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STATISTICAL PROPERTIES AND APPLICATIONS OF TRANSMUTED SKEW STUDENT t DISTRIBUTION
DAVID Ikwuoche John, MATHEW Stephen
Department of Mathematics and Statistics, Federal University Wukari, Nigeria davidij@fuwukari.edu.ng, matsteve231@gmail.com https://orcid.org/0000-0002-7100-5357
Abstract
In this study, a modified 2-parameter skew t distribution called the transmuted skew student t distribution (TSStD) was presented. Some statistical and reliability properties of TSStD such as the quantile function, the raw moments, and the moment generating function (among others), were derived. Through the method of maximum likelihood, the two parameters of the model were estimated. The stability of the model was studied via Montecarlo simulations utilizing bias, mean square error, and root mean square error as metrics. The results from the stability study revealed that the TSStD was well-behaved. Four datasets were modeled with the transmuted skewed student t distribution and four other probability density models. On the basis of information criteria, the results revealed that the transmuted skew student t distribution provides a better fit for all the datasets compared to the other competing models.
Keywords: Transmuted, Skew, Raw moments, Quantile, Reliability function, Hazard function
I. Introduction
Empirical probability models, or probability distributions, are an essential aspect of parametric statistical investigation. Classical distributions are more susceptible to an anomaly when characterizing various data and data generating processes, according to a prevalent reality across a variety of sectors, including finance, environmental science, biological sciences, engineering, and others. This emphasizes the need to hybridize classical probability models in order to meet this complex task [1]. Owing to the applicability of distribution theory and the availability of diverse data in today's world, the thirst for improved statistical distributions that might be used to describe and model these events have increased spontaneously [2], [3], [4], [5], and [6]. Many scholars have over the years help modified simple statistical methodologies in relation to distribution theory and these methods have been found immensely useful in statistical modeling. Several methods of modifying probability distributions have been proposed over the years by scholars to improve statistical methodology of distribution theory. Among the methods are Exponentiated Exponential Distribution [7], the Sine-G family [8], the New Sine-G Family [9], the G-families using the transformed-transformer [10], Transmuted G family by [11], and amongst others. Several distributions have been modified using the Transmuted G family of distributions over the years and some of them include Transmuted Lomax and Transmuted Exponentiated Lomax distributions [12], Transmuted Frechet distribution [13], Transmuted Exponentiated Gamma distribution [14], Transmuted additive Weibull distribution [15], Transmuted generalized Gomperte distribution [16], Kumaraswamy Transmuted Exponentiated modified Weibull
distribution [17], Transmuted Exponential power distribution [18], etc. Using the method of transmutation by [19], this paper proposes a new probability distribution called the Transmuted Skew Student t Distribution (TSStD).
II. Methods
I. The Transmuted Family of Distribution
This section presents the TSStD, some of its statistical properties, simulation study as well as application to real life data. The cumulative density function (CDF) and probability density function (PDF) of the Transmuted family of distribution generator is given by;
F (v) = (l + a) G (v)-aG (v)2 (1)
f (v) = g (v){(l + a) -2aG (v)} (2)
where, a is the transmuted parameter (shape), F(v) is the CDF and f(v) is the PDF of the baseline distribution.
II. The Skew Student t Distribution
Using the simplified version of the Skew Student t Distribution (SStD) with 2 degrees of freedom introduced by Jonson et al., [20] whose CDF and PDF are expressed as;
( M
G ( * ) = 2
g( * ) = "
i
A + *2
A
2 (A + * )3 where, A is the shape parameter.
;— œ < *<œ
(3)
III. Transmuted Skew Student t Distribution (TSStD)
On substituting G(v) and g(v) in equation (3) into (1) and (2) the PDF and CDF of TSSiD are obtained as;
f ( *; A ,«) = ■
A
2 (A + * )
3
.^2
(l + a) -2a
f4A+*+*^
2yfÄ
+ *
(4)
and
F ( v;A,a ) = (l+a )
2 + 2VA+V7
-a
1 —+
2 i4A
+v
(5)
2
1 v
v
PDF ofTSSID
CDF ofTSStD
F&rareter Values ; ;
- 3=0.01,1=0.10 !
--3=0.02,1=0.30
--3=0.03,1=0.50 ji \\
---3=0.0+,1=0.70 i: j
- 3=0.05,1=0.90 ! \ O
J; 'x.
- *■'-"' ''--Ki&îss.-.-«™,---
fàrarrefer Values
---- a=0.01,l=0.10
a=0.0£1=130
---a=0.03,l=0.50 i //
----- a=Q.04,1=0.70
------ a=0.05,1=0.93
!
Figure 1: The PDF Plot of TSStD
Figure 2: The CDF Plot of TSStD
IV. Properties of Transmuted Skew Student t Distribution
The survival [S(x)], hazard [h(x)], odd [O(x)], and quantile [Q(v)] functions, skewness, and kurtosis statistics are presented as well as the S(x) plot as follows
i N
(6)
U 2*1 A + v^ V 2 2*1 A + v* I
A
S ( " ) =1 -F *v ) =1-1(1+a)( 2+^A+7 ]2+^A+7 )
h ( x ) =
3
2\2
f ( x )_ 2 (A + v2 ) S ( x )"
(1 + a) -2a
.+ v2 + v
Y
2VA
+ v
(1+«)( 1+
O( x) =
F ( x) =_
S ( x) ( 1
(1+a )|1
—+-
v ] 1v 2VA + v2 J ""2 2*1 A + v2
1
2A
2 2*1A+v2 J L 2 ^VX+v
-a I -+-
(1 + «)[ 1 +
v )( v
2*1 A + v2 J t 2 2*1 A+ v2
2
(7)
(8)
S(v) of TSStD
- Parameter Values
\ \\ ---- a=0.01, 1=0.10 a =0.02, 1=0.30 ---a =0.03, 1=0.50 ----- a =0.04, 1=0.70 a =0.05, 1=0.90
•t;--.,______
-3
-2
Figure 3: The S(v) Plot of TSStD for varied parameter value
Quantile Function: the inverse of the CDF of TSStD gives the quantile function of TSStD and after numerous algebraic simplifications, this is expressed as;
v
v
Q ( * ) = ±
( a2A2 - 4aA V + 2aA2 + A2 )2
aAu
aA
Au*2 Au*
A
i(u 2 -u )
4(u*2-u ) 8(u*2-u ) u*2-u* u*2-u 8(u*2 -u )
(9)
Skewness: the Galton measure of skewness [GSK] which measures the presence and lack of symmetry of a probability distribution is presented for TSStD using Q(v) as follows;
GSK =
Q ( 1 )+Q ( 3 ) - Q ( 4 ) - Q ( 1
Q13 )- Q14
(10)
Kurtosis: the kurtosis measure whether or not the probability distribution is heavy-tailed. For TSStD the Moor's kurtosis measure is derived using Q(v) as follows;
MKT = -
Q ( 1 ) +Q (3 )- Q (5 )- Q ( 4
Q 1:4 )- Q [1
(11)
IV. Moments, Moment Generating Function, and Characteristic Function
Moments: A random variable V with TSStD has its moment in the form;
2
u = r 2
Gß
r - r + 2 r 2 '~2
■Hß
r + 2 r
. 2 2 _ Proof
The rth moment of a random variable v with a valid probability distribution is given by;
ur = E [ vr ] = | vrf ( v ) dv
Let v be a random variable following the TSStD. Then the moment is derived as follows;
=[* ] = J *
2 (A + *2 )T
(a+*2 )1
d*
(12)
(13)
(14)
•a *r
u = ,
r J 2
(a + *2 )3 (a + * )
d* = -2A-2
(a+*2 )2 (a+*2 )2
d*
u = A
(a + *2 )2 (a + *2 )
By transformation, let
d*
(15)
1 du 1
u = *• ; * = ur ; — = r*r-1; d* = —— du . Substituting the transformations into (15) to obtain;
d*
r*
1
u = A
r 1 lA 2
A + u
aur
A + u
-du
(16)
rw
■J(-1)
A
u = —
r r
Lu' 3du -J
A+ ur
A+ ur
du
r-r+1 r+1-r+1
i" u r , " r
;—^du -J
A + ur
A
u = — r r
A2
ur . a r ur
—Jdu - a Jr-Ti
-du
1 + — A
V y
1+— A
V y
A+ u'
-du
(17)
r-1
Also, letting k = —, ^ u = (Ak)2, ^ — = — ur , ^ du = -A—dk then by substituting for u and du in
A V / An Ar
dk 2
Ar
A
du Ar
(17), we obtain the following;
A
u, = —
r r
A
u = — r 2
A
(Ak)2
Ar
-dk
A 2
(1 + k)2 2(Ak)r2rJ A'
r2-r r r
0 I • 2 " ,2
r k 2 ,, aA r k
dk - ~J (i+k)2
(Ak)2
2u r
Ar
(1 + k)2 2(Ak)2
dk
A2 o (1 + k)
dk
GJ—T dk -H J^
(1+ k)2 0 (1 + k)
-dk
(18)
where, g = A 2 ; h = aA . Therefore, the Beta function is given as;
aA
A2
A
P[ P, ?] = J
(1+1 r
-dt
(19)
Therefore, on transforming (18) into the form presented in (19), the following is obtained;
u = — r 2
2 2 12 2 2
-dk - H
(1+k
22 2 2
dk
r - r + 2 r r + 2 r
GP -,— - HP -
2 2 _ _ 2 2 _
(20)
(1 + k)2
which completes the proof.
The mean of TSStD, that is, E[v] based on equation (20) by substituting r to be 1 is given as;
A
E[v] = u = — 1 2
GP
12-1 + 2 £ 2 ' 2
- HP
1 + 2 1
22
GP
1
1,2
HP
3 1 2' 2
(21)
Moment Generating Function (MGF): The MGF of the random variable v with TSStD is given by:
" £m
M (t) = S m. A
GP
m2 - m + 2 m2
2 'T
hp
m + 2 m
(22)
Proof
2-r
P-1
t
A
m=0
The MGF is given by:
Mv (t) = E[e* ] = J etvf (v)d
(23)
tv
By McLaurin's series expansion, e is expressed as e" =
(tv )"
then equation (23) becomes
Mv (t) = E[ete ] = E
AtvT
m!
» ±m
= X Ï-E\vm J
X m! L J
As obtained in equation (20), E [ vm ] = wm, therefore,
» ^ tm y
Mv (t) = X • y
Gß
m2 - m + 2 m2
2 'T
-Hß
m + 2 m
2 'Y
r2 - r + 2 r2 - Hß r + 2 r
Gß _ 2 '2J
L 2 2 J
Characteristic Function [CF]: for a random variable p with TSStD, the CF is given as;
x C )=lff
c=0 c • 2
Proof
The CF of a valid probability distribution is expressed as;
fv (t) = E [e,tv ] = } e,tvf (vd
^ (itv)
By Taylor's series expansion of ext° = ^—j—, therefore,
r + 2 r
fv ( t y
A L ! 2
Gß
c !
rz-r+2 r
i=0
2 „ , <"> „,2
Hß
(24)
(25)
(26)
(27)
(28)
V. Order Statistics for TSSiD
Sample values such as the smallest, largest, or middle observation from a random sample provide important information. Order Statistics could be used to determine the distribution of the smallest
(minimum) order statistic and the largest (maximum) order statistic of a given distribution. Let V1, V2, <, Vn denote n-independent random sample from a distribution function F(v) and probability
density function,f(v), then v1, v2,..., Vn represent the order sample arrangement and the pdf of vw
is given by:
L (v) = (,•-!)?(«-0!f (v)^(v)'4 ^(v)]n" ; for ' = 1 2' * * n . (29)
For simplicity, [1 - F(v)]n-' in (29) can be expressed using the sum of a binomial series as
SI n ' I ( "I)"" [F ( v )] • By substitution the following is obtained:
(v)=. n . f(v)XX|M-I1(-l)mf(v)
y ' (i-l)!(n-i)!n 'X[ m / > W
Now, making the substitution of fv) and F(v) into (30) will yield the following;
m !
n=0
(=0
fn (v) =
(i -1)!(n -i)!
: nm i(-o
(1 + a)
A
2 (A + v2 )
f4A
1-a
VA
2VA
+ v + v
+ v + v
VA
+ v + v
2VA
+v
(31)
Applying binomial expansion to the term
(1 + a)
VAh
2>/ A + v2 I I 2>/a
a/Ah
to get
m + i-1
5l q
(1
( v ) = T7
+ a
m
VA+7+v
2VA
+ v
Va + v2 + v
2VA
+ v
and when substituted into (31), the ^ (v) becomes;
(i-1) !( n - i)!
Eln;i !(-»y E
A
1-a
m=0 V m )
2 (A + v2 )
(1 + a)
VA
+v2 +v
V 2*1 A + v2 )
VA
+ v2 + v
2VA
+ v
-a
Va + v2 + v v 2*!A + v2 )
H-1-,
(32)
Expanding the term
(1 + a)
■sfA'-
2^7
binomially yields jjj
vVA
+ v
substitution into (32), f.n (v) becomes,
L ( v ) = TT
n!
( i-1) !( n - i )!
A
2 (A + v2 ) 1
m+i-1
q=0
<sin:1 (-1)m e Im +;-1ii i q (a—
1-a
A
VA
+v2 +v
2 V A + v2
rVA
+ v + v
2VA
+ v
+i-1-q+q
and on
(33)
f. ( v):
n!
(i-1)!(n - i )!
A
2 (A+v2 )
1-a
VA
—2 A +v + v
22
+v
-a)
VA
~ A
+v + v
2VA
+v
where, Rn = E n?;')(-1)mE m + '-1|i
m=0 V m ) q=0 V q ) r=0
m + i-1 q i ^
) '=o vr)
(34)
VI. Parameter Estimation Using Maximum Likelihood Estimation Technique
Let l (0) be a parameter vector for the transmuted family of distributions. Consider a random variable v ~ TSStD (A, a), then by definition, the likelihood function of v with PDF f(v) is given as;
V(A + -2)
l M =l ( f ( vA)) = n
A
-av
V(A + v2 )
2 (a + v2 )
(35)
n
m=0
-1-q
q~r
r
r=0
m=0
i=1
= A" 2'"
î
¡=1
The log of l (0) is as follows;
aXv 1--^
"
I (a + v2
(36)
log ( l(9)) = log
A"
I V(A + v2 ) "«" I
_i__i
I •KA+vr)
V '=i
-log 2"i(a + v2)2
(37)
Differentiating equation (37) with respect to the parameters A and a will yield the estimates of the parameters. The differentiation w.r.t. A is obtained as
d [log (l(d))] 1 + "«"Çv[Ç(A + v )
dA
-a" I
i=1 1_¡=1
3 X 2"'1 î (A + V )2] î (A + V Y = 0
The result obtained shows that the parameter A does not exist in a closed form. A numerical estimate for the parameter will be obtained using R-software. Differentiation w.r.t. a gives
fog №))
da
na ? v
Î^(A + v2) -Q"îv
i=1_i=1
= 0
X (A + v2 )
>2
(38)
VII. Measures of Goodness-of-Fit Adopted
This section presents the measures used in model selection. They include the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Corrected AIC (CAIC), and Hannan-Quinn Information Criterion (HQIC).
AIC = 2k -2 (ll) (39)
BIC = k ( logio (n)) - ( 2ll ) (40)
AIC ( n - k-1) + 2k ( k +1)
CAIC = -
(n - k-1) HQIC = 2k log10 (ln (n) )-2 (ll) where, IL is the log-likelihood value, n is the in-sample size, and k is the parameter.
(41)
(42)
III. Results
¡=1
v
¡=1
I. Stability Study via Monte Carlo Simulations
In this section, we examined the stability of the probability model with an increase in sample size using simulation. The parameters were fixed at (0.6, 0.5) and (0.3, 0.7). Utilizing the quantile function presented in Equation (8), a random sample of sizes—30, 75, 300, 500, and 1000—was generated. The measures used for assessing the models were the Average Absolute Bias (AAB),
Mean Square Error (MSE), and Root Mean Square Error (RMSE), respectively. These measures were calculated using the expressions given below:
AAB =
MSE = £
9=1
(43)
(44)
RMSE =V MSE (45)
In accordance with the central limit theorem, an increase in sample sizes is expected to result in the reduction of estimation errors, approaching zero. Analyzing the results of the simulation as presented in Table 1, it becomes evident that this holds true for the new model. An increase in sample sizes corresponds to a decrease in bias and mean square error, as demonstrated.
Table 1: The AAB, MSE and RMSE
Size (n) Parameter Parameter Value MLE AAB MSE RMSE
30 A 0.21 -2.929E-06 0.21000293 0.04410123 0.21000293
30 a 0.72 -0.3688692 1.08886921 1.18563616 1.08886921
100 A 0.21 0.00062679 0.20937321 0.04383714 0.20937321
100 a 0.72 -0.378557 1.09855699 1.20682746 1.09855699
300 A 0.21 0.08369549 0.12630451 0.01595283 0.12630451
300 a 0.72 0.74825993 0.02825993 0.00079862 0.02825993
500 A 0.21 0.2257876 0.0157876 0.00024925 0.0157876
500 a 0.72 0.57707861 0.14292139 0.02042652 0.14292139
500 400 300 200 100 o
Figure 4: The AAB and MSE
As illustrated in the plot of the AAB and MSE presented in Figure 4, it can be observed that the probability model is well-behaved. This was due to the decay in the value of the AAB and MSE. The larger the sizes of the sample, the better the estimates are, the smaller the error and the more consistent the parameters are.
II. Application to Real-Life Data
This section provides an application of the model to real data. Other competing models were fitted to the data, and the goodness of fit of these models was assessed using various statistical information criteria. This illustration involves four data sets. The first and second datasets were sourced from [21]. The first dataset pertains to the duration of symptom decrease or disappearance
in patients with bladder cancer, measured in months for one hundred and twenty-eight patients. The second dataset focuses on the response time of patients to treatments, measured in minutes from the moment the treatment was administered.
First data set: 0.08, 2.09, 2.73, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.22, 3.52, 4.98, 6.99, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 15.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.93, 8.65, 12.63, 22.69
Second data set: 1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, 2.0
The third and fourth dataset were both sourced from [22]. The third dataset consisted of three hundred and forty-six measures of nicotine taken as obtained from different cigarette product category. The fourth data was on the windshield of an aircraft. The data comprises of one hundred and fifty-three measurements, of which eighty-eight were categorized as failed windshields and the remaining sixty-five were service times of windshields that were in good condition at the time of the observation. 1000h is the measuring unit.
Third data set: 1.3, 1.0, 1.2, 0.9, 1.1, 0.8, 0.5, 1.0, 0.7, 0.5, 1.7, 1.1, 0.8, 0.5, 1.2, 0.8, 1.1, 0.9, 1.2, 0.9, 0.8, 0.6, 0.3, 0.8, 0.6, 0.4, 1.1, 1.1, 0.2, 0.8, 0.5, 1.1, 0.1, 0.8, 1.7, 1.0, 0.8, 1.0, 0.8, 1.0, 0.2, 0.8, 0.4, 1.0, 0.2, 0.8, 1.4, 0.8, 0.5, 1.1, 0.9, 1.3, 0.9, 0.4, 1.4, 0.9, 0.5, 1.7, 0.9, 0.8, 0.8, 1.2, 0.9, 0.8, 0.5, 1.0, 0.6, 0.1, 0.2, 0.5, 0.1, 0.1, 0.9, 0.6, 0.9, 0.6, 1.2, 1.5, 1.1, 1.4, 1.2, 1.7, 1.4, 1.0, 0.7, 0.4, 0.9, 0.7, 0.8, 0.7, 0.4, 0.9, 0.6, 0.4, 1.2, 2.0, 0.7, 0.5, 0.9, 0.5, 0.9, 0.7, 0.9, 0.7, 0.4, 1.0, 0.7, 0.9, 0.7, 0.5, 1.3, 0.9, 0.8, 1.0, 0.7, 0.7, 0.6, 0.8, 1.1, 0.9, 0.9, 0.8, 0.8, 0.7, 0.7, 0.4, 0.5, 0.4, 0.9, 0.9, 0.7, 1.0, 1.0, 0.7, 1.3, 1.0, 1.1, 1.1, 0.9, 1.1, 0.8, 1.0, 0.7, 1.6, 0.8, 0.6, 0.8, 0.6, 1.2, 0.9, 0.6, 0.8, 1.0, 0.5, 0.8, 1.0, 1.1, 0.8, 0.8, 0.5, 1.1, 0.8, 0.9, 1.1, 0.8, 1.2, 1.1, 1.2, 1.1, 1.2, 0.2, 0.5, 0.7, 0.2, 0.5, 0.6, 0.1, 0.4, 0.6, 0.2, 0.5, 1.1, 0.8, 0.6, 1.1, 0.9, 0.6, 0.3, 0.9, 0.8, 0.8, 0.6, 0.4, 1.2, 1.3, 1.0, 0.6, 1.2, 0.9, 1.2, 0.9, 0.5, 0.8, 1.0, 0.7, 0.9, 1.0, 0.1, 0.2, 0.1, 0.1, 1.1, 1.0, 1.1, 0.7, 1.1, 0.7, 1.8, 1.2, 0.9, 1.7, 1.2, 1.3, 1.2, 0.9, 0.7, 0.7, 1.2, 1.0, 0.9, 1.6, 0.8, 0.8, 1.1, 1.1, 0.8, 0.6, 1.0, 0.8, 1.1, 0.8, 0.5, 1.5, 1.1, 0.8, 0.6, 1.1, 0.8, 1.1, 0.8, 1.5, 1.1, 0.8, 0.4, 1.0, 0.8, 1.4, 0.9, 0.9, 1.0, 0.9, 1.3, 0.8, 1.0, 0.5, 1.0, 0.7, 0.5, 1.4, 1.2, 0.9, 1.1, 0.9, 1.1, 1.0, 0.9, 1.2, 0.9, 1.2, 0.9, 0.5, 0.9, 0.7, 0.3, 1.0, 0.6, 1.0, 0.9, 1.0, 1.1, 0.8, 0.5, 1.1, 0.8, 1.2, 0.8, 0.5, 1.5, 1.5, 1.0, 0.8, 1.0, 0.5, 1.7, 0.3, 0.6, 0.6, 0.4, 0.5, 0.5, 0.7, 0.4, 0.5, 0.8, 0.5, 1.3, 0.9, 1.3, 0.9, 0.5, 1.2, 0.9, 1.1, 0.9, 0.5, 0.7, 0.5, 1.1, 1.1, 0.5, 0.8, 0.6, 1.2, 0.8, 0.4, 1.3, 0.8, 0.5, 1.2, 0.7, 0.5, 0.9, 1.3, 0.8, 1.2, 0.9
Fourth data set: 0.040, 1.866, 2.385, 3.443, 0.301, 1.876, 2.481, 3.467, 0.309, 1.899, 2.610, 3.478, 0.557, 1.911, 2.625, 3.578, 0.943, 1.912, 2.632, 3.595, 1.070, 1.914, 2.646, 3.699, 1.124, 1.981, 2.661, 3.779,1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.82,3, 4.035, 1.281, 2.085, 2.890, 4.121, 1.303, 2.089, 2.902, 4.167, 1.432, 2.097, 2.934, 4.240, 1.480, 2.135, 2.962, 4.255, 1.505, 2.154, 2.964, 4.278, 1.506, 2.190, 3.000, 4.305, 1.568, 2.194, 3.103, 4.376, 1.615, 2.223, 3.114, 4.449, 1.619, 2.224, 3.117, 4.485, 1.652, 2.229, 3.166, 4.570, 1.652, 2.300, 3.344, 4.602, 1.757, 2.324, 3.376, 4.663.
Five models were fitted to the above four datasets using the Adequacy Model package in R [23], these models include; Exponentiated Ailamujia distribution (EAD), Exponentiated Exponential distribution (EED), Exponentiated Weibull (EWD), the Logistic distribution (LD) and the transmuted skew student t distribution (TSSiD) respectively. The resulting fitted models selected on the basis of Akaike Information Criterion, (AIC), Bayesian Information Criterion (BIC), Consistent Akaikes Information Criterion (AIC) and Hannan-Quinn information criterion, (HQIC).
Table 2: The AIC, CAIC, BIC, HQIC and MLE of the First data
Models AIC BIC CAIC HQIC MLE Rank
TSStD -642.6124 -636.9719 -634.9719 -627.3313 0.0006 1.1348 1
EED 799.7295 805.3700 799.8286 802.0208 1.2682 0.1041 0.4493 3
EWD 275.1219 282.4499 275.4182 278.0694 1.6494 1.7404 2
EAD 7285.8160 7291.4570 7285.9150 7288.1080 1.4819 1.8290 5
LD 1080.3360 1085.9770 1080.4350 1082.6270 1.9978 1.8904 4
Table 2 to Table 5 presents the model estimates for each of the datasets. The results revealed that
the model with the smallest measure of the entire information criterion was the TSStD. The ranks
for the performance of the models were based on the information criterions of each of the models.
From the results obtained, for the five models estimated, the TSStD was the models with the best
fit.
Table 3: The AIC, CAIC, BIC, and HQIC of the second data
Models AIC BIC CAIC HQIC MLE Rank
TSStD -642.6124 -640.621 -638.621 -634.6295 9.3568e-03 -7.6898e-16 1
EED 36.3450 38.3365 37.0509 36.7338 54.366966 2.172273 0.6234 2
EWD 46.9561 49.9433 48.4561 47.5392 1.7261 1.7733 3
EAD 48.3887 50.3802 49.0946 48.7775 1.8667 0.7558 4
LD 1204.3680 1210.0080 1204.4670 1206.6590 1.7387 1.7264 5
Table 4: The AIC, CAIC, BIC, and HQIC of the Third data
Models AIC BIC CAIC HQIC MLE Ranks
TSStD -642.6124 -634.9196 -632.9196 -623.2267 -0.1381 1.4114 1
EED 325.3694 333.0623 325.4044 328.4327 6.0022 2.3477 1.3488 4
EWD 261.0285 272.5679 261.0987 265.6235 1.7973 1.5733 2
EAD 300.5637 308.2566 300.5987 303.6271 1.9331 1.5668 3
LD 293.6612 301.3540 293.6961 296.7245 0.9789 0.2148 5
Table 5: The AIC, CAIC, BIC, and HQIC of the fourth data
Models AIC BIC CAIC HQIC MLE Ranks
TSStD -347.3415 -342.4562 -340.4562 -333.5709 -0.1613 1.1483 1
EED 292.3541 297.2394 292.5004 294.3191 3.6582 0.6240 0.4305 4
EWD 275.2268 282.5548 275.5231 278.1743 1.7537 1.3413 2
EAD 283.6236 288.5089 283.7699 285.5886 1.7293 0.5065 3
LD 412.6227 417.5080 412.7690 414.5877 0.8074 1.3006 5
IV. Conclusion
This research paper presented a novel two-parameter distribution known as the Transmuted Skew Student t distribution. Some of the statistical and reliability properties for the TSStD were derived and they included the survival function, the rth moment, the hazard function, the mean, the quantile function, the moment generating function, the characteristic function and the order statistics. Before application to real dataset, a Monte-Carlo simulation study was conducted to assess the stability of the model with more sample sizes. The results revealed that the model was consistent with increase in the number of samples. The new PDF was applied to four different real datasets. Using information criterions, it was found that TSStD performs better than other competing models.
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