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Indian Journal of Pure & Applied Biosciences (IJPAB)
Year : 2021, Volume : 9, Issue : 3
First page : (151) Last page : (155)
Article doi: : http://dx.doi.org/10.18782/2582-2845.8689
On Methods of Estimation for Generalized Logarithmic Series Distribution and Its Application to Counts of Red Mites on Apple Leaves
Fehim J Wani1* 
, T. A. Raja1, S. Maqbool2, M. Iqbal Jeelani3 and Farheen Naqash4
1Division of Agricultural Economics & Statistics, FoA, Wadura, SKUAST-Kashmir
2Division of Animal Genetics and Breeding, FVSc & AH, SKUAST-Kashmir
3Division of Statistics and Computer Science, Chatha, SKUAST-Jammu
4School of Agricultural Economics & Horti-Business Management, FoH, Shalimar, SKUAST-Kashmir
*Corresponding Author E-mail: faheemwani@skuastkashmir.ac.in
Received: 19.04.2021 | Revised: 23.05.2021 | Accepted: 1.06.2021
ABSTRACT
The Generalized Logarithmic Series Distribution (GLSD) adds an extra parameter to the usual logarithmic series distribution and was introduced by Jain and Gupta (1973). This distribution has found applications in various fields. The estimation of parameters of generalized logarithmic series distribution was studied by the methods of maximum likelihood, moments, minimum chi square and weighted discrepancies. The GLSD was fitted to counts of red mites on apple leaves and it was observed that all the estimation techniques perform well in estimating the parameters of generalized logarithmic series distribution but with varying degree of non-significance.
Keyword: GLSD, Estimation, Parameters.
Full Text : PDF; Journal doi : http://dx.doi.org/10.18782
Cite this article: Wani, F. J., Raja, T. A., Maqbool, S., Iqbal Jeelani, M., & Naqash, F. (2021). On Methods of Estimation for Generalized Logarithmic Series Distribution and Its Application to Counts of Red Mites on Apple Leaves, Ind. J. Pure App. Biosci. 9(3), 151-155. doi: http://dx.doi.org/10.18782/2582-2845.8689
INTRODUCTION
The generalized logarithmic series distribution (GLSD) characterized by two parameters and was first obtained by Jain and Gupta (1973) and its probability function is given by:
Since GLSD is generalization of logarithmic series distribution (LSD). GLSD will reduce to logarithmic series distribution (LSD) when taking.
GLSD is a member of Consul and Shenton (1972) family of Lagrangian probability distribution and also Gupta (1974) modified power series distribution (MPSD). The same distribution has been obtained by many more authors. The distribution has found applications in various fields. Jain and Gupta (1973) applied it to the William’s data on number of papers by entomologists, Rao (1981) applied it to the study of correlation between two types of children in a family, and Hansen and Willenkens (1990) used it in the risk theory in a problem related to the total claim size upto time . The estimation of GLSD has been studied by many researchers, where as Gupta (1974) and Jani (1977) examined its minimum variance unbiased (MVU) estimation. Mishra (1979) discussed its maximum likelihood (ML) estimation, Jani and Shah (1979) discussed the method of moments of its estimation. Wani et al (2016) compared lagrangian probability distributions for counts of red mites on apple leaves in Kashmir valley.
ESTIMATION OF PARAMETERS
The various methods of estimation for estimating the parameters of Generalized Logarithmic Series distribution are as follows:
Maximum likelihood Estimation of GLSD
Consider a random sample of size N taken from the GLSD and let the observed frequencies be
NUMERICAL IILUSTRATION
The GLSD was fitted to count of red mites on apple leaves.
Table 1: Comparison of Observed frequencies with Expected frequencies of GLSD for Count of red mites on Apple Leaves
No. of mites/leaf |
Observed Frequency |
Expected Frequency of GLSD |
|||
Methods of Estimation |
|||||
ML |
Moments |
MC |
WD |
||
3 |
39 |
44.10 |
41.50 |
42.20 |
41.20 |
4 |
22 |
18.30 |
19.50 |
19.40 |
19.40 |
5 |
17 |
15.20 |
15.70 |
15.25 |
16.10 |
6 |
15 |
12.70 |
11.90 |
12.85 |
12.50 |
7 |
9 |
8.30 |
9.90 |
9.70 |
9.70 |
8 |
6 |
5.40 |
5.20 |
5.00 |
5.40 |
9 |
3 |
4.10 |
4.00 |
3.50 |
4.20 |
10 |
1 |
1.90 |
2.20 |
2.00 |
1.80 |
≥11 |
0 |
2.00 |
2.10 |
2.10 |
1.70 |
Total |
112 |
112.00 |
112.00 |
112.00 |
112.00 |
Parameter (Estimates) |
0.781 |
0.715 |
0.725 |
0.703 |
|
|
3.591 |
3.185 |
3.218 |
3.062 |
|
p-value |
0.892 |
0.922 |
0.920 |
0.930 |
|
In the above table the estimates of parameters and in different methods of estimation are ML (0.781, 0.767), Moments (0.715, 0.742), MC (0.725, 0.761) and WD (0.703, 0.735). The p-value of all the methods are non-significant and in agreement with the observed frequencies. We observed that all the estimation techniques, the ML, the moment, the MC and the WD method perform well in estimating the GLSD parameters.
REFERENCES
Consul, P. C., & Shenton, L. R. (1972). Use of Lagrangian expression for generating generalized probability distributions. SIAM J. Appl. Math, 23(2), 239-249.
Gupta, R. C. (1974). Modified power series distribution and some of its applications, Sankhya. Ser. B 35, 288-298.
Hanseen, B. B., & Willekens, E. (1990). The generalized logarithmic series distribution, Statistics and Probability Letters 9, 311-316.
Jani, P. N. (1977). Minimum variance unbiased estimate for some left truncated modified power series distribution. Sankhya, Series B 3(39), 258-278.
Jain, G. C., & Gupta, R. P. (1973). A logarithmic type distribution. Trabjos Estadist, 24, 99 -105.
Jani, P. N., & Shah, S. M. (1979). On fitting of the generalized logarithmic series distribution. Journal of the Indian Statistical Association 30(3), 1-10.
Kemp, A. W. (1986). Weighted discrepancies and maximum likelihood estimation for discrete distributions. Communication in Statistics - Theory and Methods 15(3), 783-803.
Mishra, A. (1979). Generalization of some discrete distributions. J Bihar. Math. Soc 11, 12-22.
Rao, B. R. (1981). Correlation between the numbers of two types of children in a family with the mpsd for the family size. Communications in Statistics – Theory and Methods 10(3), 249-254.
Wani, F. J., Raja, T. A., Maqbool, S., Khan, I., Bhat, M. A., & Jeelani, M. I. (2016). A comparison of generalized logarithmic series distribution, generalized poisson distribution and generalized negative binomial distribution for counts of red mites on apple leaves in kashmir valley. Int. J. Agricult. Stat. Sci 12(1), 117-120.
