Browsing by Author "Kololi, Moses Mukhwana"

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Estimating Non-Smooth Functional Using Non-Parametric Procedure in the Hilbert Sample Space
(International Journal of Engineering & Mathematical Sciences., 2017-06-01) Kololi, Moses Mukhwana; Orwa, George O.; Odhiambo, Romanus O.
One of the problems in statistical inference is that of estimating functional. A functional is a mathematical relation that maps two or more functions in one number. They are either smooth or non-smooth. The smoothness properties of functional determine the quality of estimation. However, non-smooth functional lack some degree of properties traditionally relied upon in estimation. Lack of these properties highlights the reason why standard techniques fail to give sharp results. In this paper, an estimator for an arbitrary non-smooth functional is proposed in the nonparametric set-up using robust stochastic Hilbert sample spaces. The estimator is based on the Mini Max criterion where lower and upper bounds are constructed. However, while working in the context of Mini Max estimation, the lower bounds are most important. The approximation theory is used to construct an estimator that is asymptotically sharp Mini Max when the means are bounded. The procedure used allows analysis and presentation data at hand without making any assumption about the underlying distribution. Therefore, the predictions do not depend on whether or not the underlying distribution is normal
Nonparametric estimator for the standardized sum using edgeworth expansions
(International Journal of Scientific Research, 2018-02-01) Kololi, Moses Mukhwana; Orwa, George O.
Constructing an estimator for functional estimation is one of the problems in statistical inference. In this paper, the problem of constructing an estimator for a studentized sum is considered in the nonparametric set-up. In this set-up, data are used to infer to an unknown quantity while making as few assumptions as possible. This non-smooth functional lack some degree of properties traditionally relied upon in estimation. Smooth functionals converge at the rate of 𝑛−12 while non-smooth functionals converge at the rate slower than 𝑛−12 . This highlights the reason why standard techniques fail to give sharp results. A clear and accurate approximation is obtained by using an approximation that admits cumulant generating function; saddle point approximation. An optimal estimator is obtained using the MiniMax criterion where the lower and upper bounds are constructed. While working in the context of MiniMax estimation, the lower bounds are most important. The MiniMax lower bound is obtained by applying the general lower bound technique based on testing two composite hypotheses. The quality of an estimator is evaluated with the MiniMax risk. Best polynomial approximation of an absolute value function and Hermite polynomials are used to construct an optimal estimator when the means are bounded by a given value $M>0$.
On minimax risk of non-smooth functional, its asymptotic properties and polynomial estimation
(IJRDO-Journal Of Mathematics, 2015-06-01) Kololi, Moses Mukhwana; Otieno, Romanus O.; Orwa, George O.; Mung'atu, Joseph K
Nonparametric estimation of non-smooth functionals deals with highly structured problems which arise, modeled or cast differently from the ones for which mainline numerical methods have been designed. Non-smooth functional estimation problems show some features that are different from those of estimating smooth functionals. This is in terms of the optimal rates of convergence as well as the technical tools needed for the analysis of the MiniMax lower bounds and the construction of the optimal estimators. The main difficulty of estimating the non-smooth functionals is traced back to the non differentiability of the absolute value function at the origin. This is reflected both in the derivation of the lower bounds and the construction of optimal estimators. The construction of the optimal estimators of the non-smooth functionals is more complicated than those for linear and quadratic functionals. In this study we consider asymptotic properties, polynomial estimation and MiniMax risk involving non-smooth functionals.