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39
Anwar Hassan et al.
A NEW CONTINUOUS PROBABILITY MODEL BASED ON A
TRIGONOMETRIC FUNCTION: THEORY AND APPLICATIONS.
RT&A, No 3 (69)
Volume 17, September 2022
A new continuous probability model based on a trigonometric function: Theory and applications
Anwar hassan •
University of Kashmir Anwar.hassan5@gmail.com
Murtiza Ali Lone •
University of Kashmir murtazastat@gmail.com
Ishfaq Hassain Dar*
•
University of Kashmir ishfaqqh@gmail.com*
Peer Bilal Ahmad
Department of Mathematical Science, IUST peer.bilal@islamicuniversity.edu.in
Abstract
In this manuscript, we highlight a new probability distribution based on a trigonometric function, obtained by specializing the Sine-G family of distributions with exponentiated exponential distribution.
The proposed distribution is quite flexible in terms of density and hazard rate functions. Several mathematical properties of the proposed distribution are also explored. For applicability of proposed distribution, two real data sets are scrutinized and it is sensed that proposed distribution leads to a better fit than all other models taken under consideration.
Keywords: Sine-G; Exopentiated exponential distribution; Hazard rate function; Order statistic; Simulation study; Maximum likelihood estimation.
1. Introduction
In distribution theory proposing new family of distributions by incorporating an extra parameter is most common among researchers. The purpose of incorporating a new parameter is to increasing the data fitting strength of the proposed probability models. Although researchers are quite successful in doing so, however they are little concerned about the over parameterization
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TRIGONOMETRIC FUNCTION: THEORY AND APPLICATIONS.
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and complexities that arise due to addition of new parameters. To know more about such families one can go through [11],[9],[6],[3],[8], [18], [17] and [1]. So keeping in view the limitations of having extra parameters, some researchers have come up with family of distributions that not only can be used to model the complex data structures, but also are void of extra parameters. Among them lets recall [10],[12],[13],[15] and [16]. After getting motivated by aforementioned work on probability models having parsimony in parameters we have introduced a new model based on Sine-G (SG) family of distribution proposed by [15]. The introduced model, namely Sine-G exponentiated exponential (SGEE) distribution has been obtained by taking the baseline distribution as exponentiated exponential distribution. The SGEE distribution has same number of parameters as baseline distribution and has greater flexibility than some well-known two parameteric probability distributions including the baseline distribution.
The rest of the manuscript is presented as, In Section 2 a brief introduction about Sine-G family of distributions is given. In Section 3, a member of the SG family namely, SGEE distribution is examined in detail and its general properties are studied including, quantile, moments, moment generating functon, order statistic etc. The maximum likelihood estimation and simulation study are discussed in Section 4. In Section 5, to check the applicability of SGEE two data sets have been scrutinized. Finally, the paper ends in Section 6 with concluding remarks.
2. Sine-G (SG) Family Of Disrtibutions
Let G (y) be the baseline commulative distribution function (CDF) of any random variable Y. Then the CDF, F(y) of the sine-G family of distributions proposed by [15] is given by
F(y) = sin (у ОД) ; y e R
The corresponding probability density function( PDF) is given by
f(y) = fg(y) cos (fG(y)) ;ye R
The survival function S (y) for SG is given by
S(y)=1 - sin(ПОД) ; y С R
= (sin (n G(y))— cos (f G(y)))
The hazard rate function A(y) is given by
A(y) = f g(y)cos (f G(y))
(sin (f G(y)) - cos (f G(y)))2
(1)
3. Sine-G Exponentiated Exponential (SGEE) Distribution And Its
Properties
Suppose the random variable Y has exponentiated exponential distribution with CDF G (y) =
(1 — e—ey)a; y, а, в > 0 then the CDF of the SGEE distribution is given by
F(y)= sin (J2(1 — e—l9y)“) ; а в, y > 0
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(i)Density Plot For SGEE Distribution
(ii)Hazard rate Plot For SGEE Distribution
y y
Figure 1: Plots of the SGEE density and hazard function for different values of a and 9.
The corresponding PDF is
f (y) = 9e-9y(1 - e-9y)a-1 cos ^П(1 - e-9y)aj ; a,9,y >
The survival and hazard rate functions are, respectively, given by S(y) = 1 - sin ^П (1 - e-9y)a j ; a, 9, y > 0
and
л(y)
J (y) S(y)
af,9e-9y(1 - e-9y)a-1cos П(1 - e-9y)a A(y) = - - - - 2 - — ; a, 9,y > 0
(sin (f (1 - e-9y)a) - cos (n(1 - e-9y)«))2
Some important series expansions:
cos x
sin x
(2)
(3)
E j! (4)
j=0
E (-1)j|r (5)
j=0
“ y2j+1 (6)
,S(-1)' 2 +1!
E (-1)j С1. V (7)
j=0 1
3.1. Quantile Function
The quantile function of SGEE is given by
Y = - 9 '°g
1 -(2sin-1(1 - U))
0
x
e
n
x
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where U ~ (0,1) distribution. The qth quantile of SGEE distribution is given by
yq = - g l°g
1 - 0'n-1(1 - q))
The median is obtained as
У0.5 = - g log
1 -I I™’ (1) ■
3.2. Moments
The rth moment of the SGEE distribution is given by
CO
E(yr) = / Vrf ШУ 0
= 1 УГa~2gе-0У(1 - е-0У)X-1<:os (П(1 - e-0y)*) dУ
by putting e-ey = z and using the series expansions (7) and (5), we get
E<r) = a е1ё-1 j (П )2,+1 C(2j +i1) -1) Г (-'ogz)rz'dz
again putting - logz = u in (8), we get rth moment as
E(Yr) = jr E E
a(2j+1) 1 (-1)1+. n,2/+1 fa(2j + 1) - 1) r(r + 1)
j=0 k=0 2j! 2
(n )2j+1('
k
(k + 1)r+1
3.3. Moment Generating Function
The moment generating function of SGEE distribution is defined by
My(t) = I otyf (У)dУ,
again using (4), (5) and (7) we have the final expression of MGF as
M (t) = aEE*(2jE)_1 (-1)j+k(n)2j+1 (t)r M+1)-r, .t < e
My(t) “ E0j00 E 2j!(k + 1)r« W( ■ ''' <e
k
3.4. Mean Residual Life And Mean Waiting Time
The mean residual life function, say y(f), is defined by
y(t) = щ ^E(t) - / yf(y)dyj - 1
O
(8)
(9)
(10)
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RT&A, No 3 (69)
Volume 17, September 2022
Table 1: Average values ofMLEs their corresponding MSEs and Bias.
Sample size Parameter MLEs MSE Bias
n a 6 a 6 a 6 a 6
30 0.8 0.5 0.86876 0.55268 0.04713 0.02634 0.06876 0.05268
1 0.88261 1.12988 0.04109 0.11373 0.08261 0.12988
1.5 0.88609 1.72108 0.04381 0.32292 0.08609 0.22108
2 0.88888 2.31489 0.05774 0.58598 0.08888 0.31489
1 0.5 1.06555 0.53199 0.05840 0.03184 0.06555 0.03199
1 1.08661 1.13391 0.07117 0.08813 0.08661 0.13391
1.5 1.08243 1.68584 0.08902 0.28176 0.08243 0.18584
2 1.06703 2.15094 0.11414 0.50617 0.06702 0.15094
1.5 0.5 1.65896 0.54400 0.23140 0.01787 0.15896 0.04400
1 1.56549 1.04447 0.14145 0.04484 0.06549 0.04447
1.5 1.65654 1.61426 0.30000 0.16226 0.15654 0.11426
2 1.72030 2.25512 0.27791 0.36792 0.22030 0.25512
2 0.5 2.22021 0.53030 0.40033 0.01187 0.22021 0.03030
1 2.22111 1.06212 0.46756 0.05600 0.22111 0.06211
1.5 2.31075 1.69956 0.61034 0.18055 0.31075 0.19956
2 2.22600 2.17031 0.57051 0.26500 0.22600 0.17031
50 0.8 0.5 0.84206 0.54269 0.02796 0.02565 0.04206 0.04269
1 0.79948 1.00343 0.02079 0.06482 -0.00051 0.00343
1.5 0.83571 1.61962 0.01981 0.13657 0.03571 0.11962
2 0.81309 2.05569 0.02020 0.24927 0.01309 0.05569
1 0.5 1.03527 0.52180 0.03822 0.01277 0.03527 0.02180
1 1.08685 1.08659 0.04641 0.05628 0.08685 0.08659
1.5 1.06537 1.68051 0.05138 0.19718 0.06537 0.18051
2 1.01739 2.02631 0.02842 0.14193 0.01739 0.02630
1.5 0.5 1.63833 0.53194 0.14694 0.00855 0.13833 0.03194
1 1.60152 1.07540 0.15908 0.04964 0.10152 0.07540
1.5 1.53020 1.53325 0.10391 0.06558 0.03020 0.03325
2 1.56678 2.08532 0.07309 0.14002 0.06678 0.08532
2 0.5 2.15774 0.52840 0.24240 0.00829 0.15774 0.02840
1 2.09796 1.02776 0.16712 0.02854 0.09796 0.02776
1.5 2.11604 1.56729 0.21126 0.09217 0.11604 0.06729
2 2.19453 2.08430 0.35438 0.11187 0.19453 0.08430
where
E(t\ = a h ^ ’h ( 1 У+к (n)2j+1 (a(2i +!) 1
E( ) 6 h h (k + 1)22j!(2) \ к
and
} a ^ a(2i+) 1 (—1)j+k n,2i+1 (a(2j + 1) - 1\ ,n n
!xf{x)dx = 6 5 £, (kTU2j! <2 ),+ ( к )Y (6t(k + 1),2)
0 j=0 k=0
Substituting (3), (11) and (12) in (10), we get y(t) as
F(t) =
1 ______a f i-1 (—1)i+k (П)2j+1 (a(2j + 1) -1
-6t)aI 6 h h tb i 1 \2o„-i ( о )
1 — sin (П2(1 — e 6t)a) 0 •=
j=0 k=0
(k + 1)22jU 2
x [1 — Y(6t(k + 1))] — t
where Y(a, b) = f yb ke ydy is the lower incomplete gamma function.
0
(11)
(12)
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TRIGONOMETRIC FUNCTION: THEORY AND APPLICATIONS.
RT&A, No 3 (69)
Volume 17, September 2022
Table 2: Average values ofMLEs their corresponding MSEs and Bias.
Sample size Parameter MLEs MSE Bias
n а д а д а д а д
100 0.8 0.5 0.82267 0.52370 0.01321 0.01083 0.02267 0.02370
1 0.79728 0.99739 0.00716 0.02731 -0.00271 -0.00260
1.5 0.82011 1.54754 0.01276 0.06654 0.02011 0.04754
2 0.79681 1.99112 0.01290 0.12323 -0.00318 -0.00887
1 0.5 1.03990 0.51737 0.01768 0.00528 0.03990 0.01737
1 1.01765 1.03115 0.01432 0.02452 0.01765 0.03115
1.5 1.02663 1.55358 0.01549 0.04853 0.02663 0.05358
2 1.03361 2.07698 0.02073 0.10262 0.03360 0.07698
1.5 0.5 1.54919 0.51586 0.04621 0.00518 0.04919 0.01586
1 1.55740 1.02647 0.04768 0.01628 0.05740 0.02646
1.5 1.56732 1.56766 0.03581 0.03419 0.06731 0.06766
2 1.57208 2.10929 0.04142 0.08105 0.07208 0.10929
2 0.5 2.04186 0.50132 0.07637 0.00288 0.04186 0.00132
1 2.04210 1.02675 0.08509 0.01862 0.04210 0.02675
1.5 2.10625 1.54551 0.13007 0.03257 0.10625 0.04551
2 2.06746 2.04980 0.08307 0.06683 0.06746 0.04980
200 0.8 0.5 0.79883 0.50439 0.00455 0.00428 -0.00116 0.00439
1 0.80354 1.01873 0.00364 0.01240 0.00354 0.01873
1.5 0.79510 1.48724 0.00360 0.02438 -0.00489 -0.01275
2 0.79160 2.01861 0.00621 0.06748 -0.00839 0.01861
1 0.5 0.99809 1.50664 0.00558 0.01866 -0.00190 0.00664
1 1.00884 2.03002 0.00997 0.05265 0.00884 0.03002
1.5 1.00041 1.51007 0.00670 0.02119 0.00041 0.01007
2 1.00884 2.03002 0.00997 0.05265 0.00884 0.03002
1.5 0.5 1.56024 0.51511 0.02153 0.00224 0.06024 0.01511
1 1.51925 1.01414 0.01833 0.00859 0.01925 0.01414
1.5 1.53288 1.52218 0.02876 0.02280 0.03288 0.02218
2 1.52145 2.01055 0.02441 0.03646 0.02145 0.01055
2 0.5 2.03758 0.50909 0.03859 0.00208 0.03758 0.00909
1 2.07367 1.03099 0.05998 0.00920 0.07367 0.03099
1.5 2.01058 1.50662 0.04448 0.01675 0.01054 0.00662
2 2.02562 2.01343 0.03551 0.02001 0.02562 0.01343
The mean waiting time of Y, say ц(t), is given by
t
ц(t) =t — Ft) j yf (y)dy
Ц (t) =t------7
a
F a(2+)—1 (—1)j+k пч2/+1 (a(2j + 1) - 1
sin (f (1 — e et)a) д x y (9t(k + 1), 2)
—tд 5 E (k + 1)22j!v 2
_( f )2j+1(a(2j + 1) — Л j(2) v k )
3.5. Order Statistics
1
Let Y1, Y2,..., Yn be a random sample of size n, and if Yi:n denote the ith order statistic, then the
PDF of Yi:n, say fi:n (y) is given by
fi:n (y)
■k-Лу) f(y)(1 — G(y))n
(i — 1)! (n — i)!
We can write the PDF fi:n (y) of ith order statistic of SGEE distribution as
fi:n (y)
апдг—вУ (1 — г—вУ)a—1 sinn (f (1 — г—ду)a) 2B(i,n — i + 1) tan (f (1 — e—dy)a)
cosec
(f (1—е~ву г—1)
n
Where B(a, b) is the beta function.
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(i)
(ii)
15 20 25 30 35 40 45 50
У
У
Figure 2: (i) The relative histogram and the fitted SGEE distribution. (ii) The empirical survival function and fitted SGEE survival function for data set I.
4. Statistical Inference
4.1. Maximum Likelihood Estimation
Let y1,y2,..., yn be a random sample from SGEE distribution, then the logarithm of the likelihood function is given by
ап —
l =nlog( —)+ nlog9 — 9 E yi + E log(1 - e—9yi)a—1 + E log cos — (1 - e—9yi)
2 i=1 i=1 i=1
2
(13)
by differentiating partially (13) with respect to the parameters a and 9 and equating the derivatives to zero, we get
dl
da
n
+ E log(1 — e—9y) [1 — — (1 — e—9y )atan( — (1 — e—9y )a)]
i=1
0
df
d9
n
9
nn
E yi + E
i=1 i=1
yte—9yj 1 e—9yi
[(a — 1)
О— (1 — e—9yi )a tan[ — (1 — e
0
It is clear that these equations cannot be solved analytically, so the MLEs of parameters are obtained through R software.
Theorem 1: If the parameter 9 is known, then the MLE of a exists and is unique.
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(i)
(ii)
y y
Figure 3: (i) The relative histogram and the fitted SGEE distribution. (ii) The empirical survival function and fitted SGEE survival function for data set II.
(i)P-P Plot for SGEE distribution
(ii)P-P Plot for SGEE distribution
Theoretical Cumuative Distribution
Theoretical Cumuative Distribution
Figure 4: P-P plot for the SGEE distribution for data set I and data set II
Proof: Since,
da = n + E log(1 - е~ву') [1 - f (1 - е~ву' )atan (П (1 - e~eyi)a)
i=1
dl n
lim — =ro + E log(1 - e-eyi) da
i=1
П ПЛ
1 - — tan = to
2 2 J
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(i)Q-Q Plot for SGEE distribution
(ii)Q-Q Plot for SGEE distribution
Figure 5: Q-Q plot for the SGEE distribution for data set I and data set II
Also
г dl
lim —
a——ж da
n
0 + E log(1 - e-9yi) < 0,
i=1
therefore, there exists atleast one root say a(0, ж), such that ддк = 0 For uniqueness of root, we have
д21 да2
n n
a2 2
(1 - e-9yi )a E (log(1 - e-9yi)) [2sec2 (П (1 - e-9yi )a) + tan (22 (1 - e-9yi )a)] < i
Hence the proof.
Theorem 2: If the parameter a is known, then the MLE of 9 exists and is unique. Proof: Since,
dl n
д9 = 9
n n
E Vi + E
i=1 i=1
Vie 9yi
1- e-9Vi
[(a - 1) - an(1 - e-9y)atan (П(1 - e-9y)a)]
r dl
lim ^
9—0 d9
=ж
Also
dl n
9—m =0 - E y <0
Therefore, there exists atleast one root say 9(0, ж), such that dd9 = 0 For uniqueness of root, we have
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Э2/^
W
n
в2
n n
E yi + E
i=1 i=1
У^ • 1- e-eyi
2
x ^ a
2
(x) + (a
1) tan(x)j + (a — 1)e eyiJ < 0
where,
x = П (1 — e—0y)
Hence proved.
Table 3: MLEs and -2l, AIC, A1CC, BIC , K-S statistic and P-valuefor data set I.
Model a в -2l AIC AICC BIC K-S statistic p-value
SGEE 0.07832 0.11465 208.3183 212.3183 212.7468 215.1862 0.13013 0.6235
(1.29403) (0.01650)
G 0.93208 0.61445 208.3212 212.3212 212.7691 215.2292 0.13189 0.6232
(1.76667) (0.15676)
APE 1.26948 9.55912 222.5222 226.5222 226.9508 229.3902 0.1757 0.2619
(1.18703) (0.89405)
NAPTE 1.91332 9.99821 218.8185 222.8185 223.2470 225.6864 0.15716 0.3877
(1.67776) (0.70944)
EE 1.47335 0.14956 208.8087 212.8087 213.2372 215.6766 0.13588 0.5697
(0.95528) (0.01917)
R 2.36356 - 236.4447 238.4447 238.5826 239.8787 0.31888 0.0026
(0.00830)
E 0.032455 - 250.5289 252.5289 252.6668 253.9629 0.39128 0.0012
(0.00582)
4.2. Simulation Study
To acertain the consistency and stability of the estimates, the simulation study was performed by taking samples of size (n=30, 50, 100 and 200) each replicated 100 times for different parameter vectors a = (0.8,1,1.5,2), в = (0.5,1,1.5,2), were obtained from SGEE distribution. In each case, the average values of MLEs (estimates) and the corresponding empirical mean squared errors (MSEs) and bias were considered. The simulation results are displayed in table 1 and table 2. From tables 1 and 2, it is obvious that as the sample size increases the MSE and bais decreases in all the cases.
5. Applications
To justify the validity and applicability of the SGEE distribution two real data sets have been used. The data set I is the strength of glass of the aircraft window taken from [4] and was also recently
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Table 4: MLEs and -2l, AIC, A1CC, BIC , K-S statistic and P-valuefor data set II.
Model a e -2l AIC AICC BIC K-S statistic p-value
SGEE 1.75535 1.36922 106.0445 110.0445 110.2263 114.5127 0.08801 0.6590
(1.39591) (0.14288)
G 0.99694 4.82104 106.1653 110.1653 110.2250 114.6335 0.08977 0.6275
(1.68238) (1.02421)
APE 1.29960 2.04747 108.8658 112.8658 113.0476 117.3340 0.09277 0.5925
(1.68238) (1.02421)
NAPTE 0.82534 1.89659 113.3371 117.3371 117.5190 121.8054 0.11192 0.3531
(1.15467) (1.66053)
EE 0.82839 1.89659 112.5386 116.5386 116.5983 120.7727 0.10192 0.4031
(2.03418) (0.18852)
r 1.08350 - 118.8298 120.8298 120.8895 123.0639 0.31888 0.0026
(0.06521)
E 0.68902 - 189.4026 191.4026 191.4623 193.6367 0.36224 0.0001
(0.08294)
reported by [2] The data set II finds its source in [14] and was also reported by [7] It is about the tensile strength (with unit in GPa) for single carbon fibers.
For comparision purpose, we compared the proposed SGEE distribution with several other models namely, gamma (G), alpha power exponential (APE) [9], noval alpha power transformed exponential (NAPTE) [6], exponentiated exponential (EE) [5], Rayleigh (R) and Exponential (E) distributions. From table (3) and (4) it is ostensive that SGEE distribution has the smallest values of the criteria -2l, AIC, AICC, BIC, K-S statistic and maximum p-value among all other distributions. Hence, we can say that the proposed model fits better for these data sets. Figure 2(i) and 3(i) display relative histograms for data set I and II respectively. Also, the Figure 2(ii) and 3(ii) shows the plots of the fitted SGEE survival function and empirical survival function of the data set I and II, respectively.
6. Concluding Remarks
A new continuous probability model based on a trigonometric function was introduced having symmetric, decreasing and positively skewed density function. Some of the well-known mathematical properties of the introduced model were also discussed. The authenticity and applicability of the introduced model was examined by considering two real data sets, it was perceived that the SGEE distribution is more appropriate for the given data sets than all other competitive models.
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