## Elements of Computational StatisticsIn recent years developments in statistics have to a great extent gone hand in hand with developments in computing. Indeed, many of the recent advances in statistics have been dependent on advances in computer science and techn- ogy. Many of the currently interesting statistical methods are computationally intensive, eitherbecausetheyrequireverylargenumbersofnumericalcompu- tions or because they depend on visualization of many projections of the data. The class of statistical methods characterized by computational intensity and the supporting theory for such methods constitute a discipline called “com- tational statistics”. (Here, I am following Wegman, 1988, and distinguishing “computationalstatistics”from“statisticalcomputing”, whichwetaketomean “computational methods, including numerical analysis, for statisticians”.) The computationally-intensive methods of modern statistics rely heavily on the developments in statistical computing and numerical analysis generally. Computational statistics shares two hallmarks with other “computational” sciences, such as computational physics, computational biology, and so on. One is a characteristic of the methodology: it is computationally intensive. The other is the nature of the tools of discovery. Tools of the scienti?c method have generally been logical deduction (theory) and observation (experimentation). The computer, used to explore large numbers of scenarios, constitutes a new type of tool. Use of the computer to simulate alternatives and to present the research worker with information about these alternatives is a characteristic of thecomputationalsciences. Insomewaysthisusageisakintoexperimentation. The observations, however, are generated from an assumed model, and those simulated data are used to evaluate and study the model. |

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### Contents

Monte Carlo Methods for Inference | 37 |

Randomization and Data Partitioning | 67 |

4 | 83 |

Tools for Identification of Structure in Data | 97 |

Estimation of Functions | 125 |

Graphical Methods in Computational Statistics | 149 |

Estimation of Probability Density Functions Using Parametric | 193 |

Exercises | 199 |

Structure in Data | 225 |

Statistical Models of Dependencies | 291 |

Appendices 328 | 329 |

B Software for Random Number Generation | 347 |

Notation and Definitions | 359 |

Bibliography | 375 |

409 | |

415 | |

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algorithm approach approximation bias bins bivariate bootstrap called classification computational statistics confidence intervals consider convergence coordinates correlations corresponding covariance curves dataset defined density estimation depends describe determine dimensions discuss distance distribution function ECDF elements equation example factor given histogram inference integral iterations jackknife jackknife estimator kernel L2 norm least squares linear maximum likelihood mean measure methods minimal minimal spanning tree Monte Carlo methods Monte Carlo study Monte Carlo test norm normal distribution number of observations objective optimization orthogonal orthogonal polynomials outliers parallel coordinates parameter plot points polynomials principal components principal components analysis probability density function problem projection pursuit properties quantiles random number random sample random variable regression represent resampling residuals rotation S-Plus scale shown in Figure similar simple simulation smoothing space splines standard structure tessellation test statistic tion transformation univariate values variance variance-covariance matrix variation vector