Published: 08 September 2024

Generalised Poisson-new linear-exponential distribution

Binod Kumar Sah1
Suresh Kumar Sahani2
1Department of Statistics, R. R. M. Campus, Tribhuvan University, Janakpurdham, Nepal
2Department of Science and Technology, Rajarshi Janak University, Janakpurdham, Nepal
Corresponding Author:
Suresh Kumar Sahani
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Abstract

It is a compound discrete probability distribution which has two parameters and whose particular case is Poisson-New Linear-Exponential Distribution (PNLED) [1]. The statistical characteristics required for this distribution such as probability mass function (pmf), statistical moments, estimation of parameters and goodness of fit have been derived and explained nicely. To test theoretical reliability of this distribution, goodness of fit has been applied to some over-dispersed data which were used by other researchers and further, we found that this distribution looks more appropriate for statistical modelling than PNLED, generalised Poisson-Lindley distribution (GPLD) [1, 2] and Poisson-Lindley distribution (PLD) [3].

1. Introduction

A generalisation of any probability distribution is considered more appropriate when it is more reliable than its particular case as well as previously obtained generalised distribution with similar conditions in all aspects.

The proposed distribution is a compound distribution. For compounding process, at least two distributions are required. We have needed Generalised Poisson distribution (GPD) [4] and New Linear-exponential distribution (NLED) [5] to conduct compounding process. GPD [4] distribution is a discrete distribution whereas NLED [5] is a continuous distribution. GPD [4] has two parameters and in mixing process it plays role of an original distribution and its original parameter λ follows NLED [5]. The pmf of GPD [4] has been given by:

1
P1 Z; λ,θ=λ(λ+θz)z-1e-λ+θzz!,

where Z=0,1,2,; λ>0; | θ|<1.

The probability density function (pdf) of NLED [5] was given by:

2
f2z ; β=β21+πβ π+βe-βy, z>0, β>0.

Hence, Generalised Poisson-New Linear-exponential distribution (GPNLED) has been obtained by compounding GPD [4] with NLED [5]. The study of this papers mentioned in the following references [1, 2], [4], [6] have very much contributed to increase quality of the paper.

In this paper, the following works have been done. Construction and derivation of:

1) Probability Mass Function (pmf) of GPNLED.

2) Statistical Moments of GPNLED.

3) Estimation of Parameters of GPNLED.

4) Goodness of Fit and applications of GPNLED.

We have been studied the following papers to improve quality of this paper which have been mentioned in the following references [2], [7-14].

2. Results

The results obtained for this paper have been placed under the following sub-headings.

2.1. Probability mass function of GPNLED

GPD has two parameters λ and θ. λ is an original parameter of GPD which follows NLED. θ is an additional parameter of GPD which is versatile in nature and hence, GPD. Here, λ>0 and |θ| < 1. NLED has a single parameter β. The Probability mass of function of GPNLED is obtained by mixing GPD with NLED. This is how the pmf of GPNLED have been extracted:

3
P (Z;β,θ)=β2e-θzz!1+πβ0i=1z-1z-1iθz1iπλz-i+λz-i+1e-λ1+β dλ
=β2e-θzz! (1+πβ)i=0z-1(z-1)!i!(z-i-1)!θz1iπΓ(z-i+1)(1+β)z-i+1+πΓ(z-i+2)(1+β)z-i+2
=β2e-θzz(1+πβ)i=0z-1θizii!(z-i-1)!Γ(z-i+1)(1+β)z-i+1π+(z-i+1)(1+β)
=β2e-θz(1+πβ)i=0z-1θizi-1i!(z-i)(1+β)z-i+2π(1+β)+(z-i+1)
=β2e-θz(1+πβ)π(1+β)+(z+1)(1+β)z+2+β2e-θz(1+πβ)i=1z-1θiz(z-i)i-1i!π(1+β)+(z-i+1)(1+β)z-i+2
z=0,1,2, ... , λ>0, β>0.

Probability mass function of GPNLED can be obtained for each value such as z=0,1,2, ... can be obtained by using the Eq. (3) as follows:

4
Pz=0=β21+πβπ1+β+1(1+β)2,
5
Pz=1=β2e-θ1+πβπ1+β+2(1+β)3,
6
Pz=2=β2e-2θ1+πβπ1+β+3(1+β)4+θπ1+β+2(1+β)3,
7
Pz=3=β2e-3θ1+πβπ1+β+4(1+β)5+2θπ1+β+3(1+β)4+1.5θ2π1+β+2(1+β)3,
8
P(z=4)=β2e-4θ(1+πβ)π(1+β)+5(1+β)6+3θπ(1+β)+4(1+β)5+4θ2π(1+β)+3(1+β)4
+16θ3π(1+β)+26(1+β)3,
9
Pz=5=β2e-5θ1+πβπ1+β+6(1+β)7+4θπ1+β+5(1+β)6+15θ2π1+β+42(1+β)5
+50θ3π(1+β)+36(1+β)4+125θ4π(1+β)+224(1+β)3.

2.2. Statistical moments of GPNLED

Let μr' denotes rth moment about origin of GPNLED:

10
μ'r=EE(Zr/λ=β21+πβ0z=0zrλ(λ+θz)z-1e-λ+θzλzΓz+1π+λe-βλdλ,
11
μ'1=β21+πβ0z=0z1λ(λ+θz)z-1e-λ+θzλzΓz+1π+λe-βλdλ=β21+πβ0λ1+θπ+λe-βλdλ=(πβ+2)β(1+πβ)(1-θ).

Fig. 1Showing the mean of GPNLED

Showing the mean of GPNLED
12
μ'2=β21+πβ0z=0z2λ(λ+θz)z-1e-λ+θzλzΓz+1π+λe-βλdλ=β21+πβ0λ(1-θ)3+λ2(1-θ)2π+λe-βλdλ=[β(πβ+2)+2(1-θ)(πβ+3)]β2(πβ+1)(1-θ)3.
13
μ'3=β21+πβ0z=0z3λ(λ+θz)z-1e-λ+θzλzΓz+1π+λe-βλdλ=β21+πβ01+2θλ(1-θ)5+3λ2(1-θ)4+λ3(1-θ)3π+λe-βλdλ=(1+2θ)(πβ+2)β(1+πβ)(1-θ)5+2(πβ+3)β2(1+πβ)(1-θ)4+6(πβ+4)β3(1+πβ)(1-θ)5.
14
μ'4=β21+πβ0z=0z4λ(λ+θz)z-1e-λ+θzλzΓz+1π+λe-βλdλ=β21+πβ01+8θ+6θ2λ(1-θ)7+7+8θλ2(1-θ)6+6λ3(1-θ)5+λ4(1-θ)4π+λe-βλdλ=(1+8θ+6θ2)(πβ+2)β(1+πβ)(1-θ)7+2(7+8θ)(πβ+3)β2(1+πβ)(1-θ)6+(6)(6)(πβ+4)β3(1+πβ)(1-θ)5+24(πβ+5)β4(1+πβ)(1-θ)4.

Central moments of GPNLED:

15
μ2=Ez2-Ez2=βπβ+2+21-θπβ+3β2(πβ+1)(1-θ)3-πβ+2απβ+11-θ2=π2β3+π2β2+3πβ2+4π+2β+2-θπ2β3+4πβ+2[β(πβ+1)]2(1-θ)3.

Fig. 2Showing the variance of GPNLED

Showing the variance of GPNLED

Proof. Variance > Mean.

Or:

[π2β3+π2β2+3πβ2+(4π+2)β+2]-θ(π2β3+4πβ+2)[β(πβ+1)]2(1-θ)3>(πβ+2)β(1+πβ)(1-θ).

Or:

(π2β3+3πβ2+2β)+(1-θ)π2α2+4πα+2-πβ2+2β1+πβ1-θ>0.

Or:

(1-θ)π2α2+4πα+2-πβ2+2β1+πβ1-θ>0.

Or:

16
1-θπ2β2+4πβ+2π2β3+3πβ2+2β.

Hence the statement.

The third moment about the mean of this distribution can be obtained as:

17
μ3=Ez3-3Ez2Ez+2Ez3=1+2θπβ+2β(1+πβ)(1-θ)5+2πβ+3β2(1+πβ)(1-θ)4+6πβ+4β3(1+πβ)(1-θ)5-3βπβ+2+21-θπβ+3β2(πβ+1)(1-θ)3πβ+2β1+πβ1-θ+2πβ+2β1+πβ1-θ3
=[β2(1+πβ)2(1+2θ)(πβ+2)+3(1-θ)β(1+πβ)2(2πβ+6)+(1-θ)2(1+πβ)2(6πβ+24)]-[3(1-θ)β(1+πβ)(πβ+2)26(1-θ)2(1+πβ)(πβ+2)(πβ+3)+[2(πβ+2)3][β(1+πβ)]3(1-θ)5.

The third central moment (μ3) gives positive value, this distribution is positively skewed in shape.

18
μ4=Ez4-4Ez3Ez+6Ez2Ez2-3Ez4=1+8θ+6θ2πβ+2β(1+πβ)(1-θ)7+27+8θπβ+3β2(1+πβ)(1-θ)6+66πβ+4β3(1+πβ)(1-θ)5+24πβ+5β4(1+πβ)(1-θ)4-41+2θπβ+2β(1+πβ)(1-θ)5+2πβ+3β2(1+πβ)(1-θ)4
+6πβ+4β3(1+πβ)(1-θ)5πβ+2β1+πβ1-θ+6βπβ+2+21-θπβ+3β2(πβ+1)(1-θ)3πβ+2β1+πβ1-θ2-3πβ+2β1+πβ1-θ4
=[{1+8θ+6θ2β31+πβ)3πβ+2+{(7+8θ)(1-θ)β2(1+πβ)3(2πβ+6)}+{6(1-θ)2β1+πβ)36πβ+24+(1-θ)3(1+πβ)3(24πβ+120)]-4(1-θ)(πβ+2)[1+2θβ22+πβ+31-θβ2πβ+6+(1-θ)26πβ+24]+6(1-θ)2(πβ+1)(πβ+2)2[{β2+πβ+2(1-θ)3+πβ}]-3(1-θ)3(πβ+2)4[β(1+πβ)]4(1-θ)7.

The Eq. (18) is obtained to know about nature of distribution according to size and it is the fourth central moment. Co-efficient of skewness and kurtosis based on moments can be obtained as follows.

19
γ1=μ3(μ2)3/2=[β2(1+πβ)2(1+2θ)(πβ+2)+3(1-θ)β(1+πβ)2(2πβ+6)+(1-θ)2(1+πβ)2(6πβ+24)]-[3(1-θ)β(1+πβ)(πβ+2)2+6(1-θ)2(1+πβ)(πβ+2)(πβ+3)]+[2(1-θ)2(πβ+2)3] (1-θ)1/2[[π2β3+π2β2+3πβ2+(4π+2)β+2]-θ(π2β3+4πβ+2)]3/2.

From Eq. (19), it has been found that range of γ1 is (2)<γ1<). Hence, it is positively skewed:

20
β2=μ4(μ2)2=[(1+8θ+6θ2)β3(1+πβ)3(πβ+2)+(7+8θ)(1-θ)β2(1+πβ)3(2πβ+6)+6(1-θ)2β(1+πβ)3(6πβ+24)+(1-θ)3(1+πβ)3(24πβ+120)]-4(1-θ)(πβ+2)[(1+2θ)β2(2+πβ)+3(1-θ)β(2πβ+6)+(1-θ)2(6πβ+24)+6(1-θ)2(πβ+1)(πβ+2)2[β(2+πβ)+2(1-θ)(3+πβ)]-3(1-θ)3(πβ+2)4 (1-θ)1/2[[π2β3+π2β2+3πβ2+(4π+2)β+2]-θ(π2β3+4πβ+2)]3/2.

2.3. Estimation of parameters

This distribution has two parameters β and θ which can be obtained as

By using P(z=0) and μ1'. We have:

Pz=0=β21+πβπ1+β+1(1+β)2.

Solving it, we get:

21
f(β)=π(1-k)β3+(1+π-k-2πk)β2-k(2+π)β-k=0.

The Polynomial Eq. (21) can be solved by using Regula-Falshi method or Newton Rapson method, where k denote P(z=0).Substituting the estimated value of β in the following Eq. (22), we get an estimated value of θ:

22
1-θ=πβ+2μ1'β1+πβ.

μ1' and μ2': substituting the value of (1-θ) obtained from Eq. (22) in the expression of μ2', we get Polynomial Eq. (23) in β which can be solve by Newton-Rapson or Regla-Falsi method:

23
f(β)=μ2'(πβ+2)3-(μ1')2(πβ+1)(πβ+2)μ1'β2πβ+1+2πβ+3=0.

2.4. Goodness of fit and applications of GPNLED

This distribution can be used to test goodness of fit and to get possible inferential Statistics in the field related to accident proneness, risk management of production engineering, ecological sciences, agricultural fields related to events about insects, error per page, biological sciences, and other fields [15-17].

Applications of GPNLE model in the field of engineering. This distribution is applicable to fit better where Poisson, Generalised Poisson and Poisson mixtures of continuous distributions are applicable such as:

1) The count of α-particles emitted per unit of time is useful in analysis of any radio-active substances.

2) Number of telephone calls received on a given switch board per small unit of time.

3) In industrial production to find the proportion of defects per unit length, per unit area, etc.

4) In the field of reliability engineering.

In the following examples Chi-square goodness of fit have been applied by using GPNLE model.

Table 1Example (1)

Z
0
1
2
3
4+
Observed frequency
35
11
8
4
2

Table 2Example (2)

Number of accidents
0
1
2
3
4
5+
Observed frequency
447
132
42
21
3
2

Table 3Example (3)

Z
0
1
2
3
4
5
6
Observed frequency
200
57
30
7
4
0
2

The first example is related to the number of errors per page which was given by Kemp and Kemp [18]. The second example is related to the data of accidents to 647 women working on H. E. Shells in 5 weeks was given by Greenwood and Yule [19]. The data related to Class per exposure (μg/kg) which is included in example (3) was given by Catcheside et al. [20].

The first and second examples have been used in the Doctoral Thesis. The theoretical frequencies obtained by using PLD, GPLD and GPNLED have been placed in Table 4 and 5 to make comparison. In Table 6, the theoretical frequencies obtained by using PLD, PNLED and GPNLED have been placed for comparison [1-3].

Table 4Observed versus expected frequency of example (1)

Number of errors per page
Observed frequency
Expected frequency
PLD
GPLD
GPNLED
0
35
33.0
35.0
35.0
1
11
15.3
13.3
13.3
2
8
6.8
6.3
6.4
3
4
2.9
3.0
2.9
4+
2
2.0
2.4
2.4
60.0
60.0
60.0
647.0
μ1'
0.7833333
μ2'
1.85
β^
1.7434
1.9387733
1.642188744
θ^
0.1174867
0.09640955

Table 5Observed versus expected frequency of example (2)

Number of accidents
Observed frequency
Expected frequency
PLD
GPLD
GPNLED
0
447
441
447.0
447.0
1
132
143
1132.7
132.9
2
42
45
44.8
45.0
3
21
14
15.1
15.0
4
3
4
5.0
5.0
5+
2
1
2.4
2.1
647
647
647.0
647.0
μ1'
0.4652241
μ2'
0.9100646
β^
2.8563455
2.496217
θ^
0.0523222
0.0415098

Table 6Observed versus expected frequency of example (3)

Class per exposure (μg/kg)
Observed frequency
Expected frequency
PLD
PNLED
GPNLED
0
200
191.8
192.8
200.0
1
57
70.3
69.2
60.5
2
30
24.9
24.6
23.7
3
7
8.6
8.7
9.5
4
4
2.9
3.1
3.8
5
0
1.0
1.1
1.5
6
2
0.5
0.5
1.0
300
300.0
300.0
300.0
μ1'
0.55333333
μ2'
1.2533333
β^
2.353339
2.0501155
2.256957115
θ^
0.1002898421

3. Conclusions

1) This distribution will be over-dispersed if (1-θ)(π2β2+4πβ+2)(π2β3+3πβ2+2β).

2) The range of γ1 is (2)<γ1<). So, it is positively skewed in shape.

3) The range of β2 is 6<β2<. So, it is leptokurtic by size.

4) From the Tables 4 and 5, it is found that P-value obtained by using GPNLED is greater than those obtained by using GPLD [2] and PLD [3]

5) From Table 6, what we have observed that the P-value obtained by using GPNLED is bigger than those of PNLED [1] and PLD [3]

6) Hence, it is suggested to apply GPNLED instead of PLD, GPLD and PNLED in similar situation in all aspects.

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About this article

Received
15 June 2024
Accepted
27 July 2024
Published
08 September 2024
Keywords
distribution
Poisson-new linear-exponential distribution
probability distribution
moments
proposed distribution
over-dispersed
goodness of fit
Acknowledgements

The authors have not disclosed any funding.

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Author Contributions

Binod Kumar Sah: conceptualization, data curation, formal analysis, investigation, writing – original draft preparation. Suresh Kumar Sahani: conceptualization, data curation, formal analysis, investigation, supervision, writing – original draft preparation, writing – review and editing.

Conflict of interest

The authors declare that they have no conflict of interest.