Showing papers on "Cumulative distribution function published in 1974"

Simplifying the Choice between Uncertain Prospects Where Preference is Nonlinear

Abstract: This work makes analytical progress in reducing or avoiding two practical difficulties in using preference or utility theory in the analysis of decisions involving uncertainty: (1) assessing the preference curve, and (2) doing calculations with the resultant curve, which may not have an analytically-convenient functional form. The paper identifies circumstances under which simplifications can be found which overcome these difficulties, while at the same time properly reflecting attitude towards risk in the analysis. It is assumed that a decision-maker must choose between risks w1 and w2. He wishes to make decisions consistent with a preference curve u(·) which exists, but has not necessarily been assessed, so he can choose i to maximize expected preference, Eu(wi). Most results require that the cumulative probability distribution of w1 and w2 cross at most once. The results are widely but not universally applicable. Situations are identified where an easy-to-assess, easy-to-analyze preference curve w...

Estimation of the Distribution Function of a Continuous Type Random Variable through Randomized Response

Abstract: In 1965 Stanley Warner [8] illustrated a technique whereby one could estimate from a sample the proportion of persons in a population possessing some characteristic X, without pointedly asking the question, “Do you possess characteristic X?”. The present article uses a variation of these ideas suggested by Warner in a later paper [7] to estimate the distribution function of a continuous-type random variable. The technique is illustrated by estimating an income distribution from the responses of a sample of 500 individuals. The potential use of devices of this type in maintaining confidentiality of existing data files is apparent.

Sparse Data, Estimational Reliability, and Risk-Efficient Decisions

Abstract: The minimal data required for a reasonable estimation of probability distributions is investigated through a Monte Carlo study of a rule for smoothing sparse data into cumulative distribution functions. In a set of estimated distributions, risky prospects not preferred by risk-averse decision makers can be identified and discarded.

A Method for the Solution of the Distribution Problem of Stochastic Linear Programming

Abstract: A method is proposed for finding a closed form expression for the cumulative distribution function (CDF) of the maximum value of the objective function in a stochastic linear programming problem in the case where either the objective function coefficients or the right-hand side coefficients are given by known probability distributions. If the objective function coefficients are random, the CDF is obtained by integrating over the space for which a given feasible basis remains optimal. A transformation is presented using the Jacobian theorem which simplifies the regions of integration and often simplifies the integration itself. A similar analysis is presented in the case where the right-hand side coefficients are random. An example is given to illustrate the technique.

Tables of finite-mean nonsymmetric stable distributions as computed from their convergent and asymptotic series †

Abstract: Stable distributions take their interest from the fact that they are the only possible limiting distributions for sums of independent identicalloy distributed random variable. The Gaussian and Cauchy distributions are the two best-known members of the family. Though stable distributions have long been known and studied, little use has been made of the non-Gaussian ones, especially those that are nonsymmetric as well, in mathematical models. One reason for this is that closed form expressions exist for only a very few stable densities. Alothough probability theory tells us much about the general behavior of the stables, particulary their asymptotic properties, to find out their specific behavior on the inner ranges involves tedious computation. In this paper we attempt to alleviate some of the problems of using finite mean nonsymmetric stables, by computing density graphs and cumulative distribution tables for them using their convergent and asymptotic series. A rough estimate of the scale parameter is als...

Normalizing transformations of Student's t distribution

Abstract: SUMMARY The accuracies of several normalizing transformations of a t distribution with v degrees of freedom are examined for large and small values of v. Expansions of the inverse transformations in terms of powers of v-' are produced and the first few terms compared with Fisher's general expansion. For small values of v the transformations are used to determine approximate percentage points of t which are then compared with the exact percentage points. The problem of transforming a Student's t variable with v degrees of freedom to a standard normal variable has received attention for a variety of reasons. Quenouille (1953, p. 235) was concerned with the combination of results from a series of experiments and suggested analysing the treatment effects using an inverse sinh transformation. This transformation is related to Fisher's normalizing transformation of the product moment correlation coefficient and is fairly good provided v is not too small. Anscombe (1950) proposed a modified form of this transformation which is suitable for smaller values of v. Chu (1956) examined the proportional errors in using the normal cumulative distribution function as an approximation to the cumulative distribution function of t and considered a square root of a logarithmic transformation. Modified versions of this were proposed by Wallace (1959) who constructed bounds on the deviation from the exact normal deviates. Moran (1966) used an empirical approach to develop a simple transformation of a particular percentage point of the t distribution and showed that the approximation is very good even for quite small values of v. Scott & Smith (1970) used the general expansion of the percentage points of the t distribution in terms of the corresponding normal percentage points, given by Fisher (1926), to develop a simple transformation similar in form to Moran's but suitable for any percentage point of t. In the following sections the accuracies of these transformations are compared for a selection of percentage points for large and small values of v.

Random collision processes with transition probabilities belonging to the same type of distribution

Abstract: By analogy with statistical mechanics we consider a random collision process with discrete time n and continuous states x e [0, oo). We assume three conditions(i),(ii)and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments. RANDOM COLLISION PROCESS; MOMENT; CHARACTERISTIC FUNCTION; KAC'S MODEL OF A MAXWELLIAN GAS

Uncertainty in Sediment Yield from a Semi-Arid Watershed

Abstract: The paper presents a stochastic model for the prediction of sediment yield in a semi -arid watershed based on rainfall data and watershed characteristics. Uncertainty stems from each of the random variables used in the model, namely, rainfall amount, storm duration, runoff, and peak flow. Soil Conservation Service formulas are used to compute the runoff and peak flow components of the Universal Soil Loss Equation. A transformation of random variables is used to obtain the distribution function of sediment yield from the joint distribution of rainfall amount and storm duration. The model has applications in the planning of reservoirs and dams where the effective lifetime of the facility may be evaluated in terms of storage capacity as well as the effects of land management on the watershed. Experimental data from the Atterbury watershed is used to calibrate the model and to evaluate uncertainties associated with our uncertain knowledge of the parameters of the joint distribution of rainfall and storm duration. INTRODUCTION The purpose of this study is threefold. First, given a deterministic model for sediment yield, appropriate random variables are defined and a transformation which gives the distribution of sediment yield is obtained. From this distribution function, using the definition of the derivative, the probability density function (pdf) of sediment yield is calculated. Thirdly, using the pdf of sediment yield and likelihood and loss functions, a Bayesian analysis may be carried out to assess the uncertainty which arises from our uncertain knowledge about the parameters of the distribution of sediment yield. The purpose of this paper is to compare the mean and variance of the data with the mean and variance of the model. We would expect higher values of those expectations with the model than those of the data for two reasons. The first is the uncertainty in the parameters of the model and the second attributable to the randomization of the deterministic model. This randomization takes into account more of the factors involved in the process than does the deterministic model and thus more accurate estimates of, say, average sediment yield can be expected. The concern here for the mean and variance is related to the fact that in the design of dams, reservoirs, and other water storage facilities, the mean annual sediment load is the only design parameter used in determining the storage capacity with respect to sediment yield over the lifetime of the project. A variety of sediment control methods exist (Task Committee on Sedimentation, 1973); however, accurate initial planning is by far one of the easiest. The authors are, respectively, Graduate Research Assistant, Systems & Industrial Engineering, Professors, Watershed Management and Systems & Industrial Engineering, University of Arizona, Tucson, Arizona 85721. 258 Assessment of sediment yield, for the most part, has been on a deterministic level. Extensive work has been done by Wischmeier (1958, 1959, 1960, 1965) in the area with the development of the Universal Soil -Loss Equation. Other methods, such as the Area -Increment and Empirical Area Reduction methods (Borland and Miller, 1958), have been proposed. However, the Universal Equation has been used effectively to compute sediment yield and found quite acceptable. In the area of probabilistic sediment modeling, Woolhiser and Todorovic (1974) have developed a stochastic model using a counting process in which sediment yield is treated as a random number of random events among other various approaches to this problem. This exposition will take the deterministic model of Wischmeier and, using a joint distribution of rainfall and storm duration, obtain the cumulative distribution of sediment yield for a single event. A sediment event is defined as occurring whenever there is a runoff -producing type storm. The model takes into account, not only the probabilistic interaction between rainfall and storm duration, but the interaction between the components of runoff and peak flow which both depend on rainfall and duration. This work parallels to some extent the work of Duckstein, Fogel and Kisiel (1972) in that the seasonal sediment yield is computed as the sum of a random number of random events mutually independent and identically distributed. From the computed values of mean sediment yield and variance of same, along with the mean and variance of the number of events per season, the mean and variance of the seasonal distribution may be calculated directly. Using a seasonal approach to naturallyoccurring events is warranted whenever welldefined events, such as sediment yield greater than some amount Z, occur relatively infrequently, so that their effects are separated by a time interval that contains the "null event." SEDIMENT MODEL The Universal Equation modified by Williams and Hann (1973) to compute the sediment yield on a per event basis is: Z = 95 (Q q) .56 K C P L S where Z = Sediment yield in tons; q = Peak flow rate in cfs; p Q = Runoff volume in acre -ft; K = Soil erodibility factor; C = Cropping -management factor; P = Erosion control practice factor; LS = Slope length and gradient factor Values for K, C, P, L, and S may be computed using the algorithms outlined by Williams and Berndt (1972). Due to the form of the data available, a conversion constant (to be defined later) was introduced to convert Z from tons to cubic feet and Q in inches to acre -feet. The values of Q and q are computed from the Soil Conservation Service formulas: p 2 xi Q (x 1 S) (inches) (2) where x1 = Effective rainfall in inches [rainfall less a constant initial abstraction]; S = Watershed infiltration constant; 259

A Statistical Estimator in a Problem of Stochastic Dominance

Abstract: The assumed situation is as follows: two cumulative distribution functions F and G are in constant unknown ratio θ, for values of the random variables below t* assumed known; F and G may behave entirely independently above t*. Observations on a sample from F are then used to obtain unbiased maximum likelihood estimates of and of other unknown parameters of G, denoted by λ. The estimators have application in problems of stochastic dominance in the context of portfolio analysis and in other situations of intertwining cumulative probability distributions.

A Note on a New Method of Histogram Sampling

Abstract: THIS is to report an obvious (once one has thought of it) improvement in speed of histogram sampling for simulations that we have recently introduced, which has approximately doubled sampling speed and in one instance reduced run times from 11 to 8 min, although some loss of accuracy is entailed. When sampling on a computer from a standard distribution the cumulative probability density function is considered.' With histograms, sampling is normally performed by multiplying the total cumulative count by a positive random fraction and seeing in which cell of the histogram this cumulative count occurs. This procedure involves searching the histogram array, whereas our new method of sampling using the inverse cumulative histogram requires only two accesses of the histogram array and thus is much quicker. The histogram is initially converted to a cumulative distribution function Y = F(X) assuming a uniform distribution within the cells of the original histogram. The values of X for a uniform series of values of Y are stored in an array, called the inverse cumulative array. For illustration in this note assume that the array has 100 elements, the first element gives the cell value of the first percentile point of the distribution and so forth. A random fraction is obtained, multiplied by 100 and the integer part M gives the appropriate index in the inverse cumulative array. The fractional part of the random number N then gives the position in this inverse cumulative cell assuming a uniform distribution within the cell, so the sample value returned is Xm + (Xm+, Xm)*N which approximates to the original histogram. More elaborate interpolation both for the initial transformation and subsequent sampling is obviously possible. The transformation to the array Xi is non-reversible, except for special cases, such as where the original histogram is a uniform distribution, since information about the original histogram is lost in the transformation. The histogram built up by sampling in this manner from the inverted cumulative array consists of 100 rectangular blocks of varying height and width, but uniform area. Thus tails on the original histogram are reproduced as long low rectangles, introducing an undesirable bias and for this reason this method is best avoided if the original histogram has a long low tail. It is sensible to arrange that no samples are returned for cells outside the main body of the original histogram. Cells that were empty within the main body of the original

Reliability Bounds for Coherent Communications with Maximum Likelihood Phase Estimation

Abstract: This concise paper describes a new technique for upper bounding the performance of coherent binary phase-shift keyed (PSK) systems which derive a phase reference using one-shot maximum likelihood phase estimates. By performance, we mean any positive monotonically increasing function of phase error, such as error probability or power loss. A tight upper bound to the phase error cumulative distribution is shown to consist of the concave support to a previously derived Chernov bound. Performance measures appropriate for binary PSK systems are listed and evaluated with the new bound.

Bayesian Confidence Limits for the Reliability of Mixed Cascade and Parallel Independent Exponential Subsystems

Abstract: This paper deals with the theoretical problem of derving Bayesian confidence intervals for the reliability of a system consisting of both cascade and parallel subsystems where each subsystem is independent and has an exponential failure probability density function (pdf). This approach is applicable when test data are available for each individual subsystem and not for the enfire system. The Mellin integral transform is used to analyze the system in a step-by-step procedure until the posterior pdf of the system reliability is obtained. The posterior cumulative distribution function is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the accuracy of the derived pdf. A computer program has been written in FORTRAN IV to evaluate the confidence limits. An example is presented which uses the computer program.

Empirical Bayes approach to statistical estimation in the Paretian law

Abstract: The two-parameter Pareto distribution of a random variable (r.v.) X with probability density function (p.d.f.)

Computer generation of random variates

Abstract: Publisher Summary The generation of observations on random variables is based on the fact that if any random variable is operated upon by some function to produce a quantity, then a different probability distribution is obtained. If one has at his disposal a routine for generating some random variable, he might be able to generate some other random variable of interest. It is possible to generate on a computer a random variable. Another mechanism generating the negative binomial arises as a model for heterogeneity. It is assumed that the individuals of some population are randomly distributed over some region so that when the individuals in some selected area are counted, an observation from a Poisson distribution with parameter is obtained. The building block approach not only simplifies the task of generating random variables in some cases, but it can provide valuable insight into the mechanisms operating to produce a particular probability distribution as a model.

New clustering methods and a learning system

Abstract: In this paper new clustering methods are presented. The essence of these methods is based on the approximation to probability density function or to cumulative distribution function. One group of clustering methods is aimed at approximating histograms or empirical distribution functions. The other group is aimed at approximating cumulative distribution functions. Using these methods one can cluster a sample data into some cluster in the sense of minimum-error-rate decision.

Procurement cycle modeling and simulation

Abstract: The objective of this simulation was to ascertain the mean time required to process a procurement action and the variability of this time. A cumulative probability distribution was generated for each step as well as a rejection probability. A Monte Carlo computer program was used to determine the time required to process through the organization and these “make time” distributions were tested against known distribution for goodness of fit.

Communication Receiver Studies

Abstract: : In the report a family of baseband suboptimal communication systems, referred to as 'the family of predistorted replica correlation receivers,' is presented and analyzed. The resulting receiver structures estimate the reference waveform directly from the received process; whereas, the well-known matched filter or correlation receiver uses a stored reference waveform. In this report the Doppler performance of two such structures is investigated, the results indicating that such structures can be relatively insensitive to Doppler. The characteristic function for the 'decision statistic' is computed for the discrete-time systems and numerically inverted to obtain the cumulative distribution function. Chernoff bounds are computed to evaluate the behavior of the 'tail probabilities.'