| Summary: | 碩士 === 國立中興大學 === 農藝學系 === 87 === One of the objective of regression analysis is to investigate the relationship between the design points tj and expected value of response variable yj on the data set {(tj,yj)},j=1,…,n . In general, the analysis could be classified as parametric and nonparametric models. The main distinction is whether the functional form of μ is known. A parametric regression model assume that the form of μ is known.
In nonparametric regression analysis, the information of observations yj on the neighborhood of design point t are used to estimate μ(t) . The simplest way is to calculate the weighted average of these observations. Both classical Fourier series and kernel estimators are the kind of weighted average.
When the observed data exhibit periodic behavior, usually a linear model including sines and cosines is employed to estimate the regression function μ. The classical Fourier series estimator is of this type. However, if the regression function μ does not satisfy periodic boundary conditions, the criterion based on the mean squared error, i.e. CV or GCV, to choose the number of trigonometric functions in the regression would yield too many terms and the chosen function performs wiggly. A combination of low-order polynomial and trigonometric terms could alleviate the above problem and achieve a smooth curve.
As the data set on the interval [0,1] , kernel estimator may also be used to estimate the regression function. The kernel estimator is biased. Under a specified kernel function, smaller bandwidth will cause small bias, but large variance, hence the estimator will be undersmoothing. On the contrary, large bandwidth will cause large bias, but small variance, so the estimator will be oversmoothing. When the σ2 is known, the risk function criteria can be employed to trade-off the biasness and variance and used to decide a suitable smoothing parameter value. However, when the σ2 is unknown, CV and GCV criteria can be used to choose a smoothing parameter value.
Beside using on the fitting of data, kernel estimation was also applicable for the estimation of median effective dose (ED50), and constructed the approximate confidence interval for the dose-response curve in bioassay. Finney(1978) had used the data of insulin, 9 doses of s preparation with a dose of insulin were treated to mice, then recorded the numbers of mice showing the symptoms of convulsions. The fitness of probit and logit models of the data set was examined by Pearson's chi-squares test, and the ED50 estimate and 95% confidence interval using parametric method and kernel estimation were compared. The ED50 estimate using kernel estimation is larger than using parametric method, and the width of confidence interval constructed by kernel estimation is wider. Additionally, on the data of treating carbofuran to Meloidogyne incognita, after lack of fit test, it was shown should that both the probit and logit models were not appropriate. The estimate of the ED50 using trimmed Spearman-Karber method was compared with the result using kernel estimate. There is little difference between these two ED50 estimates, and the width of confidence interval of ED50 using kernel estimation is narrower.
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