Abstract: The existence of nonresponse in survey sampling has engendered inconsistencies in the estimation of population parameter. Such estimation, being characterized by nonresponse bias has become a rule rather than the exception in survey sampling, and this has been long acknowledged in the literature. Several authors have come up with different techniques such as subsampling the nonresponse, imputation, and calibration to curb this menace. An attempt to overcome the challenges faced in existing works, this study considered the estimation of finite population mean using calibration approach with subsampling the nonrespondents. Owning to the fact that calibration estimation has been found to reduce bias and improve efficiency of estimators. The classical estimator by Hansen and Hurwitz for estimating the population mean with subsampling the nonrespondents is calibrated upon using the chi square distance function, and different choices of the tunning parameter result in the calibration estimators of combined regression and ratio. Expressions for the bias, variance and mean square error (MSE) of the proposed estimators are derived and their properties studied. Again, the optimum conditions under which the suggested estimators have minimum variance and MSE are equally provided. Both efficiency and empirical comparisons are in favor of the proposed estimators, and suggest that the proposed estimators are more efficient and reliable with high precision than the existing estimators even in double sampling. In addition, expressions for optimal sample sizes with respect to the cost of the survey have been derived to validate the superiority of the proposed estimators, and the empirical investigation confirms the proposed estimators as highly preferable.Abstract: The existence of nonresponse in survey sampling has engendered inconsistencies in the estimation of population parameter. Such estimation, being characterized by nonresponse bias has become a rule rather than the exception in survey sampling, and this has been long acknowledged in the literature. Several authors have come up with different techniqu...Show More
Abstract: Fourier series are a powerful tool in applied mathematics; indeed, their importance is twofold since Fourier series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions. The idea inspiring the introduction of Fourier series is to approximate a regular periodic function, of period T, via a linear superposition of trigonometric functions of the same period T; thus, Fourier polynomials are constructed. They play, in the case of regular periodic real functions, a role analogue to that one of Taylor polynomials when smooth real functions are considered. In this thesis we will study function approximation by FS method. We will make an attempt to approximate square wave function, line function by FS, and line function by Fourier exponential and trigonometric polynomial. DFT will also be used to approximate function values from data set. We compare the accuracy and the error of Fourier approximation with the actual function and we find that the approximate function is very close to the actual function. We also study the solution of 1D heat equation and Laplace equation by Fourier series method. We compare the solution of heat equation obtained by Fourier series with BTCS. We also compare the solution of Laplace equation obtained by Fourier series with Jacobi iterative method. MATLAB codes for each scheme are presented in appendix and results of running the codes give the numerical solution and graphical solution.Abstract: Fourier series are a powerful tool in applied mathematics; indeed, their importance is twofold since Fourier series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions. The idea inspiring the introduction of Fourier series is to appro...Show More
