Graduate Thesis Or Dissertation


On Some Methods for Smoothing Distribution Functions Public Deposited

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  • Methods that are applied to smooth distribution functions are useful in many applications. Areas of application include economics, financial markets and survival analysis. The empirical cumulative distribution function (ecdf) is unbiased and its asymptotic distribution is normal. However, the jump discontinuities of $\frac{1}{n}$ are undesirable in estimation because it makes the ecdf discrete even if the underlying distrbution is continuous. As a result, various methods like kernel estimators and splines have been used to smooth distribution functions. In this thesis, the Nadaraya kernel estimator (NKE) and the smoothing spline estimator are used to smooth simulated data. A new naive estimator is also introduced to estimate the distribution function of the simulated data. Plots of bias and mean squared error (MSE) are used to evaluate and compare the performances of the methods of smoothing. The results of the simulation show that the NKE performs better than the other estimators in terms of MSE when the underlying distribution function is that of the standard normal distribution. In the case of Weibull distribution, the naive estimator performs better than the NKE in terms of bias. Also in the small sample cases of underlying standard Cauchy distribution, the naive estimator has a smaller MSE than the NKE. For all distribution functions, the spline smoother gives bias estimates and the naive estimator performs better than the ecdf when their MSEs are compared. The strong uniform consistency and the asymptotic distribution of the new estimator are also provided.
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