Algorithmic inference

Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.

Abstract: PART ONE: INTRODUCTION Traditional Parametric Statistical Inference Bootstrap Statistical Inference Bootstrapping a Regression Model Theoretical Justification The Jackknife Monte Carlo Evaluation of the Bootstrap PART TWO: STATISTICAL INFERENCE USING THE BOOTSTRAP Bias Estimation Bootstrap Confidence Intervals PART THREE: APPLICATIONS OF BOOTSTRAP CONFIDENCE INTERVALS Confidence Intervals for Statistics With Unknown Sampling Distributions Inference When Traditional Distributional Assumptions Are Violated PART FOUR: CONCLUSION Future Work Limitations of the Bootstrap Concluding Remarks

Abstract: It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid ``post-selection inference'' by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals. Simultaneity is required for all linear functions that arise as coefficient estimates in all submodels. By purchasing ``simultaneity insurance'' for all possible submodels, the resulting post-selection inference is rendered universally valid under all possible model selection procedures. This inference is therefore generally conservative for particular selection procedures, but it is always less conservative than full Scheffe protection. Importantly it does not depend on the truth of the selected submodel, and hence it produces valid inference even in wrong models. We describe the structure of the simultaneous inference problem and give some asymptotic results.