Showing papers on "Cumulative distribution function published in 1976"

Inequalities for Ek(X, Y) when the marginals are fixed

Abstract: When k(x, y) is a quasi-monotone function and the random variables X and Y have fixed distributions, it is shown under some further mild conditions that ℰ k(X, Y) is a monotone functional of the joint distribution function of X and Y. Its infimum and supremum are both attained and correspond to explicitly described joint distribution functions.

More on Estimation of Distributions Using Orthogonal Expansions

Abstract: Let f(x) be a continuous, strictly positive probability density function over an interval [a, b], and let F(x) be its associated cumulative distribution function. Suppose ϕ0(x), ϕ1(x), ϕ2(x), … form a complete orthonormal basis for L 2[a, b], the set of square-integrable functions on [a, b]. Suppose, further, that over [a, b] and that log f(x) has a similar expansion. Given a random sample X 1, X 2, …, Xn from the distribution, some aspects of the estimation of f(x) and F(x) are investigated.

A Characterization of Convoluted Poisson Distributions with Applications to Estimation

Abstract: Define X as the sum of a Poisson random variable Y with parameter θ and a nonnegative integer valued variable Z independent of Y. If the distribution of Z is known, X has a one-parameter convoluted Poisson distribution. Such distributions are here characterized as solutions to a system of differential equations specifying the rate of change of the cumulative distribution function relative to a parametric change. The characterization is used in point and interval estimation of θ. It is shown that for a certain subclass of convoluted Poisson distributions, the maximum likelihood estimate of θ is the unique solution of the likelihood equation and is easily approximated numerically.

The normal approximation to the multinomial with an increasing number of classes

Abstract: For a fixed number of classes and the number of trials increasing, the approach of the multinomial cumulative distribution function to a normal cumulative distribution function is familiar. In this paper we allow the number of classes to increase as the number of trials increases, and show that under certain circumstances the probabilities assigned to arbitrary regions by the multinomial distribution are all close to the probabilities assigned by the distribution of “rounded off” normal random variables. As the number of trials increases, the amount rounded off approaches zero. The result can be used to study the asymptotic distribution of functions of multinomial random variables.

A Homogeneous Distribution Problem with Applications to Finance

Abstract: We consider the problem of determining the cumulative distribution function and/or moments of the optimal solution value of a nonlinear program dependent upon a single random variable. This problem is difficult computationally because one must in effect determine the optimal solution to an infinite number of nonlinear programs. Bereanu [Bereanu, B., G. Peeters. 1970. A ‘Wait-and-See’ problem in stochastic linear programming. An experimental computer code. Cashiers Centre Etudes Rech. Oper. 12 (3) 133–148.] has provided an algorithm to solve the distribution problem in the linear case based on extensions of the methods of parametric linear programming. (See also [Bereanu, B. 1967. On stochastic linear programming, distribution problems: stochastic technology matrix. Z. f. Wahrscheinlichkeitstheorie u. oerw. Gerbieter 8 148–152; Bereanu, B. 1971. The distribution problem in stochastic linear programming: the Cartesian integration method. Center of Mathematical Statistics of the Academy of RSR, Bucharest, 71...

Further results on simultaneous confidence intervals for the normal distribution

Abstract: Based on a random sample from the normal cumulative distribution function ϕ(x; μ, σ) with unknown parameters μ and σ, one-sided confidence contours for ϕ(x; μ, σ), −∞

The joint probability density function of the occupation time of a three-state problem

Abstract: The joint probability density function of the occupation time of a three-state stochastic process with constant transition matrix: a,, # 0, i X j, i, j = 1,2, 3 but a,, = 0, i = 1,2, 3 is studied. STOCHASTIC PROCESS; OCCUPATION TIME; PROBABILITY DENSITY FUNCTION; RANDOM WALK

An analysis of biological random processes via optimized statistical models

Abstract: The concept of a sampled probability-density vector is defined. It is shown that a relationship may be established between this new estimation means and the random process, expressed by its moment vector. This is a linear transformation using invariant matrices i.e. matrices which are independent of the random process. Thus, in deriving biological probability-density models, instead of using an analytical model, to estimate their parameters and to check the distributional assumptions, a single probability-density vector is computed, subject to some constraints. An optimized statistical model is, thus, obtained, by minimizing a certain loss function, which expresses the inaccuracy of the model. The invariant matrices permitting to obtain the optimized model, starting from the moment vector, are given and the procedure is illustrated by examples. Then, the concept of a parametric probability-density space is defined and it is shown that each of the vectors belonging to this space may express the stationary, ergodic, random process equally well. Some typical constraints in the probability-density space are investigated. It is shown that the normal (Gaussian) law may be regarded as a very strong constraint in the probability-density space, while the integral law, expressing the cumulative distribution function, is a weak one. Between these extreme cases, the large class of the usual constraints are examined, which are determined by the prior knowledge of the process, as well as by some desired model features. Thus, the concept of a constrained probability-density vector is introduced. By using a linear-programming procedure and by observing some peak constraints as well as some slope-sign ones, an optimized model with desired shape is obtained, where a certain value of the variable has a very high probability. This leads to a procedure which enables synaptic models to be derived. In such a model, the constraints in the probability-density space may be regarded as a new expression of the information transmitted in the nervous system. Moreover, the loss function may express the “aptitude” of the random process to realize a given message. Thus, by using the optimized statistical model concept, probabilistic models with desired features for various biological processes may be obtained in a simple and general manner.

Computation of the central and noncentral f distributions

Abstract: Recursion relations suitable for rapid computation are derived for the cumulative distribution of F′ = (X/m)/(Y/n) where X is χ2(λ, m) and Y is independently χ2(n). When n is even no complicated function evaluations are needed. For n odd, a special doubly noncentral t distribution is needed to start the computation. Series representations for this t distribution are given with rigorous bounds on truncation errors. Proper recursion techniques for numerical evaluation of the special functions are given.

Reliability of Slender Columns: Comparison of Different Approximations

Abstract: Following the general lines set forth in [1] for the introduction of probabilistic concepts into the practical design of imperfection-sensitive structures, numerical results are presented pertaining to the probability of failure, under random axial load, of an elasto-plastic column studied earlier under given load [2]. Several possible approximate procedures to reduce the needed amount of calculations are proposed and discussed.

Equitable quality level and error-areas under the operating characteristic curves of normal single sampling inspection plans (with σ known)

Abstract: This article assumes a continuous prior distribution for lot quality in the case of a normal single sampling inspection plan with known standard deviation. The definition of Equitable Quality Level (EQL) as given in this paper ensures that the proportion of lots of quality better than the EQL—as obtained under the prior distribution—is equal to the average probability of acceptance. The first kind of error-area at the quality levelp′ is the joint probability of producing a lot of quality equal to or better thanp′ and getting such a lot rejected by the plan whereas the second kind of error-area is the joint probability of producing a lot of quality worse thanp′ and getting it accepted. Certain measures of producer's and consumer's risks can therefore be defined in terms of error-areas. It is noted that the OC can be viewed as the upper cumulative distribution function of a hypothetical random variableY. It is shown that the EQL and the error-areas can be expressed in terms of the derivatives of the prior distribution and the moments ofY. The latter do not depend on the prior distribution. It is hinted how this technique can be used to construct plans having certain optimum properties and also to obtain approximations to compound distribution. The case of a normal prior distribution is fully dealt with.

On the exact calculation of cumulative F probabilities

Abstract: Various approximate procedures have been employed in the tabulation of F distributions. These approximations generally perform poorly for small degrees of freedom and/or for extreme tail areas. Closed form expressions are presented in this paper for the exact calculation of cumulative probabilities of F distributions. For small degrees of freedom, the probabilities may be calculated with the aid of standard mathematical tables and a calculator. An APL computer program, also presented in this paper, may be used in calculating the cumulative probabilities according to the exact expressions.

Estimation of the Multi-Dimensional Probability Density Function.

Abstract: : Suppose X1,X2,... are independent, identically distributed (i.i.d.) (R sup d) valued random variables with common probability density function f. A problem of considerable practical importance and also of theoretical interest is the estimation of f through some statistic based on the observed sequence (X sub k). Such statistics are called empirical density functions, and the author examines the rate of convergence of the empirical densities to f. The author also considers the situation when there is 'noise' in the observations (X sub k).

An Alternative Approach to the Specification of Structural Transition Functions

Abstract: This paper forwards an alternative approach of characterizing the adjustment trajectory by a cumulative probability distribution function. Estimates of Wilton's (1975) automobile import equation over the period 1984-1973 are obtained in the case where a binomial CDF is used. Confirming Wilton's results, the use of transition functions to characterize coefficient adjustment between two regimes leads to substantial improvements over standard OLS estimation. Also the CDF transition estimates compare very favourably with the polynomial results, at the same time yielding more reliable coefficient transition paths and an improved framework for more precise hypothesis testing of structural change.

Uniform Convergence Of An Empirical Cdf Based On A Relatively Small Number Of Order Statistics

Abstract: An empirical distribution function Fm, defined on a subset of order statistics of a random sample of size n taken from the distribution of a random variable with continuous distribution function F, is shown to converge uniformly with probability one to F. Small sample distributions of the one and two sided deviations and the asymptotic normality of the standardized Fm are established. The relative efficiency of Fm as compared to the classical empirical distribution function is calculated and tabled. for n = 10, 20, 50, 100, 200.

Measurements of the stochastic nature of atmospheric spectral amplitudes

Abstract: The stochastic nature of the power spectral amplitudes of the neutral atmospheric boundary layer is examined Probability density distributions and probability distributions of longitudinal and lateral power spectra amplitudes are computed from neutral atmospheric boundary layers The statistical distributions are computed for frequencies of 0006, 001, 003, 006, 01, and 05 Hz at each of the elevations of 18, 30, 60, 90, 120, and 150 m When the probability density distributions are properly nondimensionalized, the data tend to collapse to a universal curve An empirical curve fit to the universal nondimensionalized probability density distribution is also given Probability distributions of individual frequency power spectral amplitudes are also presented for all elevations and frequencies An interesting observation from the data is that greater than 10 percent of the time the power spectral amplitude at a given frequency will genrally be more than three times the temporal mean value computed by standard Fourier techniques The standard power spectral density curves are also included in the report