Showing papers on "Cumulative distribution function published in 1987"

Sampling Turbulence in the Stratified Ocean: Statistical Consequences of Strong Intermittency

Abstract: Turbulence and turbulent mixing in the ocean are strongly intermittent in amplitude, space and time. The degree of intermittency is measured by the “intermittency factor” σ2, defined as either σ2lnϵ, the variance of the logarithm of the viscous dissipation rate ϵ, or σ2lnχ, the variance of the logarithm of the temperature dissipation rate χ. Available data suggest that the cumulative distribution functions of ϵ and χ in stratified layers are approximately lognormal with large σ2 values in the range 3–7. Departures from lognormality are remarkably similar to those for Monte Carlo generated lognormal distributions contaminated with simulated noise and undersampling effects. Confidence limits for the maximum likelihood estimator of the mean of a lognormal random variable are determined by Monte Carlo techniques and by theoretical modeling. They show that such large σ2 values cause large uncertainty in estimates of the mean unless the number of data samples is extremely large. To obtain estimates of ...

Data and theory for a new model of the horizontal structure of rain cells for propagation applications

Abstract: A large population of radar-measured ground rain cells is used to devise and assess a rain cell model for use in some of the future telecommunication applications. The model is based on cells of exponential profile (which is shown to reproduce best the point rain rate CDF); both rotational and biaxial symmetries are considered for the horizontal cross sections. Furthermore, the proposed model contains analytical expressions for the joint probability densities of the parameters which define the cell, i.e., peak rain intensity, cell size and axial ratio. Finally, an algorithm is given for adapting the model to the characteristics of any given site: this algorithm requires as input the local cumulative distribution of point rainfall and provides the spatial number densities (i.e., the average number of cells per square kilometer and per unit range of the parameters) which this distribution would produce. The model offers the possibility of predicting the statistics of many propagation parameters (such as attenuation or interference by rain scattering) which are determined by the rain cell characteristics and their frequency of occurrence.

Graphical Communication of Uncertain Quantities to Nontechnical People

Abstract: Nine pictorial displays for communicating quantitative information about the value of an uncertain quantity, x, were evaluated for their ability to communicate 2, p(x > a) and p( b > x > a) to well-educated semi- and nontechnical subjects. Different displays performed best in different applications. Cumulative distribution functions alone can severely mislead some subjects in estimating the mean. A “rusty” knowledge of statistics did not improve performance, and even people with a good basic knowledge of statistics did not perform as well as one would like. Until further experiments are performed, the authors recommend the use of a cumulative distribution function plotted directly above a probability density function with the same horizontal scale, and with the location of the mean clearly marked on both curves.

Deterministic uncertainty analysis

Abstract: This paper presents a deterministic uncertainty analysis (DUA) method for calculating uncertainties that has the potential to significantly reduce the number of computer runs compared to conventional statistical analysis. The method is based upon the availability of derivative and sensitivity data such as that calculated using the well known direct or adjoint sensitivity analysis techniques. Formation of response surfaces using derivative data and the propagation of input probability distributions are discussed relative to their role in the DUA method. A sample problem that models the flow of water through a borehole is used as a basis to compare the cumulative distribution function of the flow rate as calculated by the standard statistical methods and the DUA method. Propogation of uncertainties by the DUA method is compared for ten cases in which the number of reference model runs was varied from one to ten. The DUA method gives a more accurate representation of the true cumulative distribution of the flow rate based upon as few as two model executions compared to fifty model executions using a statistical approach. 16 refs., 4 figs., 5 tabs.

The Calculation of the Limiting Distribution of the Least Squares Estimator in Near-Integrated Model

Abstract: We Tabulate the Limiting Cumulative Distribution and Probability Density Functions of the Least Squares Estimator in a First-Order Autoregressive Regression When the True Model Is Near-Integrated in the Sense of Phillips (1986 A). the Results Are Obtained Using an Exact Numerical Method Which Integrates the Appropriate Limiting Moment Generating Function. the Adequacy of the Approximation Is Examined by Monte Carlo Methods for Various First-Order Autogressive Processes with a Root Close to Unity.

Variate generation for accelerated life and proportional hazards models

Abstract: We use accelerated life and proportional hazards lifetime models to account for the effects of covariates on a random lifetime. We find that variate generation algorithms for Monte Carlo simulation in both the renewal and nonhomogeneous Poisson process cases are a simple extension of the inverse cumulative distribution function (cdf) technique.

Distribution of order statistics with random sample size

Abstract: The distribution function of the ith order statistic in random sampling from a distribution function F is obtained when the sample size. is random.

R.A. Fisher-a True Bayesian?

Abstract: Summary R.A. Fisher's claim that his fiducial argument uses the term 'probability' in the same sense as used by the Rev. Thomas Bayes is fully justifiable. But, while probability statements concerning parameters can be made, these parameters cannot be regarded as random variables in the sense of Kolmogoroff. Fisher was not a 'Bayesian' in the main current sense of the word. In the first edition (1956) of Statistical Methods and Scientific Inference, Ch. V, ? 6, R.A. Fisher discusses the logical situation arising when data of two kinds are available, one kind such as to give a fiducial distribution for the unknown parameter, the other such as to yield only a likelihood function. He imagines a charged particle recorder capable of being switched on or off at precisely chosen times. The recorder can be set to record the time at which a particle passes through, or alternatively to record whether any particles pass through in a specific time interval. Assuming the particles form a Poisson process with unknown rate 0 particles per unit time, the time t elapsing between switching on and observing the first particle has cumulative probability P(t, 0) = exp {-tO}, while the

Probability and Statistics in Geodesy and Geophysics

Abstract: Introduction. 1. Mathematical Preliminaries. 2. Estimation Theory. 3. Testing of Hypothesis. 4. Systematic Influences and Their Elimination. 5. Design of Experiments. 6. Linear Theory of Random Functions. 7. Problems of the Linear Estimation Theory and the Theory of Random Functions with an Estimated Covariance Matrix. References. Subject Index.

A method for computer generation of variates from arbitrary continuous distributions

Abstract: If X is a continuous random variable with cumulative distribution function (cdf) $F(x)$, then the most direct means of generating a variate from this distribution is the inverse cdf transform, $X = F^{ - 1} (U)$, where U is a random uniform variate. If $f(x)$ is the corresponding density function then X and U, as defined above, must satisfy the differential equation, $X' = {1/{f(X)}}$. This differential equation is applied, first, in the high accuracy computation of distribution percentiles. The differential equation is then used to motivate an interpolatory Runge-Kutta algorithm to approximate the inverse cdf in the transform described above. Finally, the best discrete least squares cubic spline approximation to the inverse cdf is constructed for the approximation of the inverse cdf. Comparisons between inverse cdf approximations are made on the basis of accuracy, efficiency, and setup requirements for the normal, exponential, Cauchy and gamma distributions.

A maximum entropy method for lightwave system performance evaluation

Abstract: A constrained maximum entropy criterion is used to approximate the cumulative distribution function of the photocurrent generated by an avalanche photodiode in response to an incident information-bearing optical signal. The approximate distribution, derived from known moments of the photocurrent, is used to evaluate the probability of error in direct detection lightwave systems. In this application, the results of the maximum entropy method are equivalent to those of a Gauss quadrature rule method. However the maximum entropy method exhibits a relative efficiency in terms of the required number of moments of the photocurrent.

A Modified One-Sided k-S Confidence Refion Narrower In One Tail

Abstract: Based on the generalized Kolmogorov-Smirnov (k-s) and the usual k-s distances, one-side confidence regions for continuous cumulative distribution function (cdf) using empirical cdf are constructed. The band width of such regions is narrower in the right or the left tail of the distribution where we may want to have more precise infrmation about the distribution function. Finite sample and asymptomatic critical values necessary for implemention are given along with an example

An Analytical Technique for the Reliability Modeling of Generation Systems including Energy Limited Units

Abstract: This paper presents a method for the calculation of Frequency and Duration (F&D) indices for electric power generating systems with energy limited units. Several types of energy limited units are discussed. For an energy limited unit, a critical load level at which this unit is put into operation is found by "peak-shaving" method and then the cumulative probability and frequency are computed by conditional convolution. The method is illustrated by numerical examples.

Two characterizations of the exponential distribution

Abstract: A sequence Xn ≥ 1 of independent and identically distrinbuted random variable with absoiutely continuous cumulative distribution function F ( x ) is considered. X j is a record value of this sequence if .Let X L(n) n≤o with L(0)=1 be the sequence of such record values and Z n,m =X L(n)-X L(m) suppose is the with smallest order statistic in a random sample of size n from F and . Two characterizations of the exponential distribution are given based on the distributional properties of Zn,m and Tk,n.

Performance of Direct-Sequence Spread-Spectrum Systems in the Presence of Multiple-Access Interference and Jamming

Abstract: An approach for calculating upper and lower bounds for the probability of error for asynchronous multiple-access spread-spectrum communication systems employing deterministic codes is presented. The technique is then generalized to include multiple-tone jamming. The approach utilizes the cumulative distribution function of individual interference terms. The computational complexity of the technique is calculated to be polynomial like. Results showing the multiple-access performance of Gold codes of lengths 31 and 127 in the presence of jamming are shown.

The Calculation of the Limiting Distribution of the Least Squares Estimator in Near-Integrated Model

Abstract: We Tabulate the Limiting Cumulative Distribution and Probability Density Functions of the Least Squares Estimator in a First-Order Autoregressive Regression When the True Model Is Near-Integrated in the Sense of Phillips (1986 A). the Results Are Obtained Using an Exact Numerical Method Which Integrates the Appropriate Limiting Moment Generating Function. the Adequacy of the Approximation Is Examined by Monte Carlo Methods for Various First-Order Autogressive Processes with a Root Close to Unity.

A modified version of handscomb's antithetic variates theorem

Abstract: We consider a modified version of Handscomb’s Antithetic Variates theorem and prove the existence of exact minimum rather than infinimum and that this minimum is achieved by using only one random number. Practical procedures for finding the optimal cumulative distribution function are given, for estimating the expected value of the response difference of a pair of arbitrary functions of scalar arguments.

Deterministic methods for sensitivity and uncertainty analysis in large-scale computer models

Abstract: The fields of sensitivity and uncertainty analysis are dominated by statistical techniques when large-scale modeling codes are being analyzed. This paper reports on the development and availability of two systems, GRESS and ADGEN, that make use of computer calculus compilers to automate the implementation of deterministic sensitivity analysis capability into existing computer models. This automation removes the traditional limitation of deterministic sensitivity methods. The paper describes a deterministic uncertainty analysis method (DUA) that uses derivative information as a basis to propagate parameter probability distributions to obtain result probability distributions. The paper demonstrates the deterministic approach to sensitivity and uncertainty analysis as applied to a sample problem that models the flow of water through a borehole. The sample problem is used as a basis to compare the cumulative distribution function of the flow rate as calculated by the standard statistical methods and the DUA method. The DUA method gives a more accurate result based upon only two model executions compared to fifty executions in the statistical case.

A simple approximation for the simulation of continuous random variables

Abstract: A simple technique is described for simulations and analytical studies where the indication is of a unimodal, right-skewed dis tribution of a continuous random variable, the type of distribu tion often approximated by a gamma distribution. The technique is more realistic and more general than the "simple" techniques described in the simulation literature in that it does not require integer parameters.

Statistics and probability

Abstract: Introduction, Cumulative distribution functions, Geometric and exponential probability, Linear combinations of random variables, Some properties of normal probability, Hypothesis testing, Large sample distributions, The central limit theorem, Hypothesis testing with discrete variables, Errors in hypothesis testing, The t-distribution, Confidence intervals, A survey of probability distributions, Chi-squared tests, Review exercise, Appendix: some supporting mathematics, Answers, Index

A limit theorem for the separable statistic in a random assignment scheme

Abstract: One considers the following scheme of random assignment of n particles in an infinite sequence of cells. Each particle is. assigned to the k-th cell with probability pk and one assumes that pk⩾ pk+1 and pk > 0 for each k. Let Xk(n) be the number of particles in the k-th cell and let f1(x), f2(x), ... be a sequence of real-valued functions defined for x=0, 1,2, ... Under certain conditions on the distribution of the probabilities and on the sequence f1(x), f2(x), ..., one investigates the asymptotic normality of the random variable . (The random variable Zn is proper since )

Some probabilistic properties of the line-of-sight angle error to a remote object (Corresp.)

Abstract: Closed-form expressions are derived for the distribution and the expectation of the error of the line-of-sight angle to a remote object when the rectangular position coordinates are independent identically distributed normal variates Generalized results also are provided both for the cumulative distribution function for a hyperspace of N dimensions and for a spherically symmetric probability density function in three-dimensional space

Continuous Singular Probability Measures That are Almost Lattice

Abstract: On construit des mesures de probabilite singulieres continues associees a des variables aleatoires independantes

Efficient evaluation of the area under the normal curve

Abstract: The task of evaluating the cumulative distribution of a normally distributed random variable is considered. A brief introduction into this problem is given and several possible approximations are presented as APL functions. Many of these functions have been written in two versions. The first version is written in the original APL style. the companion function using an APL2 coding style is also included in the paper for purposes of comparison. Both versions have been tested using simple data vectors of varying lengths and run under APL on an IBM 3090 with and without a Vector Facility. The results are compared among the algorithms, between the coding styles, and with and without hardware support for vector processing.