Poisson area-biased Ailamujia Distribution and its applications in environmental and medical sciences

Aijaz, Ahmad; Ul Ain, S. Qurat; Afaq, Ahmad; Tripathi, Rajnee Article Poisson area-biased Ailamujia Distribution and its applications in environmental and medical sciences Statistics in Transition new series (SiTns) Provided in Cooperation with: Polish Statistical Association Suggested Citation: Aijaz, Ahmad; Ul Ain, S. Qurat; Afaq, Ahmad; Tripathi, Rajnee (2022) : Poisson area-biased Ailamujia Distribution and its applications in environmental and medical sciences, Statistics in Transition new series (SiTns), ISSN 2450-0291, Sciendo, Warsaw, Vol. 23, Iss. 3, pp. 167-184, https://doi.org/10.2478/stattrans-2022-0036 This Version is available at: https://hdl.handle.net/10419/266327 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-sa/4.0/ STATISTICS IN TRANSITION new series, September 2022 Vol. 23, No. 3, pp. 167–184, DOI 10.2478/stattrans-2022-036 Received – 11.02.2021; accepted – 14.12.2021 Poisson area-biased Ailamujia Distribution and its applications in environmental and medical sciences Ahmad Aijaz 1 , S. Qurat ul Ain 2 , Ahmad Afaq 3 , Rajnee Tripathi 4 ABSTRACT In this paper, a new Poisson area-biased Ailamujia distribution has been formulated to analyse count data. It was created by combining two distributions: the Poisson and areabiased Ailamujia distributions, using the compounding technique. Several distributional properties of the formulated distribution were studied. Its ageing characteristics were determined and expressed explicitly. A variety of diagrams were used to demonstrate the characteristics of the probability mass function (pmf) and the cumulative distribution function (cdf). The parameter of the developed model was estimated by employing the maximum likelihood estimation approach. Finally, two data sets were used to demonstrate the effectiveness of the investigated distribution. Key words: compound technique, Poisson distribution, area-biased Ailamujia distribution, reliability analysis, order statistics, maximum likelihood estimator. Mathematics subject classification: 60E05, 62E15. 1. Introduction In probability distributions, discrete distributions are very essential. Researches are focused extensively in past years to build new discrete models for assessing count data. There are a variety of procedures for developing new distributions in the statistics literature. Extensions to classical distributions can be made by adding additional parameters to them. Transmutation, discretization of continuous distributions, Marshall-Olkin method, compounding, and other approaches were examples. Classical distributions frequently fail to offer an acceptable fit to observable data. This became imperative for researchers to investigate new probability models in order to overcome 1 Corresponding author’s. Department of Mathematics, Bhagwant University, Ajmer, Rajasthan, India. E-mail: [email protected]. 2 Department of Mathematics, Bhagwant University, Ajmer, Rajasthan, India. 3 Department of Mathematical Sciences, Islamic University of Science & Technology, Awantipora, Kashmir. 4 Department of Mathematics, Bhagwant University, Ajmer, Rajasthan, India. © A. Aijaz, S. Qurat ul Ain, A. Afaq, R. Tripathi. Article available under the CC BY-SA 4.0 licence 168 A. Aijaz et al.: Poisson area-biased Ailamujia Distribution… the drawbacks of classical distributions. The compounding of distributions has attracted the attention of researchers over the last decade. The compounding approach is most commonly used when the parameter of one distribution is a random variable that follows another distribution, as in the case of count data. The compounding of distributions occurs when two separate distributions are combined. It makes no odds whether they are discrete or continuous in character. Based upon parent distribution, the resultant distribution from compounding may be continuous or discrete. The concept of weighted models can be traced back from Fisher (1935). Later on weighted models were briefly discussed by C.R. Rao (1964), when sample observations have an unequal probability of choosing. Thus, in such situation we add weights to the distribution to model bias. Suppose Y denotes random variable with pmf  yp , then pmf of weighted variable w Y is defined by     0;;  y ywE yfyw yP  where  k yyw is a non–negative weight function. For 2k we get area-biased distributions. In this study, we have used compounding approach to create a new distribution by combining Poisson and area-biased Ailamujia distribution. The newly established distribution is called “Poisson area-biased Ailamujia distribution”. Compounding distributions have extensive applications in several sectors of research such as biomedicine, insurance, engineering, and communications, among others. Researchers in this field have worked extensively, and they have made significant contributions to compounding research that has been tracked back to 1920. The inception of compounding models has been traced from Greenwood and Yule (1920). Sankaran (1970), Gerstenkorn (1993,1996), Mahmodi et al. (2010), Zamani and Ismail (2010), Gupta and Ong (2004), Shanker (2017), Shi(2012), Subhradev sen (2018), Giovani Carrara Rodrigues et al. (2018), Shanker et al. (2019), This study proposes a novel probability model known as the Poisson area-biased Ailamujia distribution, which is derived via the compounding process, and discusses its many mathematical aspects. 2. Definition of Poisson Area-Biased Ailamujia Distribution Consider a random variable Y follows Poisson distribution i:e Y ~   P and assume that the parameter of   P follows area-biased Ailamujia distribution with parameter  . The distribution obtained by compounding Poisson with area-biased Ailamujia distribution follows a discrete distribution whose probability mass function STATISTICS IN TRANSITION new series, September 2022 169 is denoted as PABAD   ,Y. The probability function of the obtained model PABAD   is given by the following theorem. Theorem 2.1. The probability mass function of a discrete Poisson area-biased Ailamujia distribution PABAD   ,Y is given as     y y yyy yYP              2 321 12 2 6 14 ; 0..2,1,0   y Proof: The probability mass function of the discrete Poisson area-biased Ailamujia distribution PABAD   ,Y may be obtained as If Y ~   P, the probability mass function (pmf) of the Poisson distribution is given by  0,...;2,1,0; !      y y e Yf y As the parameter  follows area-biased Ailamujia distribution with probability density function (pdf)   0,0; 6 2 ;23 4       eg We have     dgYfyYP ;. 0         de y ey 23 4 06 2 !         de y y12 0 3 4 !6 2        4 4 12 !3 !6 2    y y y          y yyy 12 321 12 2 6 14              ; 0,...;2,1,0   y (2.1) 170 A. Aijaz et al.: Poisson area-biased Ailamujia Distribution… The following six graphs illustrate the behaviour of pmf of the Poisson area-biased Ailamujia distribution for different values of parameter The corresponding cumulative distribution function (cdf) of the discrete Poisson area-biased Ailamujia distribution is given as   yYpyYpyYF rr  1      1 1 yw wP      4 2 232233 126 1261652 1224602083748 1               y yyy    0,..;2,1,0;   y (2.2) STATISTICS IN TRANSITION new series, September 2022 171 The following six graphs illustrate the behaviour of cdf of the Poisson area-biased Ailamujia distribution for different values of parameter 3. Statistical Measures of Poisson Area-Biased Ailamujia Distribution In this section several statistical measures of the Poisson area-biased Ailamujia distribution has been studied. They include are moments, moment generating function (mgf) and probability generation function (pgf). 3.1. Moments of Poisson Area-Biased Ailamujia Distribution. The th r factorial moment of the Poisson area-biased Ailamujia distribution is denoted as   r  and can be obtained by      r rYEE  , where     1...21  rYYYYY r       de y e y y y r23 00 4 !6 2                  de ry e ry ry r23 0 4 )!(6 2            Taking r y  in place of ywithin the bracket, we get          de y e y y r r 23 00 4 !6 2                      172 A. Aijaz et al.: Poisson area-biased Ailamujia Distribution…        0 23 4 6 2     de r r      rr rr   26 !3 2 4 6 2 4 4    (3.1) Substituting 𝑟1,2,3,4 in (3.1), the first four factorial moments can be obtained, and using the relationship between factorial moments and moments about origin, the first four moments about origin of the PABAD (2.1) are obtained as   2 1 , 2 2 25      , 3 2 3 15152       , 4 23 42 105180704      . The moments about mean of the Poisson area-biased Ailamujia distribution are obtained by using the relationship between moments about mean and moments about origin 2 2 12      3 3 3 132       4 23 42 9367028      The coefficient of variation (C.V), coefficient of skewness   1  , coefficient of kurtosis  2  , index of dispersion   of the Poisson area-biased Ailamujia distribution are determined as 2 21 .' 1     VC    3 3 2 3 2 3 125 132             2 23 2 4 2252 9367028              2 12 ' 1 2  STATISTICS IN TRANSITION new series, September 2022 173 Table1. The numerical values of the mean, variance, skewness, kurtosis, coefficient of variation and index of dispersion for some values of parameter  3.2. Generating Functions (pgf, mgf, ch.f) of Poisson Area-Biased Ailamujia Distribution In this section we study pgf, mgf and characteristics function (ch.f ) of the Poisson area-biased Ailamujia distribution. Theorem.3.2.1. If Y ~   PABAD then the probability generating function  tP Y is                        6 12 18 12 11 12 12 12 6 12 2 6 1 2 2 3 3 4 4 4 t t t t t t t t tpY     Proof: The probability generating function (pgf) of the Poisson area-biased Ailamujia distribution is defined as     yPttEtP y y Y    0   y y y t yyy 12 6116 12 2 6 123 4 0                 2  1  2  C.V  0.5 4.00 8.000 0.012 0.569 0.707 2.000 0.6 3.333 6.111 0.013 0.647 0.741 1.833 0.7 2.857 4.897 0.014 0.718 0.774 1.714 0.8 2.500 4.062 0.015 0.783 0.806 1.625 0.9 2.222 3.456 0.016 0.840 0.689 1.555 1 2.00 3.000 0.017 0.887 0.836 1.500 2 1.00 1.250 0.031 0.845 0.866 1.250 3 0.666 0.777 0.048 -0.037 1.118 1.166 4 0.500 0.562 0.064 -1.535 1.322 1.125 5 0.400 0.440 0.078 -3.468 1.658 1.100 174 A. Aijaz et al.: Poisson area-biased Ailamujia Distribution…  y y t yyy                  12 6116 12 2 6 1 0 23 4                                                           00 00 23 4 12 6 12 11 12 6 1212 2 6 1 yy yy yyyy tt y t y t y                                    t t t t tt t ttt 12 12 12 12 12 1212 12 412 12 2 6 1 23 2 2 4 23 4                                 6 12 18 12 11 12 12 12 6 12 2 6 1 2 2 3 3 4 4 4 t t t t t t t t     Theorem 3.2.2. If Y ~   PABAD then the moment generating function  tMY is                            6 12 18 12 11 12 12 12 6 12 2 6 1 2 2 3 3 4 4 4 t t t t t t t t Ye e e e e e e e tM     Proof: Since the moment generating function is a generalization of the probability generating function with the relationship given as    t YY ePtM  So that                            6 12 18 12 11 12 12 12 6 12 2 6 1 2 2 3 3 4 4 4 t t t t t t t t Ye e e e e e e e tM     Similarly, the relationship between mgf and ch.f is defined as    ittM YY                              6 12 18 12 11 12 12 12 6 12 2 6 1 2 2 3 3 4 4 4 it it it it it it it it Ye e e e e e e e it      STATISTICS IN TRANSITION new series, September 2022 181 Data set 2: Data on the macroscopic fresh-water fauna in dredge samples from the bottom of water ber Lake is due to Juday (1942) and Thomas (1949). Table 7.2. Microcalanus Nauplii Microcalnus Nauplii Observed frequency Expected frequency PABAD PSBAD PAD PD PLD PSD 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 2 4 3 5 8 16 13 12 13 15 15 9 9 7 4 4 6 2 0 2 1 0 1.13 3.17 5.60 7.81 9.76 11.07 11.67 11.76 11.42 10.75 9.86 8.86 7.82 6.71 5.82 4.93 4.13 3.43 2.83 2.31 1.88 1.51 1.21 2.03 4.63 7.06 8.96 10.24 10.92 11.09 10.86 10.35 9.64 8.81 7.92 7.05 6.20 5.40 4.66 3.99 3.40 2.88 2.42 2.03 1.70 1.40 4.46 7.38 9.16 10.11 10.46 10.39 10.03 9.49 8.83 8.12 7.40 6.68 5.99 5.34 4.79 4.18 3.68 3.22 2.81 2.45 2.13 1.85 1.60 0.02 0.10 0.47 1.50 3.60 6.10 11.05 15.15 18.18 19.39 18.62 16.24 12.99 9.60 6.59 4.21 2.53 1.43 0.77 0.39 0.19 0.09 0.04 7.09 8.67 9.56 9.94 9.97 9.71 9.29 8.75 8.15 7.57 6.88 6.25 5.65 5.07 4.54 4.05 3.60 3.19 2.82 2.48 2.18 1.9 1.68 7.79 9.4 10.30 10.58 10.49 10.13 9.60 8.96 8.26 7.54 6.83 6.14 5.50 4.90 4.34 3.83 3.37 2.96 2.59 2.26 1.97 1.70 1.48 Total 150 145.52 143.63 140.41 149.98 138.93 140.89 ML estimates (Standar d Error) 0.2083 (0.0101) 0.1562 (0.008) 0.1041 (0.006) 9.6000 (0.2529) 0.1907 (0.0120) 0.2024 (0.0125) Llog 435.85 443.80 459.20 441.62 467.25 461.16 AIC 873.71 889.60 920.41 885.24 936.51 924.33 AICC 873.74 889.63 920.44 885.27 936.53 924.36 BIC 876.72 892.61 923.42 888.25 939.52 927.34 2  23.38 32.45 57.54 109.12 75.22 71.37 df 11 12 11 9 13 12 p-value 0.31875 0.03275 3.0*10-5 1.3*10-15 9.5*10-8 4.0*10-7 182 A. Aijaz et al.: Poisson area-biased Ailamujia Distribution… The following histogram represents the number of micronuclei for the proposed model when compared with other models. From Table 1 and 2, it has been observed that the discrete Poisson area-biased Ailamujia distribution have the lesser AIC, AICC, llog2 , BIC and 2  values along with higher p-values as compared to size-biased Poisson Ailamujia distribution (PSBAD), Poisson Ailamujia distribution (PAD), Poisson distribution (PD), Poisson Lindley distribution (PLD) and Poisson Shanker distribution (PSD). It is evident from the above arguments that the proposed distribution provides better fit than the compared ones. 8. Concluding Remarks The aim of this study is to use compounding to develop a new distribution for count data termed the “Poisson area-biased Ailamujia distribution”. Different distributional features of the newly formed distribution have been obtained and analysed. The parameter of the proposed distribution has been estimated by the known method of maximum likelihood estimation. Eventually, the model's efficiency was assessed using two count data sets, and it was revealed that the Poisson area-biased Ailamujia distribution provides an appropriate fit for the two count data sets. 0 5 10 15 20 25 123456789101112131415161718192021222324 Observed Frequency PABAD PSBAD PAD PD PLD PSD STATISTICS IN TRANSITION new series, September 2022 183 References Fisher, R., (1934). The effect of methods of ascertainment, Annals Eugenics, 6, pp. 1325. Rao, C. R., (1965). On Discrete Distributions Arising Out of Methods of Ascertainment. Sankhya; the Ind. J. Statist, series A, 27(2/4), pp. 311324. Gerstenkorn, T., (1993). A compound of the generalized gamma distribution with the exponential one. Recherches surles deformations,16(1), pp. 510. Gerstenkorn, T., (1996). A compound of the Polya distribution with beta distribution. Random oper. And Stoch. 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