In this paper, a new class of skew multimodal distributions with more flexible than alpha skew normal distribution and alpha-beta skew normal distribution is proposed, which makes some important distributions become its special cases. The statistical properties of the new distribution are studied in detail, its moment generating function, skewness coefficient, kurtosis coefficient, Fisher information matrix, maximum likelihood estimators are derived. Moreover, a random simulation study is carried out for test the performance of the estimators, the simulation results show that with the increase of sample size, the mean value of maximum likelihood estimators tends to the true value. The new distribution family provides a better fit compared with other known skew distributions through the analysis of a real data set.
Over the years, the family of skew-symmetric distributions has received considerable attention by many researchers. Azzalini [
f ( z ) = 2 Φ ( α z ) ϕ ( z ) , ( z , α ) T ∈ ℝ 2 , (1)
where ϕ ( ⋅ ) and Φ ( ⋅ ) are probability density function (pdf) and cumulative distribution function (cdf) of standard normal distribution, respectively. Balakrishnan [
Recently, Elal-Olivero [
f ( z ; α ) = ( 1 − α z ) 2 + 1 α 2 + 2 ϕ ( z ) , ( z , α ) T ∈ ℝ 2 , (2)
where α controls the skewness and kurtosis, Equation (2) has at most two modes. Alpha-skew-Laplace distribution (ASL) and a generalization of alpha skew normal distribution (GASN) are studied in [
f ( z ) = z n M ϕ ( z ) , z ∈ R , (3)
where n is positive even integers and M is normalizing constant. Furthermore, Shafiei et al. [
f ( z ; α , β ) = ( 1 − α z − β z 3 ) 2 + 1 6 α β + 15 β 2 + α 2 + 2 ϕ ( z ) , ( z , α , β ) T ∈ ℝ 3 , (4)
where α , β control the skewness and kurtosis, Equation (4) has at most four modes.
The motivations for considering this new family of distributions are as follows: Firstly, the new distribution family contains some classical distributions, such as normal distribution, ASN, ABSN, etc. Secondly, the admissible intervals for the skewness and the kurtosis parameters are wider than ASN and ABSN. Lastly, the new distribution family is more flexible than some other known skew distributions through the analysis of a real data set. Consequently, the reminder of the paper is organized as follows: In Section 2, we propose the new class of skew normal distributions and discussed its properties. In Section 3, we provide the maximum likelihood estimates (MLEs) of the parameters, the performance of the estimates are verified by random simulation. The real application of the new distribution family is considered in Section 4. Section 5 presents some conclusions. Lastly, some proofs and elements of Fisher information (FI) matrix are given in the appendix.
In this section, we define a class of skew distributions that allow the fitting of multimodal data sets.
Definition 1 A random variable X follows alpha-beta-gamma skew normal distribution, denoted by ABGSN ( α , β , γ ) , if it has the pdf
f ( x ; α , β , γ ) = ( 1 − α x − β x 3 − γ x 5 ) 2 + 1 C ϕ ( x ) , ( x , α , β , γ ) T ∈ ℝ 4 , (5)
where C = 945 γ 2 + 210 γ β + 30 α γ + 6 α β + 15 β 2 + α 2 + 2 .
Remark 1 Derivative of pdf (5) with respect to x is given by
∂ f ( x ; α , β , γ ) ∂ x = ϕ ( x ) C [ − γ 2 x 11 + ( 10 γ 2 − 2 β γ ) x 9 + ( 16 β γ − β 2 − 2 α γ ) x 7 + 2 γ x 6 + ( 12 α γ + 6 β 2 − 2 α β ) x 5 + ( 2 β − 10 γ ) x 4 + ( 8 α β − α 2 ) x 3 + ( 2 α − 6 β ) x 2 + ( 2 α 2 − 2 ) x − 2 α ] , (6)
Equation (6) has at most eleven zeros, therefore the pdf up to six modes, see
Proposition 1 The cdf of X ~ ABGSN ( α , β , γ ) T is
F X ( x ; α , β , γ ) = [ − γ 2 x 9 − ( 2 β γ + 9 γ 2 ) x 7 − ( β 2 + 2 α γ + 63 γ 2 + 14 β γ ) x 5 + 2 γ x 4 − ( 5 β 2 + 10 α γ + 315 γ 2 + 2 α β + 70 β γ ) x 3 + ( 2 β + 8 γ ) x 2 − ( C − 2 ) x + 2 α + 4 β + 16 γ ] ϕ ( x ) / C + Φ ( x ) , ( x , α , β , γ ) ∈ ℝ 4 .
Remark 2 Some properties of the ABGSN ( α , β , γ ) are as follows
1) ABGSN ( 0,0,0 ) , ABGSN ( α , β ,0 ) and ABGSN ( α ,0,0 ) degenerate to N ( 0,1 ) , ABSN ( α , β ) and ASN ( α ) , respectively.
2) If α = 0 , pdf (5) reduces to
f ( x ; β , γ ) = ( 1 − β x 3 − γ x 5 ) 2 + 1 945 γ 2 + 210 γ β + 15 β 2 + 2 ϕ ( x ) , ( x , β , γ ) T ∈ ℝ 3 , (7)
in sequel, we name Equation (7) the beta-gamma skew normal distribution, denoted by BGSN ( β , γ ) .
3) If β = 0 , pdf (5) reduces to
f ( x ; α , γ ) = ( 1 − α x − γ x 5 ) 2 + 1 945 γ 2 + 30 α γ + α 2 + 2 ϕ ( x ) , ( x , α , γ ) T ∈ ℝ 3 , (8)
in sequel, we name Equation (8) the alpha-gamma skew normal distribution, denoted by AGSN ( α , γ ) .
4) If α = γ = 0 , pdf (5) reduces to
f ( x ; β ) = ( 1 − β x 3 ) 2 + 1 15 β 2 + 2 ϕ ( x ) , ( x , β ) T ∈ ℝ 2 , (9)
Equation (9) is known as the beta skew normal distribution and denoted by BSN ( β ) , see Shafiei [
5) If α = β = 0 , pdf (5) reduces to
f ( x ; γ ) = ( 1 − γ x 5 ) 2 + 1 945 γ 2 + 2 ϕ ( x ) , ( x , γ ) T ∈ ℝ 2 , (10)
in sequel, we name Equation (10) the gamma skew normal distribution, denoted by GSN ( γ ) .
6) If X ~ ABGSN ( α , β , γ ) , when α → ± ∞ , β → ± ∞ and γ → ± ∞ , then X follows BN(2), BN(6) and BN(10), respectively.
7) If X ~ ABGSN ( α , β , γ ) , then − X ~ ABGSN ( − α , − β , − γ ) .
Proposition 2 Let X ~ ABGSN ( α , β , γ ) , then for k ∈ ℕ , we have
E [ X 2 k ] = 1 C [ 2 h 2 k + α 2 h 2 k + 2 + ( 2 α γ + β 2 ) h 2 k + 6 + γ 2 h 2 k + 10 + 2 α β h 2 k + 4 + 2 β γ h 2 k + 8 ] ,
E [ X 2 k − 1 ] = − 1 C ( 2 α h 2 k + 2 β h 2 k + 2 + 2 γ h 2 k + 4 ) ,
where h 2 k + i = 1 × 3 × ⋯ × ( 2 k + i − 1 ) , i = 0 , 2 , 4 , 6 , ⋯ .
Remark 3 In particularly, we have
E [ X ] = − 1 C ( 2 α + 6 β + 30 γ ) ,
V a r [ X ] = 1 C [ ( 2 + 3 α 2 + 210 α γ + 105 β 2 + 10395 γ 2 + 30 α β + 1890 β γ ) − ( 2 α + 6 β + 30 γ ) 2 / C ] ,
We use R software to optimizing E [ X ] and V a r [ X ] with respect to α , β , γ , we can get the bounds
E [ X ] ∈ ( − 0.71,0.71 ) , V a r [ X ] ∈ ( 0.81,14.07 ) .
Remark 4 Let ξ 1 and ξ 2 stand for its skewness and kurtosis coefficients, then we have
ξ 1 = E [ X 3 ] − 3 E [ X ] E [ X 2 ] + 2 E 3 [ X ] ( E [ X 2 ] − E 2 [ X ] ) 3 / 2 = [ − 16 ( α + 3 β + 15 γ ) 3 + 12 C ( α 3 + 120 β 3 + 61425 γ 3 + 95 α 2 γ + 14 α 2 β + 75 α β 2 + 2835 β 2 γ + 5775 α γ 2 + 23730 β γ 2 + 1200 α β γ − 2 β − 20 γ ) ] / [ ( 2 + 3 α 2 + 210 α γ + 105 β 2 + 10395 γ 2 + 30 α β + 1890 β γ ) C − ( 2 α + 6 β + 30 γ ) 2 ] 3 / 2 ,
ξ 2 = E [ X 4 ] − 4 E [ X ] E [ X 3 ] + 6 E 2 [ X ] E [ X 2 ] − 3 E 4 [ X ] ( E [ X 2 ] − E 2 [ X ] ) 2 − 3 = − 3 + [ − 48 ( α + 3 β + 15 γ ) 4 + C 3 ( 15 α 2 + 945 β 2 + 135135 γ 2 + 210 α β + 1890 α γ + 20790 β γ + 6 ) + 24 ( α + 3 β + 15 γ ) C ( α 3 + 165 β 3 + 89775 γ 3 + 125 α 2 γ + 17 α 2 β + 105 α β 2 + 4095 β 2 γ + 9555 α γ 2 + 35385 β γ 2 + 1830 α β γ − 2 α − 14 β − 110 γ ) ] / [ ( 2 + 3 α 2 + 210 α γ + 105 β 2 + 10395 γ 2 + 30 α β + 1890 β γ ) C − ( 2 α + 6 β + 30 γ ) 2 ] 2 ,
calculated by R software with the package DEoptim we can get the bounds
ξ 1 ∈ ( − 1.29,1.29 ) , ξ 2 ∈ ( − 1.88,5.60 ) .
Note that, the length of the admissible intervals for the skewness and the kurtosis of ABGSN are larger than the corresponding intervals of ASN and ABSN which are (−0.81, 0.81), (−1.30, 0.75) and (−1.19, 1.19), (−1.77, 3.30), respectively.
Proposition 3 Let X ~ ABGSN ( α , β , γ ) , then the MGF of X is
M X ( t ; α , β , γ ) = e t 2 2 [ t 2 S 1 − t S 2 C + 1 ] , ( t , α , β , γ ) T ∈ ℝ 4 ,
where
S 1 = γ 2 ( 4725 + 3150 t 2 + 630 t 4 + 45 t 6 + t 8 ) + β 2 ( 45 + 15 t 2 + t 4 ) + 2 α β ( 6 + t 2 ) + 2 α γ ( 45 + 15 t 2 + t 4 ) + 2 β γ ( 420 + 210 t 2 + 28 t 4 + t 6 ) ,
S 2 = α ( 2 − α t ) + 2 β ( 3 + t 2 ) + 2 γ ( 15 + 10 t 2 + t 4 ) .
Theorem 1 If X ~ ABGSN ( α , β , γ ) , then
1) The random variable Y = X 2 has the following density function
f Y ( y ; α , β , γ ) = f χ 2 ( 1 ) ( y ) 2 + y ( α + β y + γ y 2 ) 2 C , y > 0 ,
where ( α , β , γ ) T ∈ ℝ 3 and f χ 2 ( 1 ) is the pdf of the chi-square distribution with 1 degree of freedom.
2) The density function of Z = | X | is given by
f Z ( z ; α , β , γ ) = f N H ( z ) 2 + ( α z + β z 3 + γ z 5 ) 2 C , ( z , α , β , γ ) T ∈ ℝ 4
where f N H ( ⋅ ) is the density function of standard Half-Normal distribution.
3) Let T = X | X > 0 , then its density function is
f T ( t ; α , β , γ ) = [ 2 + 2 ( 1 − α t − β t 3 − γ t 5 ) 2 ] e − t 2 2 2 π C − 4 ( α + 2 β + 8 γ ) , ( t , α , β , γ ) T ∈ ℝ 4 .
Remark 5 The pdf (5) can be separated into symmetric and asymmetric parts as follows
f ( x ; α , β , γ ) = 2 + ( α x + β x 3 + γ x 5 ) 2 C ϕ ( x ) − 2 ( α x + β x 3 + γ x 5 ) C ϕ ( x ) = f 1 ( x ; α , β , γ ) − f 2 ( x ; α , β , γ ) ,
where f 1 ( x ; α , β , γ ) = [ 2 + ( α x + β x 3 + γ x 5 ) 2 ] / ( 945 γ 2 + 210 γ β + 30 α γ + 6 α β + 15 β 2 + α 2 + 2 ) ϕ ( x ) is symmetric and the second part is asymmetric one.
Definition 2 If a random variable V has density function
f 1 ( v ; α , β , γ ) = 2 + ( α v + β v 3 + γ v 5 ) 2 C ϕ ( v ) , ( x , α , β , γ ) T ∈ ℝ 4 , (11)
then we say V follows symmetric component alpha-beta-gamma skew normal distribution, denoted by SCABGSN ( α , β , γ ) .
Remark 6 Derivative of pdf (11) with respect to v is given by
∂ f 1 ( v ; α , β , γ ) ∂ x = ϕ ( v ) C [ − γ 2 v 11 − ( 2 β γ − 10 γ 2 ) v 9 + ( 16 β γ − β 2 − 2 α γ ) v 7 + ( 12 α γ + 6 β 2 − 2 α β ) v 5 + ( 8 α β − α 2 ) v 3 + ( 2 α 2 − 2 ) v − 2 α ] , (12)
Equation (12) has only eleven zeros, therefore, pdf (11) has at most six modes.
Proposition 4 If X ~ ABGSN ( α , β , γ ) and V ~ SCABGSN ( α , β , γ ) , then we have
M V ( t ; α , β , γ ) = M X ( t ; α , β , γ ) + e t 2 2 t S 2 C , ( t , α , β , γ ) T ∈ ℝ 4 ,
where
S 2 = α ( 2 − α t ) + 2 β ( 3 + t 2 ) + 2 γ ( 15 + 10 t 2 + t 4 ) .
Sampling algorithm
For generate the random number x from ABGSN ( α , β , γ ) , we adopt the acceptance-rejection method with the following steps
1) Generate random number v from SCABGSN ( α , β , γ ) .
2) Generate random number u from the Uniform (0, 1).
3) If u < f ( v ; α , β , γ ) Δ f 1 ( v ; α , β , γ ) accept x and deliver v = x , otherwise go back to step (1) and continue the process.
Where Δ = sup x f ( x ; α , β , γ ) f 1 ( x ; α , β , γ ) = 1 + 2 2 , f ( ⋅ ; α , β , γ ) and f 1 ( ⋅ ; α , β , γ ) are pdf of ABGSN ( α , β , γ ) and SCABGSN ( α , β , γ ) , respectively.
In this section, maximum likelihood estimation and a random simulation are considered.
Definition 3 Let X ~ ABGSN ( α , β , γ ) , then Y = μ + σ X is the location and scale extension of X with the pdf
f ( y ; μ , σ , α , β , γ ) = [ 1 − α y − μ σ − β ( y − μ σ ) 3 − γ ( y − μ σ ) 5 ] 2 + 1 σ ( 945 γ 2 + 210 γ β + 30 α γ + 6 α β + 15 β 2 + α 2 + 2 ) ϕ ( y − μ σ ) , σ > 0 , (13)
where ( y , μ , α , β , γ ) T ∈ ℝ 5 , we denote Y ~ ABGSN ( μ , σ , α , β , γ ) .
Let y 1 , y 2 , ⋯ , y n be a random sample of size n drawn from ABGSN ( μ , σ , α , β , γ ) , then the log-likelihood function (LLF) for ( θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ) = ( μ , σ , α , β , γ ) is given by
l = ln L ( μ , σ , α , β , γ ; y ) = ∑ i = 1 n ln [ ( 1 − α m i − β m i 3 − γ m i 5 ) 2 + 1 ] − n ln σ − n 2 ln 2 π − n ln ( 945 γ 2 + 210 γ β + 30 α γ + 6 α β + 15 β 2 + α 2 + 2 ) − 1 2 ∑ i = 1 n m i 2 , (14)
where m i = ( y i − μ ) / σ for 1 ≤ i ≤ n . Differentiating Equation (14) above partially with respect to the parameters μ , σ , α , β , γ , the following likelihood equations are obtained
∂ l ∂ μ = 1 σ [ ∑ i = 1 n 2 ( 1 − α m i − β m i 3 − γ m i 5 ) ( α + 3 β m i 2 + 5 γ m i 4 ) ( 1 − α m i − β m i 3 − γ m i 5 ) 2 + 1 + ∑ i = 1 n m i ] = 0 ,
∂ l ∂ σ = 1 σ [ ∑ i = 1 n 2 ( 1 − α m i − β m i 3 − γ m i 5 ) ( α m i + 3 β m i 3 + 5 γ m i 5 ) ( 1 − α m i − β m i 3 − γ m i 5 ) 2 + 1 + ∑ i = 1 n m i 2 − n ] = 0 ,
∂ l ∂ α = − ∑ i = 1 n 2 m i ( 1 − α m i − β m i 3 − γ m i 5 ) ( 1 − α m i − β m i 3 − γ m i 5 ) 2 + 1 − n 2 α + 6 β + 30 γ C = 0 ,
∂ l ∂ β = − ∑ i = 1 n 2 m i 3 ( 1 − α m i − β m i 3 − γ m i 5 ) ( 1 − α m i − β m i 3 − γ m i 5 ) 2 + 1 − n 210 γ + 30 β + 6 α C = 0 ,
∂ l ∂ γ = − ∑ i = 1 n 2 m i 5 ( 1 − α m i − β m i 3 − γ m i 5 ) ( 1 − α m i − β m i 3 − γ m i 5 ) 2 + 1 − n 1890 γ + 210 β + 30 α C = 0 ,
the solutions of the above system of likelihood equations gives the MLEs for ( θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ) . For interval estimation and hypothesis testing, it is necessary to observe the Fisher information matrix, under the regular condition, the corresponding Fisher information matrix can be expressed as I = ( I i j ) 5 × 5 ,
I i j = − E [ ∂ l 2 ∂ θ i ∂ θ j ] for i , j = 1 , ⋯ , 5 , the detailed expression of FI is shown in the appendix.
In this subsection, a random simulation is conducted. The sample sizes and true values of the parameters considered is n = 50 , 80 , 100 , 200 , α = − 2 , 2 , β = − 1 , 1 and γ = − 0.5 , 0.5 , while the location and the scale parameters is set to μ = 0 and σ = 1 , the number of replications is 1000. The specific steps are as follows
1) Generate 1000 samples (simulation with N= 1000 repetitions) of size n from ABGSN ( μ , σ , α , β , γ ) .
2) Compute the maximum likelihood estimates for each of the 1000 samples.
3) Compute the mean and mean squared error (MSE) of ( μ ^ , σ ^ , α ^ , β ^ , γ ^ ) over the 1000 estimates.
From
In this section, we shall examine the application of the alpha-beta-gamma skew distribution to a real data set, and the R’s package DEoptim is used for maximizing the corresponding LLF, the MLEs of the parameters, log-likelihood values, AIC and BIC of distributions are obtained, then we compared the new distribution family with normal distribution, SN, PN, ASN, BSN, ABSN.
We consider the velocity of 82 Galaxy samples in the universe, which is first described by Roeder [
Based on the sample data, Kolmogrov-Smirnov test is carried out for the seven distributions. The test statistics is D = max | F ^ n ( x ) − F ( x ) | , where F ^ n ( x ) and F ( x ) are the empirical distribution function of sample data and the given distribution function, respectively. The test results are shown in
| Parameter | Estimates of α | Estimates of β | Estimates of γ | Estimates of μ | Estimates of σ | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| α | β | γ | n | α ^ | MSE | β ^ | MSE | γ ^ | MSE | μ ^ | MSE | σ ^ | MSE |
| −2 | −1 | −0.5 | 50 | −1.6250 | 0.8307 | −0.9264 | 0.8110 | −0.6056 | 0.1145 | 0.0062 | 0.0132 | 0.9956 | 0.0034 |
| 80 | −1.7473 | 0.7997 | −0.9687 | 0.7611 | −0.5882 | 0.0874 | −0.0046 | 0.0081 | 0.9968 | 0.0019 | |||
| 100 | −1.8362 | 0.7650 | −0.9812 | 0.7242 | −0.5773 | 0.0775 | −0.0042 | 0.0062 | 0.9989 | 0.0016 | |||
| 200 | −1.9694 | 0.7230 | −1.0291 | 0.6423 | −0.5521 | 0.0570 | −0.0000 | 0.0030 | 0.9990 | 0.0010 | |||
| 0.5 | 50 | −1.7159 | 0.7750 | −1.2875 | 0.6464 | 0.5825 | 0.0594 | 0.0030 | 0.0076 | 1.0052 | 0.0033 | ||
| 80 | −1.8546 | 0.7651 | −1.1923 | 0.5876 | 0.5649 | 0.0454 | 0.0014 | 0.0052 | 1.0017 | 0.0008 | |||
| 100 | −1.9151 | 0.7500 | −1.1902 | 0.5426 | 0.5609 | 0.0396 | −0.0012 | 0.0039 | 1.0013 | 0.0007 | |||
| 200 | −1.9814 | 0.6988 | −1.1304 | 0.3921 | 0.5479 | 0.0298 | −0.0005 | 0.0018 | 1.0011 | 0.0004 | |||
| 1 | −0.5 | 50 | −1.8446 | 0.7561 | 1.1920 | 0.6069 | −0.6383 | 0.0916 | −0.0040 | 0.0113 | 0.9939 | 0.0029 | |
| 80 | −1.8938 | 0.7294 | 1.0657 | 0.4997 | −0.6254 | 0.0774 | −0.0038 | 0.0071 | 0.9976 | 0.0016 | |||
| 100 | −1.9110 | 0.7055 | 1.0973 | 0.4694 | −0.6145 | 0.0698 | 0.0029 | 0.0054 | 0.9977 | 0.0013 | |||
| 200 | −2.0465 | 0.6953 | 1.0786 | 0.3444 | −0.5799 | 0.0459 | 0.0020 | 0.0028 | 0.9984 | 0.0006 | |||
| 0.5 | 50 | −2.1792 | 0.7217 | 0.8722 | 0.7465 | 0.7018 | 0.1497 | 0.0059 | 0.0090 | 0.9984 | 0.0026 | ||
| 80 | −2.1069 | 0.7122 | 0.8727 | 0.6510 | 0.6926 | 0.1383 | −0.0038 | 0.0057 | 0.9969 | 0.0015 | |||
| 100 | −2.1039 | 0.7109 | 0.9775 | 0.6506 | 0.6845 | 0.1304 | 0.0026 | 0.0055 | 1.0008 | 0.0014 | |||
| 200 | −2.0593 | 0.7106 | 0.9931 | 0.5216 | 0.6319 | 0.0931 | 0.0015 | 0.0023 | 0.9995 | 0.0008 | |||
| 2 | −1 | −0.5 | 50 | 2.2214 | 0.7554 | −0.7836 | 0.7496 | −0.7224 | 0.1504 | −0.0039 | 0.0107 | 0.9947 | 0.0023 |
| 80 | 2.1171 | 0.7323 | −0.9270 | 0.6784 | −0.6795 | 0.1365 | 0.0022 | 0.0060 | 1.0005 | 0.0016 | |||
| 100 | 2.1049 | 0.6859 | −0.9272 | 0.6550 | −0.6780 | 0.1311 | −0.0017 | 0.0046 | 0.9985 | 0.0014 | |||
| 200 | 2.0679 | 0.6594 | −0.9904 | 0.5187 | −0.6337 | 0.0899 | −0.0016 | 0.0023 | 0.9996 | 0.0008 | |||
| 0.5 | 50 | 1.8328 | 0.7758 | −1.1703 | 0.5936 | 0.6308 | 0.0945 | 0.0029 | 0.0116 | 0.9983 | 0.0030 | ||
| 80 | 1.9013 | 0.7269 | −1.1263 | 0.4990 | 0.6225 | 0.0780 | 0.0088 | 0.0073 | 0.9955 | 0.0019 | |||
| 100 | 1.9917 | 0.7114 | −1.0831 | 0.4608 | 0.6029 | 0.0634 | 0.0051 | 0.0053 | 0.9993 | 0.0011 | |||
| 200 | 1.9975 | 0.6890 | −1.0244 | 0.3412 | 0.5694 | 0.0407 | 0.0007 | 0.0027 | 0.9995 | 0.0006 | |||
| 1 | −0.5 | 50 | 1.7721 | 0.7584 | 1.2712 | 0.6471 | −0.6005 | 0.0635 | 0.0025 | 0.0077 | 1.0014 | 0.0012 | |
| 80 | 1.8733 | 0.7420 | 1.2314 | 0.5872 | −0.5761 | 0.0472 | 0.0015 | 0.0047 | 1.0012 | 0.0008 | |||
| 100 | 1.9147 | 0.7345 | 1.2139 | 0.5512 | −0.5706 | 0.0449 | 0.0014 | 0.0040 | 1.0005 | 0.0007 | |||
| 200 | 2.0094 | 0.7229 | 1.1614 | 0.4176 | −0.5580 | 0.0321 | 0.0009 | 0.0018 | 1.0002 | 0.0004 | |||
| 0.5 | 50 | 1.6833 | 0.8291 | 0.9294 | 0.7915 | 0.6099 | 0.1152 | −0.0091 | 0.0132 | 0.9965 | 0.0034 | ||
| 80 | 1.7691 | 0.8090 | 0.9798 | 0.7392 | 0.5947 | 0.0925 | −0.0057 | 0.0078 | 0.9967 | 0.0019 | |||
| 100 | 1.8450 | 0.7902 | 1.0418 | 0.7351 | 0.5688 | 0.0807 | −0.0053 | 0.0063 | 0.9987 | 0.0020 | |||
| 200 | 1.9351 | 0.7581 | 1.0267 | 0.6239 | 0.5539 | 0.0586 | −0.0035 | 0.0035 | 0.9984 | 0.0009 | |||
Where MSE ( θ ^ i ) = E [ θ ^ i − θ i ] 2 , i = 1, ⋯ ,5 .
| Distribution | μ ^ | σ ^ | α ^ | β ^ | γ ^ | l ^ | AIC | BIC |
|---|---|---|---|---|---|---|---|---|
| noraml | 20.83 | 4.54 | --- | --- | --- | −240.42 | 484.83 | 489.65 |
| PN | 22.34 | 4.01 | 0.67 | --- | --- | −240.28 | 486.56 | 493.78 |
| SN | 24.61 | 5.91 | -1.39 | --- | --- | −239.21 | 484.42 | 491.64 |
| BSN | 20.60 | 3.26 | --- | −0.16 | --- | −232.22 | 470.44 | 477.66 |
| ASN | 17.42 | 3.87 | −1.66 | --- | --- | −230.09 | 466.18 | 473.40 |
| ABSN | 16.76 | 4.66 | −3.14 | 0.49 | --- | −221.97 | 451.93 | 461.56 |
| ABGSN | 21.14 | 2.80 | 0.38 | −0.38 | 0.05 | −215.81 | 441.62 | 453.65 |
| N | PN | SN | BSN | ASN | ABSN | ABGSN | |
|---|---|---|---|---|---|---|---|
| Statistics | 0.1911 | 0.1886 | 0.1786 | 0.1752 | 0.1652 | 0.1123 | 0.1089 |
| P-value | 0.0043 | 0.0050 | 0.0093 | 0.0114 | 0.0200 | 0.2339 | 0.2655 |
From
This paper introduced a new class of skew normal distributions which has at most six modes and some of its properties are discussed. The admissible intervals of the skewness and kurtosis coefficients for the alpha-beta-gamma skew normal distribution are wider than skew normal distribution, alpha skew normal distribution and alpha-beta skew normal distribution. Through random simulation, the MSE of the MLEs for each parameter decreases with the increase of n. The numerical results of fitting a real-life data set considered here has shown that the ABGSN provides a better fitting in comparison to the other known distributions. Based on the distribution studied in this paper, we can further extend it. For example, combining ABGSN with the classical skew normal distribution, we can get the generalized alpha-beta-gamma skew normal distribution, which makes ABGSN as a special case of it.
Project is supported by Natural Science Foundation of Chongqing (Grant No. cstc2020jcyj-msxmX0232 and cstc2019jcyj-msxmX0386).
The authors declare no conflicts of interest regarding the publication of this paper.
Wei, Z.Y., Peng, T.K. and Zhou, X.Y. (2020) The Alpha- Beta-Gamma Skew Normal Distribution and Its Application. Open Journal of Statistics, 10, 1057-1071. https://doi.org/10.4236/ojs.2020.106060
F ( x ) = ∫ − ∞ x f ( t ) d t = 1 C ∫ − ∞ x [ γ 2 t 10 + 2 β γ t 8 + ( 2 α γ + β 2 ) t 6 − 2 γ t 5 + 2 α β t 4 − 2 β t 3 + α 2 t 2 − 2 α t + 2 ] ϕ ( t ) = 1 C [ γ 2 A 10 + 2 β γ A 8 + ( 2 α γ + β 2 ) A 6 − 2 γ A 5 + 2 α β A 4 − 2 β A 3 + α 2 A 2 − 2 α A 1 + 2 Φ ( x ) ] ,
where
A i = − x i − 1 ϕ ( x ) + ( i − 1 ) ∫ − ∞ x t i − 2 ϕ ( t ) d t = − x i − 1 ϕ ( x ) + ( i − 1 ) A i − 2 , i = 1 , 2 , 3 , ⋯ .
A2. Proof of Proposition 2When Y ~ N ( 0,1 ) , we have
E [ Y 2 k ] = ∏ j = 1 k ( 2 j − 1 ) , E [ Y 2 k − 1 ] = 0 , k = 1 , 2 , 3 , ⋯ .
then
E [ X 2 k ] = 1 C ( 2 E [ Y 2 K ] − 2 α E [ Y 2 K + 1 ] + α 2 E [ Y 2 K + 2 ] + β 2 E [ Y 2 K + 6 ] + γ 2 E [ Y 2 K + 10 ] − 2 β E [ Y 2 K + 3 ] + 2 α β E [ Y 2 K + 4 ] − 2 γ E [ Y 2 K + 5 ] + 2 α γ E [ Y 2 K + 6 ] + 2 β γ E [ Y 2 K + 8 ] ) = 1 C ( 2 E [ Y 2 K ] + α 2 E [ Y 2 K + 2 ] + β 2 E [ Y 2 K + 6 ] + γ 2 E [ Y 2 K + 10 ] + 2 α β E [ Y 2 K + 4 ] + 2 α γ E [ Y 2 K + 6 ] + 2 β γ E [ Y 2 K + 8 ] ) ,
E [ X 2 k − 1 ] = 1 C ( 2 E [ Y 2 K − 1 ] − 2 α E [ Y 2 K ] + α 2 E [ Y 2 K + 1 ] + β 2 E [ Y 2 K + 5 ] + γ 2 E [ Y 2 K + 9 ] − 2 β E [ Y 2 K + 2 ] + 2 α β E [ Y 2 K + 3 ] − 2 γ E [ Y 2 K + 4 ] + 2 α γ E [ Y 2 K + 5 ] + 2 β γ E [ Y 2 K + 7 ] ) = 1 C ( − 2 α E [ Y 2 K ] − 2 β E [ Y 2 K + 2 ] − 2 γ E [ Y 2 K + 4 ] ) .
A3. Proof of Proposition 3M X ( t ; α , β , γ ) = E [ e t x ] = 1 C ∫ − ∞ ∞ e t x ( 2 − 2 α x − 2 β x 3 − 2 γ x 5 + α 2 x 2 + ( 2 α γ + β 2 ) x 6 + γ 2 x 10 + 2 α β x 4 + 2 β γ x 8 ) ϕ ( x ) d x = 1 C ( 2 D 0 − 2 α D 1 − 2 β D 3 − 2 γ D 5 + α 2 D 2 + β 2 D 6 + γ 2 D 10 + 2 α β D 4 + 2 α γ D 6 + 2 β γ D 8 ) ,
where
D 0 = ∫ − ∞ ∞ e t x ϕ ( x ) d x = e t 2 2 ,
D i = ∫ − ∞ ∞ e t x x i ϕ ( x ) d x = ( i − 1 ) D i − 2 + t D i − 1 , i = 1 , 2 , 3 , ⋯ .
A4. Proof of Theorem 11) The density function of Y is obtained by
F Y ( y ; α , β , γ ) = ℙ ( Y ≤ y ) = F X ( y ; α , β , γ ) − F X ( − y ; α , β , γ ) ,
then
f Y ( y ; α , β , γ ) = 1 2 y [ f X ( y ; α , β , γ ) + f X ( − y ; α , β , γ ) ] = ϕ ( y ) y 2 + y ( α + β y + γ y 2 ) 2 C .
2) For z > 0 , we have
f Z ( z ; α , β , γ ) = f X ( z ) + f X ( − z ) = 2 + ( 1 + α z + β z 3 + γ z 5 ) 2 + ( 1 − α z − β z 3 − γ z 5 ) 2 C = 2 ϕ ( z ) 2 + ( α z + β z 3 + γ z 5 ) 2 C .
3) The density function of T is derived by
f T ( t ; α , β , γ ) = f X ( t ) 1 − F X ( 0 ) = 2 2 π C f X ( t ) 2 π C − 4 ( α + 2 β + 8 γ ) = [ 2 + 2 ( 1 − α t − β t 3 − γ t 5 ) 2 ] e − t 2 2 2 π C − 4 ( α + 2 β + 8 γ ) .
A5. FI Matrix Elements of A B G S N ( μ , σ , α , β , γ )I 11 = n 2 − α 2 − 75 β 2 − 8505 γ 2 − 18 α β − 150 α γ − 780 β γ + 4 G 0 σ 2 C .
I 12 = I 21 = n − 2 α + 6 β + 90 γ + 4 G 1 σ 2 C .
I 13 = I 31 = n − 2 − 4 E 1 σ C .
I 14 = I 41 = n − 6 − 4 E 3 σ C .
I 15 = I 51 = n − 30 − 4 E 5 σ C .
I 22 = n 2 α 2 − 330 β 2 − 73710 γ 2 − 36 α β − 660 α γ − 9660 β γ + 4 + 4 G 2 σ 2 C .
I 23 = I 32 = n 4 α + 24 β + 180 γ − 4 E 2 σ C .
I 24 = I 42 = n 24 α + 180 β + 1680 γ − 4 E 4 σ C .
I 25 = I 52 = n 180 α + 1680 β + 18900 γ − 4 E 6 σ C .
I 33 = n 4 C F 2 − ( 2 α + 6 β + 30 γ ) 2 C 2 .
I 34 = I 43 = n 4 C F 4 − ( 2 α + 6 β + 30 γ ) ( 6 α + 30 β + 210 γ ) C 2 .
I 35 = I 53 = n 4 C F 6 − ( 30 α + 210 β + 1890 γ ) 2 C 2 .
I 44 = n 4 C F 6 − ( 6 α + 30 β + 210 γ ) 2 C 2 .
I 45 = I 54 = n 4 C F 8 − ( 6 α + 30 β + 210 γ ) ( 30 α + 210 β + 1890 γ ) C 2 .
I 55 = n 4 C F 10 − ( 30 α + 210 β + 1890 γ ) 2 C 2 .
where
E i = E [ X i ( α + 3 β X 2 + 5 γ X 4 ) ( 1 − α X − β X 3 − γ X 5 ) 2 ( 1 − α X − β X 3 − γ X 5 ) 2 + 1 ] , i = 1 , 2 , 3 , 4 , 5 , 6 ,
F i = E [ X i ( 1 − α X − β X 3 − γ X 5 ) 2 ( 1 − α X − β X 3 − γ X 5 ) 2 + 1 ] , i = 2 , 4 , 6 , 8 , 10 ,
G i = E [ X i ( α + 3 β X 2 + 5 γ X 4 ) 2 ( 1 − α X − β X 3 − γ X 5 ) 2 ( 1 − α X − β X 3 − γ X 5 ) 2 + 1 ] , i = 0 , 1 , 2 ,
X ~ N ( 0,1 ) .