##### Comparison of the Bootstrap and Delta Method Variances of the Variance Estimator of the Bernoulli Distribution

Ying-Ying Zhang, Teng-Zhong Rong, Man-Man Li

Asian Journal of Probability and Statistics, Page 1-10
DOI: 10.9734/ajpas/2018/v1i424547

It is interesting to calculate the variance of the variance estimator of the Bernoulli distribution. Therefore, we compare the Bootstrap and Delta Method variances of the variance estimator of the Bernoulli distribution in this paper. Firstly, we provide the correct Bootstrap, Delta Method, and true variances of the variance estimator of the Bernoulli distribution for three parameter values in Table 2.1. Secondly, we obtain the estimates of the variance of the variance estimator of the Bernoulli distribution by the Delta Method (analytically), the true method (analytically), and the Bootstrap Method (algorithmically). Thirdly, we compare the Bootstrap and Delta Methodsin terms of the variance estimates, the errors, and the absolute errors in three gures for 101 parameter values in [0, 1], with the purpose to explain the di erences between the Bootstrap and Delta Methods. Finally, we give three examples of the Bernoulli trials to illustrate the three methods.

##### Computational Generalization of Mixed Models on Large-Scale Data with Applications to Genetic Studies

Samson W. Wanyonyi, Drinold A. Mbete, Emile R. Chimusa

Asian Journal of Probability and Statistics, Page 1-31
DOI: 10.9734/ajpas/2018/v1i424550

Aims: To discuss different LMM-based approaches applied in GWAS and software packages implementation and Classify different computational tools that applies LMM approaches according to their applicability and performance. To identify possible SNPs associated to a particular disease using different computational tools based on LMM approaches. To estimate genetic and residual variance parameters that account phenotypic variation of the disease.

Study Design: Case control study

Place and Duration of Study: The research was carried out in Tanzania at African Institute of Mathematical Science for six months.

Methodology: Linear Mixed Models (LMMs) are widely applied in genomic wide associations studies (GWAS) owing to their effectiveness of correcting hidden relationship, population structure and family structure. This essay is aimed at exploring different mathematical approaches of LMMs in GWAS. These approaches are linear mixed model with inclusion of all markers (LMMi) and linear mixed model with exclusion of all markers (LMMe) when calculating genetic relationship matrix. LMMi is more efficient as compared to LMMe when applied in studies of randomly ascertained quantitative traits. The LMM approaches are classified based on their applicability and performance. Two computational GWAS tools namely, PLINK and EMMAX were used which were based on LMM approaches to analyze unpublished real data from West Africa (Gambia and Ghana). Genetic and residual variance parameters were estimated that accounted for the phenotypic variation of the disease to be 0.0594 and 0.0723. A total of 338408 variants and 959 people (484 males, 405 females and 70 missing phenotypes) pass filters and quality control using PLINK was used in the study. Among the remaining phenotypes, 864 are cases and 95 are controls. The performance of different mathematical approaches of LMMs and their software implementation, including EMMAX and Plink via the application to a GWAS of tuberculosis (TB) in 959 individuals in West Africa (Ghana and Gambia) was compared. Of these 864 cases of TB and 95 healthy individuals retained after quality control (QC) using Plink, and 329601 autosome single nucleotide polymorphisms (from chromosome 1 to chromosome 22) included in the analysis after 288 duplicands ID individuals removed after QC. The LMM approaches are classified based on their applicability and performance. Two computational GWAS tools, namely Plink and EMMAX were used in the analysis of data. Genetic and residual variance parameters were estimated that accounted for the phenotypic variation of the disease to be 0.0594 and 0.0723.

Results: Result showed that SNPs associated with tuberculosis were  on chromosome 17 and SNP  on chromosome 13 with both having  false discovery rate with step up significance value. Plink failed to correct hidden relatedness. Although EMMAX reduced the false positive rate, it still exhibited very low presence of stratification.

Conclusion: This study aimed at understanding and exploring different approaches of mixed models as applied in genetic studies. Overview of genetic variation, advantages, successes and application of mixed models and current challenges of mixed models in GWAS were discussed. Moreover, the study showed that SNPs was associated with a particular disease using computational tools that applies LMM approaches. The summary statistics from PLINK and EMMAX found two causal SNPs associated with the TB. These SNPs were rs7225581 on chromosome 17 and SNP rs4941412 on chromosome 13 with both having 0.69% FDR H. However, PLINK failed to correct hidden relatedness. This phenotypic variation showed that all common single nucleotide polymorphisms (SNPs) expressed approximately 18.52% of phenotypic variation of the disease.

##### On the Properties and Applications of Transmuted Odd Generalized Exponential-Exponential Distribution

Jamila Abdullahi, Umar Kabir Abdullahi, Terna Godfrey Ieren, David Adugh Kuhe, Adamu Abubakar Umar

Asian Journal of Probability and Statistics, Page 1-14
DOI: 10.9734/ajpas/2018/v1i424552

This article proposed a new distribution referred to as the transmuted odd generalized exponential-exponential distribution (TOGEED) as an extension of the popular odd generalized exponential- exponential distribution by using the Quadratic rank transmutation map (QRTM) proposed and studied by [1]. Using the transmutation map, we defined the probability density function (pdf) and cumulative distribution function (cdf) of the transmuted odd generalized Exponential- Exponential distribution. Some properties of the new distribution were extensively studied after derivation. The estimation of the distribution’s parameters was also done using the method of maximum likelihood estimation. The performance of the proposed probability distribution was checked in comparison with some other generalizations of Exponential distribution using a real life dataset.

##### Establishing the Rainfall Trend over Bungoma Region in Kenya, During the Short and Long Rain Seasons

Davis Mwenda Marangu, Samson W. Wanyonyi

Asian Journal of Probability and Statistics, Page 1-12
DOI: 10.9734/ajpas/2018/v1i424554

This paper is aimed at presenting the temporal and spatial variability of rainfall over Bungoma region between 2000 and 2015. Rainfall data was obtained for 5 stations in the region. The data was subjected to analyses by use of Microsoft Excel and SPSS. The results from analysis of the data showed that the March, April, and May (MAM) and October, November, and December (OND) rainfall totals showed a general increasing trend in all the stations. The MAM season was found to have the most reliable rainfall in all stations while the OND season was reliable in some of the stations in the study region. Kanduyi region is seen to receive the highest amount of rainfall all through the years, and Tongaren the Least amount of Rainfall. The significance of these findings is that it could be used by various policymakers and development partners for planning purposes.

##### On the Properties and Applications of Lomax-Exponential Distribution

Terna Godfrey Ieren, David Adugh Kuhe

Asian Journal of Probability and Statistics, Page 1-13
DOI: 10.9734/ajpas/2018/v1i424557

The Exponential distribution is memoryless and has a constant failure rate which makes it unsuitable for real life problems. This paper introduces a new distribution powered by an exponential random variable which gives a more flexible model for modelling real-life data. This new extension of the Exponential Distribution is called “Lomax-Exponential distribution (LED)”. The extension of the new distribution became possible with the help of a Lomax generator proposed by [1]. This paper derives and studies some expressions for various statistical properties of the new distribution including distribution function, moments, quantile function, survival function and hazard function known as reliability functions. The inference for the Lomax-Exponentially distributed random variable were investigated based on some plots of the distribution and others revealed its behaviour and usefulness in real life situations. The parameters of the distribution are estimated using the method of maximum likelihood estimation. The performance of the new Lomax-Exponential distribution has been tested and compared to the Weibull-Exponential, Transmuted Exponential and the conventional Exponential distribution using three real life data sets.