Extending normality A case of unit distribution generated from the moments of the standard normal distribution

2025-05-06 0 0 497.57KB 24 页 10玖币

侵权投诉

Extending normality: A case of unit distribution

generated from the moments of the standard

normal distribution

Miguel S. Concha-Aracena*,1, Leonardo Barrios-Blanco**,1, David

Elal-Olivero***,1, Paulo Henrique Silva ****,2 y Diego Carvalho do

Nascimento*****,1

1Departamento de Matem´

atica, Universidad de Atacama, Copiap´

o, Chile

2Department of Statistics, Federal University of Bahia, Salvador, Brazil

Resumen

This article presents an important theorem, which shows that from the moments of the

standard normal distribution one can generate density functions originating a family

of models. Additionally, we discussed that different random variable domains are

achieved with transformations. For instance, we adopted the moment of order two,

from the proposed theorem, and transformed it, which allowed us to exemplify this

class as unit distribution. We named it as Alpha-Unit (AU) distribution, which contains

a single positive parameter

(

AU(α)∈[0,1]

). We presented its properties and showed

two estimation methods for the

parameter, the maximum likelihood estimator (MLE)

and uniformly minimum-variance unbiased estimator (UMVUE) methods. In order to

analyze the statistical consistency of the estimators, a Monte Carlo simulation study

was carried out, where the robustness was demonstrated. As real-world application, we

adopted two sets of unit data, the ﬁrst regarding the dynamics of Chilean inﬂation in

the post-military period, and the other regarding the daily maximum relative humidity

of the air in the Atacama Desert. In both cases shown, the AU model is competitive,

whenever the data present a range greater than 0.4 and extremely heavy asymmetric tail.

We compared our model against other commonly used unit models, such as the beta,

Kumaraswamy, logit-normal, simplex, unit-half-normal, and unit-Lindley distributions.

Keywords:

Asymmetry accommodation; rates and proportions; single-parameter dis-

tribution; unit distribution; water monitoring;

*miguel.concha.21@alumnos.uda.cl

**leonardo.barrios.2020@alumnos.uda.cl

***david.elal@uda.cl

****paulohenri@ufba.br

*****diego.nascimento@uda.cl

arXiv:2210.09231v1 [stat.ME] 17 Oct 2022

1. Introduction

Statistical methodology plays an important role in quantitative methods, given the hypothesis

testing and inferential procedures. Nonetheless, the comparison across features is given

based on a generated function estimated from the data information. Most often, mild

suppositions are taken compromising the generalization of the results.

Under the perspective of statistical generalization (inferential method), some challenges are

found for bounded distribution estimation. For instance, the conﬁdence interval, which is of-

ten adopted from the maximum likelihood estimation approach and asymptotic supposition,

is also assumed. Specially, interval estimation can be seen off the parameter space domain.

One exempliﬁcation is the case where bounded information data are observed, nonetheless,

normality is commonly assumed to be true. This is the case of proportion/rate data, which

are double bounded in the lower limit equal to 0 and upper limit equal to 1. Relative humidity

is an example of this scenario where every decision-making should be

∈[0,1]

(Fonseca

et al. 2021, Bayer, Cintra & Cribari-Neto 2018), or commonly rates used in the ﬁeld of

ﬁnance, economics and demography, to list a few.

In the case of rates and proportions processes, as well as other processes whose variable

of interest assumes values in the range

(0,1)

, there is a well-represented class of models,

the unit distributions family, which deals with this type of double-bounded data. Among

the existing unit distributions, we can cite the power distribution, beta distribution (Ferrari

& Cribari-Neto 2004), Kumaraswamy distribution (Kumaraswamy 1980), unit-logistic

distribution (Tadikamalla & Johnson 1982), simplex distribution (Barndorff-Nielsen &

Jørgensen 1991), unit-Weibull distribution (Mazucheli et al. 2018, Mazucheli et al. 2020),

unit-Lindley distribution (Mazucheli et al. 2019), unit-half normal distribution (Bakouch

et al. 2021), unit log-log distribution (Korkmaz & Korkmaz 2021), modiﬁed Kumaraswamy

and reﬂected modiﬁed Kumaraswamy distributions (Sagrillo et al. 2021), unit-Teissier

distribution (Krishna et al. 2022), unit extended Weibull families of distributions (Guerra

et al. 2021), unit folded normal distribution (Korkmaz, Chesneau & Korkmaz 2022), unit-

Chen distribution (Korkmaz, Altun, Chesneau & Yousof 2022), and Marshall-Olkin reduced

Kies distribution (Aﬁfy et al. 2022).

Despite the applicability of the unit distributions in double-bounded variables, another

important fact is that the interval estimation for the parameter may also be limited in

a domain (like positive real number). In this manner, we also presented an inferential

alternative through the delta method.

This work starts by presenting an important theorem that transforms from a modiﬁcation of

the standard normal distribution into a class of density distributions that can be seen as unit.

Then, as an exempliﬁcation, a case of second moment was chosen to illustrate the usefulness

of this class of probabilistic models. This class of distributions shows to be competitive

for high-frequency data with range greater than 0.4, important to real-world applications,

whereas classical unit distribution fails (Santana-e Silva et al. 2022). Additionally, two

different data sets were selected to illustrate the adjustment of the proposed model. The ﬁrst

is related to Chilean inﬂation (ultimate post-military era), and the second is from the dryest

area of the planet (excluding the north and south poles).

1.1. Motivation

The normal distribution is very important in the history of statistics, where numerous

modiﬁcations to this distribution have been proposed in the literature (Stahl 2006, Limpert

& Stahel 2011). An interesting fact related to the normal distribution is that its even moments

can be used to generate new distributions, as is the case that we will show below, through a

deﬁnition and a result embodied in a theorem that accounts for the characterization of these

new distributions.

Deﬁnition 1.

A random variable

is said to be distributed according to a Bimodal Normal

(BN) distribution of order

, that is,

B∼BN(k)

, discussed in (Elal-Olivero 2010), if its

probability density function (PDF) is given by

f(b|k) = 1

cb2kφ(b),b∈R,(1.1)

where

φ(·)

is the PDF of a standard normal distribution,

c=∏k

j=1(2j−1)

and

{1,2,3,...}.

This class of distributions is always bimodal, where the observed modes move away when

the order kincreases (see Figure 1).

Figura 1: Density function of the BN distribution by varying the parameter

(displayed on

the top of each chart).

It is interesting to mention that transformations derived from the

BN(k)

distribution may

lead to other domains of interest, e.g., the unit domain. For example, let

B∼BN(k)

, then

by adding a scale parameter

, the transformation

α|B| ∈ R+

, and then the transformation

e−α|B|∈[0,1]

. Therefore, the stochastic characterization of a

BN(k)

distribution can be

obtained according to the following theorem.

Theorem 1.

Let

and

be independent random variables, where

is such that

P(W1=1) = P(W1=−1) = 1/2 and W2∼χ2

2k+1. Then,

W1√W2∼BN(k).(1.2)

So, this theorem is mainly motivated by the result that shows that if

X∼BN(k)

, then

X2∼χ2

2k+1. The demonstration is presented in Appendix A.

2. Distribution of the Second Moment of the Unit-Normal Distribution

In this section, we will discuss a new unit distribution, named Alpha-Unit, which presents a

single parameter,

. Whereas it will be presented its stochastic representations (probability

density and cumulative distribution functions), moments, characteristic function, and how

to generate random numbers from it.

By taking the general theorem presented, and considering

k=1

, that is, considering the

second moment of the standard normal distribution and its transform, a new unit distribution

called Alpha-Unit will be illustrated. However, as

increases, the concentration of the

distribution intensiﬁes.

2.1. Properties and Characterization

Deﬁnition 2.

(Alpha-Unit distribution). A random variable

follows an Alpha-Unit (AU)

distribution with parameter α>0, that is, X∼AU(α), if its PDF is given by

fX(x|α) = 2

xαln(x)

α2

φln(x)

α,0<x≤1.(2.1)

Remark 1. If X∼AU(α), then its PDF is unimodal.

Demonstration.

The maxima of the AU distribution are studied, for which the criterion of

the ﬁrst derivative is ﬁrst considered:

d fX(x|α)

dx =2

xα2

ln(x)

αφln(x)

α2

x−ln(x)

x−[ln(x)]2

xα=0.

Solving algebraically for x, we obtain:

x=









e−α2+√α4+8α2

2(i)

e−α2+√α4+8α2

2(ii)

To see if either or both expressions are solutions, it must be true that

e−h&h>0.

By working algebraically, it can be seen that this is only true for (i), therefore, the AU

distribution is unimodal. 

Proposition 1. If X∼AU(α), then its r-th order moment is given by

E[Xr] = 2er2α2

21+r2α2(1−Φ(rα)) −rαφ(rα).(2.2)

Demonstration.

E[Xr] = Z1

xr2

xαln(x)

α2

φln(x)

αdx.

By making the change of variables:











u=1

αln(x)⇒euα=x

du =1

αxdx ⇒αeuαdu =dx

then substituting into the previous equation and developing algebraically, we have:

E[Xr] = 2eα2r2

2Z0

−∞

u21

√2πe−(u−αr)2

2du.

Then, by making another change of variables:

h=u−αr

dh =du

; and replacing these

expressions in the previous equation, we have:

E[Xr] = 2eα2r2

2Z−αr

−∞(h+αr)21

√2πe−h2

2dh

=2eα2r2

2Z−αr

−∞h2+2hαr+α2r2φ(h)dh

=2eα2r2

2Z−αr

−∞

h2φ(h)dh +2αrZ−αr

−∞

hφ(h)dh +α2r2Z−αr

−∞

φ(h)dh.

Solving the integrals, we get:

E[Xr] = 2eα2r2

2αrφ(αr)+(1−Φ(αr)) −2αrφ(αr) + α2r2(1−Φ(αr)).

Then, solving algebraically, we arrive at Proposition 1. 

From Proposition 1, we obtain the mean and variance of the AU(α)model as follows:

E[X] = 2eα2

2(1+α2)(1−Φ(α)) −αφ(α),

Var[X] = E[X2]−(E[X])2

=2e2α2(1+4α2)(1−Φ(2α)) −2αφ(2α)−4eα2(1+α2)(1−Φ(α)) −αφ(α)2,

where

Φ(·)

is the cumulative distribution function (CDF) of a standard normal distribution.

Remark 2.

As illustration, Figure 2 shows the generated asymmetry and kurtosis based on

the chosen αparameter of the AU distribution.

Proposition 2. If X∼AU(α), then its CDF is given by

FX(x|α) = 2Φln(x)

α−2ln(x)

αφln(x)

α.(2.3)

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：

Extendingnormality:AcaseofunitdistributiongeneratedfromthemomentsofthestandardnormaldistributionMiguelS.Concha-Aracena*,1,LeonardoBarrios-Blanco**,1,DavidElal-Olivero***,1,PauloHenriqueSilva****,2yDiegoCarvalhodoNascimento*****,11DepartamentodeMatem´atica,UniversidaddeAtacama,Copiap´o,Chile2Departme...

展开>> 收起<<

Extending normality A case of unit distribution generated from the moments of the standard normal distribution.pdf

共24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！

Extending normality A case of unit distribution generated from the moments of the standard normal distribution

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: