Cumulative distribution function

About: Cumulative distribution function is a research topic. Over the lifetime, 6049 publications have been published within this topic receiving 145696 citations. The topic is also known as: CDF & distribution function.

Probability plotting methods for the analysis of data

Abstract: SUMMARY This paper describes and discusses graphical techniques, based on the primitive empirical cumulative distribution function and on quantile (Q-Q) plots, percent (P-P) plots and hybrids of these, which are useful in assessing a one-dimensional sample, either from original data or resulting from analysis. Areas of application include: the comparison of samples; the comparison of distributions; the presentation of results on sensitivities of statistical methods; the analysis of collections of contrasts and of collections of sample variances; the assessment of multivariate contrasts;_ and the structuring of analysis of variance mean squares. Many of the objectives and techniques are illustrated by examples. This paper reviews a variety of old and new statistical techniques based on the cumulative distribution function and its ramifications. Included in the coverage are applications, for various situations and purposes, of quantile probability plots (Q-Q plots), percentage probability plots (P-P plots) and extensions and hybrids of these. The general viewpoint is that of analysis of data by statistical methods that are suggestive and constructive rather than formal procedures to be applied in the light of a tightly specified mathematical model. The technological background is taken to be current capacities in data collection and highspeed computing systems, including graphical display facilities. It is very often useful in statistical data analysis to examine and to present a body of data as though it may have originated as a one-dimensional sample, i.e. data which one wishes to treat for purposes of analysis, as an unstructured array. Sometimes this is applicable to ' original' data; even more often such a viewpoint is useful with 'derived' data, e.g. residuals from a model fitted to the data. The empirical cumulative distribution function and probability plotting methods have a key role in the statistical treatment of one-dimensional samples, being of relevance for summarization and palatable description as well as for exposure and inference.

Probability and Random Processes

Abstract: tions. Bootstrap has found many applications in engineering field, including artificial neural networks, biomedical engineering, environmental engineering, image processing, and radar and sonar signal processing. Basic concepts of the bootstrap are summarized in each section as a step-by-step algorithm for ease of implementation. Most of the applications are taken from the signal processing literature. The principles of the bootstrap are introduced in Chapter 2. Both the nonparametric and parametric bootstrap procedures are explained. Babu and Singh (1984) have demonstrated that in general, these two procedures behave similarly for pivotal (Studentized) statistics. The fact that the bootstrap is not the solution for all of the problems has been known to statistics community for a long time; however, this fact is rarely touched on in the manuscripts meant for practitioners. It was first observed by Babu (1984) that the bootstrap does not work in the infinite variance case. Bootstrap Techniques for Signal Processing explains the limitations of bootstrap method with an example. I especially liked the presentation style. The basic results are stated without proofs; however, the application of each result is presented as a simple step-by-step process, easy for nonstatisticians to follow. The bootstrap procedures, such as moving block bootstrap for dependent data, along with applications to autoregressive models and for estimation of power spectral density, are also presented in Chapter 2. Signal detection in the presence of noise is generally formulated as a testing of hypothesis problem. Chapter 3 introduces principles of bootstrap hypothesis testing. The topics are introduced with interesting real life examples. Flow charts, typical in engineering literature, are used to aid explanations of the bootstrap hypothesis testing procedures. The bootstrap leads to second-order correction due to pivoting; this improvement in the results due to pivoting is also explained. In the second part of Chapter 3, signal processing is treated as a regression problem. The performance of the bootstrap for matched filters as well as constant false-alarm rate matched filters is also illustrated. Chapters 2 and 3 focus on estimation problems. Chapter 4 introduces bootstrap methods used in model selection. Due to the inherent structure of the subject matter, this chapter may be difficult for nonstatisticians to follow. Chapter 5 is the most impressive chapter in the book, especially from the standpoint of statisticians. It provides real data bootstrap applications to illustrate the theory covered in the earlier chapters. These include applications to optimal sensor placement for knock detection and land-mine detection. The authors also provide a MATLAB toolbox comprising frequently used routines. Overall, this is a very useful handbook for engineers, especially those working in signal processing.

Statistics for Business and Economics

Abstract: CHAPTER 1 Describing Data: Graphical 1.1 Decision Making in an Uncertain Environment 1.2 Classification of Variables 1.3 Graphs to Describe Categorical Variables 1.4 Graphs to Describe Time-Series Data 1.5 Graphs to Describe Numerical Variables 1.6 Tables and Graphs to Describe Relationships Between Variables 1.7 Data Presentation Errors CHAPTER 2 Describing Data: Numerical 2.1 Measures of Central Tendency 2.2 Measures of Variability 2.3 Weighted Mean and Measures of Grouped Data 2.4 Measures of Relationships Between Variables CHAPTER 3 Probability 3.1 Random Experiment, Outcomes, Events 3.2 Probability and Its Postulates 3.3 Probability Rules 3.4 Bivariate Probabilities 3.5 Bayes' Theorem CHAPTER 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables 4.2 Probability Distributions for Discrete Random Variables 4.3 Properties of Discrete Random Variables 4.4 Binomial Distribution 4.5 Hypergeometric Distribution 4.6 The Poisson Probability Distribution 4.7 Jointly Distributed Discrete Random Variables CHAPTER 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 5.2 Expectations for Continuous Random Variables 5.3 The Normal Distribution 5.4 Normal Distribution Approximation for Binomial Distribution 5.5 The Exponential Distribution 5.6 Jointly Distributed Continuous Random Variables CHAPTER 6 Sampling and Sampling Distributions 6.1 Sampling from a Population 6.2 Sampling Distributions of Sample Means 6.3 Sampling Distributions of Sample Proportions 6.4 Sampling Distributions of Sample Variances CHAPTER 7 Estimation: Single Population 7.1 Properties of Point Estimators 7.2 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Known 7.3 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Unknown 7.4 Confidence Interval Estimation of Population Proportion 7.5 Confidence Interval Estimation of the Variance of a Normal Distribution 7.6 Confidence Interval Estimation: Finite Populations CHAPTER 8 Estimation: Additional Topics 8.1 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Dependent Samples 8.2 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Independent Samples 8.3 Confidence Interval Estimation of the Difference Between Two Population Proportions 8.4 Sample Size Determination: Large Populations 8.5 Sample Size Determination: Finite Populations CHAPTER 9 Hypothesis Testing: Single Population 9.1 Concepts of Hypothesis Testing 9.2 Tests of the Mean of a Normal Distribution: Population Variance Known 9.3 Tests of the Mean of a Normal Distribution: Population Variance Unknown 9.4 Tests of the Population Proportion 9.5 Assessing the Power of a Test 9.6 Tests of the Variance of a Normal Distribution CHAPTER 10 Hypothesis Testing: Additional Topics 10.1 Tests of the Difference Between Two Population Means: Dependent Samples 10.2 Tests of the Difference Between Two Normal Population Means: Independent Samples 10.3 Tests of the Difference Between Two Population Proportions 10.4 Tests of the Equality of the Variances Between Two Normally Distributed Populations 10.5 Some Comments on Hypothesis Testing CHAPTER 11 Simple Regression 11.1 Overview of Linear Models 11.2 Linear Regression Model 11.3 Least Squares Coefficient Estimators 11.4 The Explanatory Power of a Linear Regression Equation 11.5 Statistical Inference: Hypothesis Tests and Confidence Intervals 11.6 Prediction 11.7 Correlation Analysis 11.8 Beta Measure of Financial Risk 11.9 Graphical Analysis CHAPTER 12 Multiple Regression 12.1 The Multiple Regression Model 12.2 Estimation of Coefficients 12.3 Explanatory Power of a Multiple Regression Equation 12.4 Confidence Intervals and Hypothesis Tests for Individual Regression Coefficients 12.5 Tests on Regression Coefficients 12.6 Prediction 12.7 Transformations for Nonlinear Regression Models 12.8 Dummy Variables for Regression Models 12.9 Multiple Regression Analysis Application Procedure CHAPTER 13 Additional Topics in Regression Analysis 13.1 Model-Building Methodology 13.2 Dummy Variables and Experimental Design 13.3 Lagged Values of the Dependent Variables as Regressors 13.4 Specification Bias 13.5 Multicollinearity 13.6 Heteroscedasticity 13.7 Autocorrelated Errors CHAPTER 14 ANALYSIS OF CATEGORICAL DATA 14.1 Goodness-of-Fit Tests: Specified Probabilities 14.2 Goodness-of-Fit Tests: Population Parameters Unknown 14.3 Contingency Tables 14.4 Sign Test and Confidence Interval 14.5 Wilcoxon Signed Rank Test 14.6 Mann--Whitney U Test 14.7 Wilcoxon Rank Sum Test 14.7 Spearman Rank Correlation CHAPTER 15 Analysis of Variance 15.1 Comparison of Several Population Means 15.2 One-Way Analysis of Variance 15.3 The Kruskal--Wallis Test 15.4 Two-Way Analysis of Variance: One Observation per Cell, Randomized Blocks 15.5 Two-Way Analysis of Variance: More Than One Observation per Cell CHAPTER 16 Time-Series Analysis and Forecasting 16.1 Index Numbers 16.2 A Nonparametric Test for Randomness 16.3 Components of a Time Series 16.4 Moving Averages 16.5 Exponential Smoothing 16.6 Autoregressive Models 16.7 Autoregressive Integrated Moving Average Models CHAPTER 17 Sampling: Additional Topics 17.1 Stratified Sampling 17.2 Other Sampling Methods CHAPTER 18 Statistical Decision Theory 18.1 Decision Making Under Uncertainty 18.2 Solutions Not Involving Specification of Probabilities 18.3 Expected Monetary Value TreePlan 18.4 Sample Information: Bayesian Analysis and Value 18.5 Allowing for Risk: Utility Analysis APPENDIX TABLES 1. Cumulative Distribution Function of the Standard Normal Distribution 2. Probability Function of the Binomial Distribution 3. Cumulative Binomial Probabilities 4. Values of e --lambda 5. Individual Poisson Probabilities 6. Cumulative Poisson Probabilities 7. Cutoff Points of the Chi-Square Distribution Function 8. Cutoff Points for the Student's t Distribution 9. Cutoff Points for the F Distribution 10. Cutoff Points for the Distribution of the Wilcoxon Test Statistic 11. Cutoff Points for the Distribution of Spearman Rank Correlation Coefficient 12. Cutoff Points for the Distribution of the Durbin--Watson Test Statistic 13 Critical Values of the Studentized Range Q (page 964 965 Applied Statistical Methods Carlson, Thorne Prentice Hall 1997) 14. Cumulative Distribution Function of the Runs Test Statistic ANSWERS TO SELECTED EVEN-NUMBERED EXERCISES INDEX I-1

A simple coding procedure enhances a neuron's information capacity.

Abstract: The contrast-response function of a class of first order interneurons in the fly's compound eye approximates to the cumulative probability distribution of contrast levels in natural scenes. Elementary information theory shows that this matching enables the neurons to encode contrast fluctuations most efficiently.

The Statistical Distribution of the Maxima of a Random Function

Abstract: This paper studies the statistical distribution of the maximum values of a random function which is the sum of an infinite number of sine waves in random phase. The results are applied to sea waves and to the pitching and rolling motion of a ship.