Submit manuscript...
eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 5 Issue 4

Discrete shanker distribution and its derived distributions

Munindra Borah, Junali Hazarika

Department of Mathematical Sciences, Tezpur University, India

Correspondence: Junali Hazarika, Department of Mathematical Sciences, Tezpur University, Napaam, Tezpur, Assam, India

Received: January 26, 2017 | Published: April 7, 2017

Citation: Borah M, Hazarika J. Discrete shanker distribution and its derived distributions. Biom Biostat Int J. 2017;5(4):146-153. DOI: 10.15406/bbij.2017.05.00140

Download PDF

Introduction

One parameter continuous Shanker distribution introduced by Shanker (2015 b) with parameter  is defined by its probability density function (pdf).

f(x:θ)= θ2θ2+1 (θ+x) eθx.  x>0. θ>0.f(x:θ)= θ2θ2+1 (θ+x) eθx.  x>0. θ>0. (1.1)

Discretization of continuous distribution

Discretization of continuous distribution can be done using different methodologies. In this paper we deal with the derivation of a new discrete distribution which takes values in {0,1, . . .} {0,1, . . .}  . This new distribution is generated by discretizing the continuous survival function of the Shanker distribution, which is may be obtained as

S(x)=xf(x:θ)dxS(x)=xf(x:θ)dx

= θ2+1+ θxθ2+1 eθx,    x>0.  θ>0.= θ2+1+ θxθ2+1 eθx,    x>0.  θ>0. (2.1)

S(x+1)= θ2+1+ θ(x+1)θ2+1 eθ(x+1),    x>0.  θ>0.S(x+1)= θ2+1+ θ(x+1)θ2+1 eθ(x+1),    x>0.  θ>0. (2.2)

The probability mass function (pmf) of discrete Shanker distribution may be obtained as

P(X=x)=S(x)S(x+1)P(X=x)=S(x)S(x+1)
=(θ2+1+ θx)(1 eθ)θ eθθ2+1e(θx),=(θ2+1+ θx)(1 eθ)θ eθθ2+1e(θx), x=0, 1,2, 3x=0, 1,2, 3 (2.3)

Probability recurrence relation 

Probability recurrence relation of discrete Shanker distribution may be obtained as

P(r+2)=eθ(2Pr+1 eθPr) , r  1P(r+2)=eθ(2Pr+1 eθPr) , r  1 (2.5)

Where P0=(θ2+1)(1 eθ)θ eθθ2+1P0=(θ2+1)(1 eθ)θ eθθ2+1 ,  and

P1=(θ2+1+ θ)(1 eθ)θ eθθ2+1 eθP1=(θ2+1+ θ)(1 eθ)θ eθθ2+1 eθ (2.6)

Factorial moment recurrence relation 

Factorial moment generating function (fmgf) may be obtained as

M(t)=G(1+t)M(t)=G(1+t)

= (θ2+1)(1 eθ)θ(θ2+1)(1eθ eθt)+θ(1 eθ)(θ2+1)(1eθ eθt)2(θ2+1)(1 eθ)θ(θ2+1)(1eθ eθt)+θ(1 eθ)(θ2+1)(1eθ eθt)2     (2.7)

 The more general form of factorial moment may also be written as

    μ'[r]=r! eθr[(θ2+1)(1eθ)+θr](θ2+1)(1eθ)r+1     (2.8)

Size- biased discrete shanker (SBDJ) distribution

If a random variable X  have discrete Shanker distribution with parameter θ then the pmf of the size-biased distribution may be derived as

fs(x;θ)=xPxμ , x=1, 2, 3,   (3.1)

Where Px  and μ  denote respectively pmf and the mean of discrete Shanker distribution. 

The pmf fs(x;θ) of size- biased discrete Shanker distribution with parameters θ may be derived from (3.1) as

fs(x,α)=xpxμ

=xeθ(x1)[(θ2+1+ θx)(1 eθ)θ eθ](1eθ)2[(θ2+1)(1eθ)+θ] x=1,2,3,  (3.2)

Recurrence relation of size- biased discrete shanker distribution

Probability generating function Gs(t) for Size- biased Discrete Shanker Distribution may be obtained as

Gs(t)=x=0txPx

     =t[(θ2+1)(1 eθ)θ eθ](1 eθ)2(1 eθt)+θ(1 eθ)3(1+eθt)[(θ2+1)(1 eθ)+θ](1 eθt)3 (3.3)

Probability recurrence relation for size- biased discrete  shanker distribution

Probability recurrence relation of Size- biased Discrete Shanker Distribution distribution may be obtained as

Pr=eθ[3Pr13eθPr2+eθPr3] for r>2  (3.4)

where

P1=[(θ2+1+ θ)(1 eθ)θ eθ](1eθ)2[(θ2+1)(1eθ)+θ] and (3.5)

P2=2e[(θ2+1+2θ)(1 eθ)θ eθ](1eθ)2[(θ2+1)(1eθ)+θ]

Factorial moment recurrence relation for size- biased discrete  shanker distribution

Factorial moment generating function Ms(t) of Size- biased discrete Shanker distribution may be obtained as

Ms(t)=Gs(1+t)

M(t)=(1+t)[(θ2+1)(1 eθ)θ eθ](1 eθ)2(1 eθeθt)+θ(1 eθ)3(1+eθ+eθt)[(θ2+1)(1 eθ)+θ](1eθ eθt)3 (3.6)

More general form  μ[r]=r! eθ(r1)[(θ2+1)(1eθ)(r+eθ)+θ(r2eθ)][(θ2+1)(1 eθ)+θ](1eθ)r    (3.7)

Factorial moment recurrence relation of Size- biased discrete Shanker distribution may be obtained as

    μ'[r]=eθA3[3A2rμ'[r1]3Ar(r1) eθ μ'[r2]+Ar(r1)(r2)e2θ μ'[r3]],  (3.7)

Where A = 1eθ.

μ[1]=[(θ2+1)(1+eθ)+θ][(θ2+1)(1 eθ)+θ]

μ[2]=2eθ[(θ2+1)(1eθ)(2+eθ)+θ(4eθ)][(θ2+1)(1 eθ)+θ](1eθ)2 (3.8)

μ[3]=6 e2θ[(θ2+1)(1eθ)(3+eθ)+θ(9eθ)][(θ2+1)(1 eθ)+θ](1eθ)3

Method of estimation of shanker distribution

The parameter θ of Shanker distribution has been estimated using Newton’s –Raphson method by considering appropriate initial guest value for θ . The function of θ can be expressed as

Fitting of discrete shanker distribution

Shanker et al.1 fitted Poisson distribution (PD), Poisson- Lindley distribution (PLD) and Poisson-Akash distribution (PAD) to eleven numbers of data sets covering ecology, genetics and thunderstorms. In this investigation discrete Shanker (DS) distribution has been fitted to all 11 data sets have been considered for a comparison (Tables 1-11).2-36

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

213

213

202.1

234

236.8

1

128

109.15

138

99.4

95.6

2

37

47.44

47.1

40.5

39.9

3

18

19

10.7

16

16.6

4

3

7.25

1.8

6.2

6.7

5

1

2.67

0.2

2.4

2.7

6

0

1.48

0.1

1.5

1.7

Total

400

400

400

400

400

 

Estimated θ

1.1621

0.6825

1.950236

2.260342

χ2

7.89

10.08

11.04

14.68

d.f.

3

2

2

2

p- vale

0.0468

0.0065

0.004

0.0006

Table 1 Observed and expected number of Homocytometer yeast cell counts per square observed by Gosset

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

38

38

25.3

35.8

36.3

1

17

22.4

29.1

20.7

20.1

2

10

11.02

16.7

11.4

11.2

3

9

4.97

6.4

6

6.1

4

3

2.13

1.8

3.1

3.2

5

2

0.88

0.4

1.6

1.6

6

1

0.36

0.2

0.8

0.8

7

0

0.14

0.1

0.6

0.7

Total

80

80

80

80

80

Estimated θ

1.0494

1.15

1.255891

1.620588

χ2

6.246

18.27

2.47

2.07

d.f.

4

2

3

3

p- vale

0.1815

0.0001

0.4807

0.558

Table 2 Observed and expected number of red mites on Apple leaves

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

188

187.98

169.4

194

196.3

1

83

85.03

109.8

79.5

76.5

2

36

33.03

35.6

31.3

30.8

3

14

11.88

7.8

12

12.4

4

2

4.07

1.2

4.5

4.9

5

1

1.99

0.2

2.7

3.1

Total

324

324

324

324

324

 

Estimated θ

1.2644

0.648148

2.043252

2.345109

χ2

0.367

15.19

1.29

2.33

d.f.

3

2

2

2

p- vale

0.9470

0.0005

0.5247

0.3119

Table 3 Observed and expected number of European corn-border of McGuire et al

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

268

268

231.3

257

260.4

1

87

87

92.8

126.7

93.4

2

26

28.23

34.7

32.8

32.1

3

9

8.01

6.3

11.2

11.5

4

4

2.18

0.8

3.8

4.1

5

2

0.58

0.1

1.2

1.4

6

1

0.15

0.1

0.4

0.5

7

3

0.05

0.1

0.2

0.3

Total

400

400

400

400

400

Estimated
θ

1.4870

0.5475

2.380442

2.659408

χ2

6.417

38.21

6.21

4.17

d.f.

3

2

3

3

p- vale

0.0930

0

0.1018

0.2437

Table 4 Distribution of number of Chromatid aberrations

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

413

413.01

374

405.7

409.5

1

124

134.97

177.4

133.6

128.7

2

42

38.92

42.1

42.6

42.1

3

15

10.49

6.6

13.3

13.9

4

5

2.71

0.8

4.1

4.6

5

0

0.68

0.1

1.2

1.5

6

2

0.22

0

0.5

0.7

Total

601

601

601

601

601

 

Estimated
θ

1.5385

0.47421

2.685373

2.915059

χ2

5.562

48.17

1.34

0.29

d.f.

3

2

3

3

p- vale

0.1350

0

0.7196

0.9619

Table 5 Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-60 μg/kg

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

200

200.01

172.5

191.8

194.1

1

57

70.01

95.4

70.3

67.6

2

30

21.51

26.4

24.9

24.5

3

7

6.17

4.9

8.6

8.9

4

4

1.69

0.7

2.9

3.2

5

0

0.45

0.1

1

1.1

6

2

0.16

0

0.5

0.6

Total

300

300

300

300

300

 

Estimated
θ

1.4798

0.55333

2.35334

2.62674

χ2

8.191

29.68

3.91

3.12

d.f.

3

2

2

2

p- vale

0.0422

0

0.1415

0.2101

Table 6 Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-75

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

155

155.01

127.8

158.3

160.7

1

83

82.63

109

77.2

74.3

2

33

37.2

46.5

35.9

35.3

3

14

15.41

13.2

16.1

16.5

4

11

6.08

2.8

7.1

7.5

5

3

2.32

0.5

3.1

3.3

6

1

1.35

0.2

2.3

2.4

Total

300

300

300

300

300

 

Estimated
θ

1.1301

0.853333

1.617611

1.963313

χ2

3.432

24.97

1.51

1.98

d.f.

4

2

3

3

p- vale

0.4883

0

0.6799

0.5766

Table 7 Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-90

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

187

187.01

155.6

185.3

187.9

1

77

87.72

117

83.5

80.2

2

40

35.21

43.9

35.9

35.3

3

17

13.07

11

15

15.4

4

6

4.62

2.1

6.1

6.6

5

2

1.58

0.3

2.5

2.7

6

1

0.79

0.1

1.7

1.9

Total

330

330

330

330

330

 

Estimated
θ

1.2345

0.751515

1.804268

2.139736

χ2

3.721

31.93

1.43

1.35

d.f.

4

2

3

3

Table 8 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for the month of June, January 1957 to December 1967

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

177

177.01

142.3

177.7

180

1

80

93.79

124.4

88

84.7

2

47

42.01

54.3

41.5

40.9

3

26

17.32

15.8

18.9

19.4

4

9

7.79

3.5

8.4

8.9

5

2

3.08

0.7

6.5

7.1

Total

341

341

341

341

341

Estimated
θ

1.1348

0.8739

1.583536

1.938989

χ2

6.972

39.74

5.15

5.02

d.f.

4

2

3

3

p- vale

0.1374

0

0.1611

0.1703

Table 9 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for the month of July, January 1957 to December 1967

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

185

184.99

151.8

184.8

187.5

1

89

92.42

122.9

87.2

83.9

2

30

39.27

49.7

39.3

38.6

3

24

15.39

13.4

17.1

17.5

4

10

6.74

2.7

7.3

7.6

5

3

2.19

0.5

5.3

5.9

Total

341

341

341

341

341

 

Estimated
θ

1.1828

0.809384

1.693425

2.038417

χ2

8.987

49.49

5.03

4.69

d.f.

4

2

3

3

p- vale

0.0414

0

0.1696

0.196

Table 10 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for the month of August, January 1957 to December 1967

No. of yeast cell per square

Observed

Expected frequency

frequency

DS

PD

PLD

PAD

0

549

549.01

449

547.5

555.1

1

246

274.27

364.8

259

249.2

2

117

117

116.54

148.2

116.9

3

67

45.67

40.1

51.2

52.3

4

25

17.04

8.1

21.9

23.2

5

7

7.16

1.3

9.2

10

6

1

2.31

0.5

6.3

7.3

Total

1012

1012

1012

1012

1012

 

Estimated
θ

1.1828

0.812253

1.68899

2.033715

χ2

16.824

119.45

9.6

9.4

d.f.

5

3

4

4

p- vale

0.0048

0

0.0477

0.0518

Table 11 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for Summer, January 1957 to December 1967

Conclusions

In this article, the discrete Shanker distribution has been introduced by discretizing the continuous Shanker distribution. We have studied some properties of the distributions. Further the applications of the distribution and goodness of fit of the distribution.

Acknowledgments

None.

Conflicts of interest

None.

References

  1. Shanker R. Amarendra distribution and its applications. American journal of Mathematics and Statistics. 2016;6(1):44–56.
  2. Adhikari TR, Srivastava RS. A size-biased poisson-lindley distribution. International Journal of Mathematical Modeling and Physical Sciences. 2013;1(3):1–5.
  3. Adhikari TR, Srivastava RS. Poisson-size-biased lindley distribution. International Journal of Scientific and Research Publication. 2014;4(3):1–6.
  4. Borah M.The Genenbauer distribution revisited: Some recurrence relation for moments, cumulants, etc., estimation of parameters and its goodness of fit. Journal of Indian Society of Agricultural Statistics. 1984;36(1):72–78.
  5. Borah M, Begum R. Some properties of Poisson-Lindley and its derive distributions. Journal of the Indian Statistical Association. 2000;4:13–25.
  6. Borah M, Deka Nath A. A study on the inflated poisson lindley distribution. Journal of Indian Soc. Ag. Statistics. 2001;54(3) 317–323.
  7. Dutta P, Bora M. Some properties and application of size-biased poisson- lindley distribution. International Journal of Mathematical Archive. 2014a;5(1):89–96.
  8. Dutta P, Borah M. Zero-modified poisson-lindley distribution. Mathematics and Computers in Simulation. (2014b);79(3):279–287.
  9. Fisher RA. The effects of methods of ascertainment upon the estimation of frequency. Ann Eugenics. 1934;6(1):13–25.
  10. Ghitany ME, Atiech B, Nadarajah S. Lindley distribution and its applications, Mathematics and Computers in Simulation. (2008a);78(4):493–506.
  11. Ghitany ME, Al-Mutari DK. Size-biased Poisson - lindley distribution and its application. Metron-International Journal of Statistics LXVI. 2008b;(3):299–311.
  12. Ghitany ME, Al-Mutairi DK. Estimation methods for the discrete Poisson-Lindley distribution. J Stat Comput Simul. 2009;79(1) 1–9.
  13. Gomez-Deniz E. Another generalization of the geometric distribution. Test. 2010;19(2):399–415.
  14. Gove HJ. Estimation and application of size-biased distributions in forestry. In: Amaro & D. Reed, editors. Modeling Forest systems A. CAB International, Wallingford, UK. 2003;201–212.
  15. Johnson NL, Kemp AW, Kotz S. Univariate discrete distributions. Hoboken NJ and John Wiley & Sons, USA; 2005..
  16. Kemp AW. The discrete half-normal distribution. MA Advances in Mathematical and Statistical Modelling. Birkhauser, Boston, USA; 2008:353–365.
  17. Krishna H, Pundir PS. Discrete Burr and discrete pareto distributions. Stat Methodol. 2009;6:177–188.
  18. Lappi J, Bailey RL. Estimation of diameter increment function or other relations using angle-count samples. Forest Science. 1987;33:725–739.
  19. Lindley DV. Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Series B. 1958;20(1):102–107.
  20. Mahmoudi E, Zakerzadeh H. Generalized poisson lindley distribution. Communications in Statistics-Theory and Methods. 2010; 39(10):1785–1798.
  21. Marshall AW, Olkin I. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997;84(3):641–652.
  22. Nakagawa T, Osaki S.The discrete Weibull distribution. IEE Trans Reliab. 1975;24(5):300–301.
  23. Patil GP, Ord JK. On size-biased sampling and related form-invariant weighted distributions. Sankhya. 1975;38(1):48–61.
  24. Patil GP, Rao CR. The weighted distributions: a survey and their applications. In Applications of Statistics. 1977;383–405.
  25. Patil GP, Rao CR. Weighted distributions and size-biased sampling with applications to wild life populations and human families. Biometrics. 1978;34:179–189.
  26. Rao CR. On discrete distributions arising out of methods on ascertainment, classical and contagious discrete distribution. In: Patil GP, editor. Statistical Publishing Society. 1965;320-332.
  27. Roy D. Discrete rayleigh distribution. IEEE Trans. Reliab. 2004;53(2):255–260.
  28. Sankaran M. The discrete poisson-lindley distribution. Biometrics. 1970;26:145–149.
  29. Shanker R, Mishra A. A two parameter poisson- lindley distribution. International journal of Statistics and Systems. 2014;9(1): 79–85.
  30. Shanker R. Shanker distribution and its Applications. International Journal of Statistics and Applications. 2015;5(6):338–348.
  31. Shanker R, Mishra A. A quasi Lindley distribution. African Journal of Mathematics and Computer Science Research. 2013; 6(4):64–71.
  32. Shanke R, Sharma S, Shanker R. A discrete two-parameter poisson-lindley distribution. Journal of the Ethiopian Statistical Association. 2012;21:15–22.
  33. Shanke R, Sharma S, Shanker R. Janardan distribution and its applications to waiting times data. Indian Journal of Applied research. 2013;3(8):500–502.
  34. Shanker R, Tekie Asehun Leonida. A new quasi poisson- lindley distribution. International Journal of Statistics and Systems. 2014;9(1):87–94.
  35. Shanke, R, Sharma S, Shanker R, et al. The discrete Poisson- Janardan distribution with applications. International Journal of Soft Computing and Engineering. 2014;4(2):31–33.
  36. Van Deusen PC. Fitting assumed distributions to horizontal point sample Diameters. Forest Science. 1986;32(1):146–148.
Creative Commons Attribution License

©2017 Borah, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.