Modified Maximum Likelihood Estimations of the Epsilon-Skew-Normal Family

Parichehr Jamshidi; Mohsen Maleki; Zahra Khodadadi

doi:10.2991/jsta.d.201208.001

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Volume 19, Issue 4, December 2020, Pages 481 - 486

Modified Maximum Likelihood Estimations of the Epsilon-Skew-Normal Family

Authors

Parichehr Jamshidi¹, Mohsen Maleki²^{, *}, Zahra Khodadadi¹

¹Department of Statistics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

²Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran

^*Corresponding author. Email: m.maleki.stat@gamil.com

Corresponding Author

Mohsen Maleki

Received 17 April 2018, Accepted 20 March 2020, Available Online 14 December 2020.

DOI: 10.2991/jsta.d.201208.001 How to use a DOI?
Keywords: Asymmetry; EM-algorithm; Epsilon-skew-normal; Maximum likelihood estimates; Two-piece distributions
Abstract: In this work, maximum likelihood (ML) estimations of the epsilon-skew-normal (ESN) family are obtained using an EM-algorithm to modify the ordinary estimation already used and solve some of its problems within issues. This family can be used for analyzing the asymmetric and near-normal data, so the skewness parameter epsilon is the most important parameter among others. We have shown that the method has better performance compared to the method in G.S. Mudholkar, A.D. Hutson, J. Statist. Plann. Infer. 83 (2000), 291–309, especially in the strong skewness and small samples. Performances of the proposed ML estimates are shown via a simulation study and some real datasets under some statistical criteria as a way to illustrate the idea.
Copyright: © 2020 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The importance of the asymmetric distributions in various applications such as meteorology, physics, economics, geology, etc., has been rapidly increasing. Also, the asymmetric distributions which contain famous distributions in the symmetric cases, such as normal distribution, are more important among all others. The epsilon-skew-normal (ESN) distribution which was introduced by Refs. [1–6], is a flexible family to model the asymmetry data and statistical models. The ESN parameters maximum likelihood (ML) estimates were found by Refs. [3,7], and their Bayesian estimates were studied by Ref. [8] as well as Ref. [9].

The classical ML estimates of this family and its application on some statistical models (as regression model by Ref. [7]; time series model by Ref. [10]; Tobit regression by Ref. [11]) were performed by an especial approach which ordered the data. Applying the variations of each produced segment and three possible likelihood function forms as well as their proposed estimates, the one with higher likelihood values was chosen as the approximation of the ML estimates (see e.g. Ref. [12]). But, there exist some problematic issues on this method, e.g., in the strong asymmetry, in the small samples, and the right/left half-normal distribution estimates in which there are not any real distributions. We have focused on an especial mixture of two right/left half-normal distributions which lead to ESN family, and have used an EM-algorithm to obtain the ML estimates of the model parameters. In addition, we have shown the modifications of the proposed ML estimates without the maintained issues (see, e.g., Ref. [13]).

The rest of this paper is organized as follows: Some properties of the ESN family and ordinary method of finding the ML estimates are considered in Section 2. The new approach of finding the ML estimates based on the mixture distributions are provided in Section 3. In Section 4, in order to show the performance of the proposed methodology, some simulation studies are provided which are later used to some real dataset. Finally, the conclusion is given in Section 5.

2. THE ESN FAMILY

The ESN distribution denoted by ESNθ,σ,ε is a unimodal distribution with mode and location parameter θ∈ℝ, scale parameter σ∈ℝ+, skewness parameter ε∈−1,1, and probability masses 1+ε/2 at below the mode and 1−ε/2 at above the mode, with the following standard density function of X~ESN0,1,ε:

fesnx;0,1,ε=12πexp−x221+ε2,x<012πexp−x221−ε2,x≥0,(1)

Note that, fesnx;θ,σ,ε=1σfesnx−θσ;0,1,ε, and it has the same range of skewness as the skew-normal distribution investigated by Refs. [14–16]. The standard random variable X~ESNθ,σ,ε has the following stochastic representation:

X=θ+σ1−U1−ε|Z1|−σU1+ε|Z2|,(2)

where U,Z1 and Z2 are independent, for which PU=1=1+ε/2=1−PU=0, and Z1 and Z2 are the standard normal distributed.

The mean and variance of the random variable X~ESNθ,σ,ε respectively are

EX=θ−4σε2π, VarX=σ2π3π−8ε2+π.(3)

To see more statistical details of the ESN distribution, refer to the Refs. [1,3].

To obtain the ML estimates of the X~ESNθ,σ,ε, Ref. [3] as well as Ref. [1] considering the sample X=X1,…,Xn⊤ and its order statistic of the sample X1≤X2≤…≤Xn, have assumed that there exists the auxiliary integer k such that the first k-th samples come from the left half-normal and the remaining samples from the right half-normal. Finally, by considering the possible values of k=0,n (corresponds to right/left half-normal) and other values between these two values which lead to n−1 half-open intervals in the form of Xj,Xj+1;j=1,…,n−1, and using the numerical method, they have obtained the n+1 plausible ML estimates and choose the one with the lowest likelihood values as the ML estimates of the ESN parameters. (See full details of this method and statistical properties of the ESN family in Ref. [3].) As previously mentioned, some problematic issues emerge within this method which is later discussed in Section 4. Moreover, we have used an especial method of constructing the ESN family with mixture models and applied an EM-algorithm to have the modified ML estimates of the ESN family parameters in the Section 3.

3. ML ESTIMATES OF THE ESN PARAMETERS USING AN EM-ALGORITHM

3.1. ML Estimates

In fact the location-scale ESN distribution is the reparameterization of a mixture of left- and right half-normal (RHN) densities with special component probabilities as follows:

fesnx|θ,σ1,σ2=2πϕx|θ,σ1I−∞,θx+21−πϕx|θ,σ2Iθ,+∞x,(4)

where π=σ1/σ1+σ2. Note that in this form, the scale parameter σ and skewness parameter ε recover as in the form of σ=σ1+σ2/2 and ε=σ1−σ2/2σ.

By using an EM-algorithm to obtain the ML estimates of the ESN parameters Θ=θ,σ1,σ2⊤, for each i.i.d. sample X=X1,…,Xn⊤~ESNΘ, by using auxiliary (latent) variables Z=Z1,…,Zn⊤ (i.e., completed data D=X,Z⊤, where in terms of the components of the mixture (4) can be equivalently represented as

Xi|Zi=1~LHNθ,σ1Xi|Zi=0~RHNθ,σ2, i=1,…,n,(5)

where LHN and RHN denotes the left- and right half-normal distribution, respectively and Zi~Binomial1,π;i=1,…,n is a multinomial (component-label) vector with probability mass function PZi=zi=πzi1−π1−zi, for which zi=0,1;i=1,…,n. So the augmented (completed) log-likelihood function is in the form of

lΘ|D=−nlogσ1+σ2−12∑i=1nZiXi−θσ12+1−ZiXi−θσ22,(6)

where Θ=θ,σ1,σ2⊤.

The conditional expectation of latent variables is z^i=EZi|Θ^,xi=I−∞,θ^xi. Now, the E-Step on the k+1th iteration of the EM-algorithm (Ref. [17]) requires the calculation of Q-function, i.e., in the form of QΘ|D=EΘlΘ|D. So,

E-Step:

QΘ|D=−nlogσ1+σ2−12∑i=1nz^ixi−θσ12+1−z^ixi−θσ22.(7)

M-steps:

M-step 1: Update θ by

θ^=∑i=1nz^iσ22+1−z^iσ12Xi∑i=1nz^iσ22+1−z^iσ12.

M-steps 2–3: Update σj,j=1,2, by solving the following stressed cubic equation:

σj3+pσj+q=0;j=1,2,

where p=−1n∑i=1nz^ijxi−θ2 and q=pσi, for which z^ij=z^iIj=1+1−z^iIj=2. Note that p<0 and q<0, so the cubic equation has unique just root in the 0,+∞ interval. The EM-algorithm must be iterated so that a sufficient convergence rule is satisfied, e.g. if ‖Θ^k+1−Θ^k≤ε‖ (see Ref. [18]).

3.2. Model Selection

In this paper we have just ESN family but with different numerical ML estimate types, therefore we compare different ESN distributions based on different estimated parameters (through mentioning different numerical approaches) to better fit on the simulated and real datasets. The Akaike information criteria (AIC; Ref. [19]) is in the form of AIC=2k−2lΘ^|x, where lΘ^|x is the maximized log-likelihood function, the Kolmogorov–Smirnov (K-S) and Anderson–Darling (A-D) statistic tests are implemented to choose more suitable models. The one-sample K-S statistic is given by Dn=supx|FnX−FX;Θ|, where supx is the supremum of the set of distances between the empirical distribution function Fn⋅ (Ref. [20]) and target ESN distribution function F⋅, and the A-D statistic is given by A2=−n−S, where S=∑i=1n2i−1nlnFXi;Θ−ln1−FXn+1−i;Θ, for the target distribution F⋅;Θ with sample X=X1,…,Xn⊤~ESNΘ. The minimum values of the mentioned criteria choose the more suitable model (and as a result, better parameter estimates).

4. NUMERICAL STUDIES

In this section, we simulate the some strongly and weakly skewed ESN samples and use the proposed ML estimates (denoted by Pr-ML) to evaluate the ordinary ML estimates (denoted by Or-ML) which correspond to Refs. [1,3]. Then we apply the both ML estimation methods to some real datasets. The implementation of the necessary algorithms is based on the R software version 3.5.2 with a core i7 760 processor 2.8 GHz, and the relative tolerance of 10−3 is used for convergence of the EM-algorithms.

4.1. Simulations

In this part, we consider 10,000 samples of size n=50, 100, and 250 from various weak and moderate skewness ε=0.5,−0.5, and strong skewness ε=0.85,−0.95, respectively, with standard location-scale parameter values. We recorded the means and standard deviations of the Pr-ML and Or-ML estimates of parameters in Table 1. The results show the performance of the Pr-ML estimates in each sample size.

Parameters	ML	n = 50		n = 100		n = 250
Parameters	ML	Mean	SD	Mean	SD	Mean	SD
θ0	Or-ML	−0.1026	0.1637	0.0837	0.0936	−0.0657	0.0746
θ0	Pr-ML	−0.1017	0.1726	0.0866	0.0894	−0.0719	0.0563
σ1	Or-ML	1.0258	0.0403	1.0137	0.0304	0.9846	0.0307
σ1	Pr-ML	1.0144	0.0397	1.0003	0.0294	0.9930	0.0308
ε0.05	Or-ML	0.0618	0.0106	0.0589	0.0095	0.0440	0.0082
ε0.05	Pr-ML	0.0585	0.0098	0.0558	0.0094	0.0528	0.0080

θ0	Or-ML	0.1134	0.2016	0.1037	0.1073	0.0589	0.0374
θ0	Pr-ML	0.1076	0.2113	0.0937	0.0783	0.0593	0.0412
σ1	Or-ML	1.0312	0.0553	1.0207	0.0494	1.0201	0.0365
σ1	Pr-ML	1.0189	0.0388	1.0054	0.0307	1.0036	0.0340
ε−0.5	Or-ML	−0.5303	0.0128	−0.5203	0.0097	−0.5176	0.0066
ε−0.5	Pr-ML	−0.5274	0.0078	−0.5148	0.0071	−0.5104	0.0061

θ0	Or-ML	0.1981	0.1803	−0.1037	0.1006	0.0579	0.0793
θ0	Pr-ML	0.1805	0.1891	−0.0937	0.0954	0.0667	0.0534
σ1	Or-ML	1.0537	0.0683	1.0365	0.0546	1.0311	0.0397
σ1	Pr-ML	1.0203	0.0442	1.0112	0.0410	1.0103	0.0385
ε0.85	Or-ML	0.9204	0.0983	0.9048	0.0784	0.8910	0.0719
ε0.85	Pr-ML	0.8399	0.0068	0.8401	0.0065	0.8411	0.0059

θ0	Or-ML	0.1907	0.1887	0.1117	0.0936	0.0794	0.0864
θ0	Pr-ML	0.1870	0.1804	0.0946	0.0911	0.0642	0.0570
σ1	Or-ML	1.0497	0.0702	1.0405	0.0528	1.0352	0.0401
σ1	Pr-ML	1.0286	0.0513	1.0201	0.0465	1.0112	0.0389
ε−0.95	Or-ML	−0.9987	0.0057	−0.9902	0.0102	−0.9893	0.0096
ε−0.95	Pr-ML	0.9601	0.0113	−0.9578	0.0094	−0.9523	0.0075

ESN, epsilon-skew-normal; ML, maximum likelihood; Pr-ML, proposed maximum likelihood; Or-ML, ordinary maximum likelihood.

Table 1

Mean and standard deviations (SDs) of the 10,000 times Or-ML and Pr-ML estimates of the ESN distribution.

4.2. Applications

In this section, considering four various real datasets, we show the performance of the proposed Pr-ML estimates in applications. All of ESN parameters estimates and criteria are given in Table 2, and the fitted ESN densities based on the two approaches Pr-ML and Or-ML estimates are curved on the histograms of the mentioned datasets in Figure 1.

Data	ML	Parameter			Criteria
Data	ML	θ	σ	ε	AIC	K-S	A-D
1st Data	Or-ML	0.0246	1.8938	−1.0000	2108.798	0.0417	1.4202
1st Data	Pr-ML	0.0246	1.8939	−1.0000	2108.841	0.0416	1.4169
2nd Data	Or-ML	43.9999	55.2043	−0.9384	6317.510	0.1047	12.8735
2nd Data	Pr-ML	40.9727	55.5884	−0.9569	6291.083	0.0945	8.3241
3rd Data	Or-ML	0.9790	0.2994	−0.0578	3643.910	0.0551	62.1247
3rd Data	Pr-ML	1.1461	0.2919	0.3249	3015.756	0.0343	13.3147
4th Data	Or-ML	4.4999	0.2726	−0.6772	4167.713	0.3357	592.7232
4th Data	Pr-ML	4.2438	0.3771	−0.6155	893.682	0.0796	4.3623

ESN, epsilon-skew-normal; ML, maximum likelihood; Pr-ML, proposed maximum likelihood; Or-ML, ordinary maximum likelihood; K-S, Kolmogorov–Smirnov; A-D, Anderson–Darling.

Table 2

The Or-ML and Pr-ML estimates of the fitted ESN distributions on four real datasets.

The first dataset is corresponds to the “Tstop” component of the “Bronchiolitis obliterans syndrome after lung transplants” called “bosms3” and available in the “flexsurv” package of R software. Both Pr-ML and Or-ML estimates satisfy the purely skewed right half-normal (ε=−1) distributions and the criteria are approximately identical on the dataset (see, e.g., the Table 2 and top-left of the Figure 1).

The second dataset is corresponds to the “weight” component of the “Weight versus age of chicks on different diets” called “ChickWeight” and it is available in the “datasets” package of R software. In this case, although the Or-ML and Pr-ML estimates are close (see, e.g., the Table 2 and top-right of the Figure 1), but all of the criteria prefer the fitted ESN distribution based on the Pr-ML to Or-ML estimates.

The third dataset is corresponds to the “Yearly Treering Data” called “treering” and it is available in the “datasets” package of R software. In this case, all of the criteria strongly prefer the fitted ESN distribution based on the Pr-ML to Or-ML estimates (see, e.g., the Table 2 and bottom-left of the Figure 1).

The fourth dataset is corresponds to the “mag” component of the “Locations of Earthquakes off Fiji” called “quakes” and it is available in the “datasets” package of R software. In this case, also all of the criteria strongly prefer the fitted ESN distribution based on the Pr-ML to Or-ML estimates (see, e.g., the Table 2 and bottom-right of the Figure 1).

5. CONCLUSION

We have proposed and implemented an EM-type algorithm to estimate the well-known ESN family parameters by applying the special stochastic representation. The proposed estimation methodology has better ESN distribution fitting, especially in the strong skewness. The performance of the proposed methodology is illustrated using the simulation studies and four real datasets. The performances of the ESN family have shown on many statistical models to cover the asymmetry, e.g., Refs. [3,7,10]. In fact, this methodology can be affronted on them to modify their parameter estimations.

CONFLICTS OF INTEREST

The authors declare that there are no conflicts of interest regarding the publication of this paper.

AUTHORS' CONTRIBUTIONS

All authors have read and agreed to the published version of the manuscript.

FUNDING STATEMENT

There is no funding of this paper.

ACKNOWLEDGMENTS

We would like to express our very great appreciation to editor and reviewer(s) for their valuable and constructive suggestions during the planning and development of this research work.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 4
Pages: 481 - 486
Publication Date: 2020/12/14
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.201208.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Parichehr Jamshidi
AU  - Mohsen Maleki
AU  - Zahra Khodadadi
PY  - 2020
DA  - 2020/12/14
TI  - Modified Maximum Likelihood Estimations of the Epsilon-Skew-Normal Family
JO  - Journal of Statistical Theory and Applications
SP  - 481
EP  - 486
VL  - 19
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.201208.001
DO  - 10.2991/jsta.d.201208.001
ID  - Jamshidi2020
ER  -

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