##### On the Riemann Hypothesis

Pathikrit Basu

Asian Journal of Probability and Statistics, Page 51-56
DOI: 10.9734/ajpas/2022/v20i4438

This paper provides an exact characterisation of the zeroes of the Riemann zeta function. The characterisation is based on a theorem about random vectors, which says that under some conditions, if a vector is always in the convex hull of the conditional expectations corresponding to any two mutually exclusive and exhaustive
events, then the unconditional expectation of the random vector is equal to that vector.

##### Useful Extensions of the Juchez Probability Distribution: Properties and Application

Udochukwu Victor Echebiri, Emwinloghosa Kenneth Guobadia, John Paul Kenechukwu Iwuchukwu, Nosakhare Liberty Osawe, Stanley Imaguomwanruo Moses

Asian Journal of Probability and Statistics, Page 1-18
DOI: 10.9734/ajpas/2022/v20i4434

This paper investigates the inverse and power Juchez distribution. Some relevant properties are studied, and the distributions can altogether model many varieties of datasets emanating from different life phenomena. And then, a generator for other generalized distributions termed Juchez-G is developed to suffice for the relevance of robust model development. Finally, IJD and PJD showed to be a better fit over both the baseline distribution and their respective counterpart distributions, with respect to the datasets used. The proposed method is interesting and the performance is superior.

##### Sentiment Analysis for Market Survey in the Framework of the Establishment of the IT Boarding School Vocational School

Asian Journal of Probability and Statistics, Page 19-27
DOI: 10.9734/ajpas/2022/v20i4435

Aims: This study aims to obtain indicators which are then compiled in a questionnaire research instrument which will later be used in further analysis namely sentiment analysis.

Methodology: Sentiment analysis is a method used to identify how a sentiment is expressed using text and how that sentiment can be categorized as positive sentiment or negative sentiment. Boarding school. So that researchers can conclude whether the sentiments or issues that arise in the tweets published by the surrounding community on social media are more supportive or even less supportive regarding the boarding school program at a boarding school.

Results: The findings of this study are the variables that can be used in further research, namely marketing communication strategies, marketing mix, and interest in learning.

Conclusion: Based on the sentiment analysis, it was found that society has a neutral tendency regarding public perceptions in Indonesia regarding the boarding school program at a boarding school.

##### Mapping Community Potential in Stunting Prevention in Sumberputih Village, Indonesia

. Solimun

Asian Journal of Probability and Statistics, Page 28-38
DOI: 10.9734/ajpas/2022/v20i4436

Purpose: This study aims to map stunting prevention behavior in the Sumberputih Village community, Malang Regency.

Methods: This research was conducted in Sumberputih Village, Wajak District, Malang Regency, East Java, with 3 lecturers and 5 students from the implementation team. With this activity, it is hoped that the community can prevent and increase knowledge about stunting in Sumberputih Village. This study uses cluster analysis for mapping community in preventing stunting.

Results: The number of clusters in the Bringin Village community grouping is divided into two parts, that are people care enough about stunting prevention and people that care less about stunting prevention. The sample in this study was determined using purposive sampling with used is 75 respondents, of which there are 47 people who are members of cluster 1 (people care about stunting prevention) and there are 28 people who are included in cluster 2 (people who don't care about stunting prevention).

Originality: The novelty of this research is the mapping of stunting prevention behavior in the Sumberputih village community, Malang Regency.

##### Appropriate Description of Probability Distribution of Prostate Specific Antigen (PSA): An Aid to Early Detection of Prostate Cancer

Chrysogonus Chinagorom Nwaigwe, Chukwudi Justin Ogbonna, Emmanuel Uchechukwu Oliwe

Asian Journal of Probability and Statistics, Page 39-50
DOI: 10.9734/ajpas/2022/v20i4437

Prostate cancer has been observed to be a worrisome ailment among adult males across the globe over some decades. Just like any other ailment, its early detection and treatment gives the patient a better chance of survival. Measuring the Prostate specific antigen (PSA) of the patients at specified intervals is one of the techniques of detecting the onset of prostate cancer. This paper attempts to obtain the most appropriate probability distribution of prostrate specific antigens (PSA) that gives the best estimates of the parameters of the distribution. The age specific probabilities of patients with prostate specific antigen (PSA)>4.0ng/ml were calculated. The results show that the most appropriate distribution among the distributions fitted is the Burr distribution. The probability of observing a PSA greater than 4.0 ng/ml in adult males above the age of 45 years was seen to be moderately high (0.68) with a median PSA of 8.30 ng/ml and an inter quartile range of 16.40. The results also show that PSA of 60.0 ng/ml and above were exceptional cases that require urgent attention. It was observed that men whose ages are between 60 and 80 years have higher chances of having PSA values greater 4.0ng/ml with men between 60 and 75 years having the highest chances. It is therefore suggested that men with PSA higher than 4.0ng/ml between 60 and 80 years of age may not be subjected to needle biopsies but be referred directly for digital rectal examinations.

##### Probability Model Based on Zero Truncation of Himanshu Distribution and Their Applications

Abhishek Agarwal, Himanshu Pandey

Asian Journal of Probability and Statistics, Page 57-67
DOI: 10.9734/ajpas/2022/v20i4439

In this paper we have developed a probability model to study the distribution of the number of male migrants from households and patterns of child mortality through zero truncated Himanshu distribution (ZTHD). The parameter of the proposed model is estimated by method of moment and method of maximum likelihood and its suitability is tested by fitting to observed migration data,collected from various surveys.

##### Bayesian Estimation of a Scale Parameter of the Gumbel-Lomax Distribution Using Informative and Non Informative Priors

Douglas Jah Pam, Kazeem Eyitayo Lasisi, Umar Farouk Abbas, Mustapha Tijani, Sheyi Mafolasire, Blessing Uke Agbor

Asian Journal of Probability and Statistics, Page 68-81
DOI: 10.9734/ajpas/2022/v20i4440

Estimating the scale parameter of the Gumbel-Lomax Distribution using the Bayesian method of estimation and evaluating the estimators by assuming two non-informative prior distributions and one informative prior distribution is very important for the general application of the Gumbel-Lomax distribution. These estimators are obtained using the squared error loss function (SELF), Quadratic loss function (QLF) and precautionary loss function (PLF). The posterior distributions of the scale parameter of the Gumbel-Lomax distribution are derived and the Estimators are also obtained using the above mentioned priors and loss functions. Furthermore, a simulation using a package in R software is carried out to assess the performance of the estimators by making use of the Mean Squared Errors of the Estimators under the Bayesian approach and Maximum likelihood method. Our results show that Bayesian Method using PLF under all priors produces the best estimators of the scale parameter compared to estimators using the Maximum Likelihood method, SELF and QLF under all the priors irrespective of the values of the parameters and the different sample sizes. It is also discovered that the other parameters have no effect on the estimators of the scale parameter.

##### Homogeneity versus Parsimony in Markov Manpower Models: A Hidden Markov Chain Approach

Everestus O. Ossai, Precious N. Ezra, Felix O. Ohanuba, Martin N. Eze

Asian Journal of Probability and Statistics, Page 82-93
DOI: 10.9734/ajpas/2022/v20i4441

We aim at tackling the problem of inadequate specification of a Markov manpower model in this paper, by formulating a procedure for validating the inclusion or non-inclusion of some transition parameters in the model. The mover-stayer principle and its extensions are employed to incorporate hidden classes in the model to achieve more homogeneity and this is compared with the model without the hidden classes, which is more parsimonious, using Likelihood ratio statistic, Akaike Information Criterion and Bayesian Information Criterion. The illustration shows a case of manpower data where, up to a certain level of hidden states, homogeneity is more important than parsimony.

##### Existence and Uniqueness of Stationary Probability Vector for Stochastic Matrices using Farkas’ Lemma

Somdeb Lahiri

Asian Journal of Probability and Statistics, Page 94-99
DOI: 10.9734/ajpas/2022/v20i4442

The purpose of this note is to provide a simple proof of existence of stationary probability vectors (fixed points) for stochastic matrices using Farkas’ lemma. This result as well as the uniqueness of stationary probability vectors also holds for a certain subclass of quasi-stochastic matrices.

##### Mixture Model on Development of Bivariate Product Distribution and its Properties

Udochukwu Victor Echebiri, Akpome Jennifer Nomuoja, Chimezie Stanley Ngene, Emwinloghosa Kenneth Guobadia, Jophet Ewere Okoh

Asian Journal of Probability and Statistics, Page 100-119
DOI: 10.9734/ajpas/2022/v20i4443

In the study, some bivariate distributions were developed from mixture model offspring, using the Independent (Product) distribution approach. These developments are categorized under the IID and IInD: where the Bivariate Exponential distribution, Bivariate Lindley distribution and Bivariate Juchez distribution are constructed as IIDs; and Bivariate Exponential-Lindley distribution, Bivariate Exponential-Juchez distribution and Bivariate Lindley-Juchez distribution as (IInDs). The properties of these distributions which involve: the shape of the bivariate PDFs, moments, moment generating function, mean, covariance and coefficient of correlation, maximum likelihood estimator, reliability analysis, renewal property and probability patterns; are studied across the distributions. Finally, under renewal properties, functions are derived which can model two-dimensional queuing and renewal processes, for events where the arrival and service times are dependent.

##### The Proportional Hazard Generalized Power Weibull Distribution: Properties, Applications and Regression Model

Kudugu Atiah, Suleman Nasiru, Salifu Katara

Asian Journal of Probability and Statistics, Page 120-145
DOI: 10.9734/ajpas/2022/v20i4444

We introduce the proportional hazard generalized power Weibull (PHGPW) model in which hazard rate function can assume increasing, decreasing, unimodal or (upside bathtub) and constant. Some of its mathematical properties are studied including the power series for the quantile function. Monte Carlo simulation was performed to determine the finite sample behaviour of the maximum likelihood estimates of the parameters. The flexibility of the PHGPW distribution compared with some other existing distributions is proved empirically by means of two sets of real data related to remission times of bladder cancer patients and strike duration of manufacturing company. A new regression model was defined based on the PHGPW distribution. The performance of the regression model is proved empirically using real data set.

##### Likelihood Ratio Search Procedure for Optimum Number of States in a Hidden Markov Manpower Model

Everestus O. Ossai, Mbanefo S. Madukaife

Asian Journal of Probability and Statistics, Page 146-156
DOI: 10.9734/ajpas/2022/v20i4445

This paper aims at obtaining the optimum number of states for a hidden Markov manpower model, which, hitherto, has been chosen arbitrarily. A search procedure that attains this optimum number after a few steps across a series of N hidden Markov manpower models is proposed. The likelihood ratio statistic is employed to conduct pairwise model comparison tests on the N hidden Markov manpower models ordered according to their level of parsimony. The illustration shows the usefulness of the procedure in choosing the right number of states for a hidden Markov manpower model to avoid wrong specification of such models. The proposed procedure can be useful in other areas of research, such as in biological, medical and social sciences, where application of hidden Markov model may require the determination of number of hidden states based on unobserved data with latent heterogeneity. The procedure has a straightforward formulation and its application in other areas requires mainly the adaptation of the model specifications for the new area’s system dynamics.

##### Proposed Nonparametric Tests for the Umbrella Alternative in a Mixed Design for Location and Scale

Eid Alotaibi, Rhonda C. Magel

Asian Journal of Probability and Statistics, Page 157-168
DOI: 10.9734/ajpas/2022/v20i4446

Aims: To develop and compare tests for a mixed design of a Randomized Complete Block Design (RCBD) with a Completely Randomized Design (CRD) with k populations in testing for the umbrella alternative with known peak, p, for both location and scale parameters. These tests were combinations of the Mack-Wolfe test, the Kim-Kim test, and both the Mack-Wolfe test, and the Kim-Kim test using the Moses technique.

Study Design:  Monte Carlo Simulation Study.

Place and Duration of Study: North Dakota State University, Department of Statistics, August 2020-December 2021.

Methodology: A simulation study was conducted to see how well the proposed tests maintained their significant levels. Powers were also estimated for different ratios of sample size in the CRD to the number of blocks in the RCBD; we assumed equal variance ratios between the CRD and the RCBD. Different parameter changes were considered to see if they would impact which test statistics had greater power. In all cases, it was assumed three observations per treatment, per block in the RCBD portion.

Conclusion: All proposed tests maintained their significance levels.  An overall test is recommended if both location and scale parameters change.

##### Modified Ratio Estimators for Population Means with Two Auxiliary Parameters Using Calibration Weights

Adubi S. Ayodeji, Oluwagbenga T. Babatunde, Ude O. Ifeoma

Asian Journal of Probability and Statistics, Page 169-183
DOI: 10.9734/ajpas/2022/v20i4447

Many researchers have used different auxiliary parameters such as coefficient of variation, coefficient of kurtosis, coefficient of skewness, quartiles, deciles etc., to improve the precision of estimators under various sampling schemes. This paper suggested a class of ratio estimators with two known auxiliary variable parameters for the estimation of population means under a simple random sample without replacement (SRSWOR) using the calibration weighting method. The calibrated weight was obtained using a new calibration constraint, which includes the known standard deviation of the auxiliary variable. The biases and mean square errors of the proposed estimators were derived and compared with the biases and mean square errors of the existing modified ratio estimators in Upadhyaya & Singh [1], Singh [2], Lu & Yan [3], and Yan & Tian [4]. Furthermore, we derived the condition for which the proposed estimators perform better than the existing estimators. The results from using real data sets showed that the suggested estimators perform better than the existing ratio estimators.

##### Modelling Mother-To-Child HIV Transmission Rate in Nigeria Using a Lindley-Lomax Distribution

Blessing Uke Agbor, Umar Farouk Abbas, Sabo Hamma, Felix Kromtit Makmun, Pam Douglas Jah, Tisloh Israel Mark

Asian Journal of Probability and Statistics, Page 184-199
DOI: 10.9734/ajpas/2022/v20i4448

In this research a new extension of the Lomax distribution known as Lindley-Lomax distribution has been proposed by adding a shape parameter to the Lomax distribution using the Lindley-G family of distributions. The proposed article considered and extensively studied some properties of the new distribution such as moments, moment generating function, the characteristics function, survival function, hazard function and the distribution of order statistics. A graphical study of the proposed distribution and the other related functions was also done in the article. Estimation of the parameters of the proposed distribution was also done using the method of maximum likelihood estimation. The performance of the Lindley-Lomax distribution has also been tested by an application to the rate of mother-to-child HIV transmission.

##### Some Theorems on the Demeanour of Probabilistic Uncertainty-Like Functional under the Bounds

Rohit Kumar Verma

Asian Journal of Probability and Statistics, Page 200-207
DOI: 10.9734/ajpas/2022/v20i4449

The resulting mean of the optimal solutions of minimization problems, whose objective functions are the uncertainty like functionals, are known as uncertainty mean. The uncertainty mean satisfies all the basic properties of the classical mean, weighted homogeneous mean as well as many others are special cases of uncertainty mean. The indeed paper deals with comparison property and asymptotic demeanour of the uncertainty mean.

##### On the Path Analysis Techniques and Decomposition of Correlation Coefficients

C. A. Awogbemi, S. A. Alagbe, F. S. S. Oloda

Asian Journal of Probability and Statistics, Page 208-219
DOI: 10.9734/ajpas/2022/v20i4450

Path analysis is an extension of multiple regression technique used to evaluate causal model by examining the relationship among a set of variables. The process of decomposition of path correlation coefficients and estimation of path coefficients was investigated in this work. The outcome of a set of interrelated variables was generated using path diagrams and path coefficients. Decomposition of correlation coefficients into different effects was also carried out through the method of structural equations using path analysis theorem. In order to show the proportion of total variation of the dependent variable, the decision coefficient of a specified pathway was constructed from direct coefficient of determination. The path coefficients were estimated from the specified structural equations by ordinary least squares regression method, and it was deduced that each of the coefficients of determination had large effect on some other variables.

##### Empirical Investigation on the Effect of the Number of Resamplings on the Distribution of Bootstrap Standard Error Using Response Time Data

I. Udobi, Joy

Asian Journal of Probability and Statistics, Page 220-230
DOI: 10.9734/ajpas/2022/v20i4451

Aims: To investigate how the number of bootstrapping B affects the values returned by the bootstrap standard error of the arithmetic mean and the α-trimmed mean of response data using bootstrap confidence intervals (CI) at 95% level; carried out to fill up observed gap for study on standard error, the tool generally employed in assessing the long run accuracy of a given statistical estimator of θ.

Study Design: This was a parametric, empirical bootstrap simulation study.

Place and Duration of the Study: Departments of Computer Science and Statistics, Federal Polytechnic Oko, 2020/2021 session.

Methodology: Response time data were generated with student customers of mobile telephone network (mtn) Nigeria and stored in SPSS. A sample n = 51 responses was selected using “Select Cases” command to increase precision and minimize bias. Bootstrap simulation study was carried out using R programming language. Four approaches for estimating bootstrap confidence intervals were used. The interval coverage and the interval lengths were determined and compared for B = 20, 50, 100, 500, 1000, 5000, and 10000.

Results: The 95% CI for 0.2266338 (the estimated sample standard error of 10% trimmed mean) returned the best interval for our skewed data set; when B = 20; the CI for  returned (0.2051, 0.2343) for the normal approach, (0.2082, 0.2371) for the basic, (0.2071, 0.2360) for the percentile and (0.2071, 0.2360) for the BCa method. As B increased to 5000, it returned (0.2259, 0.2277) for the normal approach, (0.2259, 0.2277) for the basic, (0.2260, 0.2277) for the percentile and (0.2261, 0.2279) for the BCa showing a shorter interval yet covering the estimate.

Conclusion: Thus for our response data study, increasing B in estimating standard error increases the chances of more precise and shorter confidence intervals rather than the chances for coverage.