a generalization of the slashed distribution via alpha skew normal distribution
TRANSCRIPT
Stat Methods ApplDOI 10.1007/s10260-014-0258-7
A generalization of the slashed distribution via alphaskew normal distribution
Wenhao Gui
Accepted: 16 February 2014© Springer-Verlag Berlin Heidelberg 2014
Abstract In this paper, we introduce a new class of the slash distribution, an alphaskew normal slash distribution. The proposed model is more flexible in terms of itskurtosis than the slashed normal distribution and can efficiently capture the bimodal-ity. Properties involving moments and moment generating function are studied. Thedistribution is illustrated with a real application.
Keywords Alpha skew normal distribution · Slash distribution · Kurtosis · Skewness
Mathematics Subject Classification 62E10 · 62F10
1 Introduction
A random variable S has a standard slash distribution SL(q) if S can be representedas
S = Z
U1q
(1)
where Z ∼ N (0, 1) and U ∼ U (0, 1) are independent. q > 0 is the shape parameter.The slash distribution was named by Rogers and Tukey in a paper published in Rogersand Tukey (1972).
W. Gui (B)Department of Mathematics and Statistics, University of Minnesota Duluth,Duluth, MN 55812, USAe-mail: [email protected]
W. GuiDepartment of Mathematics, Beijing Jiaotong University,100044 Beijing, China
123
W. Gui
In statistics, the well known normal distribution plays a key role but the inferencebased on that is sensitive to statistical errors that have a heavier-tailed distribution(Arslan and Genç 2009). The slash distribution, an extension of normal distribution,is more flexible and can effectively capture the features. It has been very popular inrobust statistical analysis (Rogers and Tukey 1972; Morgenthaler 1986; Jamshidian2001; Kashid and Kulkarni 2003).
The SL(q) density function is given by
f (x) = q
1∫
0
uqφ(xu)du, −∞ < x < ∞ (2)
where φ(t) = 1√2π
e− t22 denotes the standard normal density function. The mean and
variance of the slash distribution are given by E(S) = 0 for q > 1 and Var(S) = qq−2
for q > 2.For the limit case q → ∞, it yields the standard normal distribution. The standard
slash density has heavier tails than those of the normal distribution and has largerkurtosis. Let q = 1, the canonical slash distribution follows and it has the followingdensity (Johnson et al. 1995)
f (x) ={
φ(0)−φ(x)
x2 x �= 012φ(0) x = 0
(3)
Rogers and Tukey (1972) and Mosteller and Tukey (1977) studied the generalproperties of this canonical slash distribution. For the standard slash distribution,Kafadar (1982) proposed the maximum likelihood estimates of the location and scaleparameters.
Gómez et al. (2007) defined some symmetric extensions of the distribution basedon elliptical distributions and studied its general properties of the resulting families,including their moments. Arslan and Genç (2009) proposed a generalization of themultivariate symmetric slash distribution. Wang and Genton (2006) investigated themultivariate skew version of this distribution and examined its properties and infer-ences. They substituted the standard normal random variable Z by a skew normaldistribution introduced by Azzalini (1985) to define a skew extension of the slashdistribution. Arslan (2008) developed an alternative asymmetric extension of the dis-tribution by using the variance-mean mixture of the multivariate normal distribution.
In our work, we consider an alpha skew normal distribution introduced by Elal-Olivero (2010). The alpha skew normal distribution generalizes the normal distributionand allows to fit unimodal and bimodal data sets, it is naturally to use it to define anew extension of the slash distribution.
The rest of this paper is organized as follows: in Sect. 2, we propose the new distri-bution and study its properties, including the stochastic representation etc. Section 3studies the inference, moments and maximum likelihood estimation for the parame-ters. A simulation study is provided in Sect. 4. Section 5, a real dataset application isreported and illustrated. Section 6 concludes our work.
123
Generalization of the slashed distribution
2 Alpha skew normal slash distribution
2.1 Stochastic representation
Definition 1 A random variable X has an alpha skew normal distribution, denoted byX ∼ ASN(α), if X has density function given by
fX (x) = (1 − αx)2 + 1
2 + α2 φ(x), −∞ < x < ∞ (4)
where −∞ < α < ∞.
Remark 1 Elal-Olivero (2010) introduced this random variable to generate asymmetryin the normal distribution. If α = 0, we obtain the standard normal distribution Z ∼N (0, 1). If α → ±∞, it reduced to a bimodal-normal random variable. An algebraiccomputation shows that it is unimodal if −1.345 < α < 1.345 and bimodal otherwise.
Proposition 2 Let X ∼ ASN(α), then the kth non-central moments are given by
EXk =⎧⎨⎩
2+α2(2m+1)
2+α2
∏mj=1(2 j − 1) k = 2m
− 2α2+α2
∏mj=1(2 j − 1) k = 2m − 1
(5)
for m = 1, 2, 3, . . .
Proof See Elal-Olivero (2010). ��Proposition 3 Let X ∼ ASN(α), then the moment generating function of X is givenby
MX (t) =[
1 − αt
(2 − αt
2 + α2
)]e
t22 (6)
Proof See Elal-Olivero (2010). ��ASN(α) is more general than normal. Now we introduce a new distribution using theidea of the slash construction as the ratio of two independent random variables.
Definition 4 A random variable Y has an alpha skew normal slash distribution withparameters −∞ < α < ∞ and q > 0 if it can be represented as the ratio
Y = X
U1q
(7)
where U ∼ U (0, 1) and X ∼ ASN(α) are independent. We denote it as Y ∼ASNS(α, q).
Proposition 5 Let Y ∼ ASNS(α, q). Then, the density function of Y is given by
fY (y) = q
1∫
0
(1 − αyt)2 + 1
2 + α2 φ(yt)tqdt (8)
123
W. Gui
Proof The joint probability density function of X and U is given by
g(x, u) = (1 − αx)2 + 1
2 + α2 φ(x) (9)
for −∞ < x < ∞, 0 < u < 1. Using the following transformation: y = xu1/q , u = u,
the joint probability density function of Y and U is given by
h(y, u) =(
1 − αyu1q
)2 + 1
2 + α2 φ(
yu1q
)u
1q (10)
where −∞ < y < ∞, 0 < u < 1. The marginal density function of Y is given by
fY (y) =1∫
0
(1 − αyu
1q
)2 + 1
2 + α2 φ(
yu1q
)u
1q du (11)
Let t = u1q , the density function (8) follows. ��
Remark 2 −Y ∼ ASNS(−α, q). If α = 0, the density function (8) reduces to (2).Y ∼ ASNS(α = 0, q) is the slashed normal distribution obtained by Rogers andTukey (1972). As q → ∞, the limit case of the alpha skew normal slash distributionY ∼ ASNS(α, q = ∞) is the alpha skew normal distribution ASN(α).
Figure 1 depicts some plots of the pdf of the alpha skew normal slash distributionwith various parameters. It contains unimodal and bimodal cases.
The cumulative distribution function of the symmetric component alpha normalslash distribution Y ∼ ASNS(α, q) is given as follows.
FY (y) =y∫
−∞fY (u)du
=y∫
−∞
⎡⎣q
1∫
0
(1 − αut)2 + 1
2 + α2 φ(ut)tqdt
⎤⎦ du
= q
1∫
0
tq
⎡⎣
y∫
−∞
(1 − αut)2 + 1
2 + α2 φ(ut)du
⎤⎦ dt
= q
1∫
0
tq−1
⎡⎣
yt∫
−∞
(1 − αs)2 + 1
2 + α2 φ(s)ds
⎤⎦ dt
= q
1∫
0
tq−1[�(yt) + 2α
2 + α2 φ(yt) − α2
2 + αytφ(yt)
]dt (12)
123
Generalization of the slashed distribution
−6 −4 −2 0 2 4 6
0.00
0.05
0.10
0.15
0.20
0.25
0.30
y
f(y|
α,q)
α = 0,q = 2α = 2,q = 2α = 8,q = 2
Fig. 1 The density function of ASNS(α, q) with various parameters
2.2 Location-scale form of the distribution
Now we apply the well-known location-scale transformation and consider the generalform of the distribution.
Y = σX
U1q
+ μ (13)
where U ∼ U (0, 1) and X ∼ ASN(α) are independent, q > 0, −∞ < μ < ∞ andσ > 0. μ is called the location parameter and σ the scale parameter. We denote itas Y ∼ ASNS(α, q;μ, σ). If μ = 0 and σ = 1, it is the standard case we discussedabove. In this case, we use Y ∼ ASNS(α, q).
The density function of the general form is given by
fY (y|θ) = q
σ
1∫
0
[1 − α
( y−μσ
)t]2 + 1
2 + α2 φ
[(y − μ
σ
)t
]tqdt (14)
where θ = (α, q, μ, σ ). The results about the moments, CDF etc are similar to thestandard case and ommitted.
Proposition 6 Let Y |U = u ∼ ASN(α;μ = 0, σ = u− 1q ) and U ∼ U (0, 1), then
Y ∼ ASNS(α, q;μ = 0, σ = 1).
123
W. Gui
Proof
fY (y) =1∫
0
fY |U (y|u) fU (u)du
=1∫
0
1
u− 1q
[1 − α
(y
u− 1
q
)]2
+ 1
2 + α2 φ
(y
u− 1q
)du let t = u
1q
= q
1∫
0
(1 − αyt)2 + 1
2 + α2 φ(yt)tqdt
��Remark 3 Proposition 6 shows that the alpha skew normal slash distribution can berepresented as a scale mixture of the alpha skew normal distribution and uniformU (0, 1) distribution. The result provides another way to generate random numbersfrom the the alpha skew normal slash distribution.
2.3 Moments and moment generating function
Proposition 7 Let Y ∼ ASNS(α, q). For k = 1, 2, . . . and q > k, the kth non-centralmoment of Y is given by
μk = EY k =⎧⎨⎩
qq−2m
2+α2(2m+1)
2+α2
∏mj=1(2 j − 1) k = 2m
− qq−(2m−1)
2α2+α2
∏mj=1(2 j − 1) k = 2m − 1
(15)
for m = 1, 2, 3, ...
Proof From the stochastic representation defined in (7) and the results in (5), we have
μk = EY k
= E(XU− 1q )k
= E(Xk) × E(U− kq )
= q
q − kE(Xk)
��The following results are immediate consequences of (15).
Corollary 8 Let Y ∼ ASNS(α, q). The mean and variance of Y are given by
EY = − q
q − 1
2α
2 + α2 , q > 1 (16)
Var(Y ) =q[(q − 1)2(4 + 4α2 + 3α4) + 4α2
]
(q − 2)(q − 1)2(2 + α2)2 , q > 2 (17)
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Generalization of the slashed distribution
Corollary 9 Let Y ∼ ASNS(α, q). The skewness and kurtosis coefficients of Y aregiven by
√β1 =
4α√
q − 2
[−3(2 + α2)2 + (24 + 6α2 − 3α4)q + 6(−2 + α2 + 3α4)q2
−5α2(2 + 3α2)q3 + α2(2 + 3α2)q4
]
√q(q − 3)
[(q − 1)2(4 + 4α2 + 3α4) + 4α2
]3/2
(18)for q > 3.
β2 =
3(q−2)
⎧⎪⎪⎪⎨⎪⎪⎪⎩
6(
2+α2)3(
2+5α2)−(
2+α2)2(
116+220α2+145α4)
q+3(
2+α2)(
152+276α2+354α4+95α6)
q2
−2(
464+976α2+1512α4+956α6+145α8)
q3+8(
64+132α2+220α4+151α6+20α8)
q4
−9(
16+32α2+56α4+40α6+5α8)
q5+(
16+32α2+56α4+40α6+5α8)
q6
⎫⎪⎪⎪⎬⎪⎪⎪⎭
q(q− v3)(q−4)[(q−1)2(4+4α2+3α4)+4α2
]2(19)
for q > 4.
Remark 4√
β1 is an odd function of α while β2 is even. In addition, as q → ∞,√β1 → 4α3(2+3α2)
(4+4α2+3α4)3/2 and β2 → 3(16+32α2+56α4+40α6+5α8)
(4+4α2+3α4)2 , which are the skewnessand kurtosis coefficients for the alpha skew normal distribution obtained by Elal-Olivero (2010).
Figures 2 and 3 show the the skewness and kurtosis coefficients with various para-meters.
4 6 8 10 12 14 16
−1.
0−
0.5
0.0
0.5
1.0
q
Ske
wne
ssβ 1
α = 0α = 2α = 5α = 10
Fig. 2 The plot for the skewness coefficient√
β1 with various parameters
123
W. Gui
4 6 8 10 12
05
1015
q
Kur
tosi
s β 2
α = 0α = 1α = 5α = 20
Fig. 3 The plot for the kurtosis coefficient β2 with various parameters
Proposition 10 Let Y ∼ ASNS(α, q), the moment generating function of Y is givenby
MY (t) = q
∞∫
1
(1 − αwt (2 − αwt)
2 + α2
)e
w2 t22 w−(1+q)dw (20)
Proof Using Propositions 3 and 6, we obtain
MY (t) = EetY
= E(E(etY |U ))
= E
[(1 − αU− 1
q t (2 − αU− 1q t)
2 + α2
)e
(U− 1
q t)22
]
=1∫
0
[1 − αu− 1
q t
(2 − αu− 1
q t
2 + α2
)]e
(u− 1
q t)22 du
= q
∞∫
1
(1 − αwt (2 − αwt)
2 + α2
)e
w2 t22 w−(1+q)dw
��
123
Generalization of the slashed distribution
3 Inference
In this section, we consider the maximum likelihood estimation about the parameterθ = (α, q, μ, σ ) of the location-scale form defined in (14).
fY (y|θ) = q
σ
1∫
0
[1 − α
( y−μσ
)t]2 + 1
2 + α2 φ
[(y − μ
σ
)t
]tqdt
Suppose Y1, Y2, . . . , Yn is a random sample of size n from the distribution Y ∼ASN S(θ). The log-likelihood function is expressed as
l(θ) = n log q − n log σ +n∑
i=1
log
1∫
0
[1 − α
( yi −μσ
)t]2 + 1
2 + α2 φ
[(yi − μ
σ
)t
]tqdt
(21)
Taking the partial derivatives of the log-likelihood function with respect to α, q, μ, σ
respectively and equalizing the obtained expressions to zero yield to likelihood equa-tions. However, they do not have explicit analytical solutions for model parameters.Thus, the estimates can be obtained by means of numerical procedures such as Newton-Raphson method. The program R provides the nonlinear optimization routine optimfor solving such problems.
It is known that as the sample size increases, the distribution of the MLE tends to thenormal distribution with mean (α, q, μ, σ ) and covariance matrix equal to the inverseof the Fisher information matrix. The score vector and Hessian matrix are providedin Appendix A.1. However, the log-likelihood function given in (21) is a complexexpression. It is not generally possible to derive the Fisher information matrix. Thus,the theoretical properties (asymptotically normal, unbiased etc) of the MLE estimatorsare not easily obtained. We study the properties of the estimators numerically.
4 Simulation study
4.1 Data generation
In this section, we present how to generate the random numbers from the alpha skewnormal slash distribution.
Proposition 11 Let T, V be two independent random variables, where T ∼ χ2(3) and
V is such that P(V = 1) = P(V = −1) = 12 , then W = √
T V ∼ w2φ(w), withw ∈ R.
Proof See Elal-Olivero (2010). ��Proposition 12 Let Z ∼ N (0, 1) and W ∼ w2φ(w), w ∈ R, be two independent
random variables, if S =√
22+α2 Z +
√α2
2+α2 W , then S ∼ 2+α2x2
2+α2 φ(x).
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W. Gui
Table 1 Empirical means and SD for the MLE estimators of α, q, μ and σ
α q μ σ n = 100 n = 200
α(SD) q(SD) μ(SD) σ (SD) α(SD) q(SD) μ(SD) σ (SD)
1 1 2 2 1.0423 1.0364 2.0543 2.1281 1.0135 1.0252 2.0132 2.0113
(0.3278) (0.4112) (0.3717) (0.30496) (0.1183) (0.2205) (0.0951) (0.0243)
1 3 2 4 1.0389 3.0405 2.0437 4.1022 1.0105 3.0288 2.0245 4.0596
(0.2803) (0.2722) (0.3693) (0.3345) (0.0860) (0.1211) (0.1745) (0.0738)
3 1 2 2 3.0355 1.2045 2.0336 2.0456 3.0102 1.0542 2.0177 2.0089
(0.2807) (0.3103) (0.2448) (0.4331) (0.1918) (0.1125) (0.1780) (0.1518)
3 3 4 4 3.0293 3.0512 4.0381 4.0268 3.0175 3.0086 4.0123 4.0095
(0.2087) (0.2136) (0.3248) (0.3139) (0.1128) (0.1769) (0.0914) (0.0174)
Proof See Elal-Olivero (2010). ��Notice that
supx
(1−αx)2+12+α2 φ(x)
2+α2x2
2+α2 φ(x)= 2 + √
2
2
The random numbers from ASNS(α, q, μ, σ ) can be generated as follows:
(1) Generate Z ∼ N (0, 1), T ∼ χ2(3), P ∼ U (0, 1), Q ∼ U (0, 1) and U ∼ U (0, 1);
(2) Let V = 1 if Q < 12 . Otherwise V = −1; Set W = √
T V ;
(3) Set S =√
22+α2 Z +
√α2
2+α2 W ;
(4) If P <2[(1+αS)2+1]
(2+√2)(2+α2 S2)
,set X = S; otherwise return to step (1);
(5) Set Y = σ X
U1q
+ μ.
4.2 Behavior of MLE
In this section, we conduct a simulation to illustrate the behavior of the MLE estimatorsfor parameters α, q, μ and σ . We first generate 500 samples of size n = 100 andn = 200 from the ASNS(α, q, μ, σ ) distribution for fixed parameters.
The estimates are computed by the optim function which uses L-BFGS-B method insoftware R. The empirical means and standard deviations of the estimates are presentedin Table 1. From Table 1, the parameters are well estimated and the estimates areasymptotically unbiased.
5 Data analysis
In this section, we want to fit the proposed model to a real dataset. We consider thescore data set which consists of the scores of 104 students who attended a standard
123
Generalization of the slashed distribution
Table 2 Descriptive statistics for the score dataset
n Mean Median Standard deviation Skewness√
β1 Kurtosis β2
104 68.52 67.44 4.87 0.315 2.635
Table 3 Maximum likelihood parameter estimates(with (SD)) of the ASNS, ASN, Normal and MixtureNormal models for the score dataset
Model α q μ σ Loglik AIC BIC
Normal − − 68.5177 4.8454 −311.6855 627.371 632.6598
(0.4751) (0.3359)
ASN 0.4844 − 70.5657 4.7823 −311.5357 629.0714 637.0046
(0.4187) (1.36) (0.3844)
ASNS 2.1138 7.015 70.4495 2.8565 −303.7566 615.5132 626.0908
(0.3776) (3.215) (0.3676) (0.2718)
Model p μ1 σ1 μ2 σ2 Loglik AIC BIC
Mixture normal 0.3934 65.8227 2.0585 70.2663 5.3094 −305.5425 621.085 634.307
(0.1300) (0.5322) (0.4969) (1.1158) (0.5332)
exam. Table 2 shows the basic descriptive statistics for the data set. Figure 4 displaysthat the score distribution is bimodal.
We fit the score data set with the ASN(α, μ, σ ), ASNS(α, q, μ, σ ), N (μ, σ ) andmixture normal (p, μ1, σ1, μ2, σ2) distributions and examine the performances ofthe distributions. The maximum likelihood estimates of the parameters are obtainedand the results are reported in Table 3. The Akaike information criterion (AIC) andBayesian information criterion (BIC) are used to measure of the goodness of fit of themodels. AI C = 2k − 2 log L and B I C = k log(n) − 2 log L , where n is the samplesize, k is the number of parameters in the model and L is the maximized value of thelikelihood function for the estimated model.
From Table 3, for the score data set, both AIC and BIC show that the proposedASNS model is a best fit. It has smallest AIC/BIC and highest likelihood values. Itcan effectively depict the bimodality of the data (see Fig. 4). For ASN model, evenif it contains unimodal and bimodal cases. But we apply the MLE method and obtain−1.345 < α = 0.4844 < 1.345, the fitted ASN is a unimodal model. Its likelihoodvalue is just a little higher than that of Normal model and Its AIC/BIC is higher aswell. It can not efficiently capture the features of the data set.
For mixture normal distribution(p, μ1, σ1, μ2, σ2), its density function is given by
p1√
2πσ1e− (x−μ1)2
2σ21 + (1 − p)
1√2πσ2
e− (x−μ2)2
2σ22
The AIC/BIC of the fitted mixture normal model is a little higher than ASNS model.This mixture model can effectively capture the first peak but can not fit the secondpeak and the range between the two peaks.
123
W. Gui
Score
Den
sity
55 60 65 70 75 80
0.00
0.05
0.10
0.15
ASNSASNNormalMixture Normal
Fig. 4 Histogram and models fitted for the score data set
6 Concluding remarks
In this paper, we have proposed an alpha skew normal slash distribution ASNS(α, q).It is defined to be the quotient of two independent random variables, an alpha skewnormal ranodm variable ASN(α) and a power of the uniform distribution U (0, 1). Thisgeneralization contains the slashed normal etc as the special cases.
Properties involving moments and moment generating function are studied. Weapply the model to a real dataset where model fitting is implemented by maximumlikelihood procedure. The data analysis indicates that the proposed model is very usefuland flexible in real applications. It can successful capture the bimodality compared toASN model.
Appendix A: Appendix
Appendix A.1: Score vector and Hessian matrix
Suppose y1, y2, . . . , yn is a random sample drawn from the alpha skew normal slashdistribution ASNS(α, q, μ, σ ), then the log-likelihood function is given by (21). Theelements of the score vector are obtained by differentiation
lα =n∑
i=1
∫ 1
0
2α2( yi−μ
σ
)t+4α
(( yi−μσ
)2t2−1
)−4
( yi−μσ
)t
(2+α2)2 φ
[(yi −μ
σ
)t
]tq dt
∫ 1
0
[1−α
( yi−μσ
)t]2+1
2+α2 φ
[(yi −μ
σ
)t
]tq dt
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Generalization of the slashed distribution
lq = n
q+
n∑i=1
∫ 1
0
[1−α
( yi −μσ
)t]2 + 1
2 + α2 φ
[(yi − μ
σ
)t
]tq log tdt
∫ 1
0
[1 − α
( yi −μσ
)t]2 + 1
2 + α2 φ
[(yi − μ
σ
)t
]tq dt
lμ =n∑
i=1
∫ 1
0
tq+1
(2 + α2)σ
{[1 + α − α
(yi − μ
σ
)t
]2
+ 1 − α2
}φ
[(yi − μ
σ
)t
]dt
∫ 1
0
[1 − α
( yi −μσ
)t]2 + 1
2 + α2 φ
[(yi − μ
σ
)t
]tq dt
lσ = − n
σ+
n∑i=1
∫ 1
0
(yi −μ)tq+1
(2+α2)σ 2
{2α
[1−α
(yi −μ
σ
)t
]+(
[1 − α
(yi −μ
σ
)t
]2
+1)(yi −μ
σ)
}φ
[(yi −μ
σ
)t
]dt
∫ 1
0
[1 − α
( yi−μσ
)t]2+1
2+α2 φ
[(yi −μ
σ
)t
]tq dt
The Hessian matrix, second partial derivatives of the log-likelihood, is given by
H =
⎛⎜⎜⎝
lαα lαq lαμ lασ
lqα lqq lqμ lqσ
lμα lμq lμμ lμσ
lσα lσq lσμ lσσ
⎞⎟⎟⎠
where
lαα =n∑
i=1
∫ 1
0
⎛⎜⎜⎝
2tq+2(yi−μ)2φ[( yi−μσ )t]
(α2+2)σ2 +8αtq+1(yi−μ)φ[( yi−μ
σ )t](
1−αt(yi−μ)σ
)
(α2+2)2σ
+8α2 tq φ[( yi−μ
σ )t]((
1−αt(yi−μ)σ
)2+1
)
(α2+2)3 −
2tq φ[( yi−μσ )t]
((1−αt(yi−μ)
σ
)2+1
)
(α2+2)2
⎞⎟⎟⎠ dt
∫ 1
0
tqφ[( yi−μσ
)t]((
1− αt(yi−μ)σ
)2+1
)(α2+2
) dt
−n∑
i=1
⎛⎝∫ 1
0
⎛⎝−
2tq+1 (yi −μ) φ[( yi−μσ
)t](
1− αt(yi−μ)σ
)(α2+2
)σ
−2αtqφ[( yi−μ
σ)t]((
1− αt(yi−μ)σ
)2+1
)(α2+2
)2
⎞⎠ dt
⎞⎠ 2
⎛⎝∫ 1
0
tqφ[( yi−μσ
)t]((
1− αt(yi−μ)σ
)2+1
)(α2+2
) dt
⎞⎠ 2
lαq = lqα
=n∑
i=1
∫ 1
0
⎛⎝−
2tq+1 log(t) (yi −μ)(
1− αt(yi−μ)σ
)φ(
t(yi−μ)σ
)(α2+2
)σ
−2αtq log(t)
((1− αt(yi−μ)
σ
)2+1
)φ(
t(yi−μ)σ
)(α2+2
)2
⎞⎠ dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
−n∑
i=1
∫ 1
0
⎛⎜⎜⎝
−2tq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−2αtq
((1−αt(yi−μ)
σ
)2+1
)φ
(t(yi−μ)
σ
)
(α2+2)2
⎞⎟⎟⎠ dt
⎛⎝∫ 1
0
tq log(t)((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
123
W. Gui
lαμ = lμα
=n∑
i=1
∫ 1
0
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
2tq+2(yi−μ)
(1−αt(yi−μ)
σ
)φ′(
t(yi−μ)σ
)
(α2+2)σ2 −2αtq+2(yi−μ)φ
(t(yi−μ)
σ
)
(α2+2)σ2
+2αtq+1
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)2σ
+2tq+1
(1−v
αt(yi−μ)σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−4α2 tq+1
(1−αt(yi−μ)
σ
)φ
(t(yi −μ)
σ
)
(α2+2)2σ
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
−n∑
i=1
⎛⎜⎜⎜⎜⎜⎝∫ 1
0
⎛⎜⎜⎜⎜⎜⎝
−2tq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−2αtq
((1−αt(yi−μ)
σ
)2+1
)φ
(t(yi−μ)
σ
)
(α2+2)2
⎞⎟⎟⎟⎟⎟⎠
dt
⎞⎟⎟⎟⎟⎟⎠∫ 1
0
⎛⎜⎜⎜⎜⎝
2αtq+1(
1−αt(yi−μ)σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−tq+1
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ
⎞⎟⎟⎟⎟⎠ dt
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
lασ = lσα
=n∑
i=1
∫ 1
0
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
2tq+2(yi−μ)2(
1−αt(yi−μ)σ
)φ′(
t(yi−μ)σ
)
(α2+2)σ3 −2αtq+2(yi−μ)2φ
(t(yi−μ)
σ
)
(α2+2)σ3
+2αtq+1(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)2σ2
+2tq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ2
−4α2 tq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)2σ2
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
−n∑
i=1
∫ 1
0
⎛⎜⎜⎜⎜⎝
−2tq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−2αtq
((1−αt(yi−μ)
σ
)2+1
)φ
(t(yi−μ)
σ
)
(α2+2)2
⎞⎟⎟⎟⎟⎠ dt
∫ 1
0
⎛⎜⎜⎜⎜⎝
2αtq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ2
−tq+1(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ2
⎞⎟⎟⎟⎟⎠ dt
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
lqq =−n∑
i=1
⎛⎝∫ 1
0
tq log(t)((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2v +1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
− n
q2
n∑i=1
∫ 1
0
tq log2(t)((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
123
Generalization of the slashed distribution
lqμ = lμq
=n
−∑i=1
∫ 1
0
⎛⎜⎜⎝
2αtq+1(
1−αt(yi−μ)σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−tq+1
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ
⎞⎟⎟⎠ dt
⎛⎝∫ 1
0
tq log(t)((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
n∑i=1
∫ 1
0
⎛⎜⎜⎜⎜⎝
2αtq+1 log(t)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−tq+1 log(t)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ
⎞⎟⎟⎟⎟⎠ dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
lqσ = lσq
=−n∑
i=1
∫ 1
0
⎛⎜⎜⎜⎜⎝
2αtq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ2
−tq+1(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ2
⎞⎟⎟⎟⎟⎠ dt
⎛⎝∫ 1
0
tq log(t)((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
n∑i=1
∫ 1
0
⎛⎜⎜⎜⎜⎝
2αtq+1 log(t)(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ2
−tq+1 log(t)(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ2
⎞⎟⎟⎟⎟⎠ dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
lμμ =−n∑
i=1
⎛⎜⎜⎝∫ 1
0
⎛⎜⎜⎝
2αtq+1(
1−αt(yi−μ)σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−tq+1
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ
⎞⎟⎟⎠ dt
⎞⎟⎟⎠ 2
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
+n∑
i=1
∫ 1
0
⎛⎜⎜⎝
tq+2((
1−αt(yi−μ)σ
)2+1
)φ′′
(t(yi−μ)
σ
)
(α2+2)σ2
−4αtq+2
(1−αt(yi−μ)
σ
)φ′(
t(yi−μ)σ
)
(α2+2)σ2 +2α2 tq+2φ
(t(yi−μ)
σ
)
(α2+2)σ2
⎞⎟⎟⎠ dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
123
W. Gui
lμσ = lσμ
=n
−∑i=1
⎛⎜⎜⎝∫ 1
0
⎛⎜⎜⎝
2αtq+1(
1−αt(yi−μ)σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ
−tq+1
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ
⎞⎟⎟⎠ dt
⎞⎟⎟⎠∫ 1
0
⎛⎜⎜⎝
2αtq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ2
−tq+1(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ2
⎞⎟⎟⎠ dt
⎛⎝∫ 1
0
tq((
1− αt(yi −μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
⎞⎠ 2
n∑i=1
∫ 1
0
⎛⎜⎜⎜⎜⎜⎜⎜⎝
tq+2(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′′
(t(yi−μ)
σ
)
(α2+2)σ3 −4αtq+2(yi−μ)
(1−αt(yi−μ)
σ
)φ′(
t(yi−μ)σ
)
(α2+2)σ3
+2α2 tq+2(yi−μ)φ
(t(yi−μ)
σ
)
(α2+2)σ3 +tq+1
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ2
−2αtq+1
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ2
⎞⎟⎟⎟⎟⎟⎟⎟⎠
dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
lσσ = n
σ 2 −n∑
i=1
⎛⎜⎜⎝∫ 1
0
⎛⎜⎜⎝
2αtq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ2
−tq+1(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ2
⎞⎟⎟⎠ dt
⎞⎟⎟⎠ 2
⎛⎝∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi −μ)σ
)
α2+2dt
⎞⎠ 2
+n∑
i=1
∫ 1
0
⎛⎜⎜⎜⎜⎜⎜⎜⎝
tq+2(yi−vμ)2((
1−αt(yi−μ)σ
)2+1
)φ′′
(t(yi−μ)
σ
)
(α2+2)σ4 −4αtq+2(yi−μ)2
(1−αt(yi−μ)
σ
)φ′(
t(yi−μ)σ
)
(α2+2)σ4
+2α2 tq+2(yi−μ)2φ
(t(yi−μ)
σ
)
(α2+2)σ4 +2tq+1(yi−μ)
((1−αt(yi−μ)
σ
)2+1
)φ′(
t(yi−μ)σ
)
(α2+2)σ3
−4αtq+1(yi−μ)
(1−αt(yi−μ)
σ
)φ
(t(yi−μ)
σ
)
(α2+2)σ3
⎞⎟⎟⎟⎟⎟⎟⎟⎠
dt
∫ 1
0
tq((
1− αt(yi−μ)σ
)2+1
)φ(
t(yi−μ)σ
)
α2+2dt
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