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A TRUNCATED TWO PARAMETER PRANAV DISTRIBUTION AND ITS Volume 15 Se tember 2020 APPLICATIONS 0 Ume ' eP em er
A Truncated Two Parameter Pranav Distribution and its
Applications
Kamlesh Kumar Shukla
Department of Statistics, College of Science, Mainefhi, State of Eritrea (N.E. Africa) Email: [email protected]
Abstract
In this paper, two parameter truncated Pranav distribution has been proposed. Some statistical properties including moments, coefficient of variation, skewness and index of dispersion have been derived and presented graphically. Survival and Hazard functions are derived and its behaviors are presented graphically. Maximum likelihood method of estimation has been used to estimate the parameters of proposed model. Simulation study has also been carried out. A proposed distribution has been applied on two data sets and compares its superiority over two parameter and one parameter classical distributions.
Keywords: Truncated, Pranav distribution, Lindley distribution, Skewness, Kurtosis
I. Introduction
In the recent past decades, life time modeling has been becoming popular in distribution theory, where many statisticians are involved in introducing new models. Some of the life time models are very popular and applied in biological, engineering and agricultural areas, such as Lindley distribution; Lindley (1958), weighted Lindley distribution; Gitany et al. (2008), Akash distribution; Shanker (2015), Ishita distribution; Shanker and Shukla (2017), Pranav distribution; Shukla (2018) etc and extension of above mentioned distribution has also been becoming popular in different areas of statistics.
Shukla (2018) has introduced a Pranav distribution which is mixture of exponential distribution having scale parameter 6 and gamma distribution having shape parameter 4 and scale parameter 6, is defined by its pdf and cdf :
64
f(y6) = 16— (6+y)e-6y ;y >0, 6>0 (1.1)
F2 (y;6) = 1 -
64 + 6
6y (62 y2 + 36y + 6)'
64 + 6
-6y
; y > 0,6> 0
(1.2)
Shukla (2018) has discussed in details about its mathematical and statistical properties. Estimation of parameter using both the method of moment and the maximum likelihood estimation have mentioned in his paper and its application to model lifetime data from engineering and biomedical sciences.
Recently, Umeh and Ibenegbu (2019) have introduced extension of Pranav distribution and named as two parameter Pranav (TPPD) distribution and its pdf and cdf are defined as follows:
A TRUNCATED TWO PARAMETER PRANAV DISTRIBUTION AND ITS Volume 15 Se tember 2020 APPLICATIONS 0 Ume ' eP em er
ft
f2 (y; ft) = (aft+y) ; y > 0, ft > 0, a > 0 (1-3)
~ Gy (ft2 y2 + 3fty + 6)" + (aft4 + 6)
F2 (y; ft,a) = 1 -
efty ; y > 0,ft> 0,a > 0 (1.4)
Umeh and Ibenegbu (2019) have discussed in details about its mathematical and statistical properties. Estimation of parameter using both the method of moment and the maximum likelihood estimation including its application have mentioned in their paper.
Truncated type of distribution are more effective in application to modeling life time data because its limits used as bound either upper or lower or both according to given data. Truncated normal distribution is proposed by Johnson et al (1994). It has wide application in economics and statistics. Many researchers have been proposed truncated type of distribution and applied in different areas of statistics, especially in censor data such as truncated Wiebull distribution; Zange and Xie (2010) , truncated Lomax distribution; Aryuyuen and Bodhisuwan (2018), truncated Pareto distribution; Janinetti and Ferraro (2008), truncated Lindley distribution; Singh et al (2014). Truncated version of distribution can be defined as:
Definitionl. Let X be a random variable that is distributed according to some pdf g (x; 0..) and cdf G (x; 0..)' where 0 is a parameter vector of X.
Let X lies within the interval [a,b] where —x> <a <x <b then the conditional on a <x <b is distributed. We have the pdf of truncated distribution as reported by Singh et al (2014) defined by:
f (x; 0..) = g(x / a < x < b; 0..) =-g^-—b ^^ — 4aC (1.5)
7 5V 7 G(b; 0. .) — G(a; 0. .) 2a ' )
where f (x; 0..) = g(x; 0..) for all a < x < b and f (x; 0) = 0 elsewhere.
Notice that f (x; ft) in fact f (x; ft) is a pdf of X on interval [a, b].
b , b
f (x; 0..) = f f (x; 0..)dx =-1-f g (x; 0..)dx
jjk, ^ G(b; 0..) — G(a; 0..)J
1 G(b; 0..) — G(a; 0..) = 1 (1.6)
G(b; 0..) — G(a; 0..)
The cdf of truncated distribution is given by
F (x; 0.) = Jf (x 0.)dx = G(x 0.)—G(a- 0.) (1.7)
a G(b; 0..)—G(a; 0..)
The main objectives of this paper are (i) to propose new truncated distribution using two parameter Pranav distribution, which is called as Truncated Two Parameter Pranav distribution (TTPPD) (ii) to know statistical mathematical properties and its suitability, it has been compared with classical distributions of two parameter as well as one parameter using two lifetime datasets. The study has been divided in eight sections. Introduction about the paper is described in the first section. In the second section, TTPPD has been derived. Mathematical and statistical properties including its moment have been discussed in third section. Behavior of hazards rate has been presented mathematically as well as graphically in fourth section. Moments and its related expression have been discussed in fifth section. Simulation study of the presented distribution has been discussed to check estimation parameters using Bias and Mean square error in sixth section.
A TRUNCATED TWO PARAMETER PRANAV DISTRIBUTION AND ITS Volume 15 Se tember 2020
APPLICATIONS o ume , ep em er
Estimation of parameter of proposed distribution has been discussed in seven section where its applications and comparative study of other classical two parameter life time distributions as well as one parameter distributions have been illustrated using life time data. In the last, conclusions have been drawn according to studied of behavior and properties of TTPPD.
II. Truncated Two Parameter Pranav distribution
In this section, pdf and cdf of new truncated distribution is proposed and named Truncated two parameter Pranav distribution(TTPPD) , using (1.5) & (1.7) of definition1 and from (1.3) & (1.4) , which is defined as :
Definition 2: Let X be random variable which is distributed as TTPPD with scale parameter 6 location parameter a & b ,and shape parameter a , will be denoted by TTPPD (a,b, 6, a). The pdf and cdf of X are respectively:
f(6 a) -_64(*3 +6a)e~6x_
' ' ((63a3 + a6A + 3a262 + 6a6 + 6)e-a - (63b3 + a6A + 3b262 + 6b6 + 6)e-b)
(1.8)
((6V + a6A + 3a262 + 6a6 + 6)e6 - (63x3 + a6A + 3x262 + 6x6 + 6)e-x)
F (x; 6, a) — -,-
((63a3 + a64 + 3a262 + 6a6 + 6)e~6a - (63tf + a6A + 3b262 + 6b6 + 6)e- )
(1.9)
Where —a < x < b , and 6> 0,a> 0
Following conditions can be categories:
(i) When a = 0 and b = œ it reduces to two parameter Pranav distribution.
(i) When a = 0 it is known as right truncated two parameter Pranav distribution.
(ii) When b = œ it is known as left truncated two parameter Pranav distribution.
(iii) When and it reduces to two parameter Pranav distribution.
(iv) When a = 0 , b = œ and a = 1 it reduces to Pranav distribution.
Performance of pdf of TTPPD for varying values of parameter has been illustrated in the fig.1. From the figure1, it is clearly indicates that parameters a & b are the location parameter, 6 scale parameter
and a is the shape parameter, and value of pdf is decreasing as increased value of 6 when (6 <1)
at fixed values of a, b and a , whereas pdf is increasing as increased value of 6 when (6 >1) at fixed values of a, b and a .
pdf at alpha-2, a-O.b-lft pdf at alpha=2, a=0, b=l 0
pdf at a=0,b=10 , alpha=2 pdf at a=5, alpha=5, b-l()
Figure 1. pdf plots of TTPPD for varying values of parameters
Figure 2. cdf plots of TTPPD for varying values of parameter
III. Survival and Hazard function
S(x) and h(x) are the survival function and hazard function respectively, which are defined as: S(x) — 1- F(x)
S (x; 6, a) —
((63x3 + a64 + 3x262 + 6x6 + 6)e6 - (63b3 + a64 + 3b262 + 6b6 + 6)e-b) ((6W + a64 + 3a262 + 6a6 + 6)e'6a - (63b3 + a64 + 3b262 + 6b6 + 6)e-b)
h(x) —
fix)
S (x)
h( x) — ■
64( x3 +6a) e
-6x
(6x3 + a64 + 3x262 + 6x6 + 6)e-6x - {63b3 + a64 + 3b262 + 6b6 + 6)e-b) From the equation, It is notice that h(x) is independent of parameter 'a'
Behavior of hazard function of TTPPD for varying values of parameter is presented in figure3:
Figure 3. h(x) plots of TTPPD for varying values of parameter
IV. Moments and Mathematical Properties
Theorem: Suppose Xfollows doubly TTPPD(d,a,a,b) . Then the rth moment about originjuj of TTPPD is
U =
, d4a{/( r + \,0b )-/( r +1, da )} + {/( r + 4,0b )-/( r + 4, da )} = \a3d3 + 3a2d2 + 6ad + ad4 + 6)ed — ; T =
(b3d3 + 3b2d2 + 6bd + ad4 + 6)e-db
dr
= 1,2,3,...
f (a3d3 + 3a2d2 + 6ad + ad4 + 6)ed -1 Proof: Considering K — <
[(b3d3 + 3b2d2 + 6bd + ad4 + 6)ed
in (2.1), we have
u' = — J xr (da + x3 ) e dxdx
K a
~b b fade-dxxrdx + f edxr+3dx
K
u — dx, x — Taking d
K
da
dr+1
da
d
Je "xrdu-Je "xrdu 1 +
1 fdb da
\ J e-uur+3 du -J e~"xr+3 du I
Where
z
/(a, z) = J e~xxa-1dx, a> 0, x > 0
.ddL
" K
1 K
is the lower incomplete gamma function.
a{r(r + 1,db) -y(r + 1,da)} y(r + 4,db) -y(r + 4, da) d + d d4a {r(r +1, db) - r(r +1, da)} + {r(r + 4, db) - r(r + 4, da)}
d
d4a {r(r +1, db) - r(r + 1,da)} + { r(r + 4, db) - r(r + 4, da)}
dr
(a3d3 + 3a2d2 + 6ad + ad4 + 6)e-da -(b3d3 + 3b2 d2 + 6bd + ad4 + 6)ed
(4.1)
Now taking r —1,2, mean and variance can be obtained as
Mi =
M2 =
d4a {r(2, db) - r(2, da)} + {r(5, db) - r(5, da)}
d
da \
(a3d3 + 3a2d2 + 6ad + ad4 + 6)e-da -(b3d3 + 3b2d2 + 6bd + ad4 + 6)e
d4a{r(3, db) - r(3, da)} + {r(6, db) - r(6, da)}
d2
\a3d3 + 3a2d2 + 6ad + ad4 + 6)e'da -(b3d3 + 3b2d2 + 6bd + ad4 + 6)e-db
Variance M M (Ml') Similarly rest two moment of origin as well as coefficient of variation, coefficient of skewness, coefficient of kurtosis and Index of dispersion can be obtained, substituting r = 3,4 in the equation (4.1), which are as follows:
M3 =
M4 =
d4a {r(4, db) - r(4, da)} + {y(l, db) - r(7, da)}
d3
(a3d3 + 3a2d2 + 6ad + ad4 + 6)e'da -(b3d3 + 3b2 d2 + 6bd + ad4 + 6)e-db
d4a{r(5, db) - r(5, da)} + {r(8, db) - r(8, da)}
d4
(a3d3 + 3a2d2 + 6ad + ad4 + 6)e'da -{b3d3 + 3b2 d2 + 6bd + ad4 + 6)e-db
„ ,, , M - (Mif ) ^ , ni M + 3Mi Mi - M)2 )
Coefficient of Variation=-;-, Coefficient of Skweness=-
Mi
'\2\3/2
M - (Ml) )
Kamlesh Kumar Shukla RT&A, No 3 (58) A TRUNCATED TWO PARAMETER PRANAV DISTRIBUTION AND ITS Volume 15 Se tember 2020 APPLICATIONS_o ume , ep em er
(a4— 4AA + 6A2(A1)2 — 3(A)4 )
Coefficient of Kurtosis=-;-—r—--,
(A — (A) )
(A — (A)2 )
Index of dispersion=-;-, graph of above measures are presented in figure 4'5'6'7'8 &9.
A
Expressions of other central moments are not being given here because they have lengthy expressions. However, they can be easily obtained.
Figure4. Mean of TTPPD on varying value of parameters
5 10 15 20 25 30 5 10 15 20 25 30
Figure5. Variance of TTPPD on varying value of parameters
Figure7 .Variance of TTPPD on varying value of parameters
Figure8. Variance of TTPPD on varying value of parameters
Figure9. Variance of TTPPD on varying value of parameters
V. Maximum likelihood Method of Estimation
Let (x,x2,x3, ... ,xn) be a random sample of size n from (1.1). The likelihood function, L of Pranav distribution is given by
(
L =
((03a3 + a04 + 3a2 02 + 6a0 + 6)e'0a - (03b3 +a04 + 3b202 + 6b0 + 6)e-ffb )
fl (a0 + x3 ) e
and its log likelihood function is thus obtained as
(
ln L = n ln
04
((d3a3 + ad4 + 3a2d2 + 6ad + 6)e~da - (0b + a04 + 3b202 + 6b0 + 6)e-b )
n
^ln {ad + x3 ) - ndx
i=1
Taking a = min {x^x2,x3, ... ,xn ), b = max(xx,x2,x3, ... ,xn ), the maximum likelihood
n
estimate ft of parameter ft is the solution of the log-likelihood equation-= 0. It is obvious
8ft
8 log L
that-= 0 will not be in closed form and hence some numerical optimization technique can
8ft
be used e the equation for ft. In this paper the nonlinear method available in R software has been used to find the MLE of the parameter ft.
VI. Simulation Study
In this section, simulation of study of (2.1) has been carried out. Acceptance and Rejection method has been used to generate random number. Bias Error and Mean square Error have been calculated for varying values parameters ft and a whereas parameter a and b kept constant.
Table 1. Simulation of TSD at a=10, b=100, ft = 0.1 and a = 1
Sample Size (n) ft a Bias Error( ft ) MSE( ft ) Bias Error( a ) mse( a )
20 0.1 1.0 0.035225 0.024817 1.121145 25.13931
0.5 2.0 0.015225 0.004636 1.071145 22.94702
1.0 3.0 -0.00977 0.001911 1.021145 20.85473
1.5 4.0 -0.03477 0.024186 0.971145 18.86244
40 0.1 1.0 0.016613 0.01104 1.228047 60.32398
0.5 2.0 0.006613 0.001749 1.203047 57.89288
1.0 3.0 -0.00589 0.001386 1.178047 55.51179
1.5 4.0 -0.01839 0.013523 1.153047 53.18069
60 0.1 1.0 0.011075 0.00736 0.818698 40.21598
0.5 2.0 0.004409 0.001166 0.802031 38.59525
1.0 3.0 -0.00392 0.000924 0.785365 37.00786
1.5 4.0 -0.01226 0.009016 0.768698 35.4538
80 0.1 1.0 0.013449 0.014469 0.010876 0.009463
0.5 2.0 0.008449 0.00571 -0.00162 0.000211
1.0 3.0 0.002199 0.000387 -0.01412 0.01596
1.5 4.0 -0.00405 0.001313 -0.02662 0.056708
Table 2. Simulation of TSD at a=10, b=100, ft = 0.5 and a = 2
Sample Size (n) ft a Bias Error( ft ) MSE( ft ) Bias Error( a ) mse( a )
20 0.1 1.0 0.031624 0.020001 1.096027 24.0255
0.5 2.0 0.011624 0.002702 1.046027 21.88345
1.0 3.0 -0.01338 0.003578 0.996027 19.84139
1.5 4.0 -0.03838 0.029454 0.946027 17.89934
40 0.1 1.0 0.016796 0.011285 0.764069 23.35205
0.5 2.0 0.006796 0.001848 0.739069 21.84891
1.0 3.0 -0.0057 0.001301 0.714069 20.39577
1.5 4.0 -0.0182 0.013255 0.689069 18.99264
60 0.1 1.0 0.015861 0.015095 0.059787 0.214469
0.5 2.0 0.009195 0.005073 0.04312 0.111562
1.0 3.0 0.000861 4.45E-05 0.026454 0.041988
1.5 4.0 -0.00747 0.00335 0.009787 0.005747
80 0.1 1.0 0.012453 0.012405 0.027456 0.060306
0.5 2.0 0.007453 0.004443 0.014956 0.017894
1.0 3.0 0.001203 0.000116 0.002456 0.000483
1.5 4.0 -0.00505 0.002038 -0.01004 0.008071
Table 3. Simulation of TSD at a=10, b=100, 6 — 1 and a — 3
Sample e a Bias Error( 6 ) MSE(6 ) Bias Error( a ) mse( a )
Size (n)
20 0.1 1.0 0.03261 0.021269 0.989088 19.56588
0.5 2.0 0.01261 0.00318 0.939088 17.63771
1.0 3.0 -0.01239 0.00307 0.889088 15.80953
1.5 4.0 -0.03739 0.02796 0.839088 14.08136
40 0.1 1.0 0.01753 0.012292 0.631985 15.97622
0.5 2.0 0.00753 0.002268 0.606985 14.73725
1.0 3.0 -0.00497 0.000988 0.581985 13.54828
1.5 4.0 -0.01747 0.012208 0.556985 12.40931
60 0.1 1.0 0.015636 0.014669 0.071319 0.305184
0.5 2.0 0.008969 0.004827 0.054652 0.179212
1.0 3.0 0.000636 2.43E-05 0.037986 0.086574
1.5 4.0 -0.0077 0.003555 0.021319 0.02727
80 0.1 1.0 0.012021 0.011561 0.044151 0.155947
0.5 2.0 0.007021 0.003944 0.031651 0.080144
1.0 3.0 0.000771 4.76E-05 0.019151 0.029342
1.5 4.0 -0.00548 0.002401 0.006651 0.003539
Table 4. Simulation of TSD at a=10, b=100 , 6 — 1.5 and a — 4
Sample e a Bias Error( 6 ) MSE( 6 ) Bias Error( a ) mse( a )
Size (n)
20 0.1 1.0 0.052018 0.054117 0.279555 1.563022
0.5 2.0 0.032018 0.020503 0.229555 1.053912
1.0 3.0 0.007018 0.000985 0.179555 0.644801
1.5 4.0 -0.01798 0.006467 0.129555 0.335691
40 0.1 1.0 0.035047 0.04913 0.055135 0.121594
0.5 2.0 0.025047 0.025093 0.030135 0.036325
1.0 3.0 0.012547 0.006297 0.005135 0.001055
1.5 4.0 4.66E-05 8.68E-08 -0.01987 0.015785
60 0.1 1.0 0.02594 0.040373 0.017838 0.019092
0.5 2.0 0.019273 0.022288 0.001172 8.24E-05
1.0 3.0 0.01094 0.007181 -0.01549 0.014406
1.5 4.0 0.002607 0.000408 -0.03216 0.062062
80 0.1 1.0 0.016822 0.022637 0.059504 0.283254
0.5 2.0 0.011822 0.01118 0.047004 0.176747
1.0 3.0 0.005572 0.002483 0.034504 0.09524
1.5 4.0 -0.00068 3.68E-05 0.022004 0.038733
VII. Applications on Life time data
In this section, TTPPD has been applied on following three data sets, where maximum likelihood method of estimation has been used for estimation of its parameter. Parameter 6 is estimated whereas another parameters a, and b are considered as lowest and highest values of data. i. e. a — min(x) and b — max(x:), where x is data set. Goodness of fit has been decided using Akaike information criteria (AIC), Bayesian Information criteria (BIC) and Kolmogorov Simonov test (KS ) values respectively, which are calculated for each distribution and also compared with p-value. As we know that best goodness of fit of the distribution can be decide on the basis minimum value of
Kamlesh Kumar Shukla rt&A, No 3 (58) A TRUNCATED TWO PARAMETER PRANAV DISTRIBUTION AND ITS Volume 15 Se tember 2020 APPLICATIONS_o ume , ep em er
KS, AIC and BIC and maximum p-value for K.S.
Data Set 1: The data is given by Birnbaum and Saunders (1969) on the fatigue life of 6061 - T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 100 observations with maximum stress per cycle 31,000 psi. The data (x 10—3) are presented below (after subtracting 65).
43
25 31 32 34 35 38 39 39 40 42
43 43 44 44 47 47 48 49 49 49 51 54
55 55 55 56 56 56 58 59 59 59 59 59
63 63 64 64 65 65 65 66 66 66 66 66
67 67 67 68 69 69 69 69 71 71 72 73
73 73 74 74 76 76 77 77 77 77 77 77
79 79 80 81 83 83 84 86 86 87 90 91
92 92 92 92 93 94 97 98 98 99 101 103
105 109 136 147
Data Set 2: This data set is the strength data of glass of the aircraft window reported by Fuller et al (1994):
18.83 20.8 21.657 23.03 23.23 24.05 24.321 25.5 25.52 25.8 26.69 26.77 26.78 27.05 27.67 29.9 31.11 33.2 33.73 33.76 33.89 34.76 35.75 35.91 36.98 37.08 37.09 39.58 44.045 45.29 45.381
Table 5: MLE's, - 2ln L, AIC, K-S and p-values of the fitted distributions for data set-1
Distributions ML Estimates —2ln L AIC BIC K-S p-value
TTPPD 0 = 0.05527 927.37 931.37 930.14 0.136 0.056
a = 2.02011
TTPLD
TPPD
TPWD
TAD
TLD
Pranav
Lindley
0 = 0.02238 a = 15.80197 0 = 0.05853 a = 2.13206 0 = 0.00272 0 = 1.39558 0 = 0.03917
957.94 961.94 967.15 0.191 0.001
934.06 938.06 940.93 0.173 0.005
989.35 993.35 998.56 0.294 0.000
939.13 941.13 942.05 0.153 0.017
0 = 0.02199 958.88 960.88 962.31 0.186 0.001 0 = 0.05 853 934.06 936.06 937.49 0.167 0.007
0 = 0.02886 983.10 985.10 986.54 0.252 0.000
Exponential 0 = 0 01463 1044.87 1046.87 1048.30 0.336 0.000
Table 6: MLE's, - 2ln L, AIC, K-S and p-values of the fitted distributions for data set-2
Distributions ML Estimates
-2ln L AIC BIC K-S p-value
TTPPD
TTPLD
TPPD
TPWD
TAD
6 — 0.12066 a — 0.5999 6 — 0.12981 a — 2.17254 6 — 0.12981 a —1.9946 6 — 0.00203 6 —1.80566 6 — 0.08776
201.80 205.80 204.57 0.107 0.829
232.77 236.77 239.64 0.282 0.011
232.77 236.77 239.64 0.282 0.011
241.61 245.61 247.61 0.353 0.000
201.96 203.96 205.58 0.112 0.786
TLD
6 — 0.05392
202.18 204.18 205.61 0.117 0.738
Pranav
6 — 0.12981
232.77 234.77 236.20 0.267 0.019
Lindley
6 — 0.06299
253.98 255.98 256.98 0.365 0.000
Exponential 6 — 0.032452 274.52 276.52 277.52 0.458 0.000
VIII. Conclusions
In this paper, Truncated Two Parameter Distribution (TTPPD) has been proposed. Its mathematical and statistical properties including coefficient of variation, skewness, kurtosis and Index of dispersion have been derived and presented graphically. Maximum likelihood method has been used for estimation of its parameters. Goodness of fit of TTPPD has been discussed with two life time data sets and compared with truncated two parameter Lindley distribution (TTPLD), two parameter Weibull distribution (TPWD), two parameter Pranav distribution (TPPD), truncated Akash distribution (TAD), truncated Lindley distribution (TLD), Pranav, Lindley and Exponential distribution. It has been observed from above results, TTPPD gives good fit over above mentioned distributions on both the data sets.
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Received: June 21, 2020 Accepted: September 01, 2020
