International Journal of Scientific & Engineering Research Volume 4, Issue 12, December 2013
ISSN 2229-5518
1084
Abstract
This paper presents a new weighted distribution which is known as the double weighted rayleigh distribution (DWRD). This distribution is constructed and studied. The statistical properties of this distribution are discussed and obtained, including the mean, variance, coefficient of variation, harmonic mean, moments, mode, coefficient of skewness, coefficient of kurtosis, reliability function, hazard function and the reverse hazard function. Also the parameters of this distribution are estimated by the method of moment and the maximum likelihood estimation method.
Weighted distribution Theory gives a unified approach to dealing with model specification and data interpretation problems (biased data). Weighted distributions occur frequently in studies related to reliability, survival analysis, analysis of family data, Meta
To introduce the concept of a weighted distribution, suppose that X is a nonnegative random variable with its probability density function (pdf) f(x), then the pdf of the weight random variable XRwR is given by
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analysis, analysis of intervention data, biomedicine, ecology and
several other areas. (See Stene (1981), Gupta and Keating
(1985),Patil and Taillie (1989), and Oluyede and George (2002)).
w(X)f (x)
f w(x) = E(W(X)) , x ≥ 0
(1)
Many authors have presented important results on weighted
Where w(x) is a nonnegative weight function andE (w(x)) =
∞
distributions, Rao (1965) extended the basic ideas of the methods
of ascertainment upon estimation of frequencies by Fisher (1934)
∫ w(x)f(x)dx
0
, o <E (w(x)) <∞. The random variable XRwR is called
and introduced a unified concept of weighted distribution and
identified various sampling situations that can modeled by weighted distributions, These situations occur when the recorded observations can not be considered as a random sample from the original distributions, This mean that sometimes it is not possible to work with a truly random sample from population of interest. Zelen (1974) introduced weighted distribution to represent what he broadly perceived as a length-biased sampling. Patil and Ord (1976) studied a size biased sampling and related invariant weighted distributions. Statistical applications of weighted
the weight version of X and its distribution is related to that of X
and is called the weighted distribution with weight function w(x). Note that the weight function w(x) gave a different practical examples: such as when w(x) = xα , α> 0, Then the resulting distribution is called is a size biased version of X and the pdf of a
size random variable XRsR is defined as
α
distributions related to human population and ecology can be
x f s(x) =
f (x) , x ≥0
، α > 0
(2)
found in Patil and Rao (1978). Gupta and Tripathi (1996) studied
the weighted version of the bivariate logarithmic series
E(x α )
∞
distribution, which has applications in many fields such as:
ecology, social and behavioral sciences and species abundance
Where
E(xα) = ∫ xαf(x)dx
0
studies. Shaban and Boudrissa (2007) presented the weibull
length biased distribution with its properties and estimated of its parameters. Jing (2010) introduced the weighted inverse weibull and a beta-inverse weibull distribution throughout studying
In equation (2) when α = 1, then the weight function w(x) = x and the resulting distribution is called a length-biased distribution and the pdf of a length biased random variable XRLR is take the following form:
properties. Das and Roy (2011) discussed the length biased weighted generalized rayleigh distribution with its properties. Shi et al (2012) studied the theoretical properties of weighted
xf (x)
f L(x) = E(x)
, x ≥ 0
(3)
generalized rayleigh and related distributions. Rashwan (2013) presented the generalized gamma length biased distribution with its properties.
Where E(x)=µ is the mean of the original distribution and equal
∞
E(x)= ∫ x f (x)dx. (to know more details about different forms of a
0
weight function see Rao (1985) and Hewa (2011)).
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International Journal of Scientific & Engineering Research Volume 4, Issue 12, December 2013
ISSN 2229-5518
Rayleigh distribution is an important distribution in statistics and
= 2λx
2e−λx 2
(1 −e
−λα2x 2 )
operation research. It is applied in several areas such as health, agriculture, biology and other science. Also it is considered to be a very useful lifetime distribution in the reliability theory. If a
f D(x)
1 π / λ [1 − (
2
1
1 +α2
3
) 2 ]
random variable X follow the rayleigh distribution with a scale parameterλ, then the probability density function is given by
2
f(x) = 2 λx e-λx ،x≥ 0 ،λ> 0 (4)
= 4λ
3 2 x 2e−λx 2
(1 −e
1
−λα2x 2 )
3
with the cumulative distribution function (pdf) is
π[1 − (
) 2 ]
1 +α2
x 2 f(x) = ∫ 2λte−λt
o
2 dt = 1 −e−λx
2 2
let
1 k =
1 +α
2 , then
and f(αx) = 1 – e-λα x (5)
This paper develops a new weighted distribution which is known
f (x) =
3
4λ 2 x
2e−λx 2
(1 −e−λα2x 2 )
، x ≥0
، λ, α > 0
(8)
the double weighted distribution especially, the double weighted D
rayleigh distribution (DWRD). Properties of DWRD are
discussed and the parameters of this distribution are estimated
and obtained by using moment method and the maximum
3
π (1 −k 2 )
likelihood method. A numerical example is introduced for illustration.
The double weighted distribution is defined as follow
w(x)f (x)F(αx)
In this section, we present the statistical properties of DWRD throughout computing the mean, variance, coefficient of variation, harmonic mean, moments, mode, coefficient of skewness, coefficient of kurtosis, cumulative distribution function, reliability function, hazard function and the reverse
hazard function as follow:
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f D(x) = ،
D
x ≥0
، α > 0
(6)
- The mean of this distribution is
Where
∞
w D= ∫
0
w (x) f(x) F (αx) dx and the first weight is
∞
E(x) = µ = ∫
xf D(x)dx =
3
4λ 2
3
x3e−λx 2 (1 −e−λα2 x 2 )dx
w(x)and the second weight is(αx),F(αx) depend on the original
0 π (1 −k
2 ) 0
2 2 2
distribution f(x), f(x) = 2λxe-λx ،F(αx) = 1 – e-λα x . Taking
weight function as w(x) = x, Then
Let y = λ x2، x 2=
3
λ / λ
، dx = dy
2 λy
، y ≥ 0
∞ 2 −λx 2
−λα2x 2
∴ µ =
4λ 2
1
Γ(2) −
1 Γ(2)( 1 )2
w D= ∫
0
2λx e
(1 −e
)dx
3
π (1 −k 2 )
2λ2
2λ2
1 +α2 
Let y = λ x2، x 2= y ، x =
λ
y / λ
، dx = dx
2 λy
، y ≥ 0
2(1 −k 2)
= 3
(9)
1 
= Γ( 3 )1 − (
λ
1
(1 +α2)
3
) 2
λπ (1 −k 2 )
- The variance
1
∴WD= 2
π / λ 1 − (
1 ) 3 2
(1 +α2)
(7)
σ2 = E (x2) – E2 (x)
3 5
E(x
2) =
4λ 2
3
x 4e−λx 2
(1 −e
−λα2x 2
)dx =
3(1 −k 2 )
3
Using the equation 4,5 and 7, and substituting in equation 6 after considering w(x)= x, we obtain the probability density function of the double weighted rayleigh distribution as follow:
π (1 −k
2 ) 0
5 3
2λ(1 −k 2 )
Then
σ2=
3π(1 −k
2 )(1 −k
2λπ(1 −k
2 ) − 8(1 −k 2)2
3
2 )2
(10)
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International Journal of Scientific & Engineering Research Volume 4, Issue 12, December 2013
ISSN 2229-5518
- The coefficient of variation is
The following table gives the mode value of DWRD with
CV = σ =
µ
5 [3π(1 −k
2 )(1 −k
3 1
2 ) − 8(1 −k 2)2] 2
2
(11)
different values of the parameters of λ and α
2
- The harmonic Mean is
3
π (1 −k )
∞
1 = E( 1 ) = 4λ 2
−λx 2
− −λα2x 2
H x
π (1 −k
3 ∫ xe
2 ) 0
(1 e
)dx
3
and the following Table shows the mean, variance, coefficient of
∴ 1 = 2
λ (1 − k)
∴ H = 2
π (1 −k 2 )
(12)
variation (CV) and the mode with different parameters values of λ and α
- The shape
π (1 −k 2 )
The shape of the pdf given in equation (8) can be clarified by studying this function defined over the interval [0, ∞] and the behavior of its derivatives as follows:The limits of the pdf of DWRD is given by
Lim f
D(x) =
4λ 2
3
Lim x
2e−λx 2
(1 −e−λα2 x 2 ) = 0
(13)
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π (1 −k 2 )
4 2 0.59498
3 0.57694
0.05047
0.053169
0.37758
0.39967
0.50034
0.500001
Because
Lim e
x→∞
−λx 2
=0 and
Lim e
x→∞
−λα2 x 2 = 1
6 0.56644
10 0.56483
8 2 0.42071
3 0.40795
6 0.40053
0.055778
0.056389
0.025236
0.026584
0.027889
0.41694
0.42042
0.37759
0.39968
0.41695
0.49988
0.48799
0.35419
0.353539
0.35346
by taking the logarithm of fRDR(x), Then
10 0.39939
0.028164
0.42019
0.32299
Ln f D(x) = ln
3
4λ 2
3
2 −λα
+ 2 ln x − λx + ln(1 −e
2 x 2 )
From the above table note that:
- Value of the mean and the mode decreases at fixed λ and increasingα.
π (1 −k
2 )
- Value of the variance and the coefficient of variation (CV)
increases at fixed λ and increasingα.
Differentiating lnfRDR (x) with respect to x, we obtain
- Value of the mean is more than value of the mode (mean >
mode), this refers to the distribution is skewed to the right.
f ′(x) =f D(x) 2
− 2λx +
=2λα x
e−λα x
(14)
- The rth
1 − λα x
E(x r ) =
3
4λ 2
∞
∫ x r+2e−λx
(1 −e−λα2x 2
)dx
and equating this derivative to zero gives:
3
π (1 −k 2 ) 0
2 − 2λx +
2λα2x
e−λα2x 2
= 0
(15) dy
x 1 − λα2x 2
Let y = λ x2، x 2=
3
y / λ
، dx =
2 λy
، y ≥ 0
r +3
by solving the nonlinear equation in (15) with respect x we
obtain the mode.
E(x r ) =
4λ 2
3
1
r +3
Γ( r + 3 ) −
2
1
r +3
Γ( r + 3 )(
2
1 ) 2
1 +α2
π (1 −k
2 ) 2 λ 2
2 λ 2
2Γ( r + 3 )(1 −k
r +3
2 )
2
=
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r 3
λ 2 π (1 −k 2 )
(16)
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International Journal of Scientific & Engineering Research Volume 4, Issue 12, December 2013
ISSN 2229-5518
For the case r = 1 ، 2 ، 3 and 4 we have μ 4
− E(x) =µ′ =
1
2(1 −k 2)
3
− E(x 2) =µ′ =
5
3(1 −k 2 )
The ku is defined by
ku = − 3
σ 4
λπ (1 −k 2 )
2 3
2λ(1 −k 2 )
The coefficient of kurtosis for the DWRD is given by:
(17)
7
7
15π2 (1 −k
2 )(1 −k
3
2 )3−128π(1 −k3)(1 −k 2)(1 −k
3
2 )2
− E(x3)
= 3 =
λ
4(1 −k3)
3
λπ (1 −k 2 )
− E(x 4) =µ′ =
4
15(1 −k 2 )
3
4λ2 (1 −k 2 )
ku =
+ 144π(1 −k 2)(1 −k
5
3
2 )(1 −k
3
5
2 ) − 192(1 −k 2)4
2 2 2
− 3 (20)
[3π(1 −k
2 )(1 −k
2 ) −
8(1 −k ) ]
The first four central moments are
- µR1R = E (x – µ) = 0
- µ = E (x – µ)2
5 3 2 2
Note that, also the ku does not depend on λ but only depend on α. The following Table shows value of the SK and Ku at different
values of the parameter α because SK and Ku does not depend on
σ2= 3π(1 −k
3
2 )(1 −k
2πλ(1 −k
2 ) − 8(1 −k )
3
2 )2
λ but only depend on α
- µR3R = E (x – µ)
2[4π(1 −k3)(1 −k
(18)
3
2 )2 −9π(1 −k 2)(1 −k
5 7
2 )(1 −k
2 ) + 16(1 −k 2)3]
=
2πλ
λπ (1 −k
3
2 )3
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- µR4R = E (x – µ)
7 3 3
15π2(1 −k 2 )(1 −k 2 )3−128π(1 −k3)(1 −k 2)(1 −k 2 )2
3 5
+ 144π(1 −k 2)2(1 −k 2 )(1 −k 2 ) − 192(1 −k 2)4
3
From the above table note that
- The value of SK decreases when α increases but still more than zero. This means that, this distribution is positively skewed (skewed to the right).
- The value of Ku decreases when α increases but still more than
2
4λ π(1 −k
2 )4
zero. This means that, this distribution is a leptokurtic distribution (more peaked than normal curve distribution).
- The cumulative distribution function (cdf) is
Skewness is a measure of whether the distribution under study is
symmetric or asymmetric. Asymmetric mean that, the distribution is skewed to the right (positively skewed) or skewed
FD (x) =
3
4λ 2
3
x t 2e−λt 2
(1 −e
−λα2t 2
)dt
to the left (negatively skewed), this mean that the sign of the
coefficient indicates the direction of the skew. The formula for
SK is given
SK = μ3
π (1 −k
3
=
2 ) 0
Γ ( 3
, λx 2) −
3
Γ ( 3
, λx 2(1 +α2 )
4λ 2 1 k 2
I 2 I 2
σ3
Where µR3R is the third central moment and σ is the standard
deviation of DWRD. Then
3
π (1 −k 2 )
3
2λ 2
3
3
2λ 2
Then
2 2 [4π(1 −k3)(1 −k
3
2 )2 −9π(1 −k 2)(1 −k
5
2 )(1 −k
3
2 ) + 16(1 −k 2)3]
∴FD (x) =
2[ I ( 3 2 , λx 2) −k
2 ΓI ( 3 2 , λx 2(1 +α2)]
3
(21)
SK =
[3π(1 −k
5
2 )(1 −k
3 3
2 ) − 8(1 −k 2)2 ] 2
π (1 −k 2 )
Note that the SK does not depend on λ but only depend on α.
(19)
Where ΓRIR
is a generalized incomplete gamma function.
Kurtosis is a measure of whether the distribution is flatness or peakedness relative to a normal distribution. The ku may be equal to zero, positive and negative. A zero value indicates the
- The reliability function or survival function R(x).
This function can be derived using the cumulative distribution function and given by
R(x) = 1 – FRDR(x)
possibility of a mesokurtic distribution (that is normal high), a positive value indicates the possibility of aleptokurtic distribution
π (1 −k
=
3
2 ) − 2[ΓI ( 3 2 , λx 2) −k
3
3
2 ΓI ( 3 2 , λx 2(1 +α2 )]
(22)
(that is too tall) and a negative value indicates the possibility of
aplatykurtic distribution (that is too flat).
π (1 −k 2 )
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International Journal of Scientific & Engineering Research Volume 4, Issue 12, December 2013
ISSN 2229-5518
- The hazard or instantaneous rate function H(x).
The hazard function of x can be interpreted as instantaneous rate or the conditional probability density of failure at time x, given that the unit has survived until x. the hazard function is defined to be
exists for the MLE and the normal equations needs to be solved iteratively.
Note that, from equation (25) when α is known we obtain estimate
for λ, That is
4(1 −k 2)2
f (x)
λˆ m = 3
(27)
H(x) = D
R (x)
x 2π(1 −k
2 )2
4λ 3 2 x 2e−λx 2
(1 −e
−λα 2 x 2 )
- From equation (25) when λ is known, the estimate of α,
=
π (1 −k
3
2 ) − 2[ΓI ( 3 2 , λx 2) −k
3
2 ΓI ( 3 2 , λx 2(1 +α2 )]
(23)
can be obtained by numerical methods.
Let xR1R، xR2R، … ،xRnR be an independent random sample from the
- The reverse hazard function (φ (x))
The reverse hazard function can be interpreted as an
approximate probability of failure in [x ، x + dx], given that the
DWRD, then the likelihood function of DWRD is given by
3
n 4λ 2 x 2e−λx (1 −e−λα x ) 1
failure had occurred in [0, x]. The reverse hazard function φ
(x) is defined to be (Finkelstein (2002)).
f D(x1,x 2,...,x n ; λ, α) = ∏
i=1
3
π (1 −k 2 )
، k =
1 +α2
(28)
φ(x) = f D(x) =
FD (x)
4λα
3 2 x 2e−λx 2
3
(1 −e
−λα2x 2 )
Using the above equation, the log-likelihood function (Ln) is obtained
2[ΓI ( 3 2 , λx 2) −k
2 ΓI ( 3 2 , λx 2(1 +α2)]
3 n 3 n n 2
(24)
Ln(λ, α) = nLn4 +
n ln λ −
ln π − n ln(1 −k
2 ) + 2 ∑ xi−λ ∑ x i
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In this section, estimates of the two parameters (λ,α) of the double weighted rayleigh distribution are estimated and obtained. Method of moment (MOM) and maximum likelihood estimators
2
n
+ ∑
i=1
2
ln(1 −e−λα2 x 2 )
i=1
i=1
(29)
(MLE) are presented.
Let xR1R، xR2R، … ،xRnR be an independent random sample from the
By taking derivatives of the Ln (λ, α) with respect to the
parameters λ and α, we obtain the following equations:
−λα2x 2
∂ ln(λ, α) 3n
n 2 n
α2x 2e
DWRD with parameters λand α. The method of moment
= − ∑ x i+ ∑ 2 2
(30)
estimators are obtained by computing the population moments by using
∂λ 2λ
i=1
i=1
1 −e−λα x
E(x r ) = 2Γ( r + 3 )(1 −k
r +3
2
r
) /λ 2
3
π (1 −k 2 ), r = 1,2
and
2
∂ ln(λ, α) =
3α n
+ ∑
2λαx 2e
−λα2x 2
(31)
and equating to the sample moments
Mr = [ 1 ∑
Xr , r = 1,2].
∂α (1 +α2)[(1 +α2) 3 2
−1]
i=1
1 −e
−λα2x 2
n i=1
The following equations are obtained using the first and second sample moments.
Equating these equations to zero, then we get the normal equations as follow
2(1 −k 2)
1 n 
= ∑ Xi = X
(25)
3n
n 2 n
α2x 2e
−λα2x 2
(32)
3
π (1 −k 2 )
n i=1
− ∑ X + ∑
λ
2 2 = 0
and
5
3(1 −k 2 )
1 n X2
(26)
2
and
i=1
i=1
1 −e−λα x
3
2λ(1 −k 2 )
=
n i=1 i
3α n
+ ∑
2λαx 2e
−λα2 x 2
(33)
Solving the two equations (25) and (26) simultaneously
3
(1 +α2)[(1 +α2) 2 −1]
i=1
1 −e−λα2 x 2
(numerical method), we will get
λˆ m
and
αˆ m
as estimate of λ
and α respectively. These estimates are generally used as initial
values for the maximum likelihood method when no closed form
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International Journal of Scientific & Engineering Research Volume 4, Issue 12, December 2013
ISSN 2229-5518
This nonlinear equations system must be solved for λ and
αsimultaneously since a closed form solution is not known an
[4] Gupta, A.K., and Nadarajah, S. (1985), "relation for reliability measures under length biased sampling", Scandinavian
iterative technique is required to compute the estimators
λˆ Land
Journal of Statistics, Vol.13,49-56.
[5] Gupta, A.K., and Tripathi, R.C., (1996), "Weighted bivariate
αˆ L The system of equations is solved by netwon-raphson
iteration method. (See Jeffery (1992), vinod and Gaurav (2010)).
In this section, we analyze a data set from Gupta and Akman (1995) to illustrate the estimation methods of the parameters that used in this paper (Shaban and Boudrissa (2007)). These data are:
logarithmic series distributions: commun. Statist.Theory Meth. Vol.25, 1099-1117.
[6] Hewa, A.P., (2011), "Statistical properties of weighted generalized gamma distribution", Master
Dissertation, Statesboro, Georgia.
[7] Jeffery, D.H., (1992), "Maximum Liklihood estimation of dietary intake distributions", This paper prepared for
the human nutrition information service of USDA, IowaStateUniversity.
[8] Jing, X.K., (2010), "Weighted Inverse weibull and Beta-
inverse weibull distributions", Master Dissertation,
Statesboro, Georgia.
These data refer to millions of revaluations to failure for 23 ball bearings in fatigue test. Analysis of these data involved two steps:
The first a descriptive summary of a sample data is computed and presented in the following table:
Measure | Mean | Median | Mode | Variance | SK | Ku |
Value | 72.2243 | 67.8 | 68.64 | 1405.402 | 1.008 | 0.926 |
This summary shows a positively skewed and a leptokurtic
[9] Rashwan, N.I., (2013), "A Length-biased version of the
generalized Gamma distribution", Advances and
Applications in Statistics, Vol.32, 119-137.
[10] Oluyede, B.O., and George, E.O., (2002) "on stochastic
inequalities and comparisons of Reliability
measures for weighted distributions", Mathematical problems in Engineering, Vol.8, 1-13.
[11] Patil, G.P., and Ord, J.K., (1976), "On size biased sampling and related form invariant weighted distribution", The Indian Journal of Statistics, Vol.39, 48-61.
[12] Patil, G.P., and Rao, G.R., (1978), "Weighted distributions
IJSEanRd size biased sampling with applications to
distribution. The second, the parameters of the DWRD were
estimated numerically since there was no closed form for them. By using the moment estimates method, the system non linear
equations in (25) and (26) was solved numerically and yields
wildlife populations and human families", Biometrics, Vol.34, 179-189.
[13]Patil, G.P., and Taillie, C., (1989), "Probing encountered data, meta analysis, and weighted distribution
methods in statistical data analysis and inference", Y. Dodge, ed, Elsevier, Amsterdam.
estimators y parameters λ and α as follows:
λˆ m = 3.806 x10-4
[14] Rao, C.R., (1965), "on discrete distributions arising out of
and
αˆ m
= 0.123. these values used as initial values for the
methods of a ascertainment in classical and
contagious discrete distributions", G.P. Patil, ed.,
normal equations in (32) and (33) to obtain the maximum
pergamon press and statistical publishing society,
likelihood estimates, then for 18 iterated.
λˆ L = 0.005299 and
αˆ L = 1.540409 × 10−8
Calcutta.
[15]Rao, C.R., (1985), "Weighted distributions arising out of methods of ascertainment in a celebration of
statistics, A.C. Atkinson and S.E. Fienberg, eds., Springer-Verlag, New York.
This paper developed a new weighted distribution which is known as the double weighted Rayleigh distribution. Some statistical properties of this distribution are discussed and studies. It also estimates the parameters of this distribution using the method of moment and the maximum likelihood estimates method. The calculations are illustrated with the help of numerical example.
[1] Das, K.K. and Ray, T.D. (2011): on some Length-biased weighted weibull distribution", Pelagia Research Library, Advances in Applied Science Research, Vol.2, 465-475.
[2] Finkelstein, M.S., (2002). "on the reverse hazard rate", Reliability Engineering and System Safety, Vol.78,
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[3] Fisher, R.A., (1934), "The effects of methods of ascertainment
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[16] Shaban, S.A., and Boudrissa, N.A., (2007), "The weibull Length
biased distribution-properties and estimation
[17] | htt Shi, | p/interstat.statjournals.net/index.php. X., oluyeede, B.O., and Pararai, M., (2012), |
"Theoretical Properties of weighted generalized | ||
Rayleigh and related distributions", Theoretical | ||
Mathematics and Applications, Vol.2, 45-62. |
[18] Stene, J. (1981), "Probability distributions arising from the ascertainment and the analysis of data on human familes and other groups", statistical distributions in scientific work, applications in physical, Social and life Sciences, Vol.6, 233-244.
[19] Vinod, K., and Gaurav, S., (2010), "Maximum Likelihood
estimation in generalized gamma type model",
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[20] Zelen, M., (1974), "Problems in cell kinetics and the early
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