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THE INVERSE BURR LOG-LOGISTIC DISTRIBUTION: PROPERTIES, APPLICATIONS AND DIFFERENT METHODS OF ESTIMATION
Festus C. Opone1 Kadir Karakaya2 Francis E.U. Osagiede3 1 Department of Statistics, Delta State University of Science and Technology, Ozoro, Nigeria.
2 Department of Statistics, Selcuk University, Konya, Turkey.
3 Department of Mathematics, University of Benin, Benin City, Nigeria.
[email protected] [email protected] [email protected]
Abstract
Lifetime distributions have played a significant role in lifetime data analysis. Despite the numerous distributions in literature, there have been several motivations for developing new ones. In this paper, a new lifetime distribution is proposed. Some important functions of the new distribution, such as probability density, cumulative distribution, survival, hazard, and quantile are derived in closed form. Some distributional properties such as moments, moment generating function, linear representation, probability weighted moments, etc. are obtained. Some estimators such as the least square estimator (LSE), the weighted least square estimator (WLSE), the Anderson-Darling estimator (ADE) and the Cramer-von Mises estimator (CvME) are investigated for three unknown parameters. The efficiency of the estimators is checked via Monte Carlo simulation based on the bias and mean square error criteria. The usability of the new distribution is investigated with two real data sets and empirical results obtained reveal that the new distribution offers a promising fit for the data sets under study.
Keywords: Bur distribution, log-logistic distribution, parameter estimation, quantile
1. INTRODUCTION
Statistical distributions have played a significant role in lifetime data analysis. Despite the numerous distributions in literature, there have been several motivations for developing new ones. In all, the central goal has remained to develop a more flexible and tractable distribution in fitting real-world problems. In the last decades, researchers have introduced different methodologies for generating new statistical distributions which are hoped to provide a better fit than the existing distributions in lifetime data analysis. Some of these methods are the Beta-G family by [7], Marshall-Olkin extended family by [11], Transmuted-G family by [14], Kumaraswamy-G family by [5], Transformer (T-X) family by [2], Weibull-G family by [4], Odd Burr-G family by [1], Type II Topp-Leone generated family by [6], etc.
Festus Opone, Kadir Karakaya, Francis Osagiede RT&A, No 1 (72) THE INVERSE BURR LOG-LOGISTIC DISTRIBUTION_Volume 18, March 2023
Recently, [13] introduced the Inverse Burr-G family of distributions using the idea of [15]. By
considering the inverse Burr as the generator, they defined the cumulative distribution function of
the inverse Burr-G family of distribution as
Jo V ' (1) = [l + {-log[l-G(xX)]\a T, x > 0, a,b> 0,
where a and b are the shape parameters and G(x, X) is the baseline distribution which depends on a parameter vector X .
The corresponding density function associated with (1) is given by
f(xX) = abg(xX)[l-G(x,X)]-1 {-log[l-G(x,X)]}-(a+l)[l + {-log[l-G(xX)]}-a IT- (2)
In this paper, we employed the technique defined in (1) and consider in particular, the case where the baseline distribution G(x, X) follows the log-logistic distribution.
The cumulative distribution function (cdf) and probability density function (pdf) of the log-logistic distribution with shape parameter l > 0 are respectively defined as
and
G(x) = 1 - (l + xA)~\ (3)
g(x) = Ax1-1 (l + x1)-2, x > 0, 1 > 0. (4)
Inserting (3) and (4) into (1) and (2), we define the cdf and pdf of a new statistical distribution as
F(x) = ll + {log(l + x1)}- r x > 0, a,1, b> 0, (5)
and
f(x) = ablx1-1 (l + x1)- {log(l + iMl + M + lY i(b+l)
(x) = a p 1x [1 + x^ [\og{1 + x")y [1 + [\og{1 + x")} J . (6)
Suppose a random variable X has the density function in (6), then we say that X follows the Inverse Burr Log-Logistic ("IBLL" for short) distribution with shape parameters a, b and l. The motivation of this paper is to develop a tractable distribution that spans all the various forms of the hazard rate properties and provides a consistently better fit than most available statistical distribution in the literature.
The rest sections of the paper are structured as follows. In Section 2, we discuss in detail, some basic mathematical properties of the proposed distribution. Section 3 presents some methods of estimation of the unknown parameters of the proposed distribution. The asymptotic behavior of unknown parameters through a Monte Carlo simulation study are investigated in Section 4. In Section 5, we illustrate the applicability of the proposed distribution in lifetime data analysis two data sets and compared its fit alongside with fit attain by some existing non-nested distributions. Finally, in Section 6, we gave a concluding remark.
2. MATHEMATICAL PROPERTIES OF THE IBLL DISTRIBUTION
In this Section, some of the mathematical properties of the IBLL distribution are discussed. These include survival, hazard, quantile functions, the linear representation of the distribution, moments, moment generating function, probability weighted moment, Renyi entropy and distribution of order statistics.
2.1 Survival, Hazard and Quantile Functions
The survival, hazard and quantile functions of the IBLL distribution are respectively derived from (5) and (6) as follows.
S(x) = l - [l + {log(l + x1)}-a Y,
h(x) =
and
Qx (p ) =
aßg(x,X)[l - G(x, X)]-1 {- log
1 -
1 -G(x,X)]}-(a+1)[l + {-log[l-G(x,X)]}-a ]-
(ß+i)
1 + {log(l + x1)}- f
expI p /p -11 - 1
0 < p < 1.
(7)
(8)
(9)
The quantile function in (9) is derived by simply inverting the distribution function in (5). This is one of the most important properties of any distribution, as it allows for generating random numbers from the distribution for the simulation study. Substituting p = 0.5 in (9), we obtain the median of the IBLL distribution as
Qx (o.5) =
exp
((0.5)- if-
-1 " - 1
(10)
Some graphical presentations of the pdf and hazard function of the IBLL distribution are displayed in Figures 1 and 2 respectively.
о - 15 .0 ß - 1.0 Л - 1.0
a - 6 .0 ß - 3 .0 !. - 4 .0
a - 3 .0 ß = 1.0 ;. - 5.0
a . 4.0 p - 3.0 /. - 2.0
--a = 0.2 £-1.0 Л -3.0
------------a - 0.1 ß -3.0 Л - 1.0
--------a -8.0 ß = 0.2 ,t-3.0
"" a - 5.0 ß -0.3 Л - 2.0
Л
0.5 1.0
0.0
0.5
1.0
1.5
2.0
Figure 1: Density Plots of the IBLL Distribution
Figure 2: Hazard Plots of the IBLL Distribution
ß
Figure 1 shows that the density plot of the IBLL distribution accommodates decreasing, left-skewed, right-skewed and symmetric shapes, while the plots displayed in Figure 2 indicates that the hazard function of the IBLL distribution exhibits a decreasing, increasing, upside down bathtub and decreasing-increasing-decreasing hazard properties.
2.2 Linear Representation
The linear representation of the density and distribution functions allow for easy derivation of some properties such as the moments, probability weighted moment, moment generating function, distribution of order statistics, etc. The following lemmas will guide us in the derivation of the linear representation of the density and distribution functions of the IBLL distribution.
Lemma 1:
For any positive real non-integer s > 0, consider the generalized binomial series expansion (see [12]).
' ^ + k - D k
(1 + X)- = I
k=0
(-1)
kxk.
Lemma 2:
For any real parameter^ > 0, the convergent series holds.
(- log[l - yjr =
y
a
-
(
m =0
m
y
i-y-
s + 2
m
0 < y < 1.
Applying the result on power series raised to a positive integer, with as = (s + 2) 1 that is,
£ a.y = £ bs,mys,
V s=0 0 s=0
so that,
(- log[l - , ]r = ££
fa-
v m
b ya
s,m ✓
where bs,m = (sa0 )-1 £ {m(J + 0 - s }ajbs-q,m and b0, m = < (see [8]).
J=0
Now, applying the above two lemmas in (5),
1 + (log[l + x^f = iC+kk -'] (-1)" (log[l + x'j)-»,
k =0
(iog[i+xir = ££
ak > i - (i+x1Y.
bs m
v m 0
m+s-ak
so that (5) now becomes,
¥ ¥ ¥
F (x) = Z Z Z
fb + k -i\f-ak
k =0 m =0 x =0
k
m
] (- O'Am [l - (l + *' Y
m+s-ak
¥
Z ym+s-akHm+s-ak (x)
k ,m,s=0
where,
m
m=0 s=0
m=0 s=0
y
m+s-ak
Z
k ,m,s=0
b + k -\Y-ak ^
, k Jl m 0
(- OX
and Hm+s-ak (x) = a is the distribution function of the log-logistic distribution with
m + s -ak as the power parameter.
Differentiating (11), we obtain its associated density function as
f W _ Z ym+s-akhm+s-ak+l (x).
(12)
k ,m,s=0
where hm+s_ak(x) = l(m + s -ak +1)X1 1 (l + x1^ 2 [l - (l + x1) 1 ] is the density function of the log-logistic distribution with m + s - ak + 1 as the power parameter.
Other useful properties such as the moments and moment generating function can be directly obtained from (12).
2.3 The Moments and Moment Generating Function
Let X be a continuous random variable following a known probability distribution with density function f (x), then the rth ordinary moment of X is defined as
E[xr ] = = f¥ xrf(x) dx.
Substituting (6) into (12), the rth ordinary moment of the IBLL distribution is obtained as
[1 CO
X J = Z Vm+s-ok+l Z/hm+a 1 (x)dx'
k ,m,s=0
co / \ r / \ X
= l Z ym+s-ak+1 (m + s -ak +1)£xr+1-1 (l + x1)-2 [l - (l + x1 )-1 J
\m+s-ak
dx,
k ,m,s=0
using lemma 1,
[l - (1 + x')-' ]
1 - (1 + xTl"'" = f\m + S 1 (- ')« (l + x1)',
q=0 V '
(13)
(14)
J
Substituting this expression into (14), we have ¥ ¥ [ m + s - ak ö
k,m,s=0 q=0 \ " 0
Ex]= A I If + I(-1)"(m + s-A + l)¥„^£x-'(l + S'Y^dx, (15)
Further simplification of (15) and invoking the beta function, yields ' m + s - ak
EX ^ , q
k,m,s,q=0 \ H 0
(- l)q {m + s -ak + \)Vm+s-ak +1 B
r r 1 + —, q +1--
(16)
When r = 1 in (16) we obtain the mean of X. The variance, skewness and kurtosis of X can be computed from (16), using the following mathematical relationships.
variance
(s) = a- (A )2,
Мз - 3M2Mi + 2 (м[ )3
skewness (Sk ) _
kurtosis ( Ks)
where ^, ¡i2, and ju4 are the first four ordinary moments of the IBLL distribution. The rth incomplete moment of X is obtained from (16) as
'm + s - ak ,
(m2 - (m1 )2
¿4 - 4 ¿3 А + 6м'з (A )2 - 3 (A )4 A - (a )2
jr (0 = Z I
k,m,s,q=0 *
q
(- l)q (m + s -ak + \)ym+s-ak+1 B
i r r 1 + —, q +1--
l X
where B(a, 0)= f" xa-1 (1 + xyib+a)dx and Bz (a, b)= fV-1 (1 + xy(b+a)dx are respectively
J0 »0
the beta function of the second kind and the incomplete beta function of the second kind.
The moment generating function of X is define using the Maclaurin series expansion of the exponential function as
(17)
¥ tn
Mx (t) = E[e'<] = £ПE[^.
n=0 n-
(18)
Inserting (16) into (18), we define the moment generating function of IBLL distribution as Cm + s-ak
¥ j
Mx (t)= Z -
k ,m,s,q,n =0
n!
V
q
(- l)q (m + s -ak +1)^
m+s-ak+1
B
nn
1 + —, q +1--
X X
(19)
Table 1 presents the numerical computation of the mean, variance, skewness and kurtosis of the IBLL distribution at varying values of the parameters.
Table 1: Moments of the IBLL Distribution at varying values of the Parameters
l a b M Sk K
0.5 6 6 3.2316 11.9924 0.4073 1.5944
8 2.7794 12.6009 0.6770 1.7675
8 6 4.3219 10.0142 -0.1784 1.7717
8 4.1435 11.8485 -0.0735 1.5029
1.0 6 6 3.3973 2.5320 1.1591 5.3969
8 3.6548 2.8901 0.8689 4.6926
8 6 2.9264 1.1707 2.0539 10.003
8 3.1257 1.3177 1.8620 8.8985
з
From Table 1, we observed that the IBLL distribution is positively skewed (S^ > 0), negatively skewed (S^ < 0) and approximately symmetric (S^ » 0). This result is consistent with the plots of the density function displayed in Figure 1. Also, at some selected values of the parameters, the IBLL distribution is both leptokurtic (Ks > 3) and platykurtic (Ks < 3). Figure 3 displays the plot of the skewness and kurtosis of IBLL distribution for l = 1.
Figure 3: The Skewness and Kurtosis for IBLL Distribution (a, f, l) 2.4 The Probability Weighted Moments
The probability weighted moments (PWMs) are generally used to construct the estimator of the parameters as well as the quantiles of a known statistical distribution whose cdf is invertible. For a random variable X, [9] defined the (q, ryh PWMs as
Pqr = E[xrF(x)q ] = £ xrf(x)F(x)qdx,
(20)
combining (5) and (6), we have
f(x) F (x)q = ablx1-1 (l + x'Y {log(l + x^ril + {log(l + x1)}" \ applying the lemmas in (21), we have
11 + (logll + X1
(biq+l}+l)
(21)
(log[l + x'])-]-(",*11*1'= ^4 0 (-1)' (log[l + x'])
k =0
ak
(log[l + x1])-'^^ ±±
m=0 s=0
1 - (l + x1)
m+s-a[k+l]-1
= z
p =o
f-a[k +1] +[ / .\_i lm+s-i L J b^m 1l - (l + X*) ]
m J L J
'm + S-a[k +1] -^ (-1) p (l + xi)- p .
P 0
Substituting these expressions into (21), we have
f(X)F(Xy = abltHib +1]+'J-4' +1]+T + '-a[k +1]-^ (-irPK,mXi-i (1 + (22)
m =0 i =0 k =0 p =0 V k 0V m /V P 0
By inserting (22) into (20) and further simplification, we obtain the PWMs of the IBLL distribution
as
¥ ¥ ¥ ¥
p„ = «ßYYYY
m =0 s =0 k =0 p =0
ß[q +1] + k Y-a[k +1] + 1Y m + s-a[k +1]-10
m
(- l)k+pb B
, r r 1 + —, p — l l
(23)
From (23), we remark that the PWMs of the inverse Burr log-logistic distribution can be expressed as a linear combination of the log-logistic densities.
2.5 Distribution of Order Statistics
Let X1, X2,!, Xn be random samples of size n from a known probability distribution. Suppose n denotes the r order statistics, then the density function of n is defined by
fr : n ( *) =
1
B ( r, n - r +1) j=0
n - r V j 0
"—' vi — у
I , (-l)j f ( *) F ( *)
r+j-1
(24)
th
Inserting (5) and (6) into (24), we define the r order statistics of the density of IBLL distribution as follows.
f(x)F(xy+- = ablx11 (l + x1)- {log(l + x^ril + {log(l + x1)}* . (25)
We further simplify (25) using a similar approach in (21) as
f(x) F (x)--1 = aßl^vmx1-- (l + x1)--P+l),
(26)
Substituting (26) into (24), we have
frn(x)= n/aßA
n - r
B[r, n - r +1] j=J У j ,
(-1)J V x" 1 11 + x
j ~ vH ll I v-l
m
(l + x1
-(p+1)
(27)
where
V
¥ ¥ ¥ =T.T.T.
s =0 k =0 p =0 th
f
ß[q +1] + kY- a[k +1] +1Ym + s -a[k +1] -1
k
m
(- 1)k+Pbs
(27) is readily the r order statistics of the density function of IBLL distribution.
An expression for the obtained using (27) as
th th
An expression for the q moment of the r order statistics of the density of IBLL distribution is
k
m=0
E[xqn ]
a ß
¥ n-r
Iir-r| (-1)JvmB
B[r, n - r +1] m=o j=o ^ j
rr i +—, p — i i
(28)
Again, we show that the qth moment of the Ttt% order statistics of the density of IBLL distribution can be expressed as a linear combination of the log-logistic densities.
2.6 Renyi Entropy
The entropy of a random variable X represents the measure of randomness associated with the random variable X. The Renyi entropy of X is defined by
(g) = t^-log f fg(x)dx, g> 0, g^ 1.
1 — g
(29)
The Renyi entropy of a random variable X following the IBLL distribution is derived by inserting (6) into (29) as
r*") = I-"log[m'l + x'Y {log(l + xl)}-"""[l + {log(l + j-"""'
"
Applying the lemmas in (30), we have
(log[l + x = ±(g + f k -'] (- l)k (log[l + xlr,
k=n k )
ll + (logll + x 1|)-ar(b+l)= £
k-n V
, r -IX ¥ ¥ f
+ +g]+g) = ^^
m=0 ^=0
(log[l + xÄ]y
(1 + x -)-1
11 - (1+x ^i-1 rs-a[k+g]-g = ±
p =0
+ Km [l - (l + ^ j-
g](-1) p (l + xi)
+s-a[k+gj-
(m + s -a\k + g]-
Substituting these expressions into (30), we have
1
,(g) = --log
1 -g
(aßly^w j; xr[1-l) (l + x1)-{p+r) dx k=0
Evaluating the integral function in (31) yields,
Ar) = 7~log 1 -r
{aßjr-
k=0
r(l-1) +1 r + 1p -1 l ' l
where,
¥ ¥ ¥
w
= zzz
m=0 ^ =0 p =0
'giß +1) + k -lY- a[k + g] - gYm + s -a[k + g] -y\ pb
, k J! m J^ p 0
(30)
(31)
(32)
3. METHODS OF PARAMETER ESTIMATION
In this section, five estimators, i.e., maximum likelihood, least squares, weighted least squares, Anderson-Darling, and Cramer-von Mises, in order to estimate the unknown parameters of the IBLL
distribution are investigated. Let X1, X2,..., Xn be a random sample from the IBLL (x) distribution,,X(2),...,X(n) represent the associated order statistics and x^.^ indicates the observed values of X^ for i = 1,2,...,n, where X = (a,ß,l). The likelihood and log-likelihood functions are obtained, respectively, by,
t
R
7
t
L(x) = a" b lfix1-1 (1 + x?Y {log(1 + x?))(a+i) 1 + {log(1 + x,1)}
(b+1)
and
n
! ( X ) = nlog (a) + n log (b)+n log (l) + (l-1) j log (x )
i=1
-j log (1 + xi) -(a + 1) j log (log (1 + xi))-(b + 1) j log (1 + {log (1 + xi)}
i=i i=1 i=1 v
(33)
(34)
Then, the maximum likelihood estimator (MLE) of x is obtained by 'E! = argmaxt(E).
Let us give the following functions that give us the four different estimators:
Qls(X) = £( [1 + {log(1 + xil)}-
n + 1
-p i
Q (X) = I ^^ ([! + { log (1 + , i / )}-
Qad (X) = -n-±^[log{[1 + { log(1 + X(S )}- ]"'}
+iog {i-[i+{log (i+X( ^)}- f};
n + 1
and
QCvM (X ) = ¿ +1 [[ + { log( 1 + x(, / )}
b 2i -1
2n
(35)
(36)
(37)
(38)
(39)
Then, the least square estimator (LSE), the weighted least square estimator (WLSE), the AndersonDarling estimator (ADE) and the Cramer-von Mises estimator (CvME) of the X are achieved, respectively, by
~2 = argminQLS(S), (40)
~3 = argminQWbS(E), S4( = argminQAD(S) and
S5 = argminQCvM(S).
(41)
(42)
(43)
All of the maximization and minimization problems in Equations (35), (40), (41), (42), and (43) can be obtained by optim function in the R software.
4. SIMULATION EXPERIMENTS
In this section, the bias and mean square errors (MSEs) of the estimators are calculated with 5000 reputations based on the Monte Carlo simulation. The quantile function given in Equation (9) is used
to generate data from the IBLL (x) distribution by taking U(0,l) instead of p, where U(0, l) is the standard uniform distribution. Eight parameters setting are chosen based on Table 1 as
x = (6,6,0.5)(S1), x = (6,8,0.5)(S2), x = (8,6,0.5)(S3), x = (8,8,0.5)(S4), x = (6,6,1)(s5), x = (6,8,1)(S6), x = (8,6,1)(S7) and x = (8,8,1)(S8).
The sample size n = 50,100,150,200,250,500,1000 is selected in the simulation experiment. The simulation results are given in Tables 2-3. It can be inferred from Tables 2 and 3 that as sample the size increases, bias and MSEs for all estimators decrease and converge to zero. When the sample size increases, the bias and MSE values of the estimators converge. Although the bias and MSE of the estimators converge with each other when the sample size increases, generally the LSE in bias gives better results than the others.
Table 2: Average bias for all estimators
A
B
A
A
JL
A
JL
A
JL
A
JL
A
50 100 150 200 250 500 1000
0.9375 0.7669 0.7121 0.6581 0.6475 0.5943 0.5817
4.1957 1.2045 0.2743 -0.0165 -0.2638 -0.5326 -0.6845
0.0704 0.0116 -0.0095 -0.0167 -0.0238 -0.0327 -0.0380
0.5942 0.5610 0.4901 0.4909 0.5789 0.3437 0.2675
0.2506 0.0525 0.0965 -0.0139 -0.2129 0.1976 0.0253
0.0135 -0.0018 0.0011 -0.0032 -0.0129 0.0017 -0.0040
1.0350 0.7021 0.5846 0.5050 0.4872 0.4036 0.3801
7.8711 3.3303 1.3907 0.6594 0.2729 -0.1782 -0.3884
0.1362 0.0527 0.0236 0.0090 -0.0004 -0.0143 -0.0218
1.2633 0.8007 0.6546 0.5543 0.5208 0.4037 0.3547
1.1027 1.1953 0.7200 0.4239 0.1843 -0.1478 -0.3211
0.0256 0.0228 0.0108 0.0035 -0.0032 -0.0131 -0.0187
0.8859 0.6974 0.5870 0.5754 0.6399 0.3731 0.2826
0.2643 0.1023 0.1106 -0.0416 -0.2247 0.1995 0.0241
0.0002 -0.0068 -0.0029 -0.0077 -0.0160 0.0003 -0.0047
50 100 150 200 250 500 1000
1.1512
0.9526 0.8626 0.8322 0.8124 0.7788 0.7426
4.9371 1.6496 0.3856 -0.2357 -0.5348 -1.1148 -1.2944
0.0428 -0.0035 -0.0203 -0.0301 -0.0359 -0.0479 -0.0518
0.3964 0.5254 0.4366 0.4681 0.5519 0.3783 0.3472
0.0485 -0.1075 -0.1390 -0.3456 -0.5409 0.0574 -0.1384
0.0064 -0.0100 -0.0078 -0.0151 -0.0222 -0.0067 -0.0121
1.1587 0.8302 0.6930 0.6290 0.6234 0.5499 0.5063
10.6314
5.5053
2.9162
1.0993
0.4238
-0.5278
-0.8184
0.1139 0.0471 0.0208 0.0021 -0.0090 -0.0268 -0.0333
1.4767 0.9739 0.7759 0.6780 0.6521 0.5343 0.4556
0.3899 1.0317 0.9129 0.5895 0.2444 -0.4227 -0.6417
-0.0102 0.0011 0.0003 -0.0046 -0.0114 -0.0236 -0.0276
0.6971 0.6672 0.5498 0.5559 0.6099 0.4093 0.3623
0.1149 -0.0897 -0.1608 -0.3655 -0.5278 0.0538 -0.1365
-0.0078 -0.0162 -0.0130 -0.0193 -0.0246 -0.0083 -0.0128
50 100 150 200 250 500 1000
0.6404 0.3057 0.2074 0.1603 0.1602 0.1270 0.1233
6.6985 3.7878 2.3201 1.6798 1.1944 0.4670 0.1356
0.1097 0.0679 0.0453 0.0345 0.0254 0.0100 0.0015
0.6121
0.4339 0.4792 0.4903 0.5091 0.1819 0.1057
0.0227 0.4604 0.2701 0.1043 -0.0343 0.5339 0.5177
0.0017 0.0109 0.0040 -0.0018 -0.0059 0.0128 0.0119
1.0037 0.4177 0.2730 0.1926 0.1992 0.1055 0.0784
11.7552
7.2752
4.4954
3.0110
2.0675
0.8581
0.3467
0.1591 0.1038 0.0717 0.0533 0.0389 0.0196 0.0079
1.4091 0.6287 0.3839 0.2641 0.2532 0.1325 0.0893
0.9300 1.8584 1.9209 1.8022 1.4024 0.7466 0.3217
0.0156 0.0394 0.0406 0.0377 0.0295 0.0169 0.0071
0.9293 0.6244 0.5920 0.5702 0.5717 0.2199 0.1247
0.1141
0.5242 0.2946 0.1333 -0.0091 0.5181 0.5136
-0.0063 0.0068 0.0011 -0.0036 -0.0073 0.0112 0.0113
50 100 150 200 250 500 1000
0.7510 0.3105 0.3283 0.2099 0.2275 0.1828 0.1476
7.9644 5.8365 3.4028 2.9491 1.8875 0.7228 0.2229
0.0905 0.0692 0.0406 0.0373 0.0244 0.0081 0.0007
0.3044 0.3377 0.4856 0.4213 0.4184 0.1918 0.1720
-0.0100 0.1576 -0.0614 -0.0560 -0.1450 0.2020 0.4073
0.0053 0.0021 -0.0064 -0.0062 -0.0077 0.0015 0.0044
1.0016 0.4839 0.3546 0.2381 0.2428 0.1329 0.0791
17.0148
12.2917
8.6056
6.0182
4.3372
1.6237
0.6492
0.1599 0.1128 0.0796 0.0632 0.0472 0.0218 0.0096
1.4711
0.7972 0.5630 0.3530 0.3183 0.1614 0.0887
0.6111
1.4888 1.8400 2.4052 2.2085 1.3081 0.6102
-0.0013 0.0181 0.0227 0.0327 0.0290 0.0179 0.0088
0.6719 0.5115 0.6109 0.5157 0.4790 0.2385 0.1914
0.0439 0.2013 -0.0267 -0.0249 -0.0962 0.1780 0.4067
-0.0061 -0.0031 -0.0097 -0.0085 -0.0089 -0.0003 0.0038
50 100 150 200 250 500 1000
0.6470 0.3333 0.2436 0.1539 0.1581 0.0856 0.0604
5.5044 2.5247 1.4603 1.0055 0.6795 0.3212 0.1184
0.2278 0.1206 0.0759 0.0589 0.0412 0.0204 0.0067
0.5451 0.4569 0.4206 0.4435 0.4363 0.1461 0.0730
0.4398 0.2635 0.2336 0.0369 -0.0472 0.5693 0.3805
0.0442 0.0169 0.0163 -0.0027 -0.0088 0.0383 0.0257
0.9605 0.4849 0.3433 0.2277 0.2019 0.0978 0.0541
8.6682 4.1211 2.2477 1.4496 1.0266 0.4501 0.1913
0.3141 0.1599 0.1053 0.0796 0.0603 0.0300 0.0133
1.0958 0.5393 0.3757 0.2551 0.2300 0.1111 0.0625
1.4447 1.7448 1.3905 1.1010 0.8008 0.4035 0.1720
0.0834 0.0941 0.0772 0.0633 0.0485 0.0262 0.0115
0.8335 0.6018 0.5227 0.5127 0.4926 0.1750 0.0878
0.4091 0.3162 0.2319 0.0303 -0.0507 0.5688 0.3796
0.0121 0.0062 0.0067 -0.0101 -0.0145 0.0352 0.0242
50 100 150 200 250 500 1000
0.6140 0.3562 0.2326 0.1950 0.1257 0.1055 0.0716
7.6142 4.3665 2.6693 1.6498 1.4433 0.5230 0.2093
0.2150 0.1293 0.0876 0.0573 0.0550 0.0188 0.0065
0.3106 0.3879 0.3632 0.3831 0.3322 0.1714 0.1057
0.2057 0.1270 -0.1900 -0.2584 -0.1799 0.4174 0.5346
0.0289 0.0016 -0.0137 -0.0208 -0.0165 0.0160 0.0210
0.9251 0.5263 0.3453 0.2838 0.1810 0.1191 0.0632
12.6867
7.6389
4.6530
2.8226
2.2602
0.8336
0.3884
0.3072 0.1810 0.1209 0.0844 0.0784 0.0312 0.0160
1.1159
0.6175 0.3940 0.3148 0.2129 0.1332 0.0725
1.0707 2.0318 2.0958 1.6056 1.5243 0.7311 0.3517
0.0333 0.0689 0.0704 0.0552 0.0583 0.0268 0.0137
0.5886 0.5437 0.4666 0.4609 0.3892 0.2057 0.1214
0.2560 0.1983 -0.1442 -0.2513 -0.1762 0.3961 0.5303
0.0003 -0.0100 -0.0223 -0.0278 -0.0221 0.0118 0.0191
50 100 150 200 250 500 1000
0.5456 0.2945 0.1463 0.1257 0.0829 0.0838 0.0161
7.2943 4.1175 2.6347 1.8119 1.4882 0.6181 0.3224
0.2396 0.1439 0.1066 0.0778 0.0665 0.0287 0.0170
0.5129 0.6107 0.5751 0.5207 0.3985 0.2951 0.0801
0.1875 0.3300 0.2452 0.0253 0.1005 0.2163 0.5628
0.0157 0.0101 0.0028 -0.0082 -0.0025 0.0051 0.0267
0.8056 0.4796 0.2828 0.2434 0.1842 0.1192 0.0286
12.7427
7.3733
5.0317
2.8158
2.2312
0.8505
0.4301
0.3507 0.2060 0.1533 0.1010 0.0830 0.0375 0.0217
1.2360 0.6888 0.3928 0.2986 0.2281 0.1453 0.0454
1.1529 1.8438 1.9872 1.7793 1.5328 0.7482 0.3875
0.0444 0.0748 0.0843 0.0742 0.0649 0.0323 0.0190
0.8404 0.7686 0.6942 0.6036 0.4688 0.3361 0.0986
0.3053 0.4205 0.2156 0.0326 0.1045 0.1959 0.5609
-0.0004 0.0035 -0.0055 -0.0128 -0.0066 0.0018 0.0255
50 100 150 200 250 500 1000
0.6720 0.2777 0.1709 0.1214 0.1023 0.0812 0.0486
8.8634 6.5039 4.2887 3.1787 2.5330 1.0523 0.5072
0.2082 0.1560 0.1125 0.0867 0.0718 0.0322 0.0161
0.3249 0.4394 0.4596 0.4665 0.3430 0.2729 0.1995
-0.0013 0.2362 0.0644 -0.1327 0.1276 -0.0724 0.4641
0.0092 0.0066 -0.0083 -0.0159 -0.0047 -0.0093 0.0092
1.0153 0.5192 0.2647 0.2429 0.1813 0.1200 0.0792
17.5598
12.4466
9.4685
6.0748
4.7362
1.6726
0.7237
0.3332 0.2254 0.1790 0.1277 0.1024 0.0455 0.0212
1.3750 0.8599 0.4694 0.3481 0.2613 0.1514 0.0927
0.9431 1.3023 2.0652 2.4949 2.3733 1.3147 0.6599
0.0138 0.0289 0.0577 0.0680 0.0639 0.0370 0.0187
0.6687 0.6414 0.5743 0.5472 0.4220 0.3129 0.2190
0.0618 0.1311 0.1148 -0.0944 0.1297 -0.0768 0.4620
-0.0120 -0.0095 -0.0138 -0.0200 -0.0094 -0.0120 0.0078
LSE
WLSE
MLE
ADE
CvME
n
a
a
a
S1
S2
S3
S4
S5
S6
S7
S8
Table 3: Average MSEs for all estimators
A
100 150 200 250 500 1000
8.2297 4.1556 2.8888 2.1614 1.7531 1.0220 0.6702
164.7673
45.3542
16.3444
8.9423
5.1202
2.4029
1.4077
0.0134 0.0097 0.0054 0.0035
100 150 200 250 500 1000
8.8167 4.4242 3.1118 2.5062 2.0678 1.3012 0.9071
217.3302
90.7887
41.5761
19.6554
14.2230
5.4485
3.5782
0.0646 0.0323 0.0211 0.0143 0.0117 0.0066 0.0048
100 150 200 250 500 1000
12.8929 6.2261
3.1126 2.5849
251.5244
120.4071
58.1565
33.9692
21.0839
5.8824
2.3354
0.0902 0.0488 0.0294 0.0208 0.0155 0.0064 0.0029
100 150 200 250 500 1000
3.2894 2.7217
0.7350
323.5838 233.2069 114.9577 91.4859 49.9875 15.3314 5.9916
0.0714 0.0480 0.0294 0.0245 0.0172 0.0074
100 150 200 250 500 1000
8.1641 3.5454 2.3815 1.6541 1.3997 0.6750 0.3345
195.7027
74.3962
34.2574
15.8051
9.0560
3.4245
1.4597
0.2060 0.1209 0.0797 0.0595 0.0268 0.0126
100 150 200 250 500 1000
3.6646 2.2539 1.6717 1.3400 0.6809 0.3562
310.9795 177.4051 83.5288 41.3850 29.7434
0.3789 0.2185 0.1373 0.0887 0.0714 0.0308 0.0154
100 150 200 250 500 1000
12.9186 6.2518 4.1932
2.6473 1.3850 0.6859
286.8685 140.0123
37.2524 27.0774
2.7299
0.3886 0.2125 0.1328 0.0905 0.0716 0.0304 0.0132
100 150 200 250 500 1000
362.1085 265.6164 140.1165
68.5264 19.2085 7.1666
0.3166 0.2145 0.1392 0.1041 0.0814 0.0353 0.0165
A
5.0257 3.6519 3.2908 2.9582 2.8691 1.6475 0.8923
11.1636 7.9128 6.6964 5.8082 5.1904 6.4392
0.0284 0.0172 0.0158 0.0143 0.0127 0.0127 0.0075
3.0613 2.7898 2.3110 2.0040 2.0892 1.4503 0.9311
10.8334 0.0187
13.0720 0.0137
10.2296 0.0116
6.8737 0.0091
7.0033 0.0093
9.4806 0.0100
7.4636 0.0074
6.4170 5.0548
2.3211 1.4838
9.1597 0.0163
10.9579 0.0147
9.5840 0.0120
7.0750 0.0096
5.8694 0.0088
7.6789 0.0096
6.5431 0.0074
2.9855 2.7306
7.9723 0.0099
11.5590 0.0093
11.2110 0.0084
10.5398 0.0077
8.8887 7.0680 9.4144
0.0071 0.0054 0.0061
12.6878 0.1316 9.2761 0.0764
2.8503 2.5443 1.4751 0.7874
5.8383 5.2502 7.3287 4.6437
0.0704 0.0564 0.0516 0.0553 0.0347
3.1524 2.7496 2.1236 1.9406 1.5852 1.1749 0.8043
13.3794 14.5714 8.1668 7.3814
.7517
0.0889 0.0580 0.0425 0.0383 0.0360 0.0411 0.0362
.3044
2.2580 1.4708
9.3243 0.0684
11.6613 0.0624
10.5405 0.0526
6.8701 0.0395
5.7995 0.0334
5.8288 0.0312
6.8897 0.0303
4.3368 3.5661 3.2768 2.4830 1.6788 1.5655
9.4767 0.0446
16.3254 0.0513
14.5328 0.0375
10.2293 0.0329
10.6151 0.0310
6.0184 0.0197
10.8404 0.0273
13.5481 6.0824 3.9830 2.8287 2.2584 1.1197 0.6056
510.6877 245.3384 64.0081 29.2083 12.0601 3.8997
0.0078 0.0039
13.7269
3.0617 2.5515 1.2919 0.7359
786.3757 478.1817 249.6613 69.6146
8.5585 4.1769
0.1495 0.0770 0.0476 0.0271 0.0200 0.0084 0.0048
21.4541
6.2919 4.5631
1.9815 0.9678
828.8226 515.7800 260.9974 137.2141 88.9140 14.4027
0.1873 0.1039 0.0624 0.0415 0.0295 0.0119 0.0049
21.7192 10.0976 6.6965 5.0877 4.2953 2.0690 1.0540
1370.9550
1050.0127
786.8659
423.5659
289.5450
49.0908
11.8543
0.1690 0.1098 0.0729 0.0518 0.0391 0.0146 0.0062
14.0401 5.5268 3.6924 2.5552 2.1205 0.9738 0.4722
549.4940 251.6307 108.0385 36.9380
2.2476
0.8162 0.3654 0.2098 0.1343 0.1001 0.0412 0.0187
13.7931 5.8716 3.5175 2.6789 2.1197 0.9909 0.5063
949.0247 604.6772 351.2598 137.1735 89.1849 18.2522 6.3992
0.7276 0.4157 0.2552 0.1655 0.1257 0.0507 0.0241
20.5358 9.7176
4.9175 3.8568 1.9551 0.9413
896.3676 495.3834 326.9165 118.5219 94.0344 13.9261 4.6254
0.7858 0.4156 0.2768 0.1600 0.1251 0.0473
22.7560 10.5993 6.6212 5.2374 4.0239 2.1088 1.0989
1438.6746
1111.1386
864.3592
430.0210
343.1026
52.8106
13.4083
0.7077 0.4408 0.3178 0.2116 0.1598 0.0608 0.0267
A
5.8939 3.9870 2.8786 2.3131 1.1672 0.6217
28.9090 32.5961 21.7640 14.9217 9.9242 4.0569 1.8678
0.0530 0.0414 0.0294 0.0222 0.0166 0.0082 0.0041
12.2312 6.0499 4.1946 3.1435 2.6280 1.3441 0.7322
43.6199 31.6162 24.2292 9.0206 4.5383
0.0364 0.0340 0.0292 0.0230 0.0188 0.0090 0.0050
17.4964 8.4520 5.7657 4.3428 3.7146 1.9674 0.9659
39.1557 36.5044 28.4078 11.5138 4.1395
0.0328 0.0343 0.0315 0.0271 0.0220 0.0109 0.0048
17.0598 8.5251 5.9103 4.6558 4.0458 2.0385 1.0497
29.9717 41.0548 46.8700
62.1376 29.1442 11.2937
0.0248 0.0258 0.0239 0.0259 0.0235 0.0125 0.0061
11.6743 5.0756 3.5101 2.4107 2.0613 0.9556 0.4699
29.8464 40.0784 27.8111 19.5292 13.3906 5.2053 2.1680
0.2232 0.1980 0.1456 0.1090 0.0848 0.0389 0.0183
11.1650 5.3194 3.2827 2.5211 2.0253 0.9762 0.5032
34.4827 56.4762 57.1584 40.1409
0.1613 0.1663 0.1426 0.1089 0.0929
14.8899 0.0473
0.0234
16.6925 8.6520 5.8300 4.6726
24.0555 37.3454 40.5734 37.1928 30.0171 11.9486 4.3075
0.1260 0.1412 0.1309 0.1107 0.0914 0.0439 0.0194
8.9670 5.7875 4.8011 3.7871 2.0804 1.0918
38.2959 48.2825 66.7278 63.5251 29.9620 12.3481
0.1059 0.0998 0.0983 0.1074 0.0945 0.0515 0.0255
10.612 8.3217
3.1087
5.6329 5.0758
2.9603 1.6769 6.4718 0.9029 3.8582
A
0.0253 0.0166 0.0152 0.0137 0.0123 0.0125 0.0074
3.5605 12.515
2.9681 11.4890
2.5483 9.0402
2.1611 6.7379
2.1822 6.9496
1.4850 9.4552
0.9446 7.5026
0.0171 0.0128 0.0112 0.0090 0.0091 0.0098 0.0074
7.1741 8.4168
5.6795 11.9140
4.9602 9.3776
4.0819 7.1352
3.6651 5.9658
2.3334 7.5252
1.4899 6.4957
0.0140 0.0145 0.0117 0.0094 0.0086 0.0093 0.0073
5.0078 7.4320
3.5890 11.5626
3.8311 11.7176
3.1638 10.8964
2.8216 9.0300
1.4884 6.9479
1.4281 9.4462
0.0083 0.0088 0.0083 0.0077 0.0070 0.0052 0.0061
5.7930 11.6472
3.9402 10.0963
3.4972 7.3340
2.9013 5.6422
2.5984 5.1582
1.4930 7.3074
0.7923 4.6257
0.1094 0.0746 0.0670 0.0531 0.0496 0.0543 0.0343
3.5131 13.2201
3.0771 16.3857
2.2514 9.5506
2.0465 7.5400
1.6268 6.9089
1.1975 9.4081
0.8103 9.7011
0.0762 0.0592 0.0429 0.0381 0.0351 0.0403 0.0358
7.0636 9.8315
6.4745 13.3001
5.2672 9.7465
4.0352 6.7307
3.0889 5.7940
2.2817 5.6618
1.4775 6.8678
0.0612 0.0617 0.0491 0.0383 0.0325 0.0302 0.0300
4.8009 8.7280
4.6587 14.2683
3.7190 14.6579
3.3280 10.6299
2.6327 10.5669
1.7198 6.0311
1.5768 10.8464
0.0384 0.0449 0.0373 0.0326 0.0303 0.0195 0.0271
ADE
B
0.1856
11.8089
0.0184
0.0246
1584
0.6629
1949
6045
22.5280
0.0202
13.5950
1.4285
5. Real Data Analysis
In this section, two applications to the real data sets are examined to demonstrate the applicability of the IBLL distribution. We compare the IBLL distribution with Log-Logistic (LL), Burr III (BIII), Burr XII (BXII), Weibull (W) and Lindley (L) for two data sets. The pdfs of these distributions are given in Table 4. The MLEs of parameters and standard (SE) of MLEs are obtained and reported in Tables 5-6 for two datasets. To select the best distribution some criteria and goodness of-fit statistics such as the estimated log-likelihood values " ( x) , Akaike information criteria (AIC), Bayesian
information criteria (BIC), consistent Akaike information criteria (CAIC), Hannan-Quinn information criterion (HQIC), Kolmogorov-Smirnov (KS), Anderson-Darling (AD) and Cramer von Mises (CvM) statistics and related p values (KS-pval, AD-pval, CvM-pval) are calculated for all distributions. The fitted cdfs for two data sets are plotted in Figures 4 and 5. It is easily seen from Tables 5-6 and Figures 4-5 that the IBLL distribution gives the best modeling for both datasets, according to all criteria. The IBLL distribution can be used and be a good alternative in the literature because of its superior modeling capability.
The first data set represents the remission times (in months) of a random sample of 128 bladder cancer patients and it can be found in [10]. The first data is given by 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 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, 16.62, 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, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.
The second data set represents the survival times (in days) of guinea pigs injected with different amount of tubercle bacilli and it can be consulted detail information in [3]. The second data is given by 12, 15, 22, 24, 24, 32, 32, 33, 34, 38, 38, 43, 44, 48, 52, 53, 54, 54, 55, 56, 57, 58, 58, 59, 60, 60, 60, 60, 61, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 76, 76, 81, 83, 84, 85, 87, 91, 95, 96, 98, 99, 109, 110, 121, 127, 129, 131, 143, 146, 146, 175, 175, 211, 233, 258, 258, 263, 297, 341, 341, 376.
Table 4: The pdfs list of the all distributions Distribution Pdf Range of the
parameters
IBLL
W
BXII
LL
BIII
f (x) = p xPp(1 + x*
f ( X ) = p p2 x -( *+1>(1 + x - * ( P2+1) f ( X ) = p p x* -1 (1 + x* ( P2 +1)
f ( * ) = ( Pi 1 P2 ) ( * 1 P2 )P eXP ( - ( * 1 P2 )P )
( P2 +1)
( P2 + 1)
p^ P2 > 0 P^ P2 > 0 P^ P2 > 0
p > 0
L
f (x) = p2 (x + 1) / (1 + p ) exp (-p^x )
P > 0
Table 5: The modelling results for first data set
IBLL LL BIII BXII W L
" (X ) -409.8744 -504.8603 -426.6864 -453.5166 -414.0869 -419.5299
AIC 825.7488 1011.7206 857.3729 911.0332 832.1738 841.0598
BIC 834.3048 1014.5726 863.0769 916.7372 837.8778 843.9118
CAIC 825.9423 1011.7523 857.4689 911.1292 832.2698 841.0916
HQIC 829.2251 1012.8794 859.6905 913.3508 834.4913 842.2186
KS 0.0345 0.5260 0.1017 0.2507 0.0700 0.1164
AD 0.1184 63.3436 2.9190 13.3638 0.9577 2.7853
CvM 0.0179 13.4609 0.4508 2.7195 0.1537 0.5191
KS-pval 0.9980 0.0000 0.1413 0.0000 0.5570 0.0623
AD-pval 0.9998 0.0000 0.0302 0.0000 0.3801 0.0353
CVM-pval 0.9986 0.0000 0.0531 0.0000 0.3789 0.0355
Pi 15.8105 0.7897 1.0333 2.3349 1.0478 0.1960
P" 0.4855 4.2070 0.2337 9.5607
P# 0.2312
SE of pi 3.2570 0.0556 0.0604 0.3541 0.0676 0.0123
SE of p2 0.1236 0.4054 0.0400 0.8529
SE of p3 0.0182
Table 6: The modelling results for second data set
IBLL LL BIII BXII W L
" (X ) -389.6891 -526.9707 -395.5659 -490.5493 -397.1477 -394.5197
AIC 785.3781 1055.9415 795.1318 985.0986 798.2953 791.0394
BIC 792.2081 1058.2182 799.6852 989.6519 802.8487 793.3160
CAIC 785.7311 1055.9986 795.3057 985.2725 798.4693 791.0965
HQIC 788.0972 1056.8478 796.9445 986.9113 800.1080 791.9457
KS 0.0871 0.7210 0.1512 0.4813 0.1465 0.1326
AD 0.5375 57.3053 1.4907 23.5566 2.3730 1.8706
CvM 0.0898 12.0148 0.2500 5.0353 0.4312 0.3452
KS-pval 0.6450 0.0000 0.0745 0.0000 0.0911 0.1592
AD-pval 0.7083 0.0000 0.1787 0.0000 0.0580 0.1085
CVM-pval 0.6384 0.0000 0.1884 0.0000 0.0596 0.1011
Pi 29.2029 0.3533 1.4165 48.7608 1.3932 0.0198
P" 1.2525 286.9916 0.0047 110.5552
P# 0.1294
SE of pi 5.8536 0.0322 0.1163 18.8737 0.1184 0.0016
SE of p2 0.5107 125.6638 0.0017 9.9344
SE of p3 0.0081
Figure 4: Empirical and fitted cdf plots for the first data
Figure 5: Empirical and fitted cdf plots for the second data
6. CONCLUSION
In this article, a new flexible distribution is introduced. The density and hazard functions of the new model are illustrated by various plots that are very flexible. Many properties related to the distribution have been obtained. Renyi entropy, which is an important measure of randomness, is examined. Some estimation techniques are used to investigate the parameter estimation problem. The performances of the estimators are examined by Monte Carlo simulation. Finally, bladder cancer and survival times of guinea pig data are modeled. As a result of the modeling, it was seen that the best distribution according to all criteria is the IBLL distribution for both data.
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