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An adjusted network information criterion for model selection in statistical neural network models
(JMASM, Inc., 2016) Udomboso, C. G.; Amahia, G. N.; Dontwi, I. K.
In this paper, we derived and investigated the Adjusted Network Information Criterion (ANIC) criterion, based on Kullback’s symmetric divergence, which has been designed to be an asymptotically unbiased estimator of the expected Kullback-Leibler information of a fitted model. The ANIC improves model selection in more sample sizes than does the NIC.
Alternative goodness-of-fit test in logistic regression models
(Medwell Journals, 2011) Nja, M. E.; Enang, E. I.; Chukwu, A. U.; Udomboso, C. G.
The Deviance and the Pearson chi-square are two traditional goodness-of-fit tests in generalized linear models for which the logistic model is a special case. The effort involved in the computation of either the Deviance or Pearson chi-square statistic is enormous and this provides a reason for prospecting an alternative goodness-of-fit test in logistic regression models with discrete predictor variables. The Deviance is based on the log likelihood function while the Pearson chi-square derives from the discrepancies between observed and predicted counts. Replacing observed and predicted counts with observed proportions and predicted probabilities, respectively in a cross-classification data arrangement, the standard error of estimate is proposed as an alternative goodness-of-fit test in logistic regression models. The illustrative example returns favourable comparisons with Deviance and the Pearson chi-square statistics.
An alternative technique to ordinal logistic regression model under failed parallelism assumption
(2009-12) Adeleke, K.A .; Adepoju, A.A
Maternal health status is often measured in medical studies on an ordinal scale but data of this type arc generally reduced for analysis to a single dichotomy. Several statistical models have been developed to make full use of information in ordinal response data, but have not been much used in analyzing pregnancy outcomes. The authors discussed two of these statistical models the ordinal logistic regression model and the multinomial logistic regression model. Logistic regression models are used to analyze the dependent variable with multiple outcomes which can either be ranked or not. In this study, we described two logistic regression models for analyzing the categorical response variable. The first model uses the proportion-! odds model while the second uses the multinomial logistic regression model. The fits of these models using data on delivery from a Nigerian State hospital record/database were illustrated and compared to study the pregnancy outcomes. Analyses based on these models were carried out using STATA statistical package. The Multinomial logistic regression was found to be an important alternative to the ordinal regression technique when proportional odds assumption failed. The weight of the baby and the mother's history of disease (treated or not treated) were found to be important in determining the pregnancy outcome.
An Application of Bayesian Dynamic Linear Model to Okun’s Law
(Scienpress Ltd, 2017) Awe, O. O.; Sanusi, K.A.; Adepoju, A. A.
Many authors have used dynamic time series regression models to analyse Okun’s law. This type of models often require first differencing the dependent and independent variables, as well as investigating the maximum lag length required for the model to be efficient. In this paper, we propose a straight-forward time-varying parameter state space model for analyzing Okun’s law. In particular, as a case study, we investigate the validity and stability of Okuns law using a Bayesian Dynamic Linear Model which implicitly describes the time-varying relationship between Gross Domestic Product (GDP) and unemployment rate of a major economy in Africa for three decades. The time-varying parameters of this model are estimated via a modified recursive forward filtering, backward sampling algorithm. We find that Okuns law exhibited structural instability in Nigeria in the period 1970-2011, with the sensitivity of unemployment rate to movements in output growth loosing stability over time, which may have been a contributor to her recent economic decline
Application of ordinal logistic regression model to occupation data
(Duncan Science Company, 2009) Adepoju, A. A.; Adegbite, M.
People's occupational choices might be influenced by their parents' occupation, gender, previous experiences, ages, and their own education level. We can study the relationship of one's occupation choice with education level and father's occupation. The occupational choices will be the outcome variable which consists of categories of occupations. The regression methods are capable of allowing researchers to identify explanatory variables related to organizational programs and services that contribute to the overall staff status. These methods also permit researchers to estimate the magnitude of the effect of the explanatory variables on the outcome variable. Therefore, regression methods seem to be superior in studying the relationship between the explanatory and outcome variables. This study used ordinal logistic regression method to examine the relationship between the ordinal outcome variable, different levels of staff status in the Lagos State Civil Service of Nigeria, the explanatory variables are Gender, Indigenous status, Educational Qualification, Previous Experience and Age. The outcome variable was measured on an ordered, categorical, and three-point Likert scale as Junior staff Middle Management staff, and Senior Management staff. Within the complete models, the legit link was the better choice because of its satisfying parallel lines assumption and larger model- fitting statistics. The study revealed that two explanatory variables namely, Education Qualification and Previous Working Experience significantly predicted the probability of an individual staff being a member of any of the three levels of staff status
Application of regression type estimator in double sampling skills to students’ enrollment in Oyo State
(2019-03) Udomboso, C. G.; Akanbi, O. B.; Afolabi, S. A.
This research derived the precision using Regression Estimation technique with the application of secondary data obtained using the number of students enrollment in 2015 (Auxiliary variable “x”) and 2016 (response variable “y”) respectively in secondary schools of Ibadan, Oyo State, Nigeria for the purpose of obtaining average enrollment figures in the selected state in order to know the bright future of secondary schools in Oyo State in general and to establish the empirical comparison of the optimum variances in obtaining the most efficient estimator in order to satisfy the condition; p2≥ 122124 based on the coefficients of Variation for the validity and reliability, the relative efficiency was also determined based on the conditions attached to the supremacy in terms of the estimated mean square error (variance) whereby the regression line does not pass through the origin from the graph of Relative Efficiency (R.E) against Correlation Coefficients (p) that maintain inverse relation. Proper conclusions and recommendations are made based on findings from the analysis in terms of adequate record keeping among the contemporary states within.
ARIMA model and neural network: a comparative study of crime rate modelling
(2015) James, T. O.; Suleiman, S.; Udomboso, C. G.; Babayemi, A. W.
Crime rate is a serious issue that affects everyone in society. It affects the victims, perpetrators, their families the government and even reality of good governance. In this study forecasting of crime rate using autoregression integrated moving average (AR1MA) model was compared with feed forward neural networks. The J multi software was used for analysis of data gotten from State Police Headquarter in Kebbi State from January 2004 to December 2013 and the series was stationary at first difference and ARIMA (0, 1, 1) was obtained as the best model for the series. This was model by Neural Network using SPSS. In the training of the network, the samples were automatically partitioned in to 73.3% of training and 26.7% of testing. The computational result shows that Artificial Neural Network provides better model than ARIMA by having minimum error in the in-sample and out -of- sample in MAE, MSE, and RMSE with 3.84614, 2.00466 and 1.41586 respectively.
Assessment of simultaneous equation techniques under the influence of outliers
(2011) Oseni, B. M.; Adepoju, A. A.
Most simultaneous equations estimation techniques are based on the assumptions of normality which gives little consideration to some atypical data often called outliers which may be present in the observations. The outliers may have some obvious distorting influence on the estimates produce by these techniques. This study investigates the distorting effect of outliers on four simultaneous equation estimation techniques through Monte Carlo method. Outliers of various degrees were introduced into observations of different sizes. The estimators were ranked based on their ability to absorb the shock due to outliers in the observations. The Total Absolute Bias (TAB), Variance and Root Mean Square Error (RMSE) were used in ranking the performances of the estimators. Based on the criterion of tab, two stages least squares (2SLS) ranked the best, closely followed by three stage least squares (3SLS) and ordinary least squares (OLS) in that order, while limited information maximum likelihood (LIML) was the poorest when outliers of not more than 5% are present in the observation. It is however, not strange to observe that OLS outperformed the other estimators when variance was used. This could be misleading since variance may be measured around a wrong parameter. Based on the criterion of RMSE, ordinary least squares yields estimates with the least value of RMSE while LIML yields the greatest when outliers of not more than 10% are present in the observation. Also it was established that OLS has the greatest capacity to absorb the shock due to the presence of outliers in the observation
Autoregressive distributed lags (ARDL) modelling of the impacts of climate change on rice production in Kebbi State
(Professional Statisticians Society of Nigeria, 2018) James, T. O.; Babayemi, A. W.; Abdulmuahymin, A. S.; Udomboso, C. G.; Bello, M. L.
Autoregress Distributed Lag (ARDL) is an econometric model that determines the long run a d short run association between the Serial (Stationary/ non-stationary as well as reparameterizing them to Error correction model (EMC). Rice cultivation and production is a major source of income for millions of households around the globe especially in Nigeria. It is also a major staple food, but Climate change poses great threat to the stability and sustainability of rice production for sufficient agricultural system, since most Nigeria consumes rice more than other foods and Kebbi state, is one of the major states contributing to the total rice output of the country. Climate change is the major challenge facing rice production. This study therefore, investigates the long-run and short run effect of factors affecting rice production in Kebbi State. 1000 simulations of data were obtained from the data collected between the period of 2005 to 2016 from the state Ministry of Agriculture. The result showed that rainfall has impact both in the long run and short run; 100% increase in rainfall, will tend to give 99.98% increase in rice production in the long-run. However, temperature tends to show insignificant impact on rice production. The result of this paper facilitate understanding for government and agriculturist in the linkages between climate change variables and rice production which can boost and increases the production of rice in Kebbi State.
The Bayesian Approach to Estimation of Multi-Equation Econometric Models in the Presence of Multicollinearity
(2014) Okewole, D. M. O
The Bayesian approach conveys information not available in the data but on prior knowledge of the subject matter, which enables one to make probability statements about the parameters of interest, while the classical approaches deals solely with the data. Several researches on the classical approaches have shown them to be sensitive to multicollinearity, a violation of one of the assumptions of multi-equation models which often plagues economic variables. Studies on the performance of the Bayesian method in this context are however limited. This study was aimed at investigating the performance of the Bayesian approach in estimating multi-equation models in the presence of multicollinearity. The purely just- and over-identified multi-equation models were considered. In both cases, the normal distribution with zero mean and large variance served as locally uniform prior for the regression coefficients. Three Bayesian Method Prior Variances (BMPV) were specified as 10, 100 and 1000 in a Monte Carlo prior variance sensitivity analysis. The Wishart distribution with zero degree of freedom served as prior distribution for inverse of error variance-covariance matrix, being its conjugate. The posterior distributions for the two models were then derived from the prior distributions and the likelihood functions as a bivariate Student-t and generalized Student-t distributions respectively. The estimates were then compared with those from the classical estimators; Ordinary Least Squares (OLS), Two stage Least Squares (2SLS), Three stage Least Squares (3SLS) and Limited Information Maximum Likelihood (LIML). Samples of sizes T=20, 40, 60, and 100 in 5000 replicates were generated based on eight specified research scenario. The Mean Squared Error (MSE) of the estimates were computed and used as evaluation criteria. The BMPV 10 produced the least MSE in the prior variance sensitivity analysis for the over-identified model, whereas for the just-identified model without multicollinearity, BMPV 100 was the smallest. The Bayesian method was better in the small sample cases T  40 than the classical estimators for  (the coefficient of the exogenous variable in the just-identified model); when T=20, MSE for BMPV 10, 100 and 1000 were 0.169, 0.168 and 0.171 respectively, whereas OLS, 2SLS, 3SLS and LIML yielded same results; 0.244, when T=40, BMPV 10, 100 and 1000 were 0.1220, 0.1272, 0.1361 respectively and 0.1262 for the classical methods. The 2SLS and 3SLS estimates of (coefficient of the endogenous explanatory variable) which were the same in the over identified model had smaller MSE than the Bayesian method; when T=20, MSE for 2SLS/3SLS =0.0280, whereas BMPV 10=0.0286, BMPV 100 = 0.0300, and BMPV 1000 = 0.033. The Bayesian method was less sensitive to multicollinearity in estimating coefficients of the correlated exogenous variables; MSE (T=20) for BMPV 10, 100, and 1000 were 0.4529, 0.5220, 0.5290 respectively, while it was 0.7492 for the classical estimators. The MSE of LIML (0.0036) was similar to that of BMPV 100 (0.0036) and BMPV 1000 (0.0036) in large sample caseT 100 for. Bayesian approach was suitable for estimating the parameters of exogenous variables in the small sample cases when the model is purely just-identified, and in over identified model in the presence of multicollinearity
Bayesian approach to survival modeling of remission duration for acute leukemia
(2019) Akanbi, O. B.; Oladoja, O. M.; Udomboso, C. G.
The problem of analyzing time to event data arises in a number of applied fields like biology and medicine. This study constructs a survival model of remission duration from a clinical trial data using Bayesian approach. Two covariates; drug and remission status, were used to describe the variation in the remission duration using the Weibull proportional hazards model which forms the likelihood function of the regression vector. Using a uniform prior, the summary of the posterior distribution; Weibull regression model of four parameters ( η, µ,β1, β2, was obtained. With Laplace transform, initial estimates of the location and spread of the posterior density of the parameters were obtained. In this present study, data from children with acute leukemia was used. The information from the Laplace transform was used to find a density for the Metropolis random walk algorithm from Markov Chain Monte Carlos simulation to indicate the acceptance rate (24.55%).
Bayesian optimal filtering in dynamic linear models: an empirical study of economic time series data
(2015) Awe, O. O.; Adepoju, A. A.
This paper reviews a recursive Bayesian methodology for optimal data cleaning and filtering of economic time series data with the aim of using the Kalman filter to estimate the parameters of a specified state space model which describes an economic phenomena under study. The Kalman filter, being a recursive algorithm, is ideal for usage on time-dependent data. As an example, the yearly measurements of eight key economic time series data of the Nigerian economy is used to demonstrate that the integrated random walk model is suitable for modeling time series with no clear trend or seasonal variation. We find that the Kalman filter is both predictive and adaptive, as it looks forward with an estimate of the variance and mean of the time series one step into the future and it does not require stationarity of the time series data considered
Bio-social correlates of intention to use or not to use contraception: The case of Ghana and Nigeria
(Union for African Population Studies, 2015) Udomboso, C. G.; Amoateng, A. Y.; Doegah, P. T.
Based on the 2008 and 2013 Demographic and Health Survey data of Ghana and Nigeria respectively, statistical neural network and logit regression models were used to examine the effects of selected bio-social factors on the intention to use contraception among never married and ever married women in the two countries. The results showed that on the whole, the SNN model identified more biosocial factors affecting the intention to use contraception in the two countries than did the logic model. Educational attainment, exposure to media, and visitation to a health facility affected intention to use contraception significantly and positively in both countries. On the other hand, number of living children, infrequent sexual intercourse, postpartum amenorrhea, opposition to contraception and lack of access to contraceptives negatively affected intention to use contraception. The study findings have underscored the rational nature of the decisions women make in using contraception or not.
Bootstrap approach for estimating seemingly unrelated regressions with varying degrees of autocorrelated disturbances
(2013) Ebukuyo, O. B.; Adepoju, A. A.; Olamide, E. I.
The Seemingly Unrelated Regressions (SUR) model proposed in 1962 by Arnold Zellner has gained a wide acceptability and its practical use is enormous. In this research, two methods of estimation techniques were examined in the presence of varying degrees of _rst order Autoregressive [AR(1)] coefficients in the error terms of the model. Data was simulated using bootstrapping approach for sample sizes of 20, 50, 100, 500 and 1000. Performances of Ordinary Least Squares (OLS) and Generalized Least Squares (GLS) estimators were examined under a definite form of the variance-covariance matrix used for estimation in all the sample sizes considered. The results revealed that the GLS estimator was efficient both in small and large sample sizes. Comparative performances of the estimators were studied with 0.3 and 0.5 as assumed coefficients of AR(1) in the first and second regressions and these coefficients were further interchanged for each regression equation, it was deduced that standard errors of the parameters decreased with increase in the coefficients of AR(1) for both estimators with the SUR estimator performing better as sample size increased. Examining the performances of the SUR estimator with varying degrees of AR(1) using Mean Square Error (MSE), the SUR estimator performed better with autocorrelation coefficient of 0.3 than that of 0.5 in both regression equations with best MSE obtained to be 0.8185 using _ = 0:3 in the second regression equation for sample size of 50. Key words: Autocorrelation||Bootstrapping||Generalized least squares||Ordinary least squares||Seemingly unrelated regressions
Clinical efficacy and health implications of inconsistency in different production batches of antimycotic drugs in a developing country
(2011-03) Ogunshe, A. A. O.; Adepoju, A. A.; Oladimeji, M. E.
Objective: This study aimed at evaluating the in vitro efficacy and health implications of inconsistencies in different production batches of antimycotic drugs. Materials and Methods: In vitro susceptibility profiles of 36 Candida spp. – C. albicans (19.4%), C. glabrata (30.6%), C. tropicalis (33.3%), and C. pseudotropicalis (16.7%) – obtained from human endocervical and high vaginal swabs (ECS/HVS) to two different batches (B1 and B2) of six antimycotic drugs (clotrimazole, doxycycline, iconazole, itraconazole, metronidazole and nystatin) was determined using modified agar well-diffusion method. Results: None of the Candida strains had entirely the same (100%) susceptibility / resistance profiles in both batches of corresponding antimycotic drugs; while, different multiple antifungal susceptibility (MAS) rates were also recorded in batches 1 and 2 for corresponding antifungals. Only 14.3%, 27.3%, 16.7–33.3%, and 8.3–25.0% of C. albicans, C. glabrata, C. pseudotropicalis, and C. tropicalis strains, respectively, had similar susceptibility/resistance profiles toward coressponding antifungal agents in both batches; while up to 57.1% of C. albicans, 45.5% of C. glabrata, 66.7% of C. pseudotropicalis, and 50.0% of C. tropicalis strains were susceptible to one batch of antifungals but resistant to corresponding antifungals in the second batch. As high as about 71.4% (C. albicans), 73.0% (C. glabrata), 50.0% (C. pseudotropicalis), and 66.74% (C. tropicalis) strains had differences of ≥10.0 mm among corresponding antimycotic agents. Conclusions: Candida strains exhibited different in vitro susceptibility / resistance patterns toward two batches of corresponding antimycotic agents, which has clinical implications on the efficacy of the drugs and treatment of patients. The findings of the present study will be of benefit in providing additional information in support of submission for drug registration to the appropriate regulatory agencies.
Comparative analysis of rainfall prediction using statistical neural network and classical linear regression model
(Medwell Journals, 2011) Udomboso, C. G.; Amahia., G. N.
Different types of models have been used in modeling rainfall. Since 1990s however, interest has shifted from traditional models to ANN in rainfall modeling. Many researchers found out that the ANN performed better than such traditional models. In this study, we compared a traditional linear model and ANN in the modeling of rainfall in Ibadan, Nigeria. Ibadan is a city in West Africa, located in the tropical rainforest zone, using the data obtained from the Nigeria Meteorological (NIMET) station. Three variables were considered in this study rainfall, temperature and humidity. In selecting between the two models, we concentrated on the choice of adjusted R2 (R-2 ), Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC). Though, the MSE and R2 were also used, it was concluded from results that MSE is not a good choice for model selection. This is due to the nature of the rainfall data (which has wide variations). It was found that the Statistical Neural Network (SNN), generally performed better than the traditional (OLS).
Comparative performance of the limited information techniques in a two equation structural model
(2007) Adepoju, A. A.
The samples with which we deal in practice are rather small. seldom exceeding 80 observations and frequently much smaller. 'Thus, it is of great interest to inquire into the properties of estimators for the typical sample sizes encountered in practice. The performances of three simultaneous estimation method using a model consisting of a mixture of an identified and over identified equations with correlated error terms and compared. The result of the Monte Carlo study revealed that the Two Stage least Squares (2SLS) and the Limited Information Maximum Likelihood (LIML) estimates are similar and in most cases identical in respect of the just-identified equation. The Total Absolute Biases (TAB) of 2SLS and LIML revealed asymptotic behavior under (upper triangular matrix) P1 while those of Ordinary Least Squares (OLS) exhibited no such behavior. For both upper and lower triangular matrices (P, and P2), 2SI.S estimates showed asymptotic behavior in the middle interval. The OI.S is the only stable estimator with a stable behavior of Root Mean Square Error (RM3F.) as its estimates increase (decrease) consistently for equation 1(equation 2) for P, (for P2).
Comparing ANN and ARIMA model in predicting the discharge of River Opeki from 2010 to 2020
(John Willey & Sons Limited, 2018) Fashae, O. A.; Olusola, A. O.; Ndubuisi, I.; Udomboso, C. G.
Many attempts have been made in the recent past to model and forecast streamflow using various techniques with the use of time series techniques proving to be the most common. Time series analysis plays an important role in hydrological research. Traditionally, the class of autoregressive moving average techniques models has been the statistical method most widely used for modelling water discharge, but it has been shown to be deficient in representing nonlinear dynamics inherent in the transformation of runoff data. In contrast, the relatively newly improved and efficient soft computing technique artificial neural networks has the capability to approximate virtually any continuous function up to an arbitrary degree of accuracy, which is not otherwise true of other conventional hydrological techniques. This technique corresponds to human neurological system, which consists of a series of basic computing elements called neurons, which are interconnected together to form networks. The aim of the study is to compare the artificial neural network and autoregressive integrated moving average to model River Opeki discharge (1982–2010) and to use the best predictor to forecast the discharge of the river from 2010 to 2020. The performance of the two models was subjected to statistical test based on correlation coefficient (r) and the root‐mean‐square error. The result showed that autoregressive integrated moving average performs better considering the level of root‐mean‐square error and higher correlation coefficient.
A comparison of least squares dummy variable (LSDV) and the pooled estimator in fixed effect model
(2008) Olayiwola, O. M.; Adepoju, A.A.; Olajide, J. T
This paper examines a comparison of Least Squares Dummy Variable and a pooled estimator in a fixed effect model. The aims of the research are: to estimate the individuals firms parameters by using least squares dummy variables. To estimate parameter of the pooled observations using ordinary least squares. To estimate the behavioral relationship between individuals variables and to test for the significance différence across the groups. The framework was based on a fixed effect model. The analysis of a panel model was carried out using the Ordinary Least Squares (OLS) and Least Squares Dummy Variable (LSDV) methods. various tests were carried out to determine which of the methods to use when dealing with a panel data. The results of the analysis showed significant différence across the different groups effect. F is also significant at 95% level by using either the fixed effect or pooled model in a panel data. Also, a measure of fit of the model carried out showed that the fixed effect model significantly explained the variation in the dependent variable while pooling the model explained a very small proportion of the total variation in the dépendent variable. In the light of the above, it will be appropriate to use a fixed effect least squares dummy variable rather than pooling the data in the analysis of a panel data
Development of alternative linear estimators in complex surveys
(2011-11) Ikughur, J. A.
The estimation of multiple characteristics using Probability Proportional to Size (PPS) sampling scheme has introduced some complexities in sample surveys. It requires transformation of auxiliary information into probability measures and the utilization of correlation coefficient between study variables y and measure of size x. Existing estimators of finite population characteristics are rigidly specified by a fixed order of positive correlation between y and x and are assumed efficient for all populations. However, the assumptions break down when the study variables are negatively correlated with measure of size. In this study, a linear class of estimators that are functions of moments in positive and negative correlation coefficients were proposed. Using laws of proportions and probability measure theory, a class of alternative linear estimators〖 τ ̂〗_(g,c) were developed for use in PPS sampling schemes. Using linear regression model with slope β and well-behaved error term ε, the expectation of cth standardized moment of the study variable given by E((y-μ_y)/σ_y )^c=E[β((x-μ_x)/σ_y )+((ε-μ_ε)/σ_y ) ]^c,c=1,2,3,4 with β^c=(ρ^2 (σ_y^2)/(σ_x^2 ))^(c/2) provided a link between moments in correlation coefficient and distribution of the target population, where ρ is the correlation coefficient, 〖μ_y,μ〗_x,μ_ε and σ_(y,)^2 σ_x^2 ,σ_ε^2 are means and variances of y,x ε respectively. The minimum variance was used as optimality criterion for comparing the performance of 〖 τ ̂〗_(g,c) with the conventional estimator namely, Hansen and Hurwitz’s estimator〖 τ ̂〗_HH, and other existing alternative estimators namely, Amahia-Chaubey-Rao’s estimator (τ ̂_ACR), Grewal’s estimator (τ ̂_G), Rao’s estimator (τ ̂_R) and Ekaette’s estimator (τ ̂_E) under the PPS sampling design. Using the general super-population model with parameter g, the expected Mean Square Error (MSE) was derived for the estimators and their relative efficiencies were then computed. Empirical studies with samples drawn from four populations, namely; Population I,II,III and IV having correlation coefficients, ρ=0.16,0.39,-0.32 and-0.775 respectively were conducted. The derived transformation for generalized selection probabilities defining the class of linear estimators is p_(i,g)^*=(1-ρ^c)/N+ρ^c p_i; c=1,2,3,4 where p_i=x_i/X,X=∑_i^N▒x_i or p_i=z_i/Z,Z=∑_i^N▒z_i ,z_i=1/x_i for positive and negative correlations respectively. Provided that 〖CV〗_x<〖CV〗_y,γ_y 〖<γ〗_x,K_y