NATIONAL ANTHEM

2. NATIONAL ANTHEM 1:35 PM

3. “UNIBEN ANTHEM” “ARISE MIGHTY UNIBEN”

5. 224THINAUGURAL LECTURE SERIES UNIVERSITY OF BENIN BENIN CITY NIGERIA TOPIC: DEVELOPMENT FROM THE CRADLE: THE STATISTICIAN’S FOOTPRINT BY: PROFESSOR JOSEPH ERUNMWOSA OSEMWENKHAE Professor of Mathematical Statistics DATE: SEPTEMBER 12TH, 2019

6. PROTOCOL • The Vice Chancellor, • Deputy Vice Chancellor (Administration), • Deputy Vice Chancellor (Academics), • Deputy Vice Chancellor (Ekehuan Campus), • The Registrar, • The Bursar, • The Librarian, • Members of the University Governing Council, • Members of the University Senate, • Provost of the College of Medical Sciences, • Dean, School of Postgraduate Studies, • Dean, Faculty of PhysicalSciences, • Other Deans and Directors, • Academic and Non-teaching Staff of the University, • My Lords, Spiritual and Temporal, • Great University of Benin students, • Members of the Press, • Distinguished Guests, • Ladies and Gentlemen.

7. DEDICATION This inaugural lecture is dedicated to - God Almighty, the I am that I am, the Author and Finisher of my faith in Christ Jesus. To Him be the glory forever and ever, Amen. -To my peaceful and loving mother (Mrs. Uwubamwen Osemwenkhae) of blessed memory and my wonderful father Prince Napoloen Osemwenkhae-Ugiugo (the Odionwere of Ugo-Niyekorhionmwon) who, though, did not have a formal education, yet toiled day and night to ensure that I was educated. -To my eldest brother, Mr. Felix Osemwenkhae, who taught me Lacombe/Arithmetic (even in the farm) and instilled good mathematical skills in me. -To my eldest sister, Mrs. Janet Imudia (late), who was always there for me as a ‘second’ mother.

8. DEDICATION CONT. • To my respected M. Sc. and Ph. D supervisor and teacher, Prof S. M. Ogbonmwan, who brought me up statistically. Thank you Sir. • To my dearest, one and only loving wife, Mrs. Blessing Ehomahenmwen Osemwenkhae, for her perseverance, support and love without whom I am incomplete. • - To my only children Dr. Covenant Osemwenkhae, Promise Osemwenkhae, Osamudiamen Osemwenkhae and Osamagbe Osemwenkhae. I love you all

9. PREAMBLE Like Joseph who was noted for his dreams (Gen. 37:5-11), I have been dreaming that one day I will have the opportunity to deliver my inaugural lecture. Today that dream is fulfilled so much so that I am the first professor to give an inaugural lecture in the Department of Statistics of this great University since its inception. To God be the glory! Mr. Vice-Chancellor Sir, statistics has a biblical history. In Numbers 1:1-3, the Lord spoke to Moses that he and Aaron are to take a census of the people of Israel by clans and families. The reason for this was to make inference on the military strength of Israel. This approach to data collection and decision making is now well developed in Sampling Theory.

10. WHAT IS STATISTICS? Statistics is that branch of science that deals with the issues of data collection, organization, presentation and analysis of data for meaningful (progressive) decision(s)/inference. It is well rooted in the theory of mathematics and uses concepts in the area of differential equations, topology, measures theory and the likes. Little wonder why the Senate of many Universities made this discipline to reside in the Faculty of Physical Sciences where it extends its hand of fellowship to all branches of Physical Sciences, Engineering and Technology, Life and Medical Sciences, Agriculture and Food Security, Social and Economic Sciences, etc. In early years, statistics was supposed to be the science of kings, used for the purpose of administration.

11. Statistics as a discipline has been misunderstood, misinterpreted and misapplied. • Statistics is not just the branch of knowledge that anyone can dabble into. • Statistics is not just the branch of knowledge that entails mere collection of data, presentation of pictures and making one interpretation or the other. • Statistics is not just the branch of knowledge that is contained in mathematics.

12. Vice-Chancellor Sir and my honorable audience, at this moment, I shall give attention to my footprintsin the discipline. This lecture is divided into the following areas: - Density estimation (in particular kernel densities), - Predictive discriminant analysis, - Cluster analysis, - Social and economic statistics, - Medical and biological statistics, - Engineering statistics, - Education statistics and - Statistics in agriculture and food security. APOLOGIES

13. DENSITY ESTIMATION • The fundamental principle of density estimation is hinged on the probability density function (pdf). If X is a continuous random variable with probability density functionf(x), (1) • What is density estimation?. • These are some problemsof KDE: • - Getting the density estimate from the observed values; and • How the estimated density can be used for statistical analysis. • The following are its merits: • - It can be used for the investigation of the properties of a given data. • - The skewness and multi-modality in a dataset can be easily shown by density estimation. • -It is useful for making decisions and drawing conclusions. • Methods of estimating densities abound in literature

14. THE KERNEL DENSITY ESTIMATION METHOD • THE UNIVARIATE SYMMETRIC KERNELS • Here the dimension of the random variable X is one (univariate) and symmetric. • Mr. Vice-Chancellor Sir and amiable audience, consider the following density estimator, • (2) • where K(.) is the kernel function, h, is window width (band width or smoothing parameter), and the Xi’s are the sample points. Equation (2) is the kernel density estimator (KDE)

15. SYMMETRIC DISTRIBUTION

16. UNIVARIATE WINDOW WIDTHS SELECTIONS Since the introduction of density estimation techniques by Rosenblatt (1956), the selection ofwindow widths has been a crucial issue (Silverman (1986), Titterington (1983), Taylor (1989), Ogbonmwan (1993), Hossjer and Ruppert (1995) Ogbonmwan and Osemwenkhae (1997)), Osemwenkhae (2003), etc. This challenge was a subject of motivation for Ogbonmwan and Osemwenkhae (1999) to examine, higher order optimal window widths with the associated Mean Integrated Squared Errors (MISE) and the resulting efficiencies for any symmetric kernels.

17. UNIVARIATE WINDOW WIDTHS SELECTIONS CONT. The choice of the window width using the Least Square Cross-Validation (LSCV) has gained prominence. For a larger data set, the LSCV scheme leads to a degenerate choice of optimal window width. A theoretical verification of this degeneracy was established in Ogbonmwan and Osemwenkhae (2003). Osemwenkhae and Oyegue (2004) examined the selection of optimal window width for higher order of band widths in symmetric kernels. A generalized form of the smoothing parameter, h, in kernel density estimation was presented. This was aimed at removing the rigour(s) of specifying the order of the bias before obtaining the corresponding smoothing parameter. This generalized h was tested on some selected symmetric kernels and was found to work for all even order of the bias.

18. EFFICIENCIES OF UNIVARIATE SYMMETRIC KERNELS This is the second area of our research focus. In Ogbonmwan and Osemwenkhae (1997), the choice of kernel estimators as it relates to their efficiencies were examined. The claim that the choice of kernels poses no problem in density estimation was viewed with some caution when the efficiencies of such kernels were considered. Furthermore, Osemwenkhae and Ogbonmwan (2003a) generalized efficiencies for higher order symmetric univariate kernels. The problem of obtaining the efficiency of symmetric kernels, especially for cases where the order of the smoothing parameter is greater than two has not received much attention as at then. We provided a formula for obtaining the generalized efficiency for any higher order symmetric kernel. In some cases, the corresponding efficiencies of kernels were unstable.

19. EFFICIENCIES OF UNIVARIATE SYMMETRIC KERNELS CONT. The works of Osemwenkhae and Ishiekwene (2007) and Osemwenkhae and Okuonghae (2007a) considered the construction of asymptotically Minimum Variance Unbiased Estimators (MVUE) for symmetric univariate kernels. The works found that the MVUE is a desirable property and that (MVUE) is only possible at higher order values of the smoothing parameter, h in univariate kernels.

20. GLOBAL ERROR MEASUREMENTS & DATA DRIVEN TECHNIQUES IN SYMMETRIC KDES Generally, the (3) Osemwenkhae and Ogbonmwan (2003b) focused on the analysis and computation of the generalized global errors (MISE). Furthermore, the work of Ishiekwene et al. (2008) presented a new algorithm for boosting KDE. The algorithm enjoys the property of bias reduction like other existing boosting algorithms and also enjoys the property of fewer functionevaluations when compared to other boosting schemes. Numerical examples were used and compared with existing algorithm and the findings were comparatively exciting.

21. BALANCING THE COMPONENTS OF THE ERROR TERMS IN KDE Osemwenkhae and Oyegue (2006a) compared the Integrated Bias Squared (IBS) and Mean Integrated Squared Error (MISE) in higher order univariate kernels. The aim was to encourage practicing statisticians to apply these comparisons in judging their choices of kernels at higher order. The rectangular, the biweight and the Gaussian kernels did not show any distinctive reduction in their IBS, but significant reductions were found in their MISE.

22. Furthermore, Osemwenkhae and Oyegue (2006b) compared the Asymptotic Integrated Squared Bias (AISB) and the Asymptotic Integrated Variance (AIV) at various higher order values of the smoothing parameter, h, vis-à-vis its effect on the Asymptotic Mean Integrated Squared Error (AMISE). For the three test kernels, it was observed that both the AISB and the AIV were decreasing for successive higher order m, of the smoothing parameter, h, with a lower propagation error. The Mean Integrated Squared Error (MISE) and the Integrated Bias Squared (IBS) decreasing indicates closeness between a parameter and its estimator. Depending on the tolerance limit specified for the MISE and the percentage efficiency permitted, the extent of bias reduction required, can be monitored. (Osemwenkhae and Odiase (2006 a & b), Ishiekweneet al. (2008)).

23. GLOBAL ERROR MEASUREMENTS & DATA DRIVEN TECHNIQUES IN SYMMETRIC KDES CONT. Ishiekwene and Osemwenkhae (2006) worked on a comparison of fourth-order window width selectors in univariate kernel density estimation. In the work, the choice of an ideal window width in kernel density estimation was presented using a fourth-order derivative kernel method. The obtained results were similar to that of Jones and Signorini (1997). However, the AMISE obtained from using our optimal window width method was found to be superior to that of order-two in the literature. The uniform, the Gaussian and the Epanecknikov kernel functions were considered in Osemwenkhae et al. (2005). Rather than relying on the Mean Integrated Squared Error and efficiency properties, the study asserted that the choice of a kernel functionshould be data-driven within the descriptive goal of the experimenter (Osemwenkhae et al.2005, Osemwenkhae et al. (2015)).

24. MULTIVARIATE KDEs In practice, the dimension of the random variable X may not just be one but more than one. To this end, fundamental properties defining kernel estimators were extended. Ogbonmwan and Osemwenkhae (2000) and Osemwenkhae et al. (2005) examined higher order forms for optimal window width in multivariate kernel density estimation (MKDE). In these works, a generalized higher order forms for the window width of the multivariate kernel density estimation were obtained. This was an extension of the work of Ogbonmwan and Osemwenkhae (1999).

25. MULTIVARIATE KDEs Oyegue and Osemwenkhae (2011) examined the efficiency of the multivariate kernel density estimators. Since one of the most used measures of the global accuracy of the multivariate kernel density estimator is the Mean Integrated Squared Error (MISE), we provided a generalization of the higher order optimal bandwidth and the Asymptotic Mean Integrated Squared Error (AMISE) for the multivariate kernel density estimator. Our method has a faster rate of convergence than many in the literature. The advantage was that it enables one to know the speed at which the estimated density approaches the true density.

26. MULTIVARIATE KDEs Continues We have applied the concept of the intersection of confidence intervals approach to the multivariate density (Ogbeideet al. 2017). This was done in an attempt to correct the problem of discontinuities and boundary value problem in the density to be constructed. The quality of the estimates obtained showed reduced Asymptotic Mean Integrated Squared Error (AMISE) and faster rates of convergence. Furthermore, Ogbeideet al. (2016) considered the Modified Cluster Sampling Multivariate Kernel DensityEstimation (MCMKDE) approach. The approach was based on the relevant ideas of estimating the population clusters from the dataset. Empirical cluster samples, which were observations grouped into data cluster either via rows or columns information according to the empirical cluster they belong, were formed. The bandwidth parameters derived by these approaches based on the data set clusters were used to smooth the density. The estimates from the approach showed some improvements over many existing methods including the fixed bandwidths method. This approach compensated for discontinuities in the estimated density curve (pilot plot).

27. NON-SYMMETRIC KERNELS Symmetric Distribution Non-Symmetric Distribution

28. NON-SYMMETRIC KERNELS CONT. Osemwenkhae and Orhionkpaiyo (2006) examined the existence of non-symmetric densities. The study examined the basic defining properties of univariate kernels when the kernel in question need not be symmetric. We showed that the construction of densities corresponding to any set of data drawn from a non-symmetric population was possible. Osemwenkhae and Izevbizua (2005) and Osemwenkhae and Okuonghae (2007b) examined the problems of finding the appropriate window width, when estimating densities wherever the density in question is non-symmetric. These works generalized the window width corresponding to non-symmetric kernels at any given order of the non-symmetric kernel. Osemwenkhae and Isere (2007) and Osemwenkhae et al.(2007) worked on the generalized biases and generalized errors in non-symmetric univariate kernels. The practice of obtaining biases of any non-symmetric kernel when the order of the smoothing parameter, h, is one was seen not to be sufficient as the error size for this case was still very large. A new scheme for higher order biases in non-symmetric univariate kernels was presented. These schemes do not only achieve the possibility of reducing the size of the global error term, but also generalized the bias term for any non-symmetric kernels.

29. THE ADAPTIVE KERNELS Another area of density estimation that attracted our attention was the adaptive/adjustable/variable KDEs. The work of Osemwenkhae and Ogbeide (2010a) considered the variable kernel method in kernel density estimation as an adaptive method. This method has the advantage of enjoying adjustable smoothing parameter, h, all through the process of estimation of the kernel density. Osemwenkhae and Ogbeide (2010b) examined of the AMISE measures for both the fixed kernel estimator and the adaptive kernel estimator. It was found that the error propagation was smaller in the AMISE as well as its rate of convergence for the AMISE was faster for the adaptive kernel method than for the fixed kernel method. Ogbeideet al. (2011) and Ogbeide and Osemwenkhae (2012) focused on the IICI and the Modified Kernel Nearest Neighbourhood Estimation (MKNNE) in estimating densities. These approaches were used to estimate the performance of students in a University. The quality of the adaptive density estimates obtained showed some improvements over existing schemes in terms of reduced error rates and better rate of convergence.

30. PREDICTIVE DISCRIMINANT ANALYSIS (PDA) Improving classification accuracy is usually achieved with best subset of relevant predictors obtained by using classical variable selection methods. The goal of variable selection methods is to choose the best subset (or training sample) of relevant variables that typically reduces the complexity of a model and makes it easier to interpret as well as improve classification accuracy of the model and reduce the training time. Clinician-like professionals in artificial intelligence, criminal justice, education, medicine, neuropsychology, psychiatry, and psychology are regularly faced with the problem of making predictions. Criteria for these predictions may be diagnostic condition, disease type, graduate school success, and so on. The predictor information utilized may involve biopsy slide ratings, intelligence scores, achievement scores, and so on. When the criterion for prediction involves one or more predictor variables alongside with a categorical criterion, such prediction will call for the use of predictive discriminant analysis (PDA ). See Charles and June (1970) and Huberty and Stephen (2006). The work of Osemwenkhae and Iduseri (2011) considered an efficient data-driven rule for obtaining an optimal Predictive Discriminant Function (PDF) of a discriminant analysis.

31. In the study, we carried out a sequential-stepwise analysis on the predictor variables and percentage-N-fold cross validation on the data set obtained from students’ academic records in a University system. The hit rates, the results obtained for the optimized predictive discriminant function (PDF) by Z(OPT)caliberation on the training and the validation sets, when compared with that of PDF obtained using the conventional rule, showed a significant improvement in terms of how well each PDF classifies cases into the different categories. It was also discovered that the optimized PDF produces consistent high hit rates with little variability and thus reducing the problem of overfitting. Hit rate is the percentage of cases on the diagonal of the confusion matrix, or simply the percentage of correct classification. The need to obtain more precise estimates of discriminant weights with best matching performance from an efficient cross validation rule was studied by Iduseri and Osemwenkhae (2014a). The study data were obtained from a large international air carrier. The data set included measures of interest in outdoor activities, sociability and conservativeness of the employees in three different jobs classifications. The result showed that the discriminant weights were fine-tuned over and beyond the training and validation sets. Our method produced better estimates of the true values of the discriminant weights that would be found if the observation set comprised all members of the population.

32. Despite the remarkable successes recorded in reducing the problem of overfitting(Osemwenkhae and Iduseri, 2011; Iduseriand Osemwenkhae, 2014a) in the application of predictive discriminant analysis, ascertaining the presence of overfitting amounts to estimating the actual hit rate. In this wise, the work of Iduseri and Osemwenkhae (2014b) examined a new approach of using the percentage-N-fold cross validation rule (NFCV-P) to determine the expected value for the actual hit rate of a predictive discriminant function (PDF). The new approach produced estimates of actual hit rate that were consistent, and essentially equivalent to that of the Leave-One-Out Cross Validation (LOOCV) methodwith less computational expense. In an effort to obtain maximum hit rate for Predictive Discriminant Function (PDF) that is optimal even for the training sample that gave birth to it, and obtain shorter training time as well as enhance generalization by reducing overfitting, we developed an efficient variable selection method for obtaining a subset of predictors that will be superior to all other subsets from the same historical sample (see Iduseri and Osemwenkhae, 2015).

33. The new variable selection method which is a modification of the LOOCV method addressed the problems inherent with the all-possible subset approach. In application to real life datasets, the obtained predictive function using our method achieved an actual hit rate that was essentially equal to that of the all-possible-subset method, with a significantly less computational expense. Our results in Iduseri and Osemwenkhae (2015) showed significant performance in terms of computational expense when compared to existing classical variable selection methods. Recently, we developed a new approach for obtaining a near optimal training sample that will produce a statistically optimal hit rate using a modified winsorization with graphical diagnostic techniques (Iduseri and Osemwenkhae, 2018). By winsorization we mean a transformation of the statistics of a batch or sample by limiting extreme values. In application to real life data sets, the new approach was able to identify and remove legitimate contaminants in one or more predictors in the training sample, thereby resolving any significant differences in the variances for the groups formed by the dependent variable.

34. It is worthy to note that the graphical diagnostic technique associated with the new approach is a novel visual tool which served as an alternative to graphical test for homogeneity of variances.We have also paid attention to the simultaneous occurrence of multicollinearity and legitimate contaminant in Y-space due to non-normality of error variable in multiple linear regression as contained in Iduseri and Osemwenkhae (2016a). A new class of Modified Winsorized Shrinkage Estimators (MWSEs) was presented in the work and their performance was evaluated using Estimated Mean Squared Error (EMSE). Simulation studies reveal that the MWSEs show consistently minimum EMSE among the considered shrinkage estimators.

35. THE CLUSTERING METHODS Another area of our research is the clustering methods. Other synonyms of clustering/cluster analysis are automatic classification, numerical taxonomy, botrology, etc. Some available clustering methods are partition methods (k-means and k-mediods), model based methods, hierarchical methods, grid-based methods and density based methods (see Jain, 2010). Our focus was on the comparative study of k-means and k-medoids clustering methods. We provided a formal and organized study on the effect of the nature of data and cluster structure on the performance of k-means and k-medoids clustering methods. A cluster validation method, called Silhouette analysis, was used to assess and qualify the cluster partitions created by both methods. Results obtained revealed that the performance of k-means was at its peak with data in which clusters are of relatively uniform sizes while the k-medoids method tends to perform better than k-means when the input data have varied cluster sizes. See Ekhator and Osemwenkhae (2015) and Osemwenkhae et al.(2016).

36. THE DISTRIBUTION THEORY • Another area of our research is in distribution theory. Six methods for estimating the Weibull shape and scale parameters were considered and compared. These methods are: • the least squares method, • the weighted least squares method, • the method of moments, • the energy pattern factor method, • the method of L-moments and • the maximum likelihood method. • The maximum likelihood method produced the best method when all six methods were applied to wind speed sample by possessing the smallest mean squared error. One very useful result which was obtained from the study revealed that the weighted least squares method proved to be a very efficient method for estimating both the Weibull shape and scale parameters. This happens to be a rare incidence in many studies, see Osatohanmwenet al. (2018).

37. THE DISTRIBUTION THEORY CONT. Osemwenkhae and Iyenoma (2018). The application of the distribution was subjected to two lifetime data sets and some measures of goodness-of-fit such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Komolgorov-Smirnov (K-S) criterion. Results obtained from the two lifetime data sets, revealed that the inverse Burr distribution was an appropriate model in fitting the lifetime data sets and had superiority over other distributions (Normal, logistic, Gumbel, inverse Gamma, and inverse Weibull) considered.

38. THE SOCIAL AND ECONOMIC STATISTICS MAKING STOCK MARKET INVESTMENT DECISION The need for profit and return on investments is ever present in the business climate. Generally, investors desire to profit from their investments. To completely comprehend this, statistics can be valuable.

39. THE SOCIAL AND ECONOMIC STATISTICS CONT. Osemwenkhaeand Eguasa (2016) developed a time series model for the Nigerian Stock Exchange using the daily stock closing price of Okomu Oil Palm Company Plc. from Jan. 2010 to Dec. 2014. We applied the Box-Jenkins autoregressive integrated moving average (ARIMA) modeling methodology and our work revealed that the ARIMA (0,1,1) was best in modeling the stock price data as indicated by the AIC. This model was however improved upon by Osemwenkhae et al. (2016) as the presence of conditional variance (volatility) in the model residuals was addressed. We went ahead to formulate a hybrid of ARIMA(0,1,1)-ARCH(5) model. This model better explained and captured the dynamics of the daily stock price of Okomu oil Plc. I invite all those who wish to invest in the stock of the Company studied to use the model we have developed so as to avoid or minimize losses.

40. THE MEDICAL AND BIOLOGICAL STATISTICS Another area of application of our research is in the medical and biological statistics. Okuonghae and Osemwenkhae (2006a, b) applied both mathematical and statistical models to problems in epidemiology. The objective of the work was to model the dynamics of the disease (sleeping sickness) as it affects the human and domestic animal populations. • Vibrio cholerae

41. THE MEDICAL AND BIOLOGICAL STATISTICS CONT. Another area of application of our research is in the medical and biological statistics. Okuonghae and Osemwenkhae (2006a, b) applied both mathematical and statistical models to problems in epidemiology. The objective of the work was to model the dynamics of the disease (sleeping sickness) as it affects the human and domestic animal populations. Similarly, Osemwenkhae et al. (2009) and Isere and Osemwenkhae (2010) examined the issue of theendemicity of cholera in Nigeria. In these studies, we investigated Vibrio cholerae (V. cholerae) in its aquatic habitat using a mathematical model to describe its nature in their habitat with emphasis on its endemic nature. In other words, it is not possible for multiple endemic equilibria to exist if the reproduction number is less than one.

42. THE MEDICAL AND BIOLOGICAL STATISTICS CONT. This was further corroborated by (Isere et al. 2009) using a statistical scheme Cumulative Sum (CUSUM) for its detection by applying the WHO weekly epidemiological record of cholera cases in Nigeria between 1996 and 2005 to demonstrate the application of the techniques. From the demonstration, the technique has a good potential as a tool for detecting cholera outbreaks. In the same vein, Isere et al. (2009) investigated the trend of cholera outbreaks in Nigeria using Exponentially Weighted Moving Average (EWMA) and CUSUM control charts. The results showed that the schemes performed creditably as they both gave an out-of-control signal in the years 1999 and 2000, indicating that there were cholera epidemics in those years. However, it was found that EWMA control chart was superior to CUSUM in the investigation of the trend of cholera outbreaks in Nigeria.

43. THE MEDICAL AND BIOLOGICAL STATISTICS CONT. Other biological and medical applications were in Osagieand Osemwenkhae (2011) and Osagieet al. (2012). In Osagie and Osemwenkhae (2011), we mathematically analyzed the acute toxicity of HunteriaUmbellata, in mice in Nigeria using the logistic model. HunteriaUmbellata is a plant with therapeutic potentials in the treatment of various diseases that include yaws, peptic ulcers, diabetes, piles, infertility and inflammation. Data on the acute toxicity of seeds extract of HunteriaUmbellata was obtained via intraperitoneal route and analysed. The median lethal dose was determined and found to be 1.61g/kg of body weight with confidence limits as [1.4335g/kg, 1.7811g/kg]. This showed slight toxicity of HunteriaUmbellata and its toxicity at a high dose on acute exposure should be put into consideration when it is used as treatment for these diseases.

44. THE MEDICAL AND BIOLOGICAL STATISTICS CONT. In Osagie and Osemwenkhae (2012), we measured the acute toxicity of indomethacin in rats using the Weibull model. This was achieved by obtaining the median lethal dose (MLD) for indomethacin. The implication of the MLD value indicates that indomethacin could be fatal and toxic as a drug if not properly administered. Continuing in the direction of the application of statistics to medicine, Iduseri and Osemwenkhae (2016c) developed a model on maintaining high levels of immunization coverage in Edo state using best subset binary logistic regression. We found out that a target of 90-95% levels of immunization coverage was necessary to sustained control of vaccine-preventable diseases. The chi-squared test was used as a validation tool for this work. We found that the obtained logistic model which constituted only five key predictor variables out of eight potential predictors used had a good predictive performance. In addition, validation of the five key predictor variables using the chi-square test showed that all had significant association in relation to a child completion of immunization schedule.

45. ENGINEERING STATISTICS Another application of our research is in the area of engineeringstatistics. The work of Osemwenkhae and Orhionkpaiyo (2006) was a landmark opener inthe non-symmetric univariate kernels. This work was necessitated by the presence of non-symmetric densities in most disciplines of engineering and the sciences. The novel work of Ariavie and Osemwenkhae (2010)encouraged team work and group dynamics among foundation engineering student of the University of Benin. A novel modification of the existing Pre-Degree UBITS training to include group/team work was discussed. The rationale for using groups (teams) and the attendant dynamics were also presented. Attention to team size, composition, organization, responsibilities and evaluation (SCORE) criteria can measurably enhance the beneficial effects of teaming in any engineering practice. Osemwenkhaeand Osagie (2007 and 2010) developed a mathematical model explaining the gradual aging of systems using the two-parameterWeibull hazard function. This work examined a mathematical model for obtaining the shape and scale parameters, and the implication of these parameters in obtaining the aging coefficient of any system which is gradually aging, (using the two-parameter Weibull hazard function). It concluded that the nature of the aging property depends on the aging coefficient of the system, which makes the system to gradually age with time.

46. EDUCATION STATISTICS Another application of our research focus is in the area of education statistics. Educationis seen as life or a way of life of a people. We have applied statistical tools/models in this regard. The work of Osemwenkhae and Ishiekwene (2007a) applied the concept of adjustable/variable kernel methods in test evaluation/development.This study was motivated by the appearance of heavy tails when estimating learners’ performances in most psychological tests. Explaining this appearance of heavy tails poses a problem as the popular standard normal distribution (Z-Score) is often not tenable. Therefore, in any psychological test involving heavy tails (legitimate outliers), we observe that adjustable kernel method was better for fitting this density than its fixed kernel method. However, when heavy tails are not suspected, fixed kernel method may be easier in assessing learners’ performances. Furthermore, Iduseri and Osemwenkhae (2016b) used discriminant analysis to identify major prerequisites for successes in specific course of study in a University system. We presented a case study with forty (40) industrial mathematics majors of a University where discriminant analysis was used successfully to determine the major prerequisite for a success in industrial mathematics. Iduseriand Osemwenkhae (2017) extended this work by outlining key predictor variables in classical statistical procedures for separating the groups and thereby improving predictions/classification accuracy.

47. STATISTICS IN AGRICULTURE AND FOOD SECURITY

48. STATISTICS IN AGRICULTURE AND FOOD SECURITY Another area of our research interest was applying statistics in agriculture and food security. The work of Odjugo and Osemwenkhae (2009) on the effects of natural gas flaring on the yield of maize (zea mays) gained widespread acceptability and reference. It is common knowledge that gas flaring has a negative environmental impact in the Niger Delta. Our work reported the impact of natural gas flaring on our microclimate and maize yield in the Niger Delta, using Ovade flaring site as a case study. Experimental sites were located at 500m, 1km, 2km, and 70km (control) from the flare site. Soil physio-chemical properties, air and soil temperatures, rainfall and relative humidity were monitored. Physiological parameters used were emergence rate, growth rate, leaf area index (LAI) and yield. The experiment was carried out in the 2005, 2006 and 2007 planting seasons. Time series analysis and analysis of variance (ANOVA) were some statistical tools employed in analyzing the data. The results show that with rise in air and soil temperatures of the flare site, relative humidity, soil moisture, and all the soil chemical parameters decrease toward the flare site. These induced microclimatic conditions, impacted on the soil and reduced the yield of maize by 76.4%, 70.2% and 58.2% at 500m, 1km and 2km, respectively. Maize production is not economically viable within the first 2km from the flare site.

49. THE WAY FORWARD

50. Now that our efforts in making the subject of statistics meaningful to our society have been examined, it is now time to examine the way forward in this discipline.Every country needs good statistics, therefore, almost all countries have established specialized agencies (such as NBS and CBN in the case of Nigeria) whose job is to collect, process and disseminate good statistics.The problem is that in many countries, especially those in the developing world, the work of these agencies is under-appreciated and under-valued. Many statistical systems are caught in a vicious cycle of under-funding which ultimately leads to under-performance and in some cases falsification of results becomes inevitable. THE WAY FORWARD CONT.