Abstract
We investigated the finite properties as well as the goodness of fit test for the cubic smoothing spline selection methods like the Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV) and Mallow CP criterion (MCP) estimators for time-series observation when there is the presence of Autocorrelation in the error term of the model. The Monte-Carlo study considered 1,000 replication with six sample sizes: 30; 60; 120; 240; 480 and 960, four degree of autocorrelations; 0.1; 0.3; 0.5; and 0.9 and three smoothing parameters; lambdaGML= 0.07271685, lambdaGCV= 0.005146929, lambdaMCP= 0.7095105. The cubic smoothing spline selection methods were also applied to a real-life dataset. The Predictive mean square error, R-square, and adjusted R-square criteria for assessing finite properties and goodness of fit among competing models discovered that the performance of the estimators is affected by changes in the sample sizes and autocorrelation levels of the simulated and real-life data set. The study concluded that the Generalized Cross-Validation estimator provides a better fit for Autocorrelated time series observation. It is recommended that the GCV works well at the four autocorrelation levels and provides the best fit for time-series observations at all sample sizes considered. This study can be applied to; non –parametric regression, non –parametric forecasting, spatial, survival and econometric observations.
Highlights
Introduction interestsIt has a nonstop first and second subordinate with theA cubic spline is the most widely recognized example of the smoothing spline regression model
This paper presents a goodness-of-fit test for time series observations using three classical cubic spline nonparametric regression functions
The smoothing parameter in cubic spline smoothing is to control the smoothness of the fitted curve, to estimate the optimal value of the smoothing parameter λ, three smoothing parameter selection criteria are considered and compared in this study: Generalized Cross-Validation (GCV), Generalized Maximum Likelihood (GML) and Mallow Cp (MCP)
Summary
The smoothing parameter in cubic spline smoothing is to control the smoothness of the fitted curve, to estimate the optimal value of the smoothing parameter λ, three smoothing parameter selection criteria are considered and compared in this study: Generalized Cross-Validation (GCV), Generalized Maximum Likelihood (GML) and Mallow Cp (MCP). Where; n is the dataset (xi, yi), λ refers to the smoothing parameters and S (λ) is the ith diagonal member of a smoother matrix Generalized Maximum Likelihood (GML) selection method; [20] proposed the GML technique for correlated data with one smoothing parameter. Λ is Smoothing parameter, W is the correlation structure, S (λ) is the diagonal element of the smoother matrix, n is n1 + n2 pairs of measurements/observations and m are the number of zero eigenvalues. Mallow’s C.P. Criterion (MCP) selection method was developed by [21] to estimate the fit of a regression model dependent on Ordinary Least Square. The assumption underlying the application of the Generalized Cross-Validation (GCV), Generalized Maximum Likelihood (GML) and Mallow’s CP criterion (MCP), the observations must be well represented by the model
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