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 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 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

scholarly journals A New Discrete Analog of the Continuous Lindley Distribution, with Reliability Applications

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 603
Author(s):  
Abdulhakim A. Al-Babtain ◽  
Abdul Hadi N. Ahmed ◽  
Ahmed Z. Afify
Keyword(s):  
Goodness Of Fit ◽  
Negative Binomial ◽  
Real Data ◽  
Moment Generating Function ◽  
Discrete Analog ◽  
Bias Reduction ◽  
Discrete Distributions ◽  
Likelihood Method ◽  
Probability Mass Function ◽  
Lindley Distribution

In this paper, we propose and study a new probability mass function by creating a natural discrete analog to the continuous Lindley distribution as a mixture of geometric and negative binomial distributions. The new distribution has many interesting properties that make it superior to many other discrete distributions, particularly in analyzing over-dispersed count data. Several statistical properties of the introduced distribution have been established including moments and moment generating function, residual moments, characterization, entropy, estimation of the parameter by the maximum likelihood method. A bias reduction method is applied to the derived estimator; its existence and uniqueness are discussed. Applications of the goodness of fit of the proposed distribution have been examined and compared with other discrete distributions using three real data sets from biological sciences.

scholarly journals Weighted Negative Binomial Poisson-Lindley Distribution with Actuarial Applications

2018 ◽  
Author(s):  
Hossein Zamani ◽  
Noriszura Ismail ◽  
Marzieh Shekari
Keyword(s):  
Maximum Likelihood Method ◽  
Negative Binomial ◽  
Statistical Tests ◽  
Discrete Distribution ◽  
Likelihood Method ◽  
Weighted Version ◽  
Lindley Distribution ◽  
Weighted Distribution ◽  
Binomial Distributions ◽  
Over Dispersion

This study introduces a new discrete distribution which is a weighted version of Poisson-Lindley distribution. The weighted distribution is obtained using the negative binomial weight function and can be fitted to count data with over-dispersion. The p.m.f., p.g.f. and simulation procedure of the new weighted distribution, namely weighted negative binomial Poisson-Lindley (WNBPL), are provided. The maximum likelihood method for parameter estimation is also presented. The WNBPL distribution is fitted to several insurance datasets, and is compared to the Poisson and negative binomial distributions in terms of several statistical tests.

The statistical distribution of involved axillary lymph nodes in breast cancer: A SEER population-based study

2007 ◽  
Vol 25 (18_suppl) ◽  
pp. 11031-11031
Author(s):  
V. Vinh-Hung ◽  
A. Guern ◽  
G. Storme
Keyword(s):  
Breast Cancer ◽  
Lymph Nodes ◽  
Poisson Distribution ◽  
Negative Binomial ◽  
Likelihood Method ◽  
Axillary Lymph ◽  
Cascade Process ◽  
Nodal Involvement ◽  
Lack Of Fit ◽  
The Mean

11031 Background: The number of involved lymph nodes in patients with breast cancer is highly variable. It might be important to examine the frequency distribution (FD) of these numbers in order to characterize their variability. This study examines the FD’s statistical properties and determines what they imply in terms of statistical analyses. Methods: The data was based on the National Cancer Institute’s Surveillance, Epidemiology and Ends Results (SEER). It covered 109618 patients, in whom axillary dissection had been performed between 1973 and 2002, in 9 states or towns of the USA. The involved lymph nodes were fitted to different statistical distributions adapted to count data using the maximum likelihood method (ML). The fittings were evaluated with the Kolmogorov-Smirnov (KS) statistic (KS close to 1 indicates lack of fit and KS close to 0 indicates good fit) and with the chi2 statistic (large chi2 indicates lack of fit). Results: The FD showed a logarithmically decreasing frequency, i.e. log-concave type. The mean number of involved nodes was 1.1465 and the variance was 9.54. The fit with a Poisson distribution gave a chi2=7.18*106 and the KS was 0.389. The fit with a negative binomial distribution using the ML gave a chi2=291.7, and the KS was 0.0086, i.e. 45-fold improved fit. Discussion: The FD’s variance 9-fold larger than the mean number of involved nodes indicates important overdispersion, i.e. the variability increases when the number of involved nodes increase. The Poisson distribution is inappropriate as was shown by the chi2 and KS. Overdispersion implies that nodal metastases are not independent random events. Overdispersion might be explained by 1) subsets of patients had a higher than expected propensity of nodal involvement, and/or 2) nodal involvement is a cascade process in which the likelihood of involvement increases when more nodes are involved. The implication for further analyses is that building predictive models of nodal involvement should take into account overdispersion and use appropriate modeling distributions such as the negative binomial. Conclusion: The FD characterization indicated a log-concave type with overdispersion. This might be important to gain an insight into the mechanisms of nodal involvement and to improve their analyses. No significant financial relationships to disclose.

scholarly journals Theoretical Description Of A Reparamatrization of Discrete Two Parameter Poisson Lindley Distribution for Modeling Waiting And Survival Times Data

2016 ◽  
Vol 29 (1) ◽  
pp. 217-222
Author(s):  
Tanka Raj Adhikari
Keyword(s):  
Poisson Distribution ◽  
Method Of Moments ◽  
Research Paper ◽  
Theoretical Description ◽  
Survival Times ◽  
Lindley Distribution ◽  
Two Parameters ◽  
First Four Moments ◽  
Special Case ◽  
Two Parameter

In this research paper, the theoretical description of are paramatrization of a discrete two-parameter Poisson Lindley Distribution, of which Sankaran’s (1970) one parameter Poisson Lindley Distribution is a special case, is derived by compounding a Poisson Distribution with two parameters Lindley Distribution for modeling waiting and survival times data of Shanker et al. (2012).The first four moments of this distribution have derived. Estimation of the parameters by using method of moments and maximum likely hood method has been discussed.

scholarly journals OUTRIDER: A statistical method for detecting aberrantly expressed genes in RNA sequencing data

2018 ◽  
Author(s):  
Felix Brechtmann ◽  
Agnė Matusevičiūtė ◽  
Christian Mertes ◽  
Vicente A Yépez ◽  
Žiga Avsec ◽  
...  
Keyword(s):  
Gene Expression ◽  
Rna Sequencing ◽  
Negative Binomial ◽  
Statistical Significance ◽  
P Value ◽  
Rna Seq ◽  
Sequencing Data ◽  
Data Set ◽  
Aberrant Gene Expression ◽  
Aberrant Gene

AbstractRNA sequencing (RNA-seq) is gaining popularity as a complementary assay to genome sequencing for precisely identifying the molecular causes of rare disorders. A powerful approach is to identify aberrant gene expression levels as potential pathogenic events. However, existing methods for detecting aberrant read counts in RNA-seq data either lack assessments of statistical significance, so that establishing cutoffs is arbitrary, or rely on subjective manual corrections for confounders. Here, we describe OUTRIDER (OUTlier in RNA-seq fInDER), an algorithm developed to address these issues. The algorithm uses an autoencoder to model read count expectations according to the co-variation among genes resulting from technical, environmental, or common genetic variations. Given these expectations, the RNA-seq read counts are assumed to follow a negative binomial distribution with a gene-specific dispersion. Outliers are then identified as read counts that significantly deviate from this distribution. The model is automatically fitted to achieve the best correction of artificially corrupted data. Precision–recall analyses using simulated outlier read counts demonstrated the importance of combining correction for co-variation and significance-based thresholds. OUTRIDER is open source and includes functions for filtering out genes not expressed in a data set, for identifying outlier samples with too many aberrantly expressed genes, and for the P-value-based detection of aberrant gene expression, with false discovery rate adjustment. Overall, OUTRIDER provides a computationally fast and scalable end-to-end solution for identifying aberrantly expressed genes, suitable for use by rare disease diagnostic platforms.

Negative binomial improved second degree lindley distribution and its application

2020 ◽  
pp. 569-581
Author(s):  
R. Ashly ◽  
C. S. Rajitha
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Data Set ◽  
Factorial Moments ◽  
Lindley Distribution ◽  
Second Degree

The objective of this paper is to introduce a new two parameter mixed negative binomial distribution, namely negative binomial-improved second degree Lindley(NB-ISL) distribution. This distribution is obtained by mixing the negative binomial distribution with the improved second degree Lindley distribution. Many mixed distributions have been used in the literature for modeling the over dispersed count data, which provide a better fit compared to the Poisson and negative binomial distribution. In addition, we present the basic statistical properties of the new distribution such as factorial moments, mean and variance and the behavior of mean, variance and coefficient of variation are also discussed. Parameter estimation is implemented by using maximum likelihood estimation method. The performance of the NB-ISL distribution is shown in practice by applying it on real data set and compare it with some well-known count distributions. The result shows that the negative binomial-improved second degree Lindley distribution provides a better fit compared to Poisson, negative binomial and negative binomial-Lindley distributions.

scholarly journals Uma proposta para identificação de outliers multivariados

2018 ◽  
Vol 40 ◽  
pp. 40
Author(s):  
Josino José Barbosa ◽  
Tiago Martins Pereira ◽  
Fernando Luiz Pereira de Oliveira
Keyword(s):  
Probability Distributions ◽  
Estimation Method ◽  
Real Data ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Inverse Power ◽  
Test Statistic ◽  
Lindley Distribution ◽  
Power Lindley Distribution ◽  
Inverse Lindley Distribution

In the last years several probability distributions have been proposed in the literature, especially with the aim of obtaining models that are more flexible relative to the behaviors of the density and hazard rate functions. For instance, Ghitany et al. (2013) proposed a new generalization of the Lindley distribution, called power Lindley distribution, whereas Sharma et al. (2015a) proposed the inverse Lindley distribution. From these two generalizations Barco et al. (2017) studied the inverse power Lindley distribution, also called by Sharma et al. (2015b) as generalized inverse Lindley distribution. Considering the inverse power Lindley distribution, in this paper is evaluate the performance, through Monte Carlo simulations, with respect to the bias and consistency of nine different methods of estimations (the maximum likelihood method and eight others based on the distance between the empirical and theoretical cumulative distribution function). The numerical results showed a better performance of the estimation method based on the Anderson-Darling test statistic. This conclusion is also observed in the analysis of two real data sets.

Assessment of Susceptibility Risk Factors for ADHD in Imaging Genetic Studies

2016 ◽  
Vol 23 (7) ◽  
pp. 671-681 ◽  
Author(s):  
N. Vilor-Tejedor ◽  
S. Alemany ◽  
J. Forns ◽  
A. Cáceres ◽  
M. Murcia ◽  
...  
Keyword(s):  
Risk Factors ◽  
Statistical Power ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Significant Risk ◽  
P Value ◽  
Adhd Symptoms ◽  
Genetic Studies ◽  
Dichotomous Outcome ◽  
Independent Population

Objective: ADHD consists of a count of symptoms that often presents heterogeneity due to overdispersion and excess of zeros. Statistical inference is usually based on a dichotomous outcome that is underpowered. The main goal of this study was to determine a suited probability distribution to analyze ADHD symptoms in Imaging Genetic studies. Method: We used two independent population samples of children to evaluate the consistency of the standard probability distributions based on count data for describing ADHD symptoms. Results: We showed that the zero-inflated negative binomial (ZINB) distribution provided the best power for modeling ADHD symptoms. ZINB reveals a genetic variant, rs273342 (Microtubule-Associated Protein [MAPRE2]), associated with ADHD ( p value = 2.73E-05). This variant was also associated with perivascular volumes (Virchow–Robin spaces; p values < 1E-03). No associations were found when using dichotomous definition. Conclusion: We suggest that an appropriate modeling of ADHD symptoms increases statistical power to establish significant risk factors.

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