Showing papers on "Cumulative distribution function published in 1970"

A Characterization Based on the Absolute Difference of Two I. I. D. Random Variables

Abstract: : Let X be a nonnegative random variable with X sub 1 and X sub 2 as its two independent copies. The problem considered here is to characterize all the nonnegative distributions with the property that the distribution of the absolute difference /(X sub 1)-(X sub 2)/ is the same as that of X. It is shown that in general such a distribution has to be either purely discrete, or absolute continuous or singular and that it cannot be their mixture.

A Consistent Nonparametric Multivariate Density Estimator Based on Statistically Equivalent Blocks

Abstract: Let $x_1, x_2, \cdots, x_m$ be a random sample from a $p$-dimensional random variable $X = (X_1, X_2, \cdots, X_p)$ with probability distribution $P$. It is assumed that $P$ is absolutely continuous with respect to Lebesgue measure, and that the corresponding probability density function is denoted by $f$. If $z = (z_1, z_2, \cdots, z_p)$ is a point at which $f$ is both continuous and positive, an estimator for $f(z)$ based on statistically equivalent blocks is suggested and its consistency is shown. This estimator grew out of work on the nonparametric discrimination problem. Fix and Hodges [2] showed how density estimation could be used in this problem and demonstrated a consistent estimator at points such as $z$. Loftsgaarden and Quesenberry [4] proposed another estimator which is consistent at points such as $z$; their estimator was based on statistically equivalent blocks. Although this estimator is easier to use in practice than that suggested by Fix and Hodges, it does require separate calculations if the sample is to be used to estimate the density at two or more points, and gives complex regions on which the estimate is constant if it is desired to estimate $f$ on some subset of the entire space. The estimator suggested in this paper is consistent at all points at which the two above estimators are consistent and allows the investigator to estimate the density at every point of $p$-dimensional Euclidean space from one construction, as well as providing rectangular regions on which the estimate is constant.

Alternate forms for numerical evaluation of cumulative probability distributions directly from characteristic functions

Abstract: Alternate integral forms for the cumulative probability distribution in terms of the characteristic function are given. In particular, forms that can utilize a fast Fourier transform algorithm and special forms for one-sided probability density functions are derived. For a special class of discrete random variables, all integral evaluations are over a finite range.

General random number generator [G5]

Abstract: We now prove separately the inequality for every term of the summation. Let us assume the thesis is wrong > ([(XLk+:-Xi,k) 2 + (yLk+:-yl.~)~] t \"Jr [(X2.k+x-X2,k) 2 + (y2,k+l-Y2,k)2]t). which is contradictory. We can have equality to zero iff i.e. if all the corresponding sides of the two polygons are parallel. (e, d) We must prove that the minimal polygon has at least one vertex on the boundary of the corresponding domain. In fact, consider a minimal polygon: there will be at least a couple of adjaeent noncollinear sides. The corresponding vertex must lie on the boundary, otherwise a shorter perimeter polygon could be found (see Figure 7 (a), (b)). Furthermore, from simple geometrical considerations it can be deduced that the normal to the boundary must bisect the angle between the two adjacent sides (Figure 7 (b)). Conversely, if a constraint is not active, i.e. a vertex is not on the boundary, the adjacent sides in the minimal polygon must be collinear. (e) From the convexity properties of the domain D and of the function f it can be deduced that: (i) all local minima are global; (ii) any convex linear combination of minima is a minimum as well; (iii) Jensen's relation (3) applied to two minima obviously holds with equality. We have proved in (b) that in our ease Jensen's relation holds with equality if and only if the corresponding sides of the two polygons are parallel. Now assume to have found a (global) minimum: any vertex between two noncollineax sides will satisfy the bisection property proved in (d). If no straight line segments axe present on the boundary of domain C (e.g. if domain C is a circle), it is not possible to translate two adjacent noncollinear sides still satisfying the bisection property. Therefore, all minimal polygons have the vertices between two noncollinear sides in the same position. Thus they differ only in the position of vertices corresponding to nonactive constraints: the minimal reduced polygon is unique. * The work forms part of a research program supported by the Bundesministerium fiir wissenschaftliche Forsehung and the Fritz ter Meer-Stiftung. c o m m e n t If a Laplace transform P(s) is given in the form of a real procedure, L/nv produces an approximate value Fa of the inverse F(t) at T. Fa is evaluated according to Fa =-~-~=: N must be even. Since the V~ depend on …

The exact distribution of Votaw's criteria

Abstract: where 2\"1 is a p x p matr ix with diagonal elements equal to ~a~ and other elements to ~ , , X2 is a p x q matr ix with all elements equal to z~b and X3 is a q• matr ix with diagonal elements a~b and other elements ~bb'. Votaw [8] used Wilks' [10] method and derived the t-th moment of L, tha t is E(Lg, under the hypothesis H. Roy [6] has shown that , under H and when a simple random sample of size n is taken from N(#, X),

Consideration of power for some two sample tests with censoring based on a given order statistic

Abstract: SUMMARY This paper deals with some properties of three nonparametric tests which can be used for detecting a difference in location of two populations wlhen samples are censored on the occurrence of a given order statistic. The null distributions of the test statistics are presented together with some tables of critical values. Exact expressions are obtained for the powers of the tests under exponential and rectangular alternatives. Finally, the expected durations of the tests are compared. A number of tests which permit early termination of an experiment have been proposed for detecting a difference in the location of two populations. These tests have important applications in life-testing situations where testing time may be costly. The criterion for termination may be a preselected period of time from the start of the experiment or the occurrence of a given order statistic in one of the samples or the two samples combined. In this paper we consider three nonparametric tests which may be used to provide a one-sided test for difference in location of two populations. The common feature of the tests is that the experiment is terminated on or before the occurrence of a preselected order statistic in one of the samples. We first consider the distributions of the test statistics under the null hypothesis that the two samples have been drawn randomly from populations with identical cumulative distribution functions. Some tables of critical values are presented. Expressions are given for the test powers under the alternative of a translation in an exponential distribution and for changes in the location and scale parameters of a rectangular distribution. Finally some results are given for the expected durations of the tests.

Evaluation of cumulative probability distribution functions: Improved numerical methods

Abstract: An improved technique for computing the cumulative probability distribution function from the characteristic function is presented. Rapid convergence of the infinite integral defining the cumulative function is achieved by a nonlinear approximation.

An Indicator of the Reliability of Analytical Structural Design

Abstract: A statistical analysis of structural static test failure data for major components of aircraft is presented. The data sample is divided into the static test failures of wings, fuselage, horizontal stabilizer, vertical stabilizer, and landing gear. The analysis results in the approximate determination of the specific statistical strength density function and cumulative distribution as a function of the rupture strength. Finally, the required factor of safety is computed for the "no static test" or analytical design case for the aforementioned components over a wide range of unreliabilities. It is concluded that components require factors of safety for the no static test or analytical design approximately an order of magnitude above the usual standard and would be prohibitive if implemented. Nomenclature a — statistical parameter in truncated Weibull distribution b = statistical parameter in truncated Weibull distribution F — function m = inverse of the factor of safety P = probability distribution function R = reliability S = strength x — strength or load variable 7 = (= cr/V) scatter coefficient or coefficient of variation = normal probability density function <£ = normal cumulative probability function

Estimating the Parameters of a Linear Function of a Random Variable

Abstract: : Suppose we have a sample of independent observations which are values of a linear function of a random variable, and we wish to find maximum likelihood estimators for the two parameters of the linear function. If the support of the random variable is the whole real axis, the maximum likelihood estimators can be found by the usual methods; in other cases restricted maximization techniques must be used. We restricted attention to the cases in which the support is an infinite interval or a lattice and the density of the random variable is non-increasing over the support. Special attention is given to the exponential and geometric distributions. If the parameter of the distribution is also unknown, its maximum likelihood estimator can also be obtained, when the support of the random variable is a lattice. (Author)

Alternate Forms and Computational Considerations for Numerical Evaluation of Cumulative Probability Distributions Directly from Characteristic Functions

Abstract: : Alternate integral forms for the cumulative probability distribution in terms of the characteristic function are given. In particular, forms that can utilize a fast Fourier transform (FFT) algorithm and special forms for one-sided probability density functions are derived. For a special class of discrete random variables, all integral evaluations are over a finite range. Some computational aspects of utilizing the FFT are discussed.

Some nonparametric consistent estimates from censored samples

Abstract: In some cases it happens that only the smallest observations in each sample can be obtained (cf. David [2]). When considering the cost or the time of an experiment it also happens in some cases including life test ing tha t an observation of the minimum in a small sample can be obtained more easily than, or, at least, as easily as, an observation of a sample of size one (cf. Takahasi and Wakimoto [7]). David [2] considered the estimation of means of normal populations from n observations each of which is the minimum of m independent N(~, s variates. In this paper we shall give a method of obtaining nonparametr ic consistent estimates for some problems of estimation from n observations each of which is the kth least order statistic of a sample of size m from a population. Some properties of the estimates obtained by this method are discussed in Sections 3, 4 and 5 for the problems of estimation of the cumulative distribution, mean and quantile of populations, respectively.

Methods of Randomization of Large Files with High Volatility

Abstract: Key-to-address conversion algorithms which have been used for a large, direct access file are compared with respect to record density and access time. Cumulative distribution functions are plotted to demonstrate the distribution of addresses generated by each method. The long-standing practice of counting address collisions is shown to be less valuable in fudging algorithm effectiveness than considering the maximum number of contiguously occupied file locations.

The choice of boundary conditions in Monte Carlo calculations of noise

Abstract: Probability density functions and the corresponding cumulative distribution functions for density and Velocity fluctuations at the surface of a thermionic emitter are discussed. In particular, it is shown that as far as velocity fluctuations are concerned, the appropriate probability density function is given by f(w | r_{c}) = (2 / \pi^{3/2}) \exp (-w^{2}) , where w is the reduced velocity. Some of the misunderstandings encountered in the technical literature are also discussed. It is shown that the mean energy of the emitted electrons is 3/2kT and not 2kT, as is claimed sometimes, the quantity 2kTJ s , where J s , is the saturation current, representing the magnitude of energy flow.