Taylor Series Approach Research Articles

Modified Calibration Variance Estimators in the Presence of Non- response and Measurement Error

Samples from a population that can be divided into smaller groups are taken using a technique called stratified sampling. When subpopulations within the total population differ, it may be beneficial to sample each stratum (subpopulation) separately in sample surveys. Government organizations, independent consultants, and applied statisticians all frequently use this crucial strategy. There are many problems encountered by survey statisticians in estimating the population variance of the study variable. These problems include the presence of outliers in data collected for analysis, non-response, and measurement errors occurring during the survey. Shahzad et al. [1] developed variance estimators by addressing the problem of outliers using the L-moment and calibration approach. However, they do not consider the situation of non-response and measurement errors. This paper addresses these problems by proposing modified variance estimators in the presence of non-response and measurement errors. The properties (Biases and MSEs) were derived up to the first order of approximation using the Taylor series approach. The efficiency conditions of the modified estimators over the existing estimators considered in the study were established. The result of simulation studies revealed that the estimators are efficient.

POWER MEDIAN-BASED ESTIMATORS OF FINITE POPULATION MEAN

In this paper, median based mean estimators for estimating finite population mean are proposed. The proposed estimators were obtained by transforming estimators in literature utilizing mean of auxiliary variable into median based estimators with the aim of obtaining estimators with higher efficiency. The mean square error of the proposed estimators was obtained up to the first order of approximation using Taylor series approach and the optimum values of the unknown of the estimators were obtained by means of partial derivative of the mean square error and equating to zero. A Numerical study was carried out to support the fact that the proposed estimators are more efficient as compared to the existing ones, as the proposed estimators have the least mean squared error at optimum values of the unknown constants and have higher percentage relative efficiency (PRE). This implies that the proposed estimators are more efficient than the traditional ones considered in the study.

On Modification of Some Estimators using Parameters of Auxiliary Information for the Estimation of the Population Coefficient of Variation

Several studies in the theory of sampling survey have established the fact that the use of auxiliary information at the planning and estimation stages helps in enhancing the efficiency of estimators for estimating population parameters like population mean, population variance, standard deviation etc. as compared to the estimators which use not auxiliary information. In the present study, four estimators for estimating the population coefficient of variation of the study variable using auxiliary information were proposed. The properties (Biases and MSEs) of the proposed estimators were derived up to first order of approximation using Taylor series approach. Numerical analysis was conducted to justify the efficiencies of the proposed estimators and the results revealed that the proposed estimators are more efficient than the existing estimators considered in the study.

Regression-cum-ratio mean imputation class of estimators using non-conventional robust measures

Different imputation strategies have been developed by several authors to take care of missing observations during analyses. Nevertheless, the estimators involved in some of these schemes depend on known parameters of the auxiliary variable which outliers can easily influence. In this study, a new class of ratio-type imputation methods that utilize parameters that are free from outliers has been presented. The estimators of the schemes were obtained and their MSEs were derived up to first-order approximation using the Taylor series approach. Also, conditions for which the new estimators are more efficient than others considered in the study were also established. Numerical examples were conducted and the results revealed that the proposed class of estimators is more efficient.

Enhanced Variance Estimation Techniques for Addressing Non-Response and Measurement Error Situations

The use of relevant information from auxiliary variables at the estimation stage and design stage to obtain reliable and efficient estimates is a common practice in a sample survey. Several Estimators of population variance have been suggested. However, these estimators do not consider the situation of non-response due to non-availability of respondents, refusal to respond, presence of hard- core respondents or due to non-understanding of the question thereby, reducing the efficiency of the estimators and the parameters of the auxiliary variable X̄ , S2x used are sensitive to outliers or extreme values which can either lead to underestimation or overestimation. To address the aforementioned observations, classes of variance estimators under the simultaneous influence of non-response and measurement errors using outlier-free parameters as well as calibration approaches were proposed. The properties (Bias and MSE) of the modified estimators were derived up to the first order of approximation using the Taylor series approach. The efficiency conditions of the proposed estimators over the existing estimators considered in the study were established. The empirical studies were conducted using simulation and the results revealed that the proposed class of estimators have minimum MSEs and higher PREs among all the competing estimators. These imply that the proposed estimators are more efficient and can produce a better estimate of the population mean compared to other existing estimators considered in the study.

Spatial distribution and characteristics of women reporting cervical cancer screening in Malawi: An analysis of the 2020 to 2021 Malawi Population-based HIV Impact Assessment survey data.

Malawi has one of the highest incidence and mortality rates of cervical cancer in the world. Despite a national strategic plan and the roll-out of VIA and screen-and-treat services, cervical cancer screening coverage in Malawi remains far below the national target.Using a nationally representative sample of women enumerated in the Malawi Population-based Impact Assessment (MPHIA) survey we estimated the prevalence and spatial distribution of self-reported cervical cancer screening as a proxy for uptake in Malawi. MPHIA was a nationally representative household survey in Malawi, targeting adults aged 15 and above, that employed a cross-sectional, two-stage, cluster design. The primary aim of MPHIA was to assess the regional prevalence of viral load suppression and the progress towards achieving the UNAIDS 95-95-95 goals among adults aged 15 and above. The survey was carried out between January 2020 and April 2021. Prevalence of self-reported cervical cancer screening by different characteristics was estimated accounting for the survey design using the Taylor series approach. We used univariable and multivariable logistic regression approaches to examine associations between the prevalence of cervical cancer screening and demographic characteristics. A total of 13,067 adult (15 years and older) female individuals were surveyed during the MPHIA 2020 to 2021 survey, corresponding to a weighted total of 5,604,578. The prevalence of self-reported cervical cancer screening was 16.5% (95% CI 15.5-18.0%), with women living with HIV having a higher prevalence of 37.8% (95% CI 34.8-40.9) compared to 14.0% (95% CI 13.0-15.0) in HIV negative women. The highest prevalence of screening was reported in the Southwest zone (SWZ) (24.1%, 95% CI 21.3-26.9) and in major cities of Blantyre (25.9%, 95% CI 22.9-29.0), and Lilongwe (19.6%, 95% CI 18.0-21.3). Higher self-reported screening was observed in women who resided in urban regions ((22.7%; 95% CI 21.4-24.0) versus women who resided in rural areas (15.2%; 95% CI 14.0-16.8). Cervical cancer screening was strongly associated with being HIV positive (aOR 2.83; 95% CI 2.29-3.50), ever having been pregnant (aOR 1.93; 95% CI 1.19-3.14), attaining higher education level than secondary education (aOR 2.74; 95% CI 1.67-4.52) and being in the highest wealth quintile (aOR 2.86; 95% CI 2.01-4.08). The coverage of cervical cancer screening in Malawi remains low and unequal by region and wealth/education class. Current screening efforts are largely being focussed on women accessing HIV services. There is need for deliberate interventions to upscale cervical cancer screening in both HIV negative women and women living with HIV.

A New Approximate Analytical Expression of Non-Isothermal Diffusion Model

In this study, we’ve addressed the Lane-Emden boundary value problem that appears in biochemical, scientific, and chemical applications. We’ve used the Taylor series approach to solve the non-isothermal reaction-diffusion equation in a planar catalyst. We’ve derived the approximate analytical expression for concentration and effectiveness factors. The collected results are illustrated using appropriate graphs. The presented analysis proves the applicability of the utilized method's dependability and effectiveness. We’ve also solved the equation numerically by using MATLAB software to compare our approximate analytical solutions. Our analytical results of concentration and effectiveness factor are most appropriately matched with the numerical results. We’ve also discussed the influence of the parameters on concentration and effectiveness factors.

Modified robust regression-type estimators with multi-auxiliary variables using non-conventional measures of dispersion

Auxiliary variables correlated with the study variables have been identified to be useful in improving the efficiency of ratio, product and regression estimators both at planning and estimation stages. The existing regression-based estimators are functions of auxiliary variables which are sensitive to outliers. In this paper, a modified class of estimators is proposed using robust non-conventional measures of dispersion which are robust against outliers or extreme values. The properties (Biases and Mean Squared Errors (MSEs)) of the modified class of estimator were derived up to the first order of approximation using Taylor series approach. The empirical studies were conducted using stimulation to investigate the efficiency of the proposed estimators over the efficiency of the existing estimators. The results revealed that the proposed estimators have minimum MSEs and higher Percentage Relative Efficiencies (PREs) among all the competing estimators. These results implied that the proposed estimators are more efficient and can produce better estimate of the population mean compared to other existing estimators considered in the study. Therefore, it can be concluded that proposed estimators have better predictive power for estimating population mean when the study (interest) variables are characterized with outliers or extreme values.

Modified Classes of Regression-Type Estimators of Population Mean in the Presence of Auxiliary Attribute under Double Sampling Scheme

In sample survey, reliable and efficient estimates are often obtained using information from auxiliary variables during estimation and designing stages. However, there are times when the auxiliary information is attribute-based. Some authors have proposed estimators using auxiliary attribute information when the population mean of auxiliary attribute is unknown. However, the estimators are ratio- based estimators which are less efficient when the bi-serial correlation between the study variable and the auxiliary attribute is negative. In this study, regression approach was used to modify estimator d ZKi t to produce estimators that can be used for both negative and positive correlation. In addition, another existing estimator was also modified to produce an estimator that is independent of an unknown population parameter. The Biases and Mean Squared Errors (MSEs) of the modified estimators were determined using the Taylor series approach up to the first order of approximation. The proposed estimators’ efficiency conditions over some existing estimators were established. Empirical investigations were done using stimulation study and the results revealed that proposed estimators have the lowest MSEs and the highest PREs of all the competing estimators and therefore can give better estimates of the population mean. Therefore, it can be concluded that proposed estimators have better predictive power for estimating population mean under two-phase sampling scheme.

Univariate Gauss quadrature for structural modelling of tissues and materials with distributed fibres

This work presents a method for computing the averaged free energy and constitutive relations in hyperelastic material models with distributed fibres, as they apply to soft fibre-reinforced materials and biological tissues. While these models are currently implemented through either spherical cubature of the fibre free energy or its Taylor series, we here propose a new method based on a univariate Gauss quadrature rule with integration points and weights informed by the statistical moments of the distribution of fibre stretch. As an intrinsic property, the new approach separates the integration of the fibre constitutive law from the integration of the orientation distribution, the latter leading to structural tensors of even order. Provided the latter 2n−1 tensors are computed accurately up to tensorial order 2(2n−1), the method integrates exactly any polynomial of order 2n−1 that agrees with the fibre law at the n integration points. After formally introducing the quadrature method for generally non-affine fibre deformations and arbitrary order, we focus on the important special case of affine fibre kinematics and discuss the rules with n≤3 integration points, for which the corresponding positions and weights are determined analytically. At a computational cost comparable to the existing approaches, the new method does not require the fibre law to be analytic and can thus robustly deal with piece-wise definitions of the fibre energy, in contrast to Taylor-series approaches, and it does not induce additional anisotropy as it can occur with spherical cubature rules. The 3-point rule is further investigated and illustrated in numerical examples relating to soft collagenous tissues based on a Fortran implementation of the method suitable for use in finite element analyses.

On Modification of Some Ratio Estimators using Parameters of Auxiliary Variable for the Estimation of the Population Mean

Some existing estimators based on auxiliary attribute have been proposed by many authors. In this paper, we use the concept of power transformation to modify some existing estimators in order to obtain estimators that are applicable when there is positive or negative correlation between the study and auxiliary variable. The properties (Biases and MSEs) of the proposed estimators were derived up to the first order of approximation using Taylor series approach. The efficiency comparison of the proposed estimators over some existing estimators considered in the study were established. The empirical studies were conducted using existing population parameters to investigate the proficiency of the proposed estimators over some existing estimators. The results revealed that the proposed estimators have minimum Mean Square Errors and higher Percentage Relative Efficiencies than the conventional and other competing estimators in the study. These implies that the proposed estimators are more efficient and can produce better estimates of the population mean compared to the existing estimators considered in the study.

Taylor Series Approach to Grey Multiobjective Integer linear Fractional Programming Problem and A case study for environmental economic energy dispatch

Taylor Series Approach to Grey Multiobjective Integer linear Fractional Programming Problem and A case study for environmental economic energy dispatch

Semi-Local Convergence of Two Derivative-Free Methods of Order Six for Solving Equations under the Same Conditions

We propose the semi-local convergence of two derivative-free, competing methods of order six to address non-linear equations. The sufficient convergence criteria are the same, making a direct comparison between them possible. The existing convergence technique uses the standard Taylor series approach, which requires derivatives up to order seven. The novelty and originality of our work lies in the fact that in contrast to previous research works, our convergence theorems only demand the first derivative. In addition, formulas for determining the region of uniqueness for solution, convergence radii, and error estimations are suggested. Such results cannot be found in works relying on the seventh derivatives. As a consequence, we are able to broaden the utility of these productive methods. The confirmation of our convergence findings through application problems brings this research to a close.

Extended Comparison between Two Derivative-Free Methods of Order Six for Equations under the Same Conditions

Under the same conditions, we propose the extended comparison between two derivative free schemes of order six for addressing equations. The existing convergence technique used the standard Taylor series approach, which requires derivatives up to order seven. In contrast to previous researchers, our convergence theorems only demand the first derivative. In addition, formulas for determining the region of uniqueness for the solution, convergence radii, and error estimations are suggested. As a consequence, we broaden the utility of these productive schemes. Moreover, we present a comparison of attraction basins for these schemes to obtain roots of complex polynomial equations. The confirmation of our convergence findings on application problems brings this research to a close.

On the Efficiency of Modified Regression-type Estimators Using Robust Regression Slopes and Non-conventional Measures of Dispersion

Supplementary variables associated with the study variables have been identified to be helpful in improving the efficiency of ratio, product and regression estimators both at planning and estimation stages. The existing regression-based estimators are functions of regression slopes and known auxiliary variables which are sensitive to outliers. Zaman & Bulut [1] and Zaman [2] addressed the issue of regression slopes in the aforementioned estimators using robust regression slopes like Huber-M, Hampel-M, Least Trimmed Squares (LTS) and Least Absolute Deviation (LAD). However, their estimators still utilized known auxiliary functions which are also sensitive to outliers or extreme values. Similarly, Yadav and Zaman [3] suggested non-conventional robust parameters of auxiliary variable which are robust against outliers. However, the problems of effects of outliers on regression slopes were not considered. In this study, the estimators of Zaman & Bulut [1] and Zaman [2] estimators were modified using robust non-conventional measures and Yadav and Zaman [3] estimators were modified using robust regression slopes. The properties (Biases and MSEs) of the modified estimators were derived up to the first order of approximation using Taylor series approach. The efficiency conditions of the proposed estimators over the existing estimators considered in the study were established. The empirical studies were conducted using stimulation data to investigate the efficiency of the proposed estimators over the efficiency of the existing estimators. The results revealed that the proposed estimators have minimum MSEs and higher PREs among all the competing estimators for the simulated populations in the study. This implies that the proposed estimators are more efficient and can produce better estimate of the population mean compared to other existing estimators considered in the study.

Modified Classes of Regression-Type Estimators of Population Mean in the Presences of Auxiliary Attribute

The use of relevant information from auxiliary variable at the estimation stage and design stage to obtain reliable and efficient estimate is a common practice is a sample survey. But situations arise when the available auxiliary information are attribute in nature. There are some existing estimators based on auxiliary attribute in literature, however, they are less efficient when the bi-serial correlation between the study variable and auxiliary attribute is negative. Also, some depend on an unknown parameter of the study variable (Cy) which makes their applicability of the estimators in real life situations not possible unless if the value is estimated using a large sample which requires additional resources. In this work, the concept of regression base estimator was used to obtain estimators that are independent of unknown population parameter of the study variable and applicable for both negative and positive correlations. The properties (Biases and MSEs) of the modified estimators were derived up to the first order of approximation using Taylor series approach. The efficiency conditions of the proposed estimation over the existing estimator considered in the study were established. The empirical studies were conducted using both existing population parameters and stimulation to investigate the efficiency of the proposed estimators over the efficiency of the existing estimators. The results revealed that the proposed estimators have minimum MSEs and higher PREs among all the competing estimators. These imply that the proposed estimators are more efficient and can produce better estimate of the population mean compared to other existing estimators considered in the study.

Efficient sixth order iterative method free from higher derivatives for nonlinear equations

In this paper, we proposed new iterative sixth order convergence method for solving nonlinear equations. The combination of the Taylor series and composition approach is used to derive the new method. Numerous methods have been developed by many researchers whenever the function’s second and higher order derivatives exist in the neighbourhood of the root. Computing the second and higher derivative of a function is a very cumbersome and time consuming task. In terms of low computation cost, the newly proposed method finds the best approximation to the root of non-linear equations by evaluating the function and its first derivative. The proposed method has been theoretically demonstrated to have sixth-order convergence. The proposed method has an efficiency index of 1.56. Several comparisons of the proposed method with the various existing iterative method of the same order have been performed on the number of problems. Finally, the computational results suggest that the newly proposed method is efficient compared to the well-known existing methods.

A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model

<abstract> <p>This contribution proposes a numerical scheme for solving fractional parabolic partial differential equations (PDEs). One of the advantages of using the proposed scheme is its applicability for fractional and integer order derivatives. The scheme can be useful to get conditions for obtaining a positive solution to epidemic disease models. A COVID-19 mathematical model is constructed, and linear local stability conditions for the model are obtained; afterward, a fractional diffusive epidemic model is constructed. The numerical scheme is constructed by employing the fractional Taylor series approach. The proposed fractional scheme is second-order accurate in space and time and unconditionally stable for parabolic PDEs. In addition to this, convergence conditions are obtained by employing a proposed numerical scheme for the fractional differential equation of susceptible individuals. The scheme is also compared with existing numerical schemes, including the non-standard finite difference method. From theoretical analysis and graphical illustration, it is found that the proposed scheme is more accurate than the so-called existing non-standard finite difference method, which is a method with notably good boundedness and positivity properties.</p> </abstract>

2D Numerical Model of Sediment Transport Under Dam-break Flow Using Finite Element

The potential hazard of dam construction is the possibility of dam failure. Dam failure will cause damage to property and the environment, as well as loss of human life. In addition, the dam-break flow also causes erosion and sediment transport which can affect the morphology of rivers around the dam. Dam-break flow analysis is needed to minimize the potential hazards of dam construction. Dam-break flow analysis can be done by performing numerical modeling. This study develops a numerical model using the Taylor Galerkin method. The Taylor Galerkin model is used in simulating the dam-break flow along with the sediment transport that occurs. Mathematically, this flow is generally expressed by the shallow water-Exner equations. The shallow water equations describe the movement of water, and the Exner equation describes the movement of sediment. The model will use the Galerkin method for spatial derivatives and the Taylor series approach for time derivatives in this study. A numerical filter by Hansen was also added to the model to overcome the instability of the model due to numerical oscillations. To determine the performance of the Taylor Galerkin model, simulation results were compared with experimental data and other numerical results from previous studies. The Taylor Galerkin model can simulate the dam-break flow with sediment movement over a movable bed well based on this study. Studies like this are needed to reduce the high risk of dam failure.

Kinetics of the Catalytic Combustion of Ethanol and Ethyl Acetate with Estimation of Activation Energy and Rate Constants: An Analytical Study

Background: A mathematical model for the combustion of ethanol and ethyl acetate mixtures using Mn9Cu1 (mixture of manganese and copper with a weight ratio of 9:1) catalyst is discussed. The model’s kinetic mechanism is expressed in terms of nonlinear reaction-diffusion equations with common initial and boundary conditions in a finite planar, cylindrical, and spherical geometry. : A Taylor series approach is used to derive general approximate analytical expressions of ethanol, acetaldehyde, and ethyl acetate molar concentrations inside the particle and reactor phase for various values of rate constants, diffusion, and kinetic parameters. The effect of shape factor for the planar, cylindrical, and spherical geometry of dispersed particles was examined for the first time. Activation energy and rate constant at the reference temperature of ethanol, acetaldehyde, and ethyl acetate are also obtained from the rate equations. A direct comparison with numerical simulations confirms the accuracy of the derived analytical results. Objective: Derive general approximate analytical expressions of ethanol, acetaldehyde, and ethyl acetate molar concentrations inside the particle and reactor phase for various parameter values. Methods: We employ the simple and reliable Taylor series method. Results: Semi-analytic expressions of the concentration and bulk concentration of ethanol, ethyl acetate, and acetaldehyde. Conclusion: Approximate analytical expressions of the concentrations of ethanol, acetaldehyde and ethyl acetate were derived for arbitrary catalyst particle (planar, cylindrical and spherical) by using a simple, reliable, and robust method. In addition, the concentration of the species in reactor phase was also reported. The effects of the kinetic parameters, which are influenced by adsorption equilibrium constant, effective diffusivity, activation energy, on concentration, were discussed.