scholarly journals On a Generalised Exponential-Lindley Mixture of Generalised Poisson Distribution

2020 ◽  
Vol 4 ◽  
pp. 33-42
Author(s):  
Binod Kumar Sah ◽  
A. Mishra
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial ◽  
Mixture Distribution ◽  
Discrete Data ◽  
P Value ◽  
Data Sets ◽  
Mixing Distribution ◽  
Original Distribution ◽  
Compound Distribution ◽  
First Four Moments

Background: A mixture distribution arises when some or all parameters in a mixing distribution vary according to the nature of original distribution. A generalised exponential-Lindley distribution (GELD) was obtained by Mishra and Sah (2015). In this paper, generalized exponential- Lindley mixture of generalised Poisson distribution (GELMGPD) has been obtained by mixing generalised Poisson distribution (GPD) of Consul and Jain’s (1973) with GELD. In the proposed distribution, GELD is the original distribution and GPD is a mixing distribution. Generalised exponential- Lindley mixture of Poisson distribution (GELMPD) was obtained by Sah and Mishra (2019). It is a particular case of GELMGPD. Materials and Methods: GELMGPD is a compound distribution obtained by using the theoretical concept of some continuous mixtures of generalised Poisson distribution of Consul and Jain (1973). In this mixing process, GELD plays a role of original distribution and GPD is considered as mixing distribution. Results: Probability mass of function (pmf) and the first four moments about origin of the generalised exponential-Lindley mixture of generalised Poisson distribution have been obtained. The method of moments has been discussed to estimate parameters of the GELMGPD. This distribution has been fitted to a number of discrete data-sets which are negative binomial in nature. P-value of this distribution has been compared to the PLD of Sankaran (1970) and GELMPD of Sah and Mishra (2019) for similar type of data-sets. Conclusion: It is found that P-value of GELMGPD is greater than that in each case of PLD and GELMPD. Hence, it is expected to be a better alternative to the PLD of Sankaran and GELMPD of Sah and Mishra for similar types of discrete data-sets which are negative binomial in nature. It is also observed that GELMGPD gives much more significant result when the value of is negative.

scholarly journals A Generalised Exponential-Lindley Mixture of Poisson Distribution

2019 ◽  
Vol 3 ◽  
pp. 11-20
Author(s):  
Binod Kumar Sah ◽  
A. Mishra
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Mixture Distribution ◽  
Discrete Data ◽  
Likelihood Method ◽  
P Value ◽  
Data Set ◽  
Lindley Distribution ◽  
First Four Moments

Background: The exponential and the Lindley (1958) distributions occupy central places among the class of continuous probability distributions and play important roles in statistical theory. A Generalised Exponential-Lindley Distribution (GELD) was given by Mishra and Sah (2015) of which, both the exponential and the Lindley distributions are the particular cases. Mixtures of distributions form an important class of distributions in the domain of probability distributions. A mixture distribution arises when some or all the parameters in a probability function vary according to certain probability law. In this paper, a Generalised Exponential- Lindley Mixture of Poisson Distribution (GELMPD) has been obtained by mixing Poisson distribution with the GELD. Materials and Methods: It is based on the concept of the generalisations of some continuous mixtures of Poisson distribution. Results: The Probability mass of function of generalized exponential-Lindley mixture of Poisson distribution has been obtained by mixing Poisson distribution with GELD. The first four moments about origin of this distribution have been obtained. The estimation of its parameters has been discussed using method of moments and also as maximum likelihood method. This distribution has been fitted to a number of discrete data-sets which are negative binomial in nature and it has been observed that the distribution gives a better fit than the Poisson–Lindley Distribution (PLD) of Sankaran (1970). Conclusion: P-value of the GELMPD is found greater than that in case of PLD. Hence, it is expected to be a better alternative to the PLD of Sankaran for similar type of discrete data-set which is negative binomial in nature.

scholarly journals A Generalised Poisson Mishra Distribution

2018 ◽  
Vol 2 ◽  
pp. 27-36
Author(s):  
Binod Kumar Sah
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial ◽  
Mass Function ◽  
Discrete Data ◽  
P Value ◽  
Probability Mass Function ◽  
Data Sets ◽  
Estimation Of Parameters ◽  
First Moment ◽  
First Four Moments

Background: “Mishra distribution" of B. K. Sah (2015) has been obtained in honor of Professor A. Mishra, Department of Statistics, Patna University, Patna (Sah, 2015). A one parameter Poisson-Mishra distribution (PMD) of B. K. Sah (2017) has been obtained by compounding Poisson distribution with Mishra distribution. It has been found that this distribution gives better fit to all the discrete data sets which are negative binomial in nature used by Sankarn (1970) and others. A generalisation of PMD has been obtained by mixing the generalised Poisson distribution of Consul and Jain (1973) with the Mishra distribution.Materials and Methods: It is based on the concept of the generalisations of some continuous mixtures of Poisson distribution.Results: Probability density function and the first four moments about origin of the proposed distribution have been obtained. The estimation of parameters of this distribution has been discussed by using the first moment about origin and the probability mass function at x = 0 . This distribution has been fitted to a number of discrete data-sets to which earlier Poisson-Lindley distribution (PLD) and PMD have been fitted.Conclusion: P-value of generalised Poisson-Mishra distribution is greater than PLD and PMD. Hence, it provides a better alternative to the PLD of Sankarn and PMD of B. K. Sah.Nepalese Journal of Statistics, Vol. 2, 27-36

Patterns of macroparasite aggregation in wildlife host populations

Parasitology ◽  
1998 ◽  
Vol 117 (6) ◽  
pp. 597-610 ◽  
Author(s):  
D. J. SHAW ◽  
B. T. GRENFELL ◽  
A. P. DOBSON
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Goodness Of Fit ◽  
Negative Binomial ◽  
Data Sets ◽  
Frequency Distributions ◽  
System A ◽  
Host Sex ◽  
Host Parasite

Frequency distributions from 49 published wildlife host–macroparasite systems were analysed by maximum likelihood for goodness of fit to the negative binomial distribution. In 45 of the 49 (90%) data-sets, the negative binomial distribution provided a statistically satisfactory fit. In the other 4 data-sets the negative binomial distribution still provided a better fit than the Poisson distribution, and only 1 of the data-sets fitted the Poisson distribution. The degree of aggregation was large, with 43 of the 49 data-sets having an estimated k of less than 1. From these 49 data-sets, 22 subsets of host data were available (i.e. host data could be divided by either host sex, age, where or when hosts were sampled). In 11 of these 22 subsets there was significant variation in the degree of aggregation between host subsets of the same host–parasite system. A common k estimate was always larger than that obtained with all the host data considered together. These results indicate that lumping host data can hide important variations in aggregation between hosts and can exaggerate the true degree of aggregation. Wherever possible common k estimates should be used to estimate the degree of aggregation. In addition, significant differences in the degree of aggregation between subgroups of host data, were generally associated with significant differences in both mean parasite burdens and the prevalence of infection.

scholarly journals Excess False Positive Rates in Methods for Differential Gene Expression Analysis using RNA-Seq Data

2015 ◽  
Author(s):  
David M Rocke ◽  
Luyao Ruan ◽  
Yilun Zhang ◽  
J. Jared Gossett ◽  
Blythe Durbin-Johnson ◽  
...  
Keyword(s):  
Linear Model ◽  
False Positive ◽  
Negative Binomial ◽  
False Positive Rate ◽  
Real Data ◽  
False Positives ◽  
P Value ◽  
Data Sets ◽  
Rna Seq ◽  
Positive Rate

Motivation: An important property of a valid method for testing for differential expression is that the false positive rate should at least roughly correspond to the p-value cutoff, so that if 10,000 genes are tested at a p-value cutoff of 10−4, and if all the null hypotheses are true, then there should be only about 1 gene declared to be significantly differentially expressed. We tested this by resampling from existing RNA-Seq data sets and also by matched negative binomial simulations. Results: Methods we examined, which rely strongly on a negative binomial model, such as edgeR, DESeq, and DESeq2, show large numbers of false positives in both the resampled real-data case and in the simulated negative binomial case. This also occurs with a negative binomial generalized linear model function in R. Methods that use only the variance function, such as limma-voom, do not show excessive false positives, as is also the case with a variance stabilizing transformation followed by linear model analysis with limma. The excess false positives are likely caused by apparently small biases in estimation of negative binomial dispersion and, perhaps surprisingly, occur mostly when the mean and/or the dis-persion is high, rather than for low-count genes.

scholarly journals An Alternative Distribution for Modelling Overdispersion Count Data: Poisson Shanker Distribution

2021 ◽  
pp. 108-120
Author(s):  
A Meytrianti ◽  
S Nurrohmah ◽  
M Novita
Keyword(s):  
Poisson Distribution ◽  
Count Data ◽  
Continuous Distribution ◽  
Mixture Distribution ◽  
Real Data ◽  
Data Set ◽  
Mixing Distribution ◽  
A Value ◽  
Mean And Variance ◽  
Parameter Values

Poisson distribution is a common distribution for modelling count data with assumption mean and variance has the same value (equidispersion). In fact, most of the count data have mean that is smaller than variance (overdispersion) and Poisson distribution cannot be used for modelling this kind of data. Thus, several alternative distributions have been introduced to solve this problem. One of them is Shanker distribution that only has one parameter. Since Shanker distribution is continuous distribution, it cannot be used for modelling count data. Therefore, a new distribution is offered that is Poisson-Shanker distribution. Poisson-Shanker distribution is obtained by mixing Poisson and Shanker distribution, with Shanker distribution as the mixing distribution. The result is a mixture distribution that has one parameter and can be used for modelling overdispersion count data. In this paper, we obtain that Poisson-Shanker distribution has several properties are unimodal, overdispersion, increasing hazard rate, and right skew. The first four raw moments and central moments have been obtained. Maximum likelihood is a method that is used to estimate the parameter, and the solution can be done using numerical iterations. A real data set is used to illustrate the proposed distribution. The characteristics of the Poisson-Shanker distribution parameter is also obtained by numerical simulation with several variations in parameter values and sample size. The result is MSE and bias of the estimated parameter theta will increase when the parameter value rises for a value of n and will decrease when the value of n rises for a parameter value.

scholarly journals The Negative Binomial – Weighted Garima Distribution: Model, Properties and Applications

2020 ◽  
pp. 1-10
Author(s):  
Winai Bodhisuwan ◽  
Pornpop Saengthong
Keyword(s):  
Parameter Estimation ◽  
Count Data ◽  
Negative Binomial ◽  
Likelihood Estimation ◽  
Real Data ◽  
Distribution Model ◽  
Data Sets ◽  
Factorial Moments ◽  
Special Cases ◽  
First Four Moments

In this paper, a new mixed negative binomial (NB) distribution named as negative binomial-weighted Garima (NB-WG) distribution has been introduced for modeling count data. Two special cases of the formulation distribution including negative binomial- Garima (NB-G) and negative binomial-size biased Garima (NB-SBG) are obtained by setting the specified parameter. Some statistical properties such as the factorial moments, the first four moments, variance and skewness have also been derived. Parameter estimation is implemented using maximum likelihood estimation (MLE) and real data sets are discussed to demonstrate the usefulness and applicability of the proposed distribution.

scholarly journals KAJIAN SIMULASI OVERDISPERSI PADA REGRESI POISSON DAN BINOMIAL NEGATIF TERBOBOTI GEOGRAFIS UNTUK DATA BALITA GIZI BURUK

2020 ◽  
Vol 4 (3) ◽  
pp. 484-497
Author(s):  
Puput Cahya Ambarwati ◽  
Indahwati Indahwati ◽  
Muhammad Nur Aidi
Keyword(s):  
Poisson Distribution ◽  
Spatial Data ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Negative Binomial Regression ◽  
Real Data ◽  
P Value ◽  
Simulation Data ◽  
Response Variable

Geographic weighted regression (GWR) is one of the regression methods for spatial data. GWR with the response variable following the poisson distribution can use the geographic weighted poisson regression (GWPR). GWPR often does not complete the assumption of dispersion. The classic approach commonly used to overcome overdispersion is related to poisson distribution, which is the approach obtained from poisson and gamma distribution which is similar to negative binomial distribution function. GWR for the response variable following the negative binomial distribution can use the geographical weighted negative binomial regression (GWNBR). The data used in this study are simulation data and real data. The results of the simulation data are the tolerance limits that are still precisely modeled with GWPR are overdispersion approaching 1 based on significant amount and average p-value.. The results of research from real data, the GWNBR is the best model for overdispersion cases in malnourished children in East Java Province in 2017 compared to the GWPR based on comparison of the values ​​of AIC. 

A new generalization of the logarithmic series distribution

2014 ◽  
Vol 51 (1) ◽  
pp. 41-49
Author(s):  
A. Mishra
Keyword(s):  
Method Of Moments ◽  
Goodness Of Fit ◽  
Negative Binomial ◽  
Data Sets ◽  
Estimation Of Parameters ◽  
Generalized Negative Binomial Distribution ◽  
Logarithmic Series Distribution ◽  
Logarithmic Series ◽  
Limiting Case ◽  
First Four Moments

A new generalization of the logarithmic series distribution has been obtained as a limiting case of the zero-truncated Mishra’s [10] generalized negative binomial distribution (GNBD). This distribution has an advantage over the Mishra’s [9] quasi logarithmic series distribution (QLSD) as its moments appear in compact forms unlike the QLSD. This makes the estimation of parameters easier by the method of moments. The first four moments of this distribution have been obtained and the distribution has been fitted to some well known data-sets to test its goodness of fit.

scholarly journals Adolescent, Pregnant, and HIV-Infected: Risk of Adverse Pregnancy and Perinatal Outcomes in Young Women from Southern Mozambique

2021 ◽  
Vol 10 (8) ◽  
pp. 1564
Author(s):  
Clara Pons-Duran ◽  
Aina Casellas ◽  
Azucena Bardají ◽  
Anifa Valá ◽  
Esperança Sevene ◽  
...  
Keyword(s):  
Hiv Infection ◽  
Maternal Morbidity ◽  
Low Birthweight ◽  
Negative Binomial ◽  
Perinatal Outcomes ◽  
Sub Saharan Africa ◽  
P Value ◽  
Increased Risk ◽  
Adverse Pregnancy ◽  
The Impact

Sub-Saharan Africa concentrates the burden of HIV and the highest adolescent fertility rates. However, there is limited information about the impact of the interaction between adolescence and HIV infection on maternal health in the region. Data collected prospectively from three clinical trials conducted between 2003 and 2014 were analysed to evaluate the association between age, HIV infection, and their interaction, with the risk of maternal morbidity and adverse pregnancy and perinatal outcomes in women from southern Mozambique. Logistic regression and negative binomial models were used. A total of 2352 women were included in the analyses; 31% were adolescents (≤19 years) and 29% HIV-infected women. The effect of age on maternal morbidity and pregnancy and perinatal adverse outcomes was not modified by HIV status. Adolescence was associated with an increased incidence of hospital admissions (IRR 0.55, 95%CI 0.37–0.80 for women 20–24 years; IRR 0.60, 95%CI 0.42–0.85 for women >25 years compared to adolescents; p-value < 0.01) and outpatient visits (IRR 0.86, 95%CI 0.71–1.04; IRR 0.76, 95%CI 0.63–0.92; p-value = 0.02), and an increased likelihood of having a small-for-gestational age newborn (OR 0.50, 95%CI 0.38–0.65; OR 0.43, 95%CI 0.34–0.56; p-value < 0.001), a low birthweight (OR 0.40, 95%CI 0.27–0.59; OR 0.37, 95%CI 0.26–0.53; p-value <0.001) and a premature birth (OR 0.42, 95%CI 0.24–0.72; OR 0.51, 95%CI 0.32–0.82; p-value < 0.01). Adolescence was associated with an increased risk of poor morbidity, pregnancy and perinatal outcomes, irrespective of HIV infection. In addition to provision of a specific maternity care package for this vulnerable group interventions are imperative to prevent adolescent pregnancy.

An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications

2020 ◽  
Vol 70 (4) ◽  
pp. 953-978
Author(s):  
Mustafa Ç. Korkmaz ◽  
G. G. Hamedani
Keyword(s):  
Hazard Function ◽  
Mixture Distribution ◽  
Real Data ◽  
Quantile Function ◽  
Estimation Methods ◽  
Data Sets ◽  
Unknown Parameters ◽  
Lorenz Curves ◽  
Proposed Model ◽  
New Distribution

AbstractThis paper proposes a new extended Lindley distribution, which has a more flexible density and hazard rate shapes than the Lindley and Power Lindley distributions, based on the mixture distribution structure in order to model with new distribution characteristics real data phenomena. Its some distributional properties such as the shapes, moments, quantile function, Bonferonni and Lorenz curves, mean deviations and order statistics have been obtained. Characterizations based on two truncated moments, conditional expectation as well as in terms of the hazard function are presented. Different estimation procedures have been employed to estimate the unknown parameters and their performances are compared via Monte Carlo simulations. The flexibility and importance of the proposed model are illustrated by two real data sets.

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