Inversion Formulae for the Distribution of Ratios

The distribution of linear combinations of random variables arises explicitly in many areas of engineering. This has increased the need to have available the widest possible range of statistical results on linear combinations of random variables. In this note, the exact distribution of the linear combination α X+β Y is derived when X and Y are Laplace and logistic random variables distributed independently of each other. Extensive tabulations of the associated percentage points obtained by inverting the derived distribution are also given.

In Chaps. 2 and 5 we occasionally referred to a normal distribution either informally (bell-shaped distributions/histograms) or formally, as in Sect. 5.5.3, where the normal density and its moments were briefly introduced. This chapter is devoted to the normal distribution due to its importance in statistics. What makes the normal distribution so important? The normal distribution is the proper statistical model for many natural and social phenomena. But even if some measurements cannot be modeled by the normal distribution (it could be skewed, discrete, multimodal, etc.), their sample means would closely follow the normal law, under very mild conditions. The central limit theorem covered in this chapter makes it possible to use probabilities associated with the normal curve to answer questions about the sums and averages in sufficiently large samples. This translates to the ubiquity of normality – many estimators, test statistics, and nonparametric tests covered in later chapters of this text are approximately normal, when sample sizes are not small (typically larger than 20 to 30), and this asymptotic normality is used in a substantial way. Several other important distributions can be defined through a normal distribution. Also, normality is a quite stable property – an arbitrary linear combination of normal random variables remains normal. The property of linear combinations of random variables preserving the distribution of their components is not shared by any other probability law and is a characterizing property of a normal distribution.KeywordsCentral Limit TheoremDuchenne Muscular DystrophyNormal Random VariableBivariate Normal DistributionCauchy DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A solution of the completion problem of symmetric positive definite matrices. Symmetric positive definite matrices play an important role in statistics. Typical examples are the covariance and correlation matrices which capture the second order statistics between random variables. In modelling studies these matrices are usually estimated from available data. Unfortunately, this data is not always sufficient to estimate all coefficients of the matrix. For statistical modelling, a complete matrix is needed and for this reason sensible estimates are needed for undefined coefficients. The entropy is a well accepted measure for the information content of a probability density function (pdf). The values given to the undefined coefficients can be chosen such that the information content of the corresponding multi-normal distribution is minimal. In this way a pdf is obtained that honours the available data and does not impose unnecessary extra constraints. A constraint cross entropy minimisation problem has to be solved to compute the values of the missing coefficients. The cross entropy optimization problem can be easily derived from first principles and results in an equation with Lagrange multipliers. The system of equations for the Lagrange multipliers is solved with a Newton iteration. A generalisation is possible such that a complete covariance matrix can be computed from second order statistics between linear combinations of random variables. The method has for instance been used to construct a prior pdf for reservoir models or to compute a cross variogram between two correlated lateral continuous random variables.

An approach for generating dependent random variates with specified marginal distributions and measure of correlation is developed using the geometric transformation method which is based upon an analogy between correlation and force. We first examine the mathematical transformation implied by this analogy and attempt to construct a general algorithm for generating variates that have the desired properties. We than examine the limitations of this approach, noting some potentially surprising distributional consequences associated with linear combinations of random variables. We conclude with the development of an algorithm for generating correlated Uniform(0,l) random variates with (approximately) a specified product-moment correlation.

Ordinary least squares (OLS) regression is based on the fact that the variance of a linear combination of random variables can be decomposed into the contributions of the individual variables and to the contributions of the correlations among them. The fact that one can imitate this decomposition (under certain conditions) when decomposing the GMD of a linear combination of random variables enables one to take any OLS-based econometric textbook and replicate each chapter using the GMD instead of the variance. Practically, this means doubling the number of models because every OLS econometric model can be replicated by the GMD, resulting in different estimates of the parameters. Moreover, we present via examples (Chap. 21) that the estimates can differ in sign. This means that two investigators who use the same variables, the same model, and the same data may come up with contradicting results concerning the effect of one variable on the other. The only difference between the two researchers lies in the measure of variability they use—the GMD or the variance. Needless to say that in many cases of policy decisions the debate is on the magnitude of a parameter, which is much more vulnerable than the sign and not on the sign itself. And to make life even more complicated any regression model that is estimated by the GMD can be replicated with the EG. This means moving from doubling the number of possible estimates to an infinite number of estimates.KeywordsExplanatory VariableRegression CoefficientHorizontal AxisOrdinary Little SquareQuantile RegressionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Stochastic orders of scalar products with applications

Preface. Acknowledgements. PART 1: THEORY. Chapter 1: Review of Probability Theory. 1.1 Introduction. 1.2 Basic Set Theory. 1.3 Probability. 1.4 Conditional Probability. 1.5 Random Variables and Probability Distributions. 1.6 Measures of Central Tendency, Variability, and Association. 1.7 Linear Combinations of Random Variables. 1.8 Functions of Random Variables. 1.9 Common Discrete Probability Distributions. 1.10 Common Continuous Probability Distributions. 1.11 Extreme-Value Distributions. Chapter2: Discrete random Processes. 2.1 Introduction. 2.2 Discrete-Time, Discrete-State Markov Chains. 2.3 Continuous-Time Markov Chains. 2.4 Queueing Models. Chapter 3: Random Fields. 3.1 Introduction. 3.2 Covariance Function. 3.3 Spectral Density Function. 3.4 Variance Function. 3.5 Correlation Length. 3.6 Some Common Models. 3.7 Random Fields in Higher Dimensions. Chapter 4: Best Estimates, Excursions, and Averages. 4.1 Best Linear Unbiased Estimation. 4.2 Threshold Excursions in One Dimension. 4.3 Threshold Excursions in Two Dimensions. 4.4 Averages. Chapter 5: Estimation. 5.1 Introduction. 5.2 Choosing a Distribution. 5.3 Estimation in Presence of Correlation. 5.4 Advanced Estimation Techniques. Chapter 6: Simulation. 6.1 Introduction. 6.2 Random-Number Generators. 6.3 Generating Nonuniform Random Variables. 6.4 Generating Random Fields. 6.5 Conditional Simulation of Random Fields. 6.6 Monte carlo Simulation. Chapter 7: Reliability-Based Design. 7.1 Acceptable Risk. 7.2 Assessing Risk. 7.3 Background to Design Methodologies. 7.4 Load and Resistance Factor Design. 7.5 Going Beyond Calibration. 7.6 Risk-Based Decision making. PART 2: PRACTICE. Chapter 8: Groundwater Modeling. 8.1 Introduction. 8.2 Finite-Element Model. 8.3 One-Dimensional Flow. 8.4 Simple Two-Dimensional Flow. 8.5 Two-Dimensional Flow Beneath Water-Retaining Structures. 8.6 Three-Dimensional Flow. 8.7 Three Dimensional Exit Gradient Analysis. Chapter 9: Flow Through Earth Dams. 9.1 Statistics of Flow Through Earth Dams. 9.2 Extreme Hydraulic Gradient Statistics. Chapter 10: Settlement of Shallow Foundations. 10.1 Introduction. 10.2 Two-Dimensional Probabilistic Foundation Settlement. 10.3 Three-Dimensional Probabilistic Foundation Settlement. 10.4 Strip Footing Risk Assessment. 10.5 Resistance Factors for Shallow-Foundation Settlement Design. Chapter 11: Bearing Capacity. 11.1 Strip Footings on c-o Soils. 11.2 Load and Resistance Factor Design of Shallow Foundations. 11.3 Summary. Chapter 12: Deep Foundations. 12.1 Introduction. 12.2 Random Finite-Element Method. 12.3 Monte Carlo Estimation of Pile Capacity. 12.4 Summary. Chapter 13: Slope Stability. 13.1 Introduction. 13.2 Probabilistic Slope Stability Analysis. 13.3 Slope Stability Reliability Model. Chapter 14: Earth Pressure. 14.1 Introduction. 14.2 Passive Earth Pressures. 14.3 Active Earth Pressures: Retaining Wall Reliability. Chapter 15: Mine Pillar Capacity. 15.1 Introduction. 15.2 Literature. 15.3 Parametric Studies. 15.4 Probabilistic Interpretation. 15.5 Summary. Chapter 16: Liquefaction. 16.1 Introduction. 16.2 Model Size: Soil Liquefaction. 16.3 Monte Carlo Analysis and Results. 16.4 Summary PART 3: APPENDIXES. APPENDIX A: PROBABILITY TABLES. A.1 Normal Distribution. A.2 Inverse Student t -Distribution. A.3 Inverse Chi-Square Distribution APPENDIX B: NUMERICAL INTEGRATION. B.1 Gaussian Quadrature. APPENDIX C. COMPUTING VARIANCES AND CONVARIANCES OF LOCAL AVERAGES. C.1 One-Dimensional Case. C.2 Two-Dimensional Case C.3 Three-Dimensional Case. Index.

Convex orders for linear combinations of random variables

This work investigates the use of the tolerance limits on the treatment couch position to detect mistakes in patient positioning and warn users of possible treatment errors. Computer controlled radiotherapy systems use the position of the treatment couch as a surrogate for patient position, and a tolerance limit is applied against a planned position. When the couch is out of tolerance, a warning is sent to a user to indicate a possible mistake in setup. A tight tolerance may catch all positioning mistakes while at the same time sending too many warnings; a loose tolerance will not catch all mistakes. We developed a statistical model of the absolute position for the three translational axes of the couch. The couch position for any fraction is considered a random variable xi. The ideal planned couch position xp is unknown before a patient starts treatment and must be estimated from the daily positions of xi. As such, xp is also a random variable. The tolerance, tol, is applied to the difference between the daily and planned position, di=xi−xp. The di is a linear combination of random variables and therefore the density of di is the convolution of distributions of xi and xp. Tolerance limits are based on the standard deviation of di such that couch positions that are more than two standard deviations away are considered out of tolerance. Using this framework, we investigated two methods of setting xp and tolerance limits. The first, called first day acquire (FDA), is to take couch position on the first day as the planned position. The second is to use the cumulative average (CumA) over previous fractions as the planned position. The standard deviation of di shrinks as more samples are used to determine xp and, as a result, the tolerance limit shrinks as a function of fraction number when a CumA technique is used. The metrics of sensitivity and specificity were used to characterize the performance of the two methods to correctly identify a couch position as in‐ or out‐of‐tolerance. These two methods were tested using simulated and real patient data. Five clinical sites with different indexed immobilization were tested. These were whole brain, head and neck, breast, thorax, and prostate. Analysis of the head and neck data shows that it is reasonable to model the daily couch position as a random variable in this treatment site. Using an average couch position for xp increased the sensitivity of the couch interlock and reduced the chances of acquiring a couch position that was a statistical outlier. Analysis of variation in couch position for different sites allowed the tolerance limit to be set specifically for a site and immobilization device. The CumA technique was able to increase the sensitivity of detecting out‐of‐tolerance positions while shrinking tolerance limits for a treatment course. Making better use of the software interlock on the couch positions could have a positive impact on patient safety and reduce mistakes in treatment delivery.PACS number: 87.55.Ne, 87.55.Qr, 87.55.tg, 87.55.tm

IfX andY are two random variables with the same means and variances, thenX is said to be nearer normal thanY if the absolute values of its cumulants are smaller than the corresponding cumulants ofY. Using this definition, it is shown that a linear combination of a finite number of independent identically distributed random variables is always nearer normal than its constituents, but that this is not necessarily true if not-identically distributed or not-independent variables are used. Some consequences of the results are reached for the testing of normality of time series and for the assumptions frequently made by social scientists about the distribution of their data.

A new proof for the peakedness of linear combinations of random variables

CHAPTER III - NUMERICAL CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS

An approach for generating dependent random variates with specified marginal distributions and measure of correlation is developed based upon an analogy between correlation and force. We first examine the mathematical transformation implied by this analogy and attempt to construct a general algorithm for generating variates that have the desired properties. We then examine the limitations of this approach, noting some potentially surprising distributional consequences associated with linear combinations of random variables. We conclude with the development of an algorithm for generating correlated Uniform (0,1) random variates with (approximately) a specified product-moment correlation.

An approach for generating dependent random variates with specified marginal distributions and measure of correlation is developed based upon an analogy between correlation and force. We first examine the mathematical transformation implied by this analogy and attempt to construct a general algorithm for generating variates that have the desired properties. We then examine the limitations of this approach, noting some potentially surprising distributional consequences associated with linear combinations of random variables. We conclude with the development of an algorithm for generating correlated Uniform (0,1) random variates with (approximately) a specified product-moment correlation.

evaluations when there are n random variables, which is unaffordable for many engineering applications. In the previous paper by the authors [15], a variable-point PEM method was proposed to improve the efficiency and accuracy of the existing approaches. However, when it applied to the problems with large number of design variables (number of design variables > 10), the method still requires hundreds or thousands of simulation runs. This paper further improves the efficiency of the variable-point PEM based upon two fundamental concepts: 1) The Pareto principle; and 2) The Central Limit Theorem of Statistics, i.e., under common engineering conditions, a linear combination of random variables can be approximated to first order by a normal distribution. The efficiency and accuracy of the proposed method are validated with three benchmark problems