On Some Aspects of a New Class of Two-Piece Asymmetric Normal Distribution

C. Satheesh Kumar; M.R. Anusree

doi:10.2991/jsta.2018.17.1.8

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Volume 17, Issue 1, March 2018, Pages 101 - 121

On Some Aspects of a New Class of Two-Piece Asymmetric Normal Distribution

Authors

C. Satheesh Kumardrcsatheeshkumar@gmail.com

Department of Statistics, University of Kerala, Trivandrum 695 581, India

M.R. Anusreeanusreemr@yahoo.co.in

Department of Operations, Rajagiri Business School, Kochi 682 039, India

Received 27 June 2016, Accepted 18 December 2017, Available Online 31 March 2018.

DOI: 10.2991/jsta.2018.17.1.8 How to use a DOI?
Keywords: Method of maximum likelihood; Plurimodality; Probability density function; Skewness; Skew normal distribution
Abstract: Through this chapter, we introduce a new class of two-piece asymmetric normal distribution suitable for asymmetric and plurimodal situations. We study some important aspects of this distribution by deriving explicit expressions for its distribution function, characteristic function, reliability measures etc. A location-scale extension of this class of distribution is considered and carried out the maximum likelihood estimation of its parameters. Further we have fitted the distribution to a real life data set for illustrating the usefulness of the model.
Copyright: Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

The term skew normal distribution (SND) refers to a parametric class of probability distribution that extends the normal distribution by an additional shape parameter which regulates skewness. The first systematic treatment of the SND in the scalar case was done by Azzalini (1985). He defined the SND as follows:

A random variable X is said to have skew normal distribution with skewness parameter θ ∈ R = (−∞, ∞), denoted by SND(θ), if its probability density function (p.d.f.) g₁ (x; θ) is of the following form. For x ∈ R,

(1)g1(x;θ)=2f(x)F(θx)

where f(.) and F(.) are respectively the p.d.f. and cumulative distribution function (c.d.f.) of a standard normal variate. The SND has been further studied by Azzalini (1986), Henze (1986), Azzalini and Dalla-Valle (1996), Branco and Dey (2001). Arnold et al. (1993) discussed an application of SND to psychometric data, Ball and Mankiw (1995) obtained the SND as a natural choice for the distribution of the relative price changes that influence the rate of inflation. A salient feature of the SND is that it is suitable for unimodal distributions and do not include multimodal distributions. Buccianti (2005) remarked that normal and skew normal models are not adequate to describe the situations of plurimodality. He investigated the shape of the frequency distribution of the log ratio, ln(cl−=Na+) whose components are related to water composition for 26 wells. Some SND related models that accommodates significant departures from unimodality have been developed in the literature. For example see Ma and Genton (2004) or Kumar and Anusree (2011a, 2013a, 2014a, 2015a, 2015b). Kumar and Anusree (2011b) defined an asymmetric version of the normal distribution as follows:

A random variable Y is said to follow the asymmetric normal distribution if its p.d.f. takes the following form. For y ∈ R,

(2)g2(y;λ,α)=f(y)[α+2(1−α)F(λy)],

in which λ ∈ R and α∈[0,1]. A distribution with p.d.f. (2) we denoted as AND(λ, α).

For getting more flexible asymmetric normal models, some researchers recently studied two-piece versions of skew normal distributions. For example see Kim (2005), Jamalizadeh et. al (2012), Kumar and Anusree (2013) and Salehi et. al (2013). The main objective of the present article is to introduce a two-piece version of the AND(λ, α) as an asymmetric class of distribution suitable for tackling plurimodal situations. Throughout in this chapter we denote this class of distribution as “generalized two-piece asymmetric normal distribution (GTAND)”. The paper is organized as follows. In section 2 we present the definition of GTAND and derive some of its important properties. In section 3, we discuss some concepts regarding the mode of the distribution. In section 4, we obtain expression for certain reliability measures such as failure rate, reliability function and mean residual life function of the GTAND. A location-scale extension of the GTAND is considered in section 5 and obtained some of its important properties. Further, the parameters of the GTAND are estimated by method of maximum likelihood in section 6 and a numerical illustration is given in section 7.

We need the following shorter notation in the sequel. For any reals a, b and k such that bx + k > 0

(3)ξk(a;b)=∫a∞∫0bx+ke−x2+y222πdydx.

2. Definition and Properties

In this section, first we define a wide class of two-piece asymmetric normal distribution and discuss some of its important properties.

Definition 1.

A random variable Z is said to follow a two-piece asymmetric normal distribution with parameters λ₁, λ₂ ∈ R = (−∞, ∞), α∈[0, 1] if its p.d.f. h(z; λ₁, λ₂, α) is of the following form. For z ∈ R,

(4)h(z;λ1,λ2,α)={f(z)[α+δ(λ1,λ2,α)F(λ1z)],z<0f(z)[α+δ(λ1,λ2,α)F(λ2z)],z≥0

where δ(λ₁, λ₂, α) = 2π (1 − α)[π − tan⁻¹ (λ₁) + tan⁻¹ (λ₂)]⁻¹.

Note that (4) is a proper probability density in the light of the following lemma.

Lemma 1.

If U is a standard normal variable, then for any real λ and k₀,

E[F(λ)|U|+k0]=F(k01+λ2)+2ξ(k01+λ2,λ),

where for any a ∈ R and b > 0,

(5)ξ(a;b)=∫a∞∫0bxe−x2+y222πdydx.

The distribution of a random variable Z with p.d.f. (4) we denoted as GTAND(λ₁, λ₂, α). For some choices of λ₁, λ₂ and α; the p.d.f. given in (4) of GTAND is plotted in Figures 1 and 2.

Clearly, the GTAND(λ₁, λ₂, α) contains the following special cases.

1.
When λ₁ = λ, λ₂ = ρλ and α = 0, the distribution with p.d.f.(4) reduces to the two-piece skew normal distribution of Kumar and Anusree (2013)with parameters λ and ρ,
2.
when λ₁ = λ, λ₂ = λ, the distribution with p.d.f.(4) reduces to the skew normal distribution of Kumar and Anusree (2011b),
3.
when, λ₁ → −∞, λ₂ → ∞ or λ₁ → −∞, λ₂ → −∞, α = 1 or λ₁ → ∞, λ₂ → ∞, α = 1 or λ₁ = 0, λ₂ = 0 the distribution with p.d.f.(4) reduces to the standard normal distribution,
4.
when λ₁ = −λ, λ₂ = λ and α = 0, the distribution with p.d.f.(4) reduces to is the skew normal distribution of Kim(2005) and
5.
when either λ₁ → −∞, λ₂ → −∞, α = 2 or λ₁ → ∞, λ₂ → ∞, α = 0 TPSND(λ₁, λ₂, α), the distribution with p.d.f.(4) reduces to the half normal distribution.

Result 1.

If Z follows GTAND(λ₁, λ₂, α) with p.d.f. h(z; λ₁, λ₂, α), then Y₁ = −Z follows GTAND(−λ₂, − λ₁, α).

Proof.

For any y₁ ∈ R, the p.d.f. h₁ (y₁; λ₁, λ₂, α) of Y₁ is given by

h1(y1;λ1,λ2,α)=h(−y1;λ1,λ2,α)|dzdy1|={f(−y1)[α+δ(λ1,λ2,α)F(−λ1y1)],−y1<0f(−y1)[α+δ(λ1,λ2,α)F(−λ2y1)],−y1≥0={f(y1)[α+δ(λ1,λ2,α)F(−λ2y1)],y1≤0f(y1)[α+δ(λ1,λ2,α)F(−λ1y1)],y1>0

which shows that Y₁ = −Z follows GTAND(−λ₂, − λ₁, α).

Result 2.

If Z follows GTAND(λ₁, λ₂, α) with p.d.f. h(z; λ₁, λ₂, α), then Z² has p.d.f. (6).

Proof.

The p.d.f. h₂ (y₂; λ₁, λ₂, α) of Y₂ is given by

Remark 1.

Note that when λ₁ = 0 and λ₂ = 0, (6) reduces to the p.d.f. of a Chi-square variate with one degree of freedom.

Result 3.

If Z is a GTAND(λ₁, λ₂, α) variate, then for any reals d₁, d₂ such that d₁ ≤ d₂,

(7)P(d1≤Z≤d2)={α[F(d2)−F(d1)]+δ(λ1,λ2,α)2[G(d2,λ1)−G(d1,λ1)],d1≤d2<0α[F(d2)−F(d1)]+δ(λ1,λ2,α)2[G(d2,λ2)−G(d1,λ2)],0≤d1≤d2

where G(., λ) is the distribution function of the SND(λ).

Proof.

For any d₁ ≤ d₂ < 0, by definition,

(8)P(d1≤Z≤d2)=∫d1d2h(z;λ1,λ2,α)dz=∫d1d2[αf(z)+δ(λ1,λ2,α)22f(z)F(λ1z)]dz=α[F(d2)−F(d1)]+δ(λ1,λ2,α)2[G(d2,λ1)−G(d1,λ1)]

Now, for the case 0 ≤ d₁ ≤ d₂,

(9)P(d1≤Z≤d2)=∫d1d2h(z;λ1,λ2,α)dz=∫d1d2[αf(z)+δ(λ1,λ2,α)22f(z)F(λ2z)]dz=α[F(d2)−F(d1)]+δ(λ1,λ2,α)2[G(d2,λ1)−G(d1,λ2)]

Thus (8) and (9) implies (7).

Result 4.

The distribution function H (z; λ₁, λ₂, α) of a random variable Z with p.d.f. (4) is the following.

(10)H(z;λ1,λ2,α)={αF(z)+δ(λ1,λ2,α)2[F(z)−2ξ(z,λ1)],z<0αF(z)+δ(λ1,λ2,α)2[F(z)−tan−1(λ1)π+tan−1(λ2)π−2ξ(z,λ2)],z≥0

where ξ(a; b) is as defined in (5).

Proof.

Let Z be a random variable with p.d.f. (4) and H (z; λ₁, λ₂, α) be the cumulative distribution function. Then

(11)H(z)={L1,z<0L2,z≤0,

where

(12)L1=∫−∞zαf(t)dt+δ(λ1,λ2,α)2∫−∞zf(t)F(λ1t)dt=αF(z)+δ(λ1,λ2,α)2{G(0,λ1)−[G(0,λ1)−G(z,λ1)]}=αF(z)+δ(λ1,λ2,α)2(12+tan−1(λ1)π)+δ(λ1,λ2,α)2(F(z)−12−tan−1(λ1)π−2ξ(z,λ1))

and

(13)L2=α∫−∞zf(t)dt+δ(λ1,λ2,α)2∫−∞z2f(t)F(λ2t)dt=∫−∞0αf(t)dt+δ(λ1,λ2,α)∫−∞0f(t)F(λ1dt)+∫0zαf(t)dt+δ(λ1,λ2,α)∫0zf(t)F(λ2t)dt=αF(z)+δ(λ1,λ2,α)2G(0,λ1)+δ(λ1,λ2,α)2[G(z,λ2)−G(0,λ2)]=αF(z)+δ(λ1,λ2,α)2(12−tan−1(λ1)π)+δ(λ1,λ2,α)2(F(z)−12+tan−1(λ2)π−2ξ(z,λ2))

Now on substituting (12) and (13) in (11) we get (10).

In order to obtain the characteristic function of GTAND(λ₁, λ₂, α), we need the following lemma from Ellison (1964),

Lemma 2.

For any a₁, a₂ ∈ R and a standard normal variable Z with distribution function F

E{F(a1Z+a2)}=F{a(1+a12)}.

Result 5.

The characteristic function, ϕ_Z (t) of a random variable Z following GTAND(λ₁, λ₂, α) with p.d.f (4) is the following, for any t ∈ R and i=−1.

(14)φZ(t)=e−t22[α+δ(λ1,λ2,α)F(iδ1t)]−δ(λ1,λ2,α)e−t22[ξs1(−it,λ1)−ξs2(−it,λ2)]

where, δ1=λ11+λ12, s_j = itλ_j, for j=1,2 and ξ_k(a, b) is as defined in (3).

Proof:

Let Z follows GTAND(λ₁, λ₂, α) with p.d.f. (4). By the definition of characteristic function, for any t ∈ R and i=−1, we have

φZ(t)=E(eitZ)=∫−∞∞eitzh(z;λ1,λ2,α)dz=∫−∞∞eitzf(z)[α+δ(λ1,λ2,α)F(λ1z)]dz−∫0∞eitzf(z)[α+δ(λ1,λ2,α)F(λ1z)]dz+∫0∞eitzf(z)[α+δ(λ1,λ2,α)F(λ2z)]dz=e−t22{α+δ(λ1,λ2,α)∫−∞∞e−(z−it)22F(λ1z)dz2πdz−δ(λ1,λ2,α)∫0∞e−(z−it)22F(λ1z)dz2π+δ(λ1,λ2,α)∫0∞e−(z−it)22F(λ2z)dz2π}

On substituting z − it = x, ϕ_Z (t) reduces to the following.

φZ(t)=e−t22[α+δ(λ1,λ2,α)∫−∞∞e−x22F(λ1(x+it))dx2π]−δ(λ1,λ2,α)∫−it∞e−x22F(λ1(x+it))dx2π+δ(λ1,λ2,α)∫−it∞e−x22F(λ2(x+it))dx2π=e−t22{α+δ(λ1,λ2,α)F(iδ1t)−δ(λ1,λ2,α)∫−it∞f(x)F(λ1(x+it))dx+δ(λ1,λ2,α)∫−it∞f(x)F(λ2(x+it))dx}

in the light of Lemma 1. Thus we have

φZ(t)=e−t22[α+δ(λ1,λ2,α)F(iτt)]−δ(λ1,λ2,α)e−t22∫−it∞f(x)[12+∫0λ1x+s1f(u)du]dx+δ(λ1,λ2,α)e−t22∫−it∞f(x)[12+∫0λ2x+s2f(u)du]dx,

which implies (14).

3. Mode

Result 6.

The p.d.f. of GTAND(λ₁, λ₂, α) is bimodal, if

(1).
λ₁ ≤ 0 and λ₂ ≥ 0,
(2).
λ₁ ≤ 0 or λ₂ < 0 provided and k₃ (z; λ₁, λ₂, α) + k₄ (z; λ₁, λ₂, α) ≤ 0,
(3).
λ₂ ≥ 0 or λ₁ > 0 provided and k₁ (z; λ₁, λ₂, α) + k₂ (z; λ₁, λ₂, α) ≥ 0,

and
(4).
λ₁ > 0 and λ₂ < 0 such that k₁ (z; λ₁, λ₂, α) + k₂ (z; λ₁, λ₂, α) ≥ 0 and k₃ (z; λ₁, λ₂, α) + k₄ (z; λ₁, λ₂, α) ≤0,

where
(15)k1(z;λ1,λ2,α)=λ12zf(λ1z)[α+δ(λ1,λ2,α)F(λ1z)],

(16)k2(z;λ1,λ2,α)=λ1δ(λ1,λ2,α)f2(λ1z)[α+δ(λ1,λ2,α)F(λ1z)]2,

(17)k3(z;λ1,λ2,α)=λ22zf(λz)[α+δ(λ1,λ2,α)F(λ2z)]
and
(18)k4(z;λ1,λ2,α)=λ2δ(λ1,λ2,α)f2(λ2z)[α+δ(λ1,λ2,α)F(λ2z)]2.

Proof:

In order to show that there exists unimodes in regions of z ∈ (−∞, 0] and z ∈ [0, ∞), it is enough to show that the second derivative of h(z; λ₁, λ₂, α) is negative for all α, λ₁ and λ₂ in the respective region.

For z ∈ (−∞, 0), we have

(19)d2dz2{log[h(z;λ1,λ2,α)]}=−1−λ1δ(λ1,λ2,α)[k1(z;λ1,λ2,α)+k2(z;λ1,λ2,α)]

and for z ∈ [0, ∞), we have

(20)d2dz2{log[h(z;λ1,λ2,α)]}=−1−λ2δ(λ1,λ2,α)[k3(z;λ1,λ2,α)+k4(z;λ1,λ2,α)],

where k_j(z; λ₁, λ₂, α) for j=1, 2, 3, and 4 are as given in (15) to (18). Note that f(λ₁z) and F(λ₁z) are positive for all z ∈ R and hence [α + δ(λ₁, λ₂, α)F(λ₁Z)] is positive for all α > 0. For z < 0: k₁(z; λ₁, λ₂, α) is negative always and k₂(z; λ₁, λ₂, α) is positive or negative according as λ₁ > 0(< 0). For λ₁ ≤ 0 (19) is negative and hence the density is unimodal. For λ₁ > 0, (19) is negative if k₁ (z; λ₁, λ₂, α) + k₂ (z; λ₁, λ₂, α) ≤ 0. Similarly for z ≥ 0, k₃(z; λ, α) is always positive and k₄(z; λ, α) is positive or negative according as λ₂ > 0(< 0). For λ₂ ≥ 0, (20) is negative and thus the density is unimodal and for λ₂ < 0, (20) is negative if k₃(z; λ₁, λ₂, α) + k₄(z; λ₁, λ₂, α) ≤ 0. Thus the proof of the result follows.

As a consequence of Result 6 we obtain the following result.

Result 7.

The p.d.f. of GTAND(λ₁, λ₂, α) is plurimodal, if

(1)
λ₁ ≤ 0 or λ₂ < 0 provided and k₃(z; λ₁, λ₂, α) + k₄(z; λ₁, λ₂, α) > 0,
(2)
λ₂ ≥ 0 or λ₁ > 0 provided and k₁(z; λ₁, λ₂, α) + k₂(z; λ₁, λ₂, α) < 0,

and
(3)
λ₁ > 0 and λ₂ < 0 such that k₁(z; λ₁, λ₂, α) + k₂(z; λ₁, λ₂, α) > 0 and k₃(z; λ₁, λ₂, α) + k₄(z; λ₁, λ₂, α) < 0

4. Reliability aspects

Here we derive some properties of the GTAND(λ₁, λ₂, α), which are useful in reliability studies.

Let Z follows GTAND(λ₁, λ₂, α) with p.d.f.(4). Now from the definition of reliability function R(t; λ₁, λ₂, α) and failure rate r(t; λ₁, λ₂, α) of Z we obtain the following results.

Result 8.

The reliability function R(t; λ₁, λ₂, α) of Z following the GTAND(λ₁, λ₂, α) is

(21)R(t;λ1,λ2,α)={1−αF(t)−δ(λ1,λ2,α)2[F(t)−2ξ(t,λ1)],t<01−αF(t)−δ(λ1,λ2,α)2[F(t)−tan−1(λ1)π+tan−1(λ2)π−2ξ(t,λ2)],t≥0

where ξ(t,.) is as defined in (5).

Proof follows from the definition of reliability function R(t; λ₁, λ₂, α) = 1− H(t; λ₁, λ₂, α) where H(t; λ₁, λ₂, α) is as given in Result.4.

Result 9.

The failure rate r(t; λ₁, λ₂, α) of Z following the GTAND(λ, α) is

(22)r(t;λ1,λ2,α)={f(t)[α+δ(λ1,λ2,α)F(λ1t)]1−αF(t)−δ(λ1,λ2,α)2[F(t)−2ξ(t,λ1)],t<0f(t)[α+δ(λ1,λ2,α)F(λ2t)]1−αF(t)−δ(λ1,λ2,α)2[F(t)−1πtan−1(λ2)+1πtan−1(λ2)−2ξ(t,λ2)],t≥0

Proof follows from the definition of failure rate, r(t;λ,α)=h(t;λ1,λ2,α)R(t;λ1,λ2,α) where R(t; λ₁, λ₂, α) is as defined in Result 8. Further we derive the following result regarding the mean residual life function of GTAND(λ₁, λ₂, α).

Result 10.

The mean residual life function(MRLF) μ(t; λ₁, λ₂, α) of GTAND(λ₁, λ₂, α) is

(23)μ(t;λ1,λ2,α)=1R(t;λ1,λ2,α){2αe−t222π+δ(λ1,λ2,α)([F(λ1t)+F(λ1t)]f(t))−δ12π[1−F(t1−λ12)]−δ22π[1−F(t1+λ22)]}−t

in which for j=1,2 δj=λj1+λj2.

Proof.

By definition, the MRLF of Z following the TAND(λ, α) is given by

μ(t,λ1,λ2,α)=E(Z−t|Z>t)=E(Z|Z>t)−t

where,

(24)E(Z|Z>t)=αR(t;λ1,λ2,α)∫t∞zf(z)F(λ1z)dz+αR(t;λ1,λ2,α)∫t∞zf(z)F(λ2z)dz+δ(λ1,λ2,α)R(t;λ1,λ2,α)∫t∞zf(z)F(λ1z)dz+δ(λ,α)R(t;λ1,λ2,α)∫t∞zf(z)F(λ2z)dz=1R(t;λ1,λ2,α)[2αe−t222π+δ(λ1,λ2,α)(I1+I2)],

in which

(25)I1=−∫t∞f′(z)F(λ1z)dz=[F(λ1t)f(t)]−λ1∫t∞f(λ1z)f(z)dz={F(λ1t)f(t)−λ1[1−F(t1+λ12)]2π1+λ12}

and

(26)I2={F(λ2t)f(t)−λ2[1−F(t1+λ22)]2π1+λ22},

obtained in a similar way as in (25). Now on substituting (25) and (26) in (24), we get (23).

5. Location-scale extension

In practical situation, the location-scale extension of GTAND(λ₁, λ₂, α) is more relevant. So in this section, we discuss the location-scale extension of the GTAND(λ₁, λ₂, α) and present some of its important properties similar to those we obtained for GTAND(λ₁, λ₂, α).

Definition 2.

Let Z follows GTAND(λ₁, λ₂, α), then X = μ + σZ is said to have an extended generalized two-piece asymmetric normal distribution with location parameter μ, scale parameter σ and shape parameters λ₁, λ₂ and α denoted as EGTAND(μ, σ; λ₁, λ₂, α), if its p.d.f. is given by

(27)h(x;μ,σ,λ1,λ2,α)=={1σf(x−μσ)[α+δ(λ1,λ2,α)F(λ1x−μσ)],x<μ1σf(x−μσ)[α+δ(λ1,λ2,α)F(λ2x−μσ)],x>μ

in which μ, σ > 0, λ₁, λ₂ ∈ R and α∈[0, 1].

Clearly, the EGTAND(μ, σ; λ₁, λ₂, α) contains the following special cases.

1.
When λ₁ = λ, λ₂ = ρλ and α = 0, the distribution with p.d.f. (27) reduces to the location-scale extension of the two-piece skew normal distribution of Kumar and Anusree (2013),
2.
when λ₁ = λ, λ₂ = λ, the distribution with p.d.f. (27) reduces to the location-scale extension of the skew normal distribution of Kumar and Anusree (2011b),
3.
when, λ₁ → −∞, λ₂ → ∞ or λ₁ → −∞, λ₂ → −∞, α = 1 or λ₁ → ∞, λ₂ → ∞, α = 1 or λ₁ = 0, λ₂ = 0 the distribution with p.d.f. (27) reduces to normal distribution with parameters μ and σ,
4.
when λ₁ = −λ, λ₂ = λ and α = 0, the distribution with p.d.f. (27) reduces to location-scale extension of the skew normal distribution of Kim (2005) and
5.
when either λ₁ → −∞, λ₂ → −∞, α = 2 or λ₁ → ∞, λ₂ → ∞, α = 0 TPSND(λ₁, λ₂, α), the distribution with p.d.f. (27) reduces to the half normal distribution with parameters μ and σ.

Result 11.

The characteristic function ϕ_X (t) of a random variable X following EGTAND(μ, σ ; λ₁, λ₂, α) is the following, in which for each j = 1,2 and S_j = σitλ_j. For i=−1 and t ∈ R,

φX(t)=eiμt−t2σ22[α+δ(λ1,λ2,α)F(iδ1σt)]−δ(λ1,λ2,α)eiμt−t2σ22[ξs1(−itσ,λ1)−ξs2(−itσ,λ2)].

Result 12.

The distribution function H (t) = H (t; μ, σ, λ₁, λ₂, α) of a random variable X following EGTAND(μ, σ; λ₁, λ₂, α) is the following,

H(t)={αF(t−μσ)+δ(λ1,λ2,α)2[F(t−μσ)−2ξ(t−μσ,λ1)],t<μαF(t−μσ)+δ(λ1,λ2,α)2[F(t−μσ)−tan−1(λ1)π+tan−1(λ2)π−2ξ(t−μσ,λ2)],t≥μ

where ξ(t−μσ,λ) is as defined in (5).

Result 13.

The reliability function R(t) = R(t; μ, σ, λ₁, λ₂, α) of a random variable X following EGTAND(μ, σ ; λ₁, λ₂, α) is the following,

R(t)={1−αF(t−μσ)−δ(λ1,λ2,α)2[F(t−μσ)−2ξ(t−μσ,λ1)],t<μ1−αF(t−μσ)−δ(λ1,λ2,α)2[F(t−μσ)−tan−1(λ1)π+tan−1(λ2)π−2ξ(t−μσ,λ2)],t≥μ

where ξ(t−μσ,λ) is as defined in (5).

Result 5.4

The failure rate r(t) = r(t; μ, σ, λ₁, λ₂, α) of a random variable X following EGTAND(μ, σ; λ₁, λ₂, α) is the following,

r(t)={f(t−μσ)[α+δ(λ1,λ2,α)F(λ1t−μσ)]1−αF(t−μσ)−δ(λ1,λ2,α)2[F(t−μσ)−2ξ(t−μσ,λ1)],t<μf(t−μσ)[α+δ(λ1,λ2,α)F(λ2t−μσ)]1−αF(t−μσ)−δ(λ1,λ2,α)2[F(t−μσ)−tan−1(λ1)π+tan−1(λ2)π−2ξ(t−μσ,λ2)],t≥μ

6. Estimation

Let X₁, X₂ ,..., X_n be a random sample from EGTAND(μ, σ; λ₁, λ₂, α) with p.d.f. (27). Let X₍₁₎, X₍₂₎,...,X_(n) be the ordered sample. Assume X_(r) < μ < X_(r+1), for a particular r=1,2,...,n. Then log-likelihood function of the sample is the following, in which Σ_{I_j}, denote the summation over the set I_j such that

(28)I1={i:X(i)<μ,for i=1,2,…,r} and I2={i:X(i)≥μ,for i=r+1,…,n}. logL=−nlnσ+∑I1lnf(xi−μσ)[α+δ(λ1,λ2,α)F(λ1(xi−μ)σ)]+ ∑I2lnf(λ2(xi−μ)σ)[α+δ(λ1,λ2,α)F(λ2(xi−μ)σ)]

On differentiating (28) with respect to the parameters μ, σ, λ₁, λ₂ and α and equating to zero, we obtain the following likelihood equations:

(29)∑I1(xi−μσ2)−λ1δ(λ1,λ2,α)σ∑I1f(λ1σ(xi−μ))[α+δ(λ1,λ2,α)F(λ1σ(xi−μ))]∑I2(xi−μσ2)−λ2δ(λ1,λ2,α)σ∑I2f(−λ2σ(xi−μ))[α+δ(λ1,λ2,α)F(λ2σ(xi−μ))]=0

(30)−nσ2+∑I1(xi−μ)2σ4−∑I1f(λ1σ(xi−μ))[α+δ(λ1,λ2,α)F(λ1σ(xi−μ))](xi−μ)δ(λ1,λ2,α)λ1σ3+∑I2(xi−μ)2σ4−∑I2f(λ2σ(xi−μ))[α+δ(λ1,λ2,α)F(λ2σ(xi−μ))](xi−μ)δ(λ1,λ2,α)λ2σ3=0

(31)−δ(λ1,λ2,α)∑I1f(λ1(xi−μ)σ)(xi−μ)σ[α+δ(λ1,λ2,α)F(λ1σ(xi−μ))]=0

(32)δ(λ1,λ2,α)∑I2f(λ2(xi−μ)σ)(xi−μ)σ[α+δ(λ1,λ2,α)F(λ2(xi−μ)σ)]=0

(33)∑I11−2F(λ1(xi−μ)σ)[α+δ(λ1,λ2,α)F(λ1(xi−μ)σ)]+∑I21−2F(λ1(xi−μ)σ)[α+δ(λ1,λ2,α)F(λ2(xi−μ)σ)]=0

Let

w(xi)=f(λ1(xi−μ)σ)[[α+δ(λ1,λ2,α)F(λ1(xi−μ)σ)]],

and

Ω(xi)=f(λ2(xi−μ)σ)[[α+δ(λ1,λ2,α)F(λ2(xi−μ)σ)]],

W(xi)=F(λ1(xi−μ)σ)[[α+δ(λ1,λ2,α)F(λ1(xi−μ)σ)]]

and

Δ(xi)=F(λ2(xi−μ)σ)[[α+δ(λ1,λ2,α)F(λ2(xi−μ)σ)]].

Then the equations from (29) to (33) becomes

(34)∑I1(xi−μ)σ+∑I2(xi−μ)σ=δ(λ1,λ2,α)[λ1∑I1w(xi)+λ2∑I2Ω(xi)],

(35)n2σ2=12∑I1(xi−μ)2σ4+12∑I2(xi−μ)2σ4−δ(λ1,λ2,α)λ2σ3(λ1∑I1w(xi)(xi−μ)+λ2∑I2Ω(xi)(xi−μ)),

(36)δ(λ1,λ2,α)∑I1w(xi)(xi−μσ)=0,

(37)δ(λ1,λ2,α)∑I2Ω(xi)(xi−μσ)=0.

and

(38)2∑I1W(xi)+2∑I2Δ(xi)=∑I11[α+δ(λ1,λ2,α)F(λ1xi−μσ)]+∑I21[α+δ(λ1,λ2,α)F(λ2xi−μσ)].

On solving the non-linear system of equations (34) to (38) by simultaneous solution method using some mathematical softwares such as MATHCAD, MATLAB, MATHEMATICA etc. one can obtain the maximum likelihood estimates (MLE) of the parameters of EGTAND(μ, σ; λ₁, λ₂, α).

7. Numerical computation

For a numerical illustration,we consider the following real life data set on the heights (in centimeters) of 100 Australian athletes, given in Cook and Weisberg (1994).The data recorded is as given below.

148.9 149 156 156.9 157.9 158.9 162 162 162.5 163 163.9 165 166.1 166.7 167.3 167.9 168 168.6 169.1 169.8 169.9 170
170 170.3 170.8 171.1 171.4 171.4 171.6 171.7 172 172.2 172.3 172.5 172.6 172.7 173 173.3 173.3 173.5 173.6 173.7 173.8
174 174 174 174.1 174.1 174.4 175 175 175 175.3 175.6 176 176 176 176 176.8 177 177.3 177.3 177.5 177.5 177.8 177.9
178 178.2 178.7 178.9 179.3 179.5 179.6 179.6 179.7 179.7 179.8 179.9 180.2 180.2 180.5 180.5 180.9 181 181.3 182.1
182.7 183 183.3 183.3 184.6 184.7 185 185.2 186.2 186.3 188.7 189.7 193.4 195.9.

This data has been recently used by Salehi et. al (2013) for establishing that “generalized skew two-piece skew normal distribution[ GSTPSt(λ₁, λ₂, ρ) ]”fits the data better than certain existing models. We obtained the MLE of the parameters of of N(μ, σ), the location-scale extension of SND(λ)[ ESND((μ, σ ;λ) ], the location-scale extension of TPSND(λ, ρ) [ETPSND(μ, σ ; λ, ρ)] of Kumar and Anusree (2013a), the location-scale extension of GSTPSt(λ₁, λ₂, ρ) of Salehi et. al (2013) [ EGSTPSt(μ, σ ; λ₁, λ₂, ρ) ] and EGTAND(μ, σ ; λ₁, λ₂, α) with the help of equations (6.7) to (6.11) and MATHCAD software. The values of loglikelihood( l ), the Akaike’s Information Criterion (AIC), the Bayesian Information Criterion (BIC) and the corrected Akaike’s Information Criterion (AICc) are also computed and presented in Table 1.

Distribution:	Normal (μ, σ)	ESND (μ, σ, λ)	ETPSND (μ, σ; λ, ρ)	EGSTPSt (μ, σ; λ₁, λ₂, ρ)	EGTAND (μ, σ; λ₁, λ₂, α)
μ^	174.594	174.58	173.657	167.056	173.01
σ^	8.24	8.20	8.21	7.73	8.48
λ^	-	0.0016	−0.18	0.219,0.244	−2.92,0.765
α^	-	-	-	-	2.28
ρ^	-	-	0.974	−0.991	-
l	−352.318	−352.318	−349	−348	−347.64
AIC	708.64	710.64	706	706	703
BIC	713.85	718.45	716	705	704
AICc	708.76	710.89	706.49	706.68	704

Table 1.

Estimated values of the parameters and the corresponding l, AIC, BIC and AICc values for the fitted models-the N(μ, σ), the ESND((μ, σ ;λ), the ETPSND(μ, σ ; λ, ρ), the EGSTPSt(μ, σ ; λ₁, λ₂, ρ) and the EGTAND(μ, σ ; λ₁, λ₂, α).

From Table 1 we can see that EGTAND(μ, σ; λ₁, λ₂, α) gives a better fit to the data given to the existing models-N(μ, σ), ESND((μ, σ; λ), ETPSND(μ, σ; λ, ρ) and EGSTPSt(μ, σ; λ₁, λ₂, ρ).

References

1.BC Arnold, RJ Beaver, RA Groeneveld, and WQ Meeker, The non truncated marginal of a truncated bivariate normal distribution, Psychometrika, Vol. 58, No. 3, 1993, pp. 471-488.

2.A Azzalini, A class of distributions which includes the normal ones, Scand. Jour. Stat, Vol. 12, 1985, pp. 171-178.

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4.A Azzalini and A Dalla Valle, The multivariate skew-normal distribution, Biometrika, Vol. 83, No. 4, 1996, pp. 715-726.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 17 - 1
Pages: 101 - 121
Publication Date: 2018/03/31
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.2018.17.1.8 How to use a DOI?
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TY  - JOUR
AU  - C. Satheesh Kumar
AU  - M.R. Anusree
PY  - 2018
DA  - 2018/03/31
TI  - On Some Aspects of a New Class of Two-Piece Asymmetric Normal Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 101
EP  - 121
VL  - 17
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.1.8
DO  - 10.2991/jsta.2018.17.1.8
ID  - SatheeshKumar2018
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