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Bootstrap Methods and Their Deployment in SAS and R
Published in Tanya Kolosova, Samuel Berestizhevsky, Supervised Machine Learning, 2020
Tanya Kolosova, Samuel Berestizhevsky
The empirical distribution function is the maximum likelihood estimator of the distribution for the observations when no parametric assumptions are made. The bootstrap distribution for θ^−θ is the distribution created by generating θ^ values by independent sampling with replacement from the empirical distribution Fn. The bootstrap estimate of the standard error of θ^ is then the standard deviation of the bootstrap distribution for θ^−θ. Almost any parameter of the bootstrap distribution can be used as a “bootstrap” estimate of the corresponding population parameter, for example the skewness, the kurtosis, or the median of the bootstrap distribution.
The Bootstrap
Published in Tucker S. McElroy, Dimitris N. Politis, Time Series, 2019
Tucker S. McElroy, Dimitris N. Politis
Concept 12.3. The Plug-In Principle A parameter can be expressed as a function of the distribution’s CDF (Fact 12.2.2), expressed as θ(G); the plug-in principle (Paradigm 12.3.5) proposes the estimator θ(G^), where G^ is the empirical distribution function (EDF). Remark 12.2.1 contrasts the parametric and nonparametric frameworks. Empirical distribution function: Definition 12.3.1, Facts 12.3.2 and 12.3.3, Remark 12.3.4.Examples 12.2.3, 12.2.4, 12.2.5, 12.2.6, 12.2.7, 12.2.9, 12.2.9, 12.2.10, 12.3.6, 12.3.7, 12.3.8.Exercises 12.4, 12.9, 12.10.
LiDAR and Spectral Data Integration for Coastal Wetland Assessment
Published in Yuhong He, Qihao Weng, High Spatial Resolution Remote Sensing, 2018
Kunwar K. Singh, Lindsey Smart, Gang Chen
Preliminary exploratory data analysis was performed to initially identify important variables and avoid overfitting the model as well as to reduce collinearity among predictor variables. One such exploratory test was the Kolmogorov-Smirnov (KS) test, a nonparametric test of the equality of 1D probability distributions (Smirnov 1948). This test can be used to compare a sample with a reference probability distribution or to compare two samples. The KS statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The magnitude of the KS statistic helped in identifying a subset of variables from all the input variables. Cumulative distribution functions were drawn for these variables for each of the wetland types being modeled to visualize differences across types.
Morphological and Geometrical Characterization of Historical Churches of Yucatan, Mexico
Published in International Journal of Architectural Heritage, 2022
Isis Rodriguez, Graça Vasconcelos, Paulo B. Lourenço
The statistical analysis on the database required a previous quality of the data by analyzing the presence of outliers. After this, simple regression analysis was carried out to investigate the statistical dependence between two parameters. In addition, the histogram of each parameter was defined and different distribution models were studied to understand which distribution model better described the data. Normality tests were performed to examine whether the data values follow a normal distribution by both graphical and numerical methods. For each distribution model, goodness-of-fit tests were performed using the empirical distribution function (EDF). The characterization of the normal distributions is then carried out based on the measures of skewness (s) and kurtosis (k) to evaluate possible deviations from the Gauss distribution.
A statistical study on lognormal central tendency estimation in probabilistic seismic assessments
Published in Structure and Infrastructure Engineering, 2020
Mohamad Zarrin, Mohsen Abyani, Behrouz Asgarian
A goodness of fit test is a statistical procedure for investigating whether a sample of n observations obey a given specified distribution or not. In this article, Kolmogorov–Smirnov (Benjamin & Cornell, 1970) and Anderson–Darling (Anderson & Darling, 1954) tests have been utilized as the statistical hypothesis testing. Kolmogorov–Smirnov (KS) test is a nonparametric test for continuous probability distributions, which measures the deviation of the empirical Cumulative Distribution Function (CDF) from the Hypothesized CDF (F). Anderson–Darling (AD) test is one of the most powerful goodness of fit tests based on Empirical Distribution Function (EDF). The AD test statistic () is categorized as the quadratic class of the EDF statistic, which is based on the squared difference between the Empirical CDF (ECDF) and the hypothesized F as the following equation: in which, indicates a positive weight function that could be computed by different expressions such as The hypothesis is rejected if is sufficiently large.