ρ = 1
are given in Remark 6.1.
Light-tailed case (namely ρ ≥ 1). The rate functions Jθ0 and Λ∗θ0 are good and we can refer to the
comparison between
Γ2 (1 + 1/ρ)
Jθ000 (θ0 ) = 2
θ0 [Γ(1 + 2/ρ) − Γ2 (1 + 1/ρ)]


(this value is a consequence of Proposition 4.2 noting that σ 2 (θ) = θ2 Γ(1 + 2/ρ) − Γ2 (1 + 1/ρ) )
and, for the optimal value λmax = λ̃1 in Lemma 6.1(i),
Iλ̃00

1 ,θ0

(θ0 ) =

(1 − λ̃1 )(log(1 − λ̃1 ))2
(1 − λ̃1 )(2λ̃1 )2
4λ̃1 (1 − λ̃1 )
=
=
.
2
2
θ02
λ̃1 θ0
λ̃1 θ0

We remark that Jθ000 (θ0 ) is an increasing function of ρ ∈ (0, ∞); in fact, if we set
a(ρ) := Γ0 (1 + 2/ρ)Γ(1 + 1/ρ) − Γ(1 + 2/ρ)Γ0 (1 + 1/ρ),
0

0

(1+2/ρ)
(1+1/ρ)
we have a(ρ) > 0 (noting that ΓΓ(1+2/ρ)
> ΓΓ(1+1/ρ)
because the digamma function x 7→
increasing on (0, ∞)) and therefore


d
Γ2 (1 + 1/ρ)
2Γ(1 + 1/ρ)a(ρ)
=
> 0.

2
2
2
dρ θ0 [Γ(1 + 2/ρ) − Γ (1 + 1/ρ)]
Γ(1+2/ρ)
2
2
4
ρ θ0 Γ2 (1+1/ρ) − 1 Γ (1 + 1/ρ)

12

Γ0 (x)
Γ(x)

is

Thus, for all ρ ≥ 1, MM estimators converge faster than every MQ estimators because
Jθ000 (θ0 ) ≥

1
4λ̃1 (1 − λ̃1 )
>
= Iλ̃00 ,θ (θ0 )
2
1 0
θ0
θ02

noting that inf ρ≥1 Jθ000 (θ0 ) = Jθ000 (θ0 ) ρ=1 = θ12 and 4λ̃1 (1 − λ̃1 ) ' 0.6476.
0
Heavy-tailed case (namely ρ ∈ (0, 1)). The rate functions Jθ0 and Λ∗θ0 are not good (we recall
Remark 4.1 presented above). We can say that Jθ0 (θ) = 0 for θ ≥ θ0 ; thus Iλ,θ0 (θ) > Jθ0 (θ) for
θ > θ0 (for all λ ∈ (0, 1)). Then we have to compare Iλ̃1 ,θ0 (θ) and Jθ0 (θ) in a left neighborhood of
θ0 , namely when θ ∈ (θ0 − δ, θ0 ) for δ > 0 small enough. Therefore it suffices to compare Iλ̃00 ,θ (θ0 )
1

0

d2
which coincides with Jθ000 (θ0 ) presented above for the
and the left second derivative dθ
2 Jθ0 (θ−)
θ=θ0
d2
light-tailed case. We already explained that dθ
is an increasing function of ρ ∈ (0, ∞);
2 Jθ0 (θ−)
θ=θ
0

moreover

d2

Jθ (θ−)
dθ2 0

=
θ=θ0

1
θ02

h

Γ(1+2/ρ)
Γ2 (1+1/ρ)

i → 0 as ρ → 0
−1

by taking into account the asymptotic behavior of Gamma function. In conclusion there exists
ρ0 ' 0.81068 (computed numerically) such that:
1. Iλ̃00

(θ0 ) >

d2
J (θ−)
dθ2 θ0
θ=θ0

for ρ ∈ (0, ρ0 );

2. Iλ̃00

(θ0 ) <

d2
J (θ−)
dθ2 θ0
θ=θ0

for ρ ∈ (ρ0 , 1).

1 ,θ0

1 ,θ0

Thus, by taking into account Remark 2.1, MQ estimators (with λ = λ̃1 ) converge faster than MM
estimators in the first case while, in the second case, the convergence of MQ and MM estimators
cannot be compared because we cannot find δ > 0 such that Iλ̃1 ,θ0 (θ) > Jθ0 (θ) or Iλ̃1 ,θ0 (θ) < Jθ0 (θ)
for 0 < |θ − θ0 | < δ.
Remark 6.1 (The case ρ = 1, namely the exponential distribution). If ρ = 1 the rate functions
Jθ0 and Λ∗θ0 coincide (in fact Γ(1 + 1/ρ) = 1) and an explicit expression of Λ∗θ0 is available, namely
(
Jθ0 (θ) = Λ∗θ0 (θ) =

θ
θ0

− 1 − log

∞

 
θ
θ0

for θ ∈ (0, ∞)
otherwise;

then we can directly compute Jθ000 (θ0 ) = θ12 , which meets the above expression. In this case the MM
0
estimators coincide with the ML estimators, and we already expected that they converge faster than
the MQ estimators.
Analysis of MQ estimators for particular examples with (α, ω) = (−∞, ∞). Here we
present the results concerning the specific examples listed above. For all cases except the one
with the Gumbel distribution we choose η in order to have G(0) ∈ {0.25, 0.5, 0.75}, and we have
some common features: G(0) = 1/2 for η = 0 (actually the symmetry property holds); we obtain
symmetric values with respect λ = 1/2 such that the more the tails of the distributions are light,
the more the numerical values of λmax are distant from λ = 1/2. The case with the Gumbel
distribution behaves differently because the symmetry property fails for each fixed value of η. In

13

all cases we can only give numerical values.
Normal distribution (namely G in (5)). We have G(0) = Φ(−η) and, by (9),
00
(θ0 ) =
Iλ,θ
0

(ϕ(Φ−1 (λ)){η + Φ−1 (λ)})2
,
λ(1 − λ)θ02

where ϕ is the standard Normal probability density function. Moreover

η=0
η = −Φ−1 (1/4)
η = −Φ−1 (3/4)

numerical values for λmax
0.06 (and 0.94 by symmetry)
0.90
1 − 0.90 = 0.10 (by symmetry)

Cauchy distribution (namely G in (6)). We have G(0) =
00
Iλ,θ
(θ0 )
0


=

1
π

arctan(−η) +

π
2



and, by (9),

  2
λ − 12 π
.
  2
λ − 21 π
λ(1 − λ)θ02

η + tan


π 2 1 + tan2

Moreover
numerical values for λmax
0.21 (and 0.79 by symmetry)
0.65
1 − 0.65 = 0.35 (by symmetry)

η=0
η=1
η = −1

Logistic distribution (namely G in (7)). We have G(0) =
00
Iλ,θ
(θ0 )
0

1
1+eη

and, by (9),

2
λ(1 − λ) η − log( λ1 − 1)
.
=
θ02

Moreover
η=0
η = log 3
η = − log 3

numerical values for λmax
0.08 (and 0.92 by symmetry)
0.85
1 − 0.85 = 0.15 (by symmetry)

Gumbel distribution (namely G in (8)). We have G(0) = exp(−eη ) and, by (9),
00
Iλ,θ
(θ0 ) =
0

λ(log λ)2 (η − log(− log λ))2
.
(1 − λ)θ02

Some numerical inspections reveal that in general, for each fixed value of η, we can find an optimal
value λmax = λmax (η). Then, if we consider the value λ̃2 and the function f2 in Lemma 6.1(ii), we
can say that
λmax = λmax (η) → λ̃2 as |η| → ∞
00 (θ ) behaves like
because Iλ,θ
0
0

f2 (λ)η 2
θ02

when |η| is large.

MM versus MQ estimators for particular examples with (α, ω) = (−∞, ∞). The MM
estimators are well-defined only for the case with Gumbel distribution and, in the spirit of Remark
2.1, we can compare Jθ000 (θ0 ) and Iλ00max ,θ0 (θ0 ). However, for Normal and logistic distributions, it is
possible to consider suitable GMM estimators {Θ̃n : n ≥ 1} by matching empirical and theoretical
14

variances; so we present the rate function J˜θ0 which governs the LDP of {Θ̃n : n ≥ 1} and, at least
for the case of Normal distribution, we can give an expression of J˜θ000 (θ0 ) and we can compare the
convergence of MQ and GMM estimators.
Gumbel distribution. We have µ(θ) := η + θγ∗ , where γ∗ is the Euler’s constant, and therefore
−1
µ−1 (m) := m−η
γ∗ . Thus, by Proposition 4.1(i), {µ (X̄n ) : n ≥ 1} satisfies the LDP with rate
∗
function Jθ0 with c1 = γ1∗ and c0 = −η
γ∗ . The rate functions Jθ0 and Λθ0 are good (we take into
account Remark 4.1) and we can refer to the comparison between Iλ00max ,θ0 (θ0 ) and Jθ000 (θ0 ). We
remark that, by Proposition 4.2, we have Jθ000 (θ0 ) =
σ 2 (θ)

θ2 π2
6 ).

=
Lemma 6.1,

6γ∗2
θ02 π 2

for each fixed value of η (in fact we have

Then, for all η ∈ R, we have Iλ00max ,θ0 (θ0 ) > Jθ000 (θ0 ) noting
we have max{Iλ̃00 ,θ (θ0 ), Iλ̃00 ,θ (θ0 )} > Jθ000 (θ0 ) (see Figure 1).
0

2

0

−1

0

1

2

3

4

5

1

that, for λ̃1 and λ̃2 as in

−3

−2

−1

0
η

1

2

3

Figure 1: The second derivatives Iλ̃00 ,θ (θ0 ) (broken line) and Iλ̃00 ,θ (θ0 ) (dotted line) as functions of
2 0
1 0
η. The solid line represents the value of Jθ000 (θ0 ) which does not depend on η.
Normal (and logistic) distribution. The MM estimators are not well-defined because µ(θ) := η. So
it is natural to match empirical and theoretical variances, i.e.

n
1
for the Normal distribution
1X
σ 2 (θ) =
(xi − η)2 , where σ 2 (θ) = cθ2 and c =
π2
n
for the logistic distribution,
3
i=1

and we obtain the GMM estimators {Θ̃n : n ≥ 1} defined by
!1/2
n
1 X
Θ̃n :=
(xi − η)2
.
cn
i=1

15

Then, by adapting the proof of Proposition 4.1, we can consider the function
Z ωθ
n
o
2
∗
eγ(x−η) dFθ0 (x),
Λ̃θ0 (y) := sup γy − Λ̃θ0 (γ) , where Λ̃θ0 (γ) := log
γ∈R

αθ

and we can say {Θ̃n : n ≥ 1} satisfies the LDP with good rate function J˜θ0 defined by
J˜θ0 (θ) := inf{Λ̃∗θ0 (y) : (y/c)1/2 = θ}.
From now on we restrict the attention to the case with Normal distribution because we can give
explicit formulas. We have
 2 
 i
(
( h
θ02
θ0 /2
1
y
1 y
if
γ
<
log
−
1
−
log
if y > 0
2
2
2
∗
θ0 /2−γ
2 θ02
θ02
Λ̃θ0 (γ) =
(y)
=
Λ̃
θ0
θ02
∞
if y ≤ 0,
∞
if γ ≥ 2 ,
and

(
J˜θ0 (θ) =

1
2

h

θ2
θ02

− 1 − log

 2 i
θ
θ02

∞

if θ > 0
if θ ≤ 0;

thus, after some computations, we get J˜θ000 (θ0 ) = θ22 for all value of η ∈ R. We conclude with the
0
comparison between MQ and GMM estimators. Some numerical inspections reveal that in general,
for each fixed value of η, we can find an optimal value λmax = λmax (η) (their numerical values for
η = 0, η = Φ−1 (1/4) and η = Φ−1 (3/4) were presented above); moreover Iλ00max (η),θ0 (θ0 ) > J˜θ000 (θ0 )
for |η| large enough because, for each fixed λ ∈ (0, 1),
00
(θ0 ) → ∞ as |η| → ∞.
Iλ,θ
0

On the other hand we cannot say that Iλ00max (η),θ0 (θ0 ) > J˜θ000 (θ0 ) for all η ∈ R; in fact, for η = 0, we
have
2
0.6085
< 2 = J˜θ000 (θ0 )
Iλ00max (0),θ0 (θ0 ) '
2
θ0
θ0
(where λmax (0) ' 0.06 or λmax (0) ' 0.94). For completeness, following the same lines of the
particular case with Gumbel distribution, we remark that
λmax = λmax (η) → 1/2 as |η| → ∞
because, if we consider the function f (λ) :=
large, and supλ∈(0,1) f (λ) = f (1/2).

6.2

(ϕ(Φ−1 (λ)))2
,
λ(1−λ)

00 (θ ) behaves like
Iλ,θ
0
0

f (λ)η 2
θ02

when |η| is

Example 2

We start with the analysis of MQ estimators. We conclude with the MM estimators, and their
comparison with the MQ estimators.
Analysis of MQ estimators. Here we present the results concerning the specific examples listed
above. In all cases, except the one with Gumbel distribution, we can conclude that λ = 1/2 is
optimal; however we can find counterexamples (see Appendix A). A further common feature (for
d 00
all cases except the one with Gumbel distribution) is that dλ
Iλ,θ0 (θ0 )
= 0 (and obviously this
λ=1/2

16

does not guarantee that λ = 1/2 is an optimal; this will be explained in Appendix A); in fact we
have
d 00
I (θ0 )
dλ λ,θ0

=
λ=1/2

2G0 (G−1 (λ))G00 (G−1 (λ))(G−1 )0 (λ)λ(1 − λ) − (1 − 2λ){G0 (G−1 (λ))}2
λ2 (1 − λ)2
2G00 (G−1 (λ))λ(1 − λ) − (1 − 2λ){G0 (G−1 (λ))}2
=
λ2 (1 − λ)2

λ=1/2

=0
λ=1/2

noting that G−1 (1/2) = 0 (because the distribution function G has the symmetry property) and
G00 (0) = 0 (because the probability density function G0 (x) has a maximum at x = 0).
Normal distribution (namely G in (10)). By (14) we have
00
(θ0 ) =
Iλ,θ
0

ϕ2 (Φ−1 (λ))
.
s2 λ(1 − λ)

One can check numerically that we have a unique optimal λmax (for every s), namely λmax = 0.5.
Cauchy distribution (namely G in (11)). By (14) we have
00
Iλ,θ
(θ0 ) =
0

1
s2 π 2



1+

tan2

λ−

1
2

 
π

2

λ(1 − λ)

.

One can check numerically that we have a unique optimal λmax (for every s), namely λmax = 0.5.
Logistic distribution (namely G in (12)). By (14) we have
λ(1 − λ)
s2

00
Iλ,θ
(θ0 ) =
0

One can immediately check (we have a polynomial with degree 2) that we have a unique optimal
λmax (for every s), namely λmax = 0.5.
Gumbel distribution (namely G in (13)). By (14) we have
00
Iλ,θ
(θ0 ) =
0

λ(log λ)2
s2 (1 − λ)

Then we have a unique optimal value λmax = λ̃2 (for every s), where λ̃2 is the value in Lemma
00 (θ ) = f2 (λ) .
6.1(ii); in fact, if we consider the function f2 in that lemma, we have Iλ,θ
0
s2
0
MM versus MQ estimators. The MM estimators are well-defined in all cases except the one
with Cauchy distribution. Moreover, by taking into account Remark 4.1, we can always refer to
the comparison between Jθ000 (θ0 ) and Iλ00max ,θ0 (θ0 ).
Normal distribution. In this case µ(θ) := θ. Thus, by Proposition 4.1(i), {µ−1 (X̄n ) : n ≥ 1} =
{X̄n : n ≥ 1} satisfies the LDP with rate function Jθ0 defined by
Jθ0 (θ) = Λ∗θ0 (θ) =

(θ − θ0 )2
2s2

(Jθ0 and Λ∗θ0 coincide noting that c1 = 1 and c0 = 0). Then, since λmax = 1/2 we have to compare
Jθ000 (θ0 ) = s12 (which meets the expression provided by Proposition 4.2 noting that σ 2 (θ) = s2 ) and
00
00
I1/2,θ
(θ0 ) = πs2 2 and, obviously, we have Jθ000 (θ0 ) > I1/2,θ
(θ0 ) (for every s). Thus MM estimators
0
0
converge faster than every MQ estimators; in some sense we already expected this noting that the
17

MM estimators coincide with the ML estimators.
Logistic distribution. In this case µ(θ) := θ. Thus, by Proposition 4.1(i), {µ−1 (X̄n ) : n ≥ 1} =
{X̄n : n ≥ 1} satisfies the LDP with rate function Jθ0 , which coincides with Λ∗θ0 (we have again
c1 = 1 and c0 = 0); in this case we cannot provide an explicit expression of Λ∗θ0 . Then, since
λmax = 1/2 we have to compare Jθ000 (θ0 ) = π23s2 (this is a consequence of Proposition 4.2 noting
2 2

00
00
that σ 2 (θ) = π 3s ) and I1/2,θ
(θ0 ) = 4s12 and, obviously, we have Jθ000 (θ0 ) > I1/2,θ
(θ0 ) (for every s).
0
0
Thus MM estimators converge faster than every MQ estimators but, differently from what happens
for the case with Normal distribution, they do not coincide with ML estimators.
Gumbel distribution. In this case µ(θ) := θ + sγ∗ , where γ∗ is the Euler’s constant. Thus, by
Proposition 4.1(i), {µ−1 (X̄n ) : n ≥ 1} = {X̄n : n ≥ 1} satisfies the LDP with rate function Jθ0
(with c1 = 1 and c0 = −sγ∗ ); in this case we cannot provide an explicit expression of Λ∗θ0 . The rate
functions Jθ0 and Λ∗θ0 are good and we can refer to the comparison between

Jθ000 (θ0 ) =

6
π 2 s2

(this value is a consequence of Proposition 4.2 noting that σ 2 (θ) =
λmax = λ̃2 defined in Lemma 6.1(ii),
Iλ̃00

2 ,θ0

(θ0 ) =

π 2 s2
6 )

and, for the optimal value

λ̃2 (log λ̃2 )2
λ̃2 (2λ̃2 − 2)2
4λ̃2 (1 − λ̃2 )
=
=
.
2
2
s2
s (1 − λ̃2 )
s (1 − λ̃2 )

We can check numerically that Iλ̃00 ,θ (θ0 ) > Jθ000 (θ0 ) (for every s); in fact we have 4λ̃2 (1−λ̃2 ) ' 0.6476
2 0
(we get a numerical value obtained for the statistical model with Weibull distributions because
4λ̃2 (1 − λ̃2 ) = 4λ̃1 (1 − λ̃1 ) by Lemma 6.1(iii)) and π62 ' 0.6079. Thus MQ estimators with the
optimal value λmax converge faster than MM estimators.

6.3

Example 3

Here we analyze the MQ estimators for the specific examples listed above. In all cases we can only
give numerical values; such values depend on the unknown parameter θ0 , and therefore we do not
discuss the comparison with the MM estimators (as we do for the other examples). We have the
same feature highlighted for Example 1 with (α, ω) = (−∞, ∞), namely the more the tail of the
distributions are light, the more the numerical values of λmax are distant from λ = 1/2.
Normal distribution (namely G in (10)). By (15) we have

h
i2
1
−1 ( λ ))Φ−1 ( λ ) − λ
0

ϕ(Φ
for λ ∈ (0, 1+θ
1+θ0
1+θ0
1+θ0
2 ]
λ(1−λ)
00
h
i
Iλ,θ
(θ
)
=
0
2
0

1
λ−1
−1 λ−θ0
−1 λ−θ0
0
for λ ∈ ( 1+θ
2 , 1).
λ(1−λ) −ϕ(Φ ( 1−θ0 ))Φ ( 1−θ0 ) + 1−θ0
Moreover
θ0 = 0
θ0 = 1/2
θ0 = −1/2

numerical values for λmax
0.15 (and 0.85 by symmetry)
0.94
1 − 0.94 = 0.06 (by symmetry)

Cauchy distribution (namely G in (11)). By (15) we have

"
#2
λ

tan(( 1+θ
− 12 )π)

1
λ
0




 λ(1−λ) π 1+tan2 (( λ − 1 )π) − 1+θ0
1+θ0
2
00
"
#2
Iλ,θ0 (θ0 ) =
λ−θ0
1

tan((
−
)π)

1−θ0
2

1
 + λ−1
 λ(1−λ)
− 

λ−θ0
1−θ0
π 1+tan2 ((
− 1 )π)
1−θ0

18

2

0
for λ ∈ (0, 1+θ
2 ]

0
for λ ∈ ( 1+θ
2 , 1).

Moreover
θ0 = 0
θ0 = 1/2
θ0 = −1/2

numerical values for λmax
0.39 (and 0.61 by symmetry)
0.84
1 − 0.84 = 0.16 (by symmetry)

Logistic distribution (namely G in (12)). By (15) we have
h
i2
1+θ0
λ
λ
λ
(1
−
)
log(
−
−
1)
−
1+θ0
1+θ0
λ
1+θ0
00
h
i2
(θ0 ) =
Iλ,θ
0

λ−θ0
1−θ0
λ−θ0
λ−1
1
(1
−
)
log(
−
1)
+
1−θ0
λ−θ0
1−θ0
λ(1−λ) 1−θ0



1
λ(1−λ)

0
for λ ∈ (0, 1+θ
2 ]
0
for λ ∈ ( 1+θ
2 , 1).

Moreover
θ0 = 0
θ0 = 1/2
θ0 = −1/2

6.4

numerical values for λmax
0.22 (and 0.78 by symmetry)
0.92
1 − 0.92 = 0.08 (by symmetry)

Example 4

Here we analyze Example 4. For MQ estimators we have the same rate function presented in
Example 1 with (α, ω) = (0, ∞) when G is as in (4) and ρ = 1. Thus we have a unique optimal
λmax which does not depend on θ0 , namely λmax = λ̃1 where λ̃1 is defined in Lemma 6.1(i).
Now we briefly discuss the MM estimators for Example 4. We recall that µ(θ) is finite only if
θ ∈ Θ̃ := (0, 1), where Θ̃ ⊂ Θ = (0, ∞). So we could consider the mean function on the restricted
parameter space Θ̃, i.e.
1/θ
1
µ(θ) =
=
for θ ∈ Θ̃.
1/θ − 1
1−θ

Then, if we consider the restricted parameter space Θ̃, the MM estimators µ−1 (X̄n ) : n ≥ 1
are defined by µ−1 (X̄n ) = 1 − X̄n−1 , and the function µ−1 (·) is continuous on (α, ω) = (1, ∞).
Unfortunately we cannot apply Proposition 4.1 because we cannot consider neither the hypotheses
of Proposition 4.1(i) (obvious) nor the hypotheses of Proposition 4.1(ii) because Pareto distributions
are heavy-tailed and Λ∗θ0 is not good (see Remark 4.1).

6.5

Example 5

Here we analyze Example 5. As far as the MQ estimators are concerned, we can say that we
00 (θ ) is an increasing function; in fact, by (17), the derivative
cannot find an optimal λ because Iλ,θ
0
0
00 (θ ) with respect to λ is
of Iλ,θ
0
0
d 00
(G0 (θ0 ))2
Iλ,θ0 (θ0 ) =
.
dλ
(1 − λ)2 G2 (θ0 )
We can also say that the larger is λ the faster is the convergence of the MQ estimators.
In the remaining part we deal with the MM estimators, and we discuss their comparison with
the MQ estimators. Obviously the rate functions Jθ0 and Λ∗θ0 are good (we take into account
00 (θ ) (for λ ∈ (0, 1)); for
Remark 4.1) and we can refer to the comparison between Jθ000 (θ0 ) and Iλ,θ
0
0
completeness we remark that we cannot obtain an explicit expression of Λ∗θ0 (even for the simplest

19

case with the uniform distributions, i.e. the case G(x) = x for all x ∈ (0, ∞)). It is easy to check
that, if we consider λ0 defined by
λ0 :=
we have

(µ0 (θ0 ))2 G2 (θ0 )
,
(µ0 (θ0 ))2 G2 (θ0 ) + σ 2 (θ0 )(G0 (θ0 ))2

Jθ000 (θ0 ) =

(µ0 (θ0 ))2
λ(G0 (θ0 ))2
00
(θ0 ) for λ < λ0
>
= Iλ,θ
0
σ 2 (θ0 )
(1 − λ)G2 (θ0 )

Jθ000 (θ0 ) =

(µ0 (θ0 ))2
λ(G0 (θ0 ))2
00
(θ0 ) for λ > λ0
<
= Iλ,θ
0
σ 2 (θ0 )
(1 − λ)G2 (θ0 )

and

(18)

by Proposition 4.2 and (17).
Thus the MQ estimators converge faster than MM estimators if λ is close to 1; this is not
surprising because the case λ = 1 concerns the case of ML estimators {Xn:n : n ≥ 1}. For
completeness we can say that, if {Xn : n ≥ 1} are i.i.d. with distribution function Fθ0 as in
Example 5, the LDP in Proposition 2.1 with λ = 1 is governed by a good rate function; thus we
can consider a version of Proposition 3.1 with λ = 1, and we have the LDP of {Xn:n : n ≥ 1} with
good rate function I1,θ0 defined by
(
0)
log hλ,θ1 (θ) = log G(θ
for θ ∈ Θ such that θ ∈ (0, θ0 )
G(θ)
0
I1,θ0 (θ) :=
∞
otherwise.
We also remark that in general the threshold value λ0 in (18) depends on θ0 . In fact, for
θ
(eθ −1)2 −θ2 eθ
2
, and
G(x) := ex − 1, after some computations we have µ(θ) = eθe
θ −1 − 1, σ (θ) =
(eθ −1)2
therefore
e2θ0 − 2eθ0 (1 + θ0 ) + (1 + θ0 )2
λ0 = 2θ
.
2e 0 − eθ0 (4 + 2θ0 + θ02 ) + 2 + 2θ0 + θ02
Interestingly we can say that λ0 does not depend on θ0 if G(x) := xy for some y > 0; in fact we
yθ2
yθ
, σ 2 (θ) = (y+2)(y+1)
have µ(θ) = y+1
2 , and therefore
λ0 =

y 2 y 2
( y+1
) (θ0 )
y 2 y 2
) (θ0 )
( y+1

+

yθ02
(yθ0y−1 )2
(y+2)(y+1)2

=

1
y+2
.
y =
1 + y+2
2y + 2

For instance, for the specific case of uniform distributions cited in Example 5 (for which we have

θ2
µ(θ) = 2θ and σ 2 (θ) = 12
for all θ ∈ (0, ∞); so the sequence µ−1 (X̄n ) : n ≥ 1 in Proposition 4.1
is defined by µ−1 (X̄n ) = 2X̄n ) we have G(x) := x, and therefore we get λ0 = 3/4 by setting y = 1.
Finally we remark that, in general, we cannot find δ > 0 such that Iλ0 ,θ0 (θ) > Jθ0 (θ) or
Iλ0 ,θ0 (θ) < Jθ0 (θ) for 0 < |θ − θ0 | < δ; for instance (see Figure 2 where θ0 = 1) this happens for the
statistical model with uniform distributions cited above (where G(x) := x and λ0 = 3/4).

A

A class of counterexamples

In Section 6.2, for all the examples where the distribution G is symmetric, we find that G00 (0) = 0
and that there is a unique optimal value λmax , namely λmax = 0.5.
Here we show that this is not necessarily the case, indeed we present a procedure to construct
another function G̃ with the symmetry property and such that G̃00 (0) = 0; this function will be
20

0.005
0.004
0.003
0.002
0.001
0.000

0.96

0.98

1.00

1.02

1.04

θ

Figure 2: The rate functions I3/4,θ0 (θ) (broken line) and Jθ0 (θ) (solid line) in a neighborhood of
θ0 = 1 for the statistical model with uniform distributions.

determined starting from a function G with the properties cited above (for instance it could be one
of the choices illustrated in Example 2 except the one concerning Gumbel distribution). The aim
is to illustrate that, for such a function G̃, λ = 0.5 cannot be an optimal value.
The function G̃ is defined by

G(1+x)

for x ≤ −1

 2G0 (0)+10
G
(0)x
1
G̃(x) :=
for |x| < 1
2 + 2G0 (0)+1


 2G0 (0)+G(x−1)
for x ≥ 1.
0
2G (0)+1

One can check that, if G is twice differentiable, then G̃ is also twice differentiable (and in particular
the condition G00 (0) = 0 is needed to say that G̃ is twice differentiable); the details are omitted.
Moreover, if we consider
G(−1) =
(we recall that G(0) =

1
4G0 (0) + 1
and
G(1)
=
2(2G0 (0) + 1)
2(2G0 (0) + 1)

1
2

by the symmetry property of G), we have

−1
0

for λ ≤ G(−1)
 G (λ(2G0 (0) + 1)) − 1
−1
1 2G (0)+1
G̃ (λ) :=
(λ − 2 ) G0 (0)
for G(−1) < λ < G(1)

 G−1 (λ(2G0 (0) + 1) − 2G0 (0)) + 1 for λ ≥ G(1).
21

Then, around λ = 1/2 (more precisely for λ ∈ (G(−1), G(1)) because G(−1) ∈ (0, 21 ) and G(1) ∈
 0
2
G (0)
1
00 (θ ) =
by (14); thus λ = 0.5 cannot be an optimal value
( 12 , 1)), we have Iλ,θ
0
0
λ(1−λ) 2G (0)+1
0
00 (θ ) is locally minimized at λ = 1/2 (in fact λ = 1/2 maximizes the denominator
because Iλ,θ
0
0
λ(1 − λ)).

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