Showing papers on "Probability distribution published in 1979"

Bootstrap Methods: Another Look at the Jackknife

Abstract: We discuss the following problem given a random sample X = (X 1, X 2,…, X n) from an unknown probability distribution F, estimate the sampling distribution of some prespecified random variable R(X, F), on the basis of the observed data x. (Standard jackknife theory gives an approximate mean and variance in the case R(X, F) = \(\theta \left( {\hat F} \right) - \theta \left( F \right)\), θ some parameter of interest.) A general method, called the “bootstrap”, is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.

Probability and Measure

Abstract: Probability. Measure. Integration. Random Variables and Expected Values. Convergence of Distributions. Derivatives and Conditional Probability. Stochastic Processes. Appendix. Notes on the Problems. Bibliography. List of Symbols. Index.

A versatile Markovian point process

Abstract: : A versatile class is introduced of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper. Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

Production Function Estimation and Related Risk Considerations

Abstract: There has been considerable interest in estimations of input effects on the probability distribution of output. Most empirical and theoretical analyses utilize multiplicative stochastic specifications which are analyzed and found unduly restrictive, particularly since inputs that marginally reduce risk are not allowed. A more general stochastic specification is proposed, free of these a priori restrictions. The proposed functional form estimation is discussed and demonstrated with nitrogen‐response data and common log‐linear production functions. Though nitrogen is risk‐increasing, the marginal variance contribution is smaller when compared to estimates based upon multiplicative specification. Finally, stochastic specification error effects are analyzed.

A Probability Distribution and its Uses in Fitting Data

Abstract: A four-parameter probability distribution, which includes a wide variety of curve shapes, is presented. Because of the flexibility, generality, and simplicity of the distribution, it is useful in the representation of data when the underlying model is unknown. A table based on the first four moments, which simplifies parameter estimation, is given. Further important applications of the distribution include the modeling and subsequent generation of random variates for simulation studies and Monte Carlo sampling studies of the robustness of statistical procedures.

Quadratic limit states in structural reliability

Abstract: Second-moment methods are widely applied in structural reliability. Recently, so-called first-order reliability methods have been developed that are capable of producing reliable estimates of the failure probability for arbitrary design situations and distributional assumptions for the uncertainity vector. In essence, nonlinear functional relationships or probability distribution transformations are approximated by linear Taylor expansions so that the simple second-moment calculus is retained. Failure probabilities are obtained by evaluating the standard normal integral, which is the probability content of a circular normal distribution in a domain bounded by a hyperplane. In this paper second-order expansions are studied to approximate the failure surface and some results of the statistical theory of quadratic forms in normal variates are used to calculate improved estimates of the failure probability.

Reception Through Nakagami Fading Multipath Channels with Random Delays

Abstract: The Nakagami fading distribution is shown to fit empirical results more generally than other distributions. A statistical model for a noisy, Nakagami fading multipath channel is given, following Turin's delay-line model. Optimal receivers are derived for two states of knowledge of the channel-known path delays and random path delays. Upper bounds on the probability of error are computed, for binary, equal-energy, equiprobable signals, which are uniformly orthogonal and have equal, triangular, autocorrelation moduli. Results are graphically displayed and show the dependence of error probability on number of paths, amount of fading and spread of path delays.

''Relaxation'' of initial probability density functions in the turbulent convection of scalar fields

Abstract: The evolution of an initially binary (zero unity) scalar field undergoing turbulent and molecular mixing is studied in terms of conservation equations for the probability density function of the scalar property. Attention is focused on the relaxation of the dynamic system to a state independent of the intial conditions. A few existing methods are discussed and evaluated and a new mechanistic model is proposed. Classical iteration techniques are used to obtain an equation for the single point probability density and the unperturbed Green’s function. It is suggested that use of the true Green’s function or perturbed propagator of the system might be necessary in order to obtain the correct evolution of the probability density function.

Asymptotic properties of Burgers turbulence

Abstract: The asymptotic properties of Burgers turbulence at extremely large Reynolds numbers and times are investigated by analysing the exact solution of the Burgers equation, which takes the form of a series of triangular shocks in this situation. The initial probability distribution for the velocity u is assumed to decrease exponentially as u → ∞. The probability distribution functions for the strength and the advance velocity of shocks and the distance between two shocks are obtained and the velocity correlation and the energy spectrum function are derived from these distribution functions. It is proved that the asymptotic properties of turbulence change qualitatively according as the value of the integral scale of the velocity correlation function J, which is invariant in time, is zero, finite or infinite. The turbulent energy per unit length is shown to decay in time t as t−1 (with possible logarithmic corrections) or according as J = 0 or J ≠ 0.

Locations of Medians on Stochastic Networks

Abstract: The definition of network medians is extended to the case where travel times on network links are random variables with known discrete probability distributions. Under a particular set of assumptions, it is shown that the well known “facilities at nodes” theorems of Hakimi and of Levy can be extended to such stochastic networks. The concepts are further extended to the case of stochastic oriented networks. A particular set of applications, as well as mathematical programming formulations of the problem, are also discussed briefly.

Absolute continuity and singularity of locally absolutely continuous probability distributions. i

Abstract: The second part of this study deals with criteria for the absolute continuity of measures for various classes of random processes, starting out from the general results in Part I. We consider processes with independent increments, semimartingales, multivariate point processes, Gaussian processes, Markov chains, and processes with a countable number of states. Bibliography: 33 titles.

A Fuzzy Safety Measure

Abstract: In deciding the chances of failure of a design, two types of information are available: objective and subjective. In dealing with objective information in the form of countable data, unbiased probability distributions of loading and structural capacity are obtained by employing the principle of maximum entropy. Subjective information is considered in a linguistic form and requires a transformation from the verbal to the numerical. This is attained by the use of fuzzy set theory. The fuzzy calculus is then employed to fuzzify the objective probability in light of the subjective insight. The result is a fuzzy safety measure that expresses the confidence in the objective probability and also in other, larger values of the probability of failure. In this way objective, numerical, frequency type information can be augmented by professional insight and experience.

Computational methods for solving two-stage stochastic linear programming problems

Abstract: Approximating a given continuous probability distribution of the data of a linear program by a discrete one yields solution methods for the stochastic linear programming problem with complete fixed recourse. For a procedure along the lines of [8], the reduction of the computational amount of work compared to the usual revised simplex method is figured out. Furthermore, an alternative method is proposed, where by refining particular discrete distributions the optimal value is approximated.

Conjecture on the exact transition point of the random Ising ferromagnet

Abstract: Presents a conjecture on the exact value of the transition point of the random Ising ferromagnet on the square lattice. Only the bond problem is treated; each neighbouring pair of spins has an interaction of strength J(>or=0), where J is a random variable with an arbitrarily given probability distribution. To solve the problem an exact duality transformation and the replica method are used. In order to deduce the transition point from the duality relation only it is necessary to make an unconfirmed assumption concerning the symmetry of the distribution of singularities of the free energy.

The Application of Weibull Statistics to Insulation Aging Tests

Abstract: The results of accelerated aging tests on solid electrical insulation are difficult to evaluate objectively, primarily due to the inherently large variability of the test data. This variability is often represented by the Weibull or other extreme-value probability distributions. This paper demonstrates an hypothesis test procedure which permits the objective and unambiguous evaluation of comparative dielectric tests on two different sets of data. The computation techniques are facilitated through the use of a Fortran computer program. A significant difference must be established at low probabilities of failure. Analysis of typical aging tests from the literature indicate that many experiments performed to date may not be statistically significant at utilization levels. The number of tests required to achieve unambiguous significance at low probability levels may render meaningful accelerated aging tests uneconomic.

Some Analysis Techniques for Asynchronous Multiprocessor Algorithms

Abstract: Efficient algorithms for asynchronous multiprocessor systems must achieve a balance between low process communication and high adaptability to variations in process speed. Algorithms that employ problem decomposition may be classified as static (in which decomposition takes place before execution) and dynamic (in which decomposition takes place during execution). Static and dynamic algorithms are particularly suited for low process communication and high adaptability, respectively. For static algorithms the following analysis techniques are presented: finding the probability distribution of execution time, deriving bounds on mean execution time using order statistics, finding asymptotic mean speedup, and using approximations. For dynamic algorithms the technique of modeling using a queueing system is presented. For each technique, an example application to parallel sorting is given.

Memoryless discrete-time detection of a constant signal in m-dependent noise

Abstract: The problem of designing memoryless detectors for known signals in stationary m-dependent noise processes is considered. Applying the criterion of asymptotic relative efficiency, the optimal such detector is shown to be characterized by the solution to a Fredholm integral equation whose kernel depends only on the second-order probability distributions of the noise. General expressions are derived for this solution and for the asymptotic efficiency of the optimal detector relative to other memoryless detectors. To illustrate the analysis, specific results are given for the particular case where the noise process is derived by memoryless nonlinear transformation of a Gaussian process. In addition, an extension of the analytical results to the more general case of \phi -mixing noise processes is discussed.

Calculation of Univariate and Bivariate Normal Probability Functions

Abstract: Mill's ratio is expressed as a convergent series in orthogonal polynomials. Truncation of the series provides an approximation for the complemented normal distribution function $Q(x)$, with its maximum error at a finite value of $x$. The analogous approximation for $xQ(x)$ is used to obtain a new method of calculating the bivariate normal probability function.

A Framework For Probabilistic Vehicle Routing

Abstract: The stochastic vehicle routing problem is a problem of considerable current importance in operations research and industrial engineering. The basic problem is to determine a fixed set of vehicle routes of minimal expected total distance. All vehicles must leave from and eventually return to a central depot, and capacity constraints and probabilistic customer demands must be satisfied. In previous work, we assumed that the demand at each node i could be modeled by a Poisson distribution with mean Λi and that demands at nodes are mutually independent. We then developed an efficient heuristic solution procedure which was quite effective in generating an excellent set of fixed vehicle routes, as evidenced by extensive computational results. With this previous work as a starting point, in this paper we investigate solution procedures for the case where other appropriate probability distributions are assumed. In addition, we present analytical results describing the various relationships between design...

A Rational Method of Determining Probability Distributions in Turbulent Reacting Flows

Abstract: A method is presented for determining the joint probability distribution of scalar quantities in turbulent flows. Given a limited amount of statistical information — all mean values and second moments, say — the most likely distribution is shown to maximize the entropy, H. Consequently, the calculus of variations can be used to determine the most likely (i. e., maximum-entropy) distribution that is compatible with the given information. H is a function of the a priori probability which, it is shown, is uniform for passive scalars. In general, the a priori probability is a function of the reaction rates, and this functional dependence is determined. Introduction Local-mean properties in a turbulent reacting flow can be determined from the mean equations representing the conservation of mass, momentum, energy and chemical species concentration. In these equations, terms arise which are not directly related to the mean values and must be modelled, therefore, in order to provide a determinate set of equations. These terms can be considered in two groups, the first comprising velocity-velocity correlations (Reynolds stresses) and velocity-energy or velocity-species correlations (scalar fluxes). These correlations represent the transport of the quantity in question due to turbulent velocity fluctuations and are non-zero even in inert flows. Modelled transport equations for the Reynolds stresses have been proposed by Launder, Reece and Rodi [1] and Lumley and Kajeh-Nouri [2], while modelled scalar-flux equations are discussed by Launder [3]. Although these equations are not completely satisfactory there is no difficulty in principle in using this approach to determine the velocity correlations responsible for turbulent transport (at least in reasonably simple flows). The second group of terms comprises the mean reaction rates of chemical species and, clearly, does not occur in inert flows. Only in the special case of a constant rate (i. e. isothermal) reaction can these terms be represented directly as correlations of the species concentrations and so, in general, an alternative statistical representation 0340-0204/79/0340-0320$02.00 ©Copyright by Walter de Gruyter & Co. · Berlin · New York

Probabilistic PERT

Abstract: A solution is offered to the problem of determining a probability distribution for the length of the longest path from source (start) to sink (finish) in an arbitrary PERT network (directed acyclic graph), as well as determining associated probabilities that the various paths are critical ("bottleneck probabilities"). It is assumed that the durations of delays encountered at a node are random variables having known but arbitrary probability distributions with finite expected values. The solution offered is, in a certain sense, a worst-case bound over all possible joint distributions of delays for given marginal distributions for delays. This research was motivated by the engineering problem of the timing analysis of computer hardware logic block graphs where randomness in circuit delay is associated with manufucturing variations. The probability distribution of the critical pathlength turns out to be a solution of an unconstrained minimization problem, which can be recast as a convex programming problem with linear constraints. The probability that a given path is critical turns out to be the Lagrange multiplier associated with the constraint determined by the path. The discrete version of the problem can be solved numerically by means of various parametric linear programming formulations, in particular by one which is effciently solved by Fulkerson's network flow algorithm for project cost curves.

Composite stochastic processes

Abstract: Certain problems in physics and chemistry lead to the definition of a class of stochastic processes. Although they are not Markovian they can be treated explicitly to some extent. In particular, the probability distribution for large times can be found. It is shown to obey a master equation. This demonstrates how a non-Markovian process can be approximated on a course time scale by a Markov description. The conditions for this approximation to be valid are discussed.

Necessary and Sufficient Conditions for Optimal Search Plans for Moving Targets

Abstract: Let {Xt, t ≥ 0} be a stochastic process representing the motion of a target in Euclidean n-space. Search effort is applied at the rate mt > 0 for 0 ≤ t ≤ T, and it is assumed that this effort can be distributed as finely as desired over space. We seek an optimal search plan, i.e., an allocation of effort in time and space which maximizes the probability of detecting the target by time T. When the detection function is concave, theorem 1' of this paper gives necessary and sufficient conditions for an optimal search plan for a class of target motion processes {Xt, t ≥ 0} which includes virtually any reasonable model of target motion. In the special case of a discrete-time target motion process and an exponential detection function, the necessary and sufficient conditions have the following intuitive interpretation: For t = 0, 1, 2,..., T, let gIƒt, be the probability distribution of the target's location at time t given that the effort at all times before and after t failed to detect the target. The optimal plan allocates the effort for time t so as to maximize the probability of detecting a stationary target with distribution gIƒt within the effort constraint mt. This special case is a generalization of Brown's Brown, S. S. 1978. Optimal search for a moving target in discrete space and time. Submitted for publication. result for discrete time and space target motion.

Distribution-free regression analysis of grouped survival data.

Abstract: Summary Methods based on regression modelsfor logarithmic hazardfunctions, Cox models, are given for analysis of grouped and censored survival data. By making an approximation it is possible to obtain explicitly a maximum likelihood function involving only the regression parameters. This likelihood function is a convenient analog to Cox's partial likelihood for ungrouped data. The method is applied to data from a toxicological experiment. Methods are given here for relating the probability distribution of response times to one or more explanatory variables, or classifications, without assuming a parametric model for the form of the distributions. The analysis is very similar to that suggested by Cox (1972), but is specifically intended for situations in which there is substantial grouping of the response times into class intervals. Related papers on this subject are mentioned in Section 2. Even when the data are available as essentially exact response times, the analysis suggested here may be quite attractive, using grouping to simplify computations and data handling. Such grouping can also be a substantial aid in diagnosis of model inadequacies, that is, testing goodness-of-fit. Although it is not completely crucial to the basic approach here, we will also assume that the data are grouped according to values of the explanatory variables. For example, the observational units may be cross-classified according to levels of several qualitative or quantitative factors. In this case the regression model is employed in essentially the standard manner, with parameters corresponding to main effects and possibly interactions. If it is considered crucial to use covariables without grouping on their values, then some modification of the method will be required, as pointed out in Section 3. By grouping on both response time and explanatory variables, it is of course possible to analyze extensive data sets quite easily. The calculations for the example below were done on a desk-top microcomputer. Also, such grouping reduces the problem to one involving a finite number of parameters, a product-multinomial model, and thus the model might be called "parametric." Nevertheless, because of the nature of the development of the model, it

The helical-scan magnetic tape recorder as a digital communication channel

Abstract: The properties of a magnetic tape recorder are viewed in terms of a digital magnetic recording/playback channel which exhibits fading (reduction of playback-signal level) and nonlinear behavior. A method is presented whereby channel nonlinearity may be quantified in a format useful for signal and receiver design. Measurements show this nonlinearity to be relatively small for symmetric two-level signals. Deep fades (dropouts) are the most significant source of errors in digital tape recordings. Fading is considered as multiplicative noise on an essentially linear channel, and measurements are made of the fade probability distribution and an associated additional time dispersion. While the fading process appears to occur relatively slowly (compared with the bit period), neither its probability distribution nor its associated dispersion encourage the use of a receiver which is able to adapt to the changing channel characteristics. Finally an attempt is made to ascribe the fading process to repeatable variations in head-tape separation, and a corresponding probability distribution for this separation is obtained. A helical-scan video recorder was used throughout the measurements because of its low cost and its potential as a high-density storage facility.

A Random Walk Procedure for Texture Discrimination

Abstract: We consider the problem of texture discrimination Random walks are performed in a plain domain D bounded by an absorbing boundary ? and the absorption distribution is calculated Measurements derived from such distributions are the features used for discrimination Both problems of texture discrimination and edge segment detection can be solved using the same random walk approach The border distributions and their differences with respect to a homogeneous image can classify two different images as having similar or dissimilar textures The existence of an edge segment is concluded if the boundary distribution for a given window (subimage) differs significantly from the boundary distribution for a homogeneous (uniform grey level) window The random walk procedure has been implemented and results of texture discrimination are shown A comparison is made between results obtained using the random walk approach and the first-or second-order statistics, respectively The random walk procedure is intended mainly for the texture discrimination problem, and its possible application to the edge detection problem (as shown in this paper) is just a by-product

Toward self-organizing linear search

Abstract: We consider techniques for adapting linear lists so that the more frequently accessed elements are found near the front, even though we are not told the probabilities of various elements being accessed. The main results are discussed in two sections. Perhaps the most interesting deals with techniques which move an element toward the front only after it has been requested k times in a row. The other, technically more difficult, section deals with the analysis of the heuristic which moves an element to the head of the list each time it is accessed. The behaviour of this scheme under a number of interesting probability distributions is discussed. Two basic approaches to the technique of moving an element forward after it has been accessed k times in a row are discussed. The first performs the transformation after any k identical requests. The second essentially groups requests into batches of at least k, and performs the action only if the last k requests of a batch are the same. Adopting as the transformation, the moving of the requested element to the front of the list, the second approach is shown to lead to faster average search time under all nontrivial probability distributions for k ≥2. It is also shown that the "periodic" approach, with k = 2, never leads to an average search time greater than 1.21.. times that of the optimal ordering. For the more direct approach, a ratio of 1.36.. is shown under the same constraints. In studying the simple move to front heuristic (i.e. k = 1), it is shown that for a particular distribution this scheme can lead to an average number of probes π/2 times that of the optimal order. Within an interesting class of distributions, this is shown to be the worst average behaviour.