Inference in parametric models with many L-moments Luis A. F. AlvarezChang ChiannPedro A. Morettin May 29 2024

2025-05-05 0 0 1.26MB 63 页 10玖币

侵权投诉

Inference in parametric models with many L-moments

Luis A. F. Alvarez∗Chang Chiann†Pedro A. Morettin ‡

May 29, 2024

Abstract

L-moments are expected values of linear combinations of order statistics that provide ro-

bust alternatives to traditional moments. The estimation of parametric models by matching

sample L-moments has been shown to outperform maximum likelihood estimation (MLE) in

small samples from popular distributions. The choice of the number of L-moments to be used

in estimation remains ad-hoc, though: researchers typically set the number of L-moments equal

to the number of parameters, as to achieve an order condition for identiﬁcation. This approach

is generally ineﬃcient in larger samples. In this paper, we show that, by properly choosing the

number of L-moments and weighting these accordingly, we are able to construct an estimator

that outperforms MLE in ﬁnite samples, and yet does not suﬀer from eﬃciency losses asymptot-

ically. We do so by considering a “generalised” method of L-moments estimator and deriving its

asymptotic properties in a framework where the number of L-moments varies with sample size.

We then propose methods to automatically select the number of L-moments in a given sample.

Monte Carlo evidence shows our proposed approach is able to outperform (in a mean-squared

error sense) MLE in smaller samples, whilst working as well as it in larger samples. We then

consider extensions of our approach to conditional and semiparametric models, and apply the

latter to study expenditure patterns in a ridesharing platform in Brazil.

Keywords: L-statistics; generalised method of moments; tuning parameter selection methods;

higher-order expansions.

∗Department of Economics, University of S˜ao Paulo. E-mail address: luis.alvarez@usp.br

†Department of Statistics, University of S˜ao Paulo. E-mail address: chang@ime.usp.br.

‡Department of Statistics, University of S˜ao Paulo. E-mail address: pam@ime.usp.br.

arXiv:2210.04146v3 [stat.ME] 27 May 2024

1 Introduction

L-moments, expected values of linear combinations of order statistics, were introduced by Hosking

(1990) and have been successfully applied in areas as diverse as computer science (Hosking,2007;

Yang et al.,2021), hydrology (Wang,1997;Sankarasubramanian and Srinivasan,1999;Das,2021;

Boulange et al.,2021), meteorology (Wang and Hutson,2013;ˇ

Simkov´a,2017;Li et al.,2021) and

ﬁnance (Gourieroux and Jasiak,2008;Kerstens et al.,2011). By appropriately combining order

statistics, L-moments oﬀer robust alternatives to traditional measures of dispersion, skewness and

kurtosis. Models ﬁt by matching sample L-moments (a procedure labeled “method of L-moments”

by Hosking (1990)) have been shown to outperfom maximum likelihood estimators in small samples

from ﬂexible distributions such as generalised extreme value (Hosking et al.,1985;Hosking,1990),

generalised Pareto (Hosking and Wallis,1987;Broniatowski and Decurninge,2016), generalised

exponential (Gupta and Kundu,2001) and Kumaraswamy (Dey et al.,2018).

Statistical analyses of L-moment-based parameter estimators rely on a framework where the

number of moments is ﬁxed (Hosking,1990;Broniatowski and Decurninge,2016). Practitioners

often choose the number of L-moments equal to the number of parameters in the model, so as

to achieve the order condition for identiﬁcation. This approach is generally ineﬃcient.1It also

raises the question of whether overidentifying restrictions, together with the optimal weighting of

L-moment conditions, could improve the eﬃciency of “method of L-moments” estimators, as in the

framework of generalised-method-of-moment (GMM) estimation (Hansen,1982). Another natural

question would be how to choose the number of L-moments in ﬁnite samples, as it is well-known

from GMM theory that increasing the number of moments with a ﬁxed sample size can lead to

substantial biases (Koenker and Machado,1999;Newey and Smith,2004). In the end, one can only

ask if, by correctly choosing the number of L-moments and under an appropriate weighting scheme,

it may not be possible to construct an estimator that outperforms maximum likelihood estimation

in small samples and yet achieves the Cram´er-Rao bound assymptotically. Intuitively, the answer

appears to be positive, especially if one takes into account that Hosking (1990) shows L-moments

characterise distributions with ﬁnite ﬁrst moments.

The goal of this paper lies in answering the questions outlined in the previous paragraph.

Speciﬁcally, we propose to study L-moment-based estimation in a context where: (i) the number

of L-moments varies with sample size; and (ii) weighting is used in order to optimally account for

overidentifying conditions. In this framework, we introduce a “generalised” method of L-moments

estimator and analyse its properties. We provide suﬃcient conditions under which our estimator

is consistent and asymptotically normal; we also derive the optimal choice of weights and intro-

duce a test of overidentifying restrictions. We then show that, under independent and identically

distributed (iid) data and the optimal weighting scheme, the proposed generalised L-moment esti-

1In the generalised extreme value distribution, there can be asymptotic root mean-squared error losses of 30%

with respect to the MLE when the target estimand are the distribution parameters (Hosking et al.,1985;Hosking,

1990). In our Monte Carlo exercise, we verify root mean squared errror losses of over 10% when the goal is tail

quantile estimation.

mator achieves the Cram´er-Rao lower bound. We provide simulation evidence that our L-moment

approach outperforms (in a mean-squared error sense) MLE in smaller samples; and works as well as

it in larger samples. We then construct methods to automatically select the number of L-moments

used in estimation. For that, we rely on higher order expansions of the method-of-L-moment es-

timator, similarly to the procedure of Donald and Newey (2001) and Donald et al. (2009) in the

context of GMM. We use these expansions to ﬁnd a rule for choosing the number of L-moments

so as to minimise the estimated (higher-order) mean-squared error. We also consider an approach

based on ℓ1-regularisation (Luo et al.,2015). We provide computational code to implement both

methods,2and evaluate their performance through Monte Carlo simulations. With these tools,

we aim to introduce a fully automated procedure for estimating parametric density models that

improves upon maximum likelihood in small samples, and yet does not underperform in larger

datasets.

We also consider two extensions of our main approach. First, we show how the generalised

method-of-L-moment approach introduced in this paper can be extended to the estimation of con-

ditional models. Second, we show how our approach may be used in the analysis of the “error term”

in semiparametric models, and apply this extension to study the tail behaviour of expenditure pat-

terns in a ridesharing platform in S˜ao Paulo, Brazil. We provide evidence that the heavy-tailedness

in consumption patterns persists even after partialing out the eﬀect of unobserved time-invariant

heterogeneity and observable heterogeneity in consumption trends. With these extensions, we hope

more generally to illustrate how the generalised-mehtod-of-L-moment approach to estimation may

be a convenient tool in a variety of settings, e.g. when a model’s quantile function is easier to

evaluate than its likelihood. The latter feature has been explored in followup work by one of the

authors (Alvarez and Orestes,2023;Alvarez and Biderman,2024).

Related literature This paper contributes to two main literatures. First, we relate to a couple

of papers that, building on Hosking’s original approach, propose new L-moment-based estimators.

Gourieroux and Jasiak (2008) introduce a notion of L-moment for conditional moments, which is

then used to construct a GMM estimator for a class of dynamic quantile models. As we argue

in more detail in Section 6, while conceptually attractive, their estimator is not asymptotically

eﬃcient (vis-`a-vis the conditional MLE), as it focuses on a ﬁnite number of moment conditions and

does not optimally explore the set of overidentifying restrictions available in the parametric model.

In contrast, our proposed extension of the generalised method-of-L-moment estimator to condi-

tional models is able to restore asymptotic eﬃciency. In an unconditional setting, Broniatowski

and Decurninge (2016) propose estimating distribution functions by relying on a ﬁxed number of

L-moments and a minimum divergence estimator that nests the empirical likelihood and generalized

empirical likelihood estimators as particular cases. Even though these estimators are expected to

perform better than (generalized) method-of-L-moment estimators in terms of higher-order prop-

erties (Newey and Smith,2004), both would be ﬁrst-order ineﬃcient (vis-`a-vis the MLE) when

2The repository https://github.com/luisfantozzialvarez/lmoments_redux contains Rscript that implements

our main methods, as well as replication code for our Monte Carlo exercise and empirical application.

the number of L-moments is held ﬁxed. In this paper, we thus focus on improving L-moment-

based estimation in terms of ﬁrst-order asymptotic eﬃciency, by suitably increasing the number

of L-moments with sample size and optimally weighting the overidentifying restrictions, while re-

taining its known good ﬁnite-sample behaviour. We do note, however, that one of our suggested

approaches to select the number of L-moments aims at minimising an estimate of the higher-order

mean-squared error, which may be useful in improving the higher-order behaviour of estimators

even when a bounded (as a function of sample sizes) number of L-moments is used in estimation.

Secondly, we contribute to a literature that seeks to construct estimators that, while retaining

asymptotic (ﬁrst-order) unbiasedness and eﬃciency, improve upon maximum likelihood estimation

in ﬁnite samples. The classical method to achieve ﬁnite-sample improvements over the MLE is

through (higher-order) bias correction (Pfanzagl and Wefelmeyer,1978). However, analytical bias

corrections may be diﬃcult to implement in practice, which has led the literature to consider

jaccknife and bootstrap corrections (Hahn et al.,2002). More recently, Ferrari and Yang (2010)

introduced a maximum Lq-likelihood estimator for parametric models that replaces the log-density

in the objective function of the MLE with f(x)1−q−1

1−q, where q > 0 is a tuning parameter. They

show that, by suitably choosing qin ﬁnite samples, one is able to trade-oﬀ bias and variance,

thus enabling MSE improvements over the MLE. Moreover, if q→1 asymptotically at a rate,

the estimator is asymptotically unbiased and achieves the Cr´amer-Rao lower bound. There are

some important diﬀerences between our approach and maximum Lq-likelihood estimation. First,

we note that the theoretical justiﬁcation for our construction is distinct from their method. Indeed,

for a ﬁxed number of L-moments, our proposed estimator is ﬁrst-order asymptotically unbiased,

whereas the maximum Lq-likelihood estimator is inconsistent in an asymptotic regime with qﬁxed

and consistent but ﬁrst-order biased if q→1 slowly enough. Therefore, whereas the choice of the

tuning parameter qis justiﬁed as capturing a tradeoﬀ between ﬁrst-order bias and variance; the

MSE-optimal choice of L-moments in our setting concerns a tradeoﬀ between the ﬁrst-order variance

of the estimator and its higher-order terms. This is precisely what we capture in our proposal to

select the number of L-moments by minimising an estimator of the higher-oder MSE; whereas

presently no general rule for choosing the tuning parameter q > 0 in maximum Lq-likelihood

estimation exists (Yang et al.,2021).

Structure of paper The remainder of this paper is organised as follows. In the next section, we

brieﬂy review L-moments and parameter estimation based on these quantities. Section 3works out

the asymptotic properties of our proposed estimator. In Section 4we conduct a small Monte Carlo

exercise which showcases the gains associated with our approach. Section 5proposes methods to

select the number of L-moments and assesses their properties in the context of the Monte Carlo

exercise of Section 4. Section 6presents the extensions of our main approach, as well as the

empirical application. Section 7concludes.

2 L-moments: deﬁnition and estimation

Consider a scalar random variable Ywith distribution function Fand ﬁnite ﬁrst moment. For

r∈N,Hosking (1990) deﬁnes the r-th L-moment as:

λr:=Z1

QY(u)P∗

r−1(u)du , (1)

where QY(u):= inf{y∈R:F(y)≥u}is the quantile function of Y, and the P∗

r(u) = Pr

k=0(−1)r−kr

kr+k

kuk,

r∈ {0}∪N, are shifted Legendre polynomials.3Expanding the polynomials and using the quantile

representation of a random variable (Billingsley,2012, Theorem 14.1), we arrive at the equivalent

expression:

λr=r−1

r−1

k=0

(−1)kr−1

kE[˜

Y(r−k):r],(2)

where, ˜

Yj:lis the j-th order statistic of a random sample from Fwith lobservations. Equation

(2) motivates our description of L-moments as the expected value of linear combinations of order

statistics. Notice that the ﬁrst L-moment corresponds to the expected value of Y.

To see how L-moments may oﬀer “robust” alternatives to conventional moments, it is instructive

to consider, as in Hosking (1990), the second L-moment. In this case, we have:

λ2=1

2E[˜

Y2:2 −˜

Y1:2] = 1

2Z Z (max{y1, y2} − min{y1, y2})F(dy1)F(dy2) = 1

2E|˜

Y1−˜

Y2|,

where ˜

Y1and ˜

Y2are independent copies of Y. This is a measure of dispersion. Indeed, comparing

it with the variance, we have:

V[Y] = E[(Y−E[Y])2] = E[Y2]−E[Y]2=1

2E[( ˜

Y1−˜

Y2)2],

from which we note that the variance puts more weight to larger diﬀerences.

Next, we discuss sample estimators of L-moments. Let Y1, Y2. . . YTbe an identically distributed

sample of Tobservations, where each Yt,t= 1, . . . , T , is distributed according to F. A natural

estimator of the r-th L-moment is the sample analog of (1), i.e.

λr=Z1

QY(u)P∗

r−1(u)du , (3)

where ˆ

QYis the empirical quantile process:

3Legendre polynomials are deﬁned by applying the Gramm-Schmidt orthogornalisation process to the polynomials

1, x, x2, x3. . . deﬁned on [−1,1] (Kreyszig,1989, p. 176-180). If Prdenotes the r-th Legendre polynomial, shifted

Legendre polynomials are related to the standard ones through the aﬃne transformation P∗

r(u) = Pr(2u−1) (Hosking,

1990).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

InferenceinparametricmodelswithmanyL-momentsLuisA.F.Alvarez∗ChangChiann†PedroA.Morettin‡May29,2024AbstractL-momentsareexpectedvaluesoflinearcombinationsoforderstatisticsthatprovidero-bustalternativestotraditionalmoments.TheestimationofparametricmodelsbymatchingsampleL-momentshasbeenshowntooutperform...

展开>> 收起<<

Inference in parametric models with many L-moments Luis A. F. AlvarezChang ChiannPedro A. Morettin May 29 2024.pdf

共63页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Inference in parametric models with many L-moments Luis A. F. AlvarezChang ChiannPedro A. Morettin May 29 2024

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: