Showing papers on "Probability distribution published in 1973"

A Bayesian Analysis of Some Nonparametric Problems

Abstract: The Bayesian approach to statistical problems, though fruitful in many ways, has been rather unsuccessful in treating nonparametric problems. This is due primarily to the difficulty in finding workable prior distributions on the parameter space, which in nonparametric ploblems is taken to be a set of probability distributions on a given sample space. There are two desirable properties of a prior distribution for nonparametric problems. (I) The support of the prior distribution should be large--with respect to some suitable topology on the space of probability distributions on the sample space. (II) Posterior distributions given a sample of observations from the true probability distribution should be manageable analytically. These properties are antagonistic in the sense that one may be obtained at the expense of the other. This paper presents a class of prior distributions, called Dirichlet process priors, broad in the sense of (I), for which (II) is realized, and for which treatment of many nonparametric statistical problems may be carried out, yielding results that are comparable to the classical theory. In Section 2, we review the properties of the Dirichlet distribution needed for the description of the Dirichlet process given in Section 3. Briefly, this process may be described as follows. Let $\mathscr{X}$ be a space and $\mathscr{A}$ a $\sigma$-field of subsets, and let $\alpha$ be a finite non-null measure on $(\mathscr{X}, \mathscr{A})$. Then a stochastic process $P$ indexed by elements $A$ of $\mathscr{A}$, is said to be a Dirichlet process on $(\mathscr{X}, \mathscr{A})$ with parameter $\alpha$ if for any measurable partition $(A_1, \cdots, A_k)$ of $\mathscr{X}$, the random vector $(P(A_1), \cdots, P(A_k))$ has a Dirichlet distribution with parameter $(\alpha(A_1), \cdots, \alpha(A_k)). P$ may be considered a random probability measure on $(\mathscr{X}, \mathscr{A})$, The main theorem states that if $P$ is a Dirichlet process on $(\mathscr{X}, \mathscr{A})$ with parameter $\alpha$, and if $X_1, \cdots, X_n$ is a sample from $P$, then the posterior distribution of $P$ given $X_1, \cdots, X_n$ is also a Dirichlet process on $(\mathscr{X}, \mathscr{A})$ with a parameter $\alpha + \sum^n_1 \delta_{x_i}$, where $\delta_x$ denotes the measure giving mass one to the point $x$. In Section 4, an alternative definition of the Dirichlet process is given. This definition exhibits a version of the Dirichlet process that gives probability one to the set of discrete probability measures on $(\mathscr{X}, \mathscr{A})$. This is in contrast to Dubins and Freedman [2], whose methods for choosing a distribution function on the interval [0, 1] lead with probability one to singular continuous distributions. Methods of choosing a distribution function on [0, 1] that with probability one is absolutely continuous have been described by Kraft [7]. The general method of choosing a distribution function on [0, 1], described in Section 2 of Kraft and van Eeden [10], can of course be used to define the Dirichlet process on [0, 1]. Special mention must be made of the papers of Freedman and Fabius. Freedman [5] defines a notion of tailfree for a distribution on the set of all probability measures on a countable space $\mathscr{X}$. For a tailfree prior, posterior distribution given a sample from the true probability measure may be fairly easily computed. Fabius [3] extends the notion of tailfree to the case where $\mathscr{X}$ is the unit interval [0, 1], but it is clear his extension may be made to cover quite general $\mathscr{X}$. With such an extension, the Dirichlet process would be a special case of a tailfree distribution for which the posterior distribution has a particularly simple form. There are disadvantages to the fact that $P$ chosen by a Dirichlet process is discrete with probability one. These appear mainly because in sampling from a $P$ chosen by a Dirichlet process, we expect eventually to see one observation exactly equal to another. For example, consider the goodness-of-fit problem of testing the hypothesis $H_0$ that a distribution on the interval [0, 1] is uniform. If on the alternative hypothesis we place a Dirichlet process prior with parameter $\alpha$ itself a uniform measure on [0, 1], and if we are given a sample of size $n \geqq 2$, the only nontrivial nonrandomized Bayes rule is to reject $H_0$ if and only if two or more of the observations are exactly equal. This is really a test of the hypothesis that a distribution is continuous against the hypothesis that it is discrete. Thus, there is still a need for a prior that chooses a continuous distribution with probability one and yet satisfies properties (I) and (II). Some applications in which the possible doubling up of the values of the observations plays no essential role are presented in Section 5. These include the estimation of a distribution function, of a mean, of quantiles, of a variance and of a covariance. A two-sample problem is considered in which the Mann-Whitney statistic, equivalent to the rank-sum statistic, appears naturally. A decision theoretic upper tolerance limit for a quantile is also treated. Finally, a hypothesis testing problem concerning a quantile is shown to yield the sign test. In each of these problems, useful ways of combining prior information with the statistical observations appear. Other applications exist. In his Ph. D. dissertation [1], Charles Antoniak finds a need to consider mixtures of Dirichlet processes. He treats several problems, including the estimation of a mixing distribution, bio-assay, empirical Bayes problems, and discrimination problems.

Mathematical statistics with applications

Abstract: This calculus-based book presents a blend of theory and application. It focuses on inference making as the goal of studying probability and statistics, and features an emphasis on real-life applications of the mathematically divided techniques. Exercises based on documented data sets and other realistic experimental situations motivate the theoretical foundation. Updated and expanded coverage includes: the negative binomial probability distribution (optional); and explicit models for the analysis of variance of completely randomized block designs.

Statistics for Business and Economics

Abstract: CHAPTER 1 Describing Data: Graphical 1.1 Decision Making in an Uncertain Environment 1.2 Classification of Variables 1.3 Graphs to Describe Categorical Variables 1.4 Graphs to Describe Time-Series Data 1.5 Graphs to Describe Numerical Variables 1.6 Tables and Graphs to Describe Relationships Between Variables 1.7 Data Presentation Errors CHAPTER 2 Describing Data: Numerical 2.1 Measures of Central Tendency 2.2 Measures of Variability 2.3 Weighted Mean and Measures of Grouped Data 2.4 Measures of Relationships Between Variables CHAPTER 3 Probability 3.1 Random Experiment, Outcomes, Events 3.2 Probability and Its Postulates 3.3 Probability Rules 3.4 Bivariate Probabilities 3.5 Bayes' Theorem CHAPTER 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables 4.2 Probability Distributions for Discrete Random Variables 4.3 Properties of Discrete Random Variables 4.4 Binomial Distribution 4.5 Hypergeometric Distribution 4.6 The Poisson Probability Distribution 4.7 Jointly Distributed Discrete Random Variables CHAPTER 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 5.2 Expectations for Continuous Random Variables 5.3 The Normal Distribution 5.4 Normal Distribution Approximation for Binomial Distribution 5.5 The Exponential Distribution 5.6 Jointly Distributed Continuous Random Variables CHAPTER 6 Sampling and Sampling Distributions 6.1 Sampling from a Population 6.2 Sampling Distributions of Sample Means 6.3 Sampling Distributions of Sample Proportions 6.4 Sampling Distributions of Sample Variances CHAPTER 7 Estimation: Single Population 7.1 Properties of Point Estimators 7.2 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Known 7.3 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Unknown 7.4 Confidence Interval Estimation of Population Proportion 7.5 Confidence Interval Estimation of the Variance of a Normal Distribution 7.6 Confidence Interval Estimation: Finite Populations CHAPTER 8 Estimation: Additional Topics 8.1 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Dependent Samples 8.2 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Independent Samples 8.3 Confidence Interval Estimation of the Difference Between Two Population Proportions 8.4 Sample Size Determination: Large Populations 8.5 Sample Size Determination: Finite Populations CHAPTER 9 Hypothesis Testing: Single Population 9.1 Concepts of Hypothesis Testing 9.2 Tests of the Mean of a Normal Distribution: Population Variance Known 9.3 Tests of the Mean of a Normal Distribution: Population Variance Unknown 9.4 Tests of the Population Proportion 9.5 Assessing the Power of a Test 9.6 Tests of the Variance of a Normal Distribution CHAPTER 10 Hypothesis Testing: Additional Topics 10.1 Tests of the Difference Between Two Population Means: Dependent Samples 10.2 Tests of the Difference Between Two Normal Population Means: Independent Samples 10.3 Tests of the Difference Between Two Population Proportions 10.4 Tests of the Equality of the Variances Between Two Normally Distributed Populations 10.5 Some Comments on Hypothesis Testing CHAPTER 11 Simple Regression 11.1 Overview of Linear Models 11.2 Linear Regression Model 11.3 Least Squares Coefficient Estimators 11.4 The Explanatory Power of a Linear Regression Equation 11.5 Statistical Inference: Hypothesis Tests and Confidence Intervals 11.6 Prediction 11.7 Correlation Analysis 11.8 Beta Measure of Financial Risk 11.9 Graphical Analysis CHAPTER 12 Multiple Regression 12.1 The Multiple Regression Model 12.2 Estimation of Coefficients 12.3 Explanatory Power of a Multiple Regression Equation 12.4 Confidence Intervals and Hypothesis Tests for Individual Regression Coefficients 12.5 Tests on Regression Coefficients 12.6 Prediction 12.7 Transformations for Nonlinear Regression Models 12.8 Dummy Variables for Regression Models 12.9 Multiple Regression Analysis Application Procedure CHAPTER 13 Additional Topics in Regression Analysis 13.1 Model-Building Methodology 13.2 Dummy Variables and Experimental Design 13.3 Lagged Values of the Dependent Variables as Regressors 13.4 Specification Bias 13.5 Multicollinearity 13.6 Heteroscedasticity 13.7 Autocorrelated Errors CHAPTER 14 ANALYSIS OF CATEGORICAL DATA 14.1 Goodness-of-Fit Tests: Specified Probabilities 14.2 Goodness-of-Fit Tests: Population Parameters Unknown 14.3 Contingency Tables 14.4 Sign Test and Confidence Interval 14.5 Wilcoxon Signed Rank Test 14.6 Mann--Whitney U Test 14.7 Wilcoxon Rank Sum Test 14.7 Spearman Rank Correlation CHAPTER 15 Analysis of Variance 15.1 Comparison of Several Population Means 15.2 One-Way Analysis of Variance 15.3 The Kruskal--Wallis Test 15.4 Two-Way Analysis of Variance: One Observation per Cell, Randomized Blocks 15.5 Two-Way Analysis of Variance: More Than One Observation per Cell CHAPTER 16 Time-Series Analysis and Forecasting 16.1 Index Numbers 16.2 A Nonparametric Test for Randomness 16.3 Components of a Time Series 16.4 Moving Averages 16.5 Exponential Smoothing 16.6 Autoregressive Models 16.7 Autoregressive Integrated Moving Average Models CHAPTER 17 Sampling: Additional Topics 17.1 Stratified Sampling 17.2 Other Sampling Methods CHAPTER 18 Statistical Decision Theory 18.1 Decision Making Under Uncertainty 18.2 Solutions Not Involving Specification of Probabilities 18.3 Expected Monetary Value TreePlan 18.4 Sample Information: Bayesian Analysis and Value 18.5 Allowing for Risk: Utility Analysis APPENDIX TABLES 1. Cumulative Distribution Function of the Standard Normal Distribution 2. Probability Function of the Binomial Distribution 3. Cumulative Binomial Probabilities 4. Values of e --lambda 5. Individual Poisson Probabilities 6. Cumulative Poisson Probabilities 7. Cutoff Points of the Chi-Square Distribution Function 8. Cutoff Points for the Student's t Distribution 9. Cutoff Points for the F Distribution 10. Cutoff Points for the Distribution of the Wilcoxon Test Statistic 11. Cutoff Points for the Distribution of Spearman Rank Correlation Coefficient 12. Cutoff Points for the Distribution of the Durbin--Watson Test Statistic 13 Critical Values of the Studentized Range Q (page 964 965 Applied Statistical Methods Carlson, Thorne Prentice Hall 1997) 14. Cumulative Distribution Function of the Runs Test Statistic ANSWERS TO SELECTED EVEN-NUMBERED EXERCISES INDEX I-1

A selfconsistent theory of localization

Abstract: A new basis has been found for the theory of localization of electrons in disordered systems. The method is based on a selfconsistent solution of the equation for the self energy in second order perturbation theory, whose solution may be purely real almost everywhere (localized states) or complex everywhere (nonlocalized states). The equations used are exact for a Bethe lattice. The selfconsistency condition gives a nonlinear integral equation in two variables for the probability distribution of the real and imaginary parts of the self energy. A simple approximation for the stability limit of localized states gives Anderson's 'upper limit approximation'. Exact solution of the stability problem in a special case gives results very close to Anderson's best estimate.

Weak convergence of probability measures and random functions in the function space D[0,∞)

Abstract: This paper extends the theory of weak convergence of probability measures and random functions in the function space D[0,1] to the case D [0,∞), elaborating ideas of C. Stone and W. Whitt. 7)[0,∞) is a suitable space for the analysis of many processes appearing in applied probability.

The Prediction of Systematic and Specific Risk in Common Stocks

Abstract: Ex ante predictions of the riskiness of returns on common stocks — or, in more general terms, predictions of the probability distribution of returns — can be based on fundamental (accounting) data for the firm and also on the previous history of stock prices. In this article, we attempt to combine both sources of information to provide efficient predictions of the probability distribution of returns. We predict two parameters of the distribution of returns for each security in each year: the response to the overall market return (β), and the variance of the part of risk, specific to the security, that is uncorrelated with the market return. A cross section of time series data on returns and accounting variables, taken primarily from the Compustat tape, is used. Several recent developments in statistical methodology are applied.

Optimization of k nearest neighbor density estimates

Abstract: Nonparametric density estimation using the k -nearest-neighbor approach is discussed. By developing a relation between the volume and the coverage of a region, a functional form for the optimum k in terms of the sample size, the dimensionality of the observation space, and the underlying probability distribution is obtained. Within the class of density functions that can be made circularly symmetric by a linear transformation, the optimum matrix for use in a quadratic form metric is obtained. For Gaussian densities this becomes the inverse covariance matrix that is often used without proof of optimality. The close relationship of this approach to that of Parzen estimators is then investigated.

Man-Made Noise in Urban Environments and Transportation Systems: Models and Measurements

Abstract: Analytically tractable statistical-physical models of man-made noise environments have been constructed [1]-[3]. These permit quantitative description of the various types of electromagnetic interference appearing in typical radio receivers and, in particular here, for the communication links employed in mobile transportation systems and urban environments generally. This paper presents a summary of some of the principal analytical results obtained to date [1], [4], and includes some suggested next steps for joint theoretical and experimental study of these increasingly important phenomena. First-order probability density functions (pdf's) and probability distributions (pd's) are obtained explicitly; (higher order pdf's and pd's may also be found by similar methods) [2]. These models are based on a Poisson distribution of sources in space. The approach is canonical, in that the results are, in form, independent of particular emitted waveforms, propagation conditions, source distributions, beam patterns, and specific system parameters, as long as the interference is narrow-band following (at least) the aperture and/or the RF stages of a typical receiver. Considered here only are the cases of communication interference, where source and receiver bandwidths are comparable. The paper concludes with a short discussion of some features of suggested future interaction between theory and experiment.

The Maximum-Entropy Distribution of the Future Market Price of a Stock

Abstract: This paper uses the principle of maximum entropy to construct a probability distribution of future stock price for a hypothetical investor having specified expectations. The result obtained is in good agreement with observations recorded in the literature. Thus, the paper concludes that the hypothetical individual investor is representative of a large class of investors. This new derivation of the well known random-walk theory of stock-price movements leads to an improved understanding of the model parameters by relating the variance of the random-walk process to the risk aversion of the investors. A practical use of the model is proposed to help the investor form an objective opinion of his skill.

Probability distributions and correlations in a turbulent boundary layer

Abstract: One‐dimensional and joint probability density distributions for longitudinal components of turbulent velocities as well as higher‐order correlations are measured in a turbulent boundary layer on a flat plate using hot‐wire anemometry and high‐speed computing methods. The effect of the nonlinear response of the hot‐wire is taken into account. Data pertaining to the general nature of the turbulent boundary layer are presented and comparison is made between the measured correlations and those corresponding to a Gaussian probability distribution of turbulent velocities as well as to non‐Gaussian distributions of the Gram‐Charlier type. Similar comparisons are made of the measured one‐dimensional and joint probability distributions. Probability distributions in the boundary layer are also compared to those measured downstream of a grid. The closure of the tails of the probability distribution and its effect on the accuracy of the measurements of higher‐order moments is considered.

Probabilities for Highest Wave in Hurricane

Abstract: The function, which is called the extremal Rayleigh distribution, is found to be an accurate approximation to the probability distribution function for the maximum pro a hurricane with time-varying intensity A formula is developed for the maximum wave distribution function in time-varying storms and computed for a large number of historical hurricanes The extremal Rayleigh function was found to be a very accurate approximation in all cases Two parameters are shown to be interpretable in terms of an equivalent uniform intensity storm as the reciprocal of the logarithm square wave height and the logarithm of the number of waves in the storm, respectively Data from Hurricane Carla was analyzed to determine the parameters which best described the probability behavior of the larger waves in each 20 min record Statistical formulas and tables are given to facilitate probability use of the extremal Rayleigh probability law in coastal engineering and hurricane research

Consistency of an estimate of tree-dependent probability distributions (Corresp.)

Abstract: We demonstrate in this correspondence that the procedure introduced by Chow and Liu for estimating tree-dependent probability distributions is consistent.

Response of the impact damper to stationary random excitation

Abstract: The response of an impact damper system to an excitation with approximately white‐power spectral density and Gaussian probability distribution is determined, using two independent methods: digital computer and electronic‐analog techniques. Results are given for mean‐squared level, power spectral density, probability density, probability distribution, and amplitude probability density of the response. The impact damper is found to be a practical and efficient device for reducing the response amplitude of systems subjected to random excitation.

Man-Made Noise in Urban Environments and Transportation Systems: Models and Measurements

Abstract: Analytically tractable statistical-physical models of man- made noise environments have been constructed (l) -(3). These per- mit quantitative description of the various types of electromagnetic interference appearing in typical radio receivers and, in particular here, for the communication links employed in mobile transportation sys- tems and urban environments generally. This paper presents a sum- mary of some of the principal analytical results obtained to date (l) , (4), and includes some suggested next steps for joint theoretical and experimental study of these increasingly important phenomena. First- order probability density functions (pdfs) and probability distributions (pd's) are obtained explicitly; (higher order pdfs and pd's may also be found by similar methods) (2). These models are based on a Poisson distribution of sources in space. The approach is canonical, in that the results are, in form, independent of particular emitted waveforms, propagation conditions, source distniutions, beam patterns, and spe- cific system parameters, as long as the interference is narrow-band fol- lowing (at least) the aperture and/or the RF stages of a typical receiver. Considered here only are the cases of communication interference, where source and receiver bandwidths are comparable. The paper concludes with a short discussion of some features of suggested future interaction between theory and experiment.

Statistical properties of two sine waves in Gaussian noise

Abstract: A detailed study is presented of some statistical properties of the stochastic process, that consists of the sum of two sine waves of unknown relative phase and a normal process. Since none of the statistics investigated seem to yield a closed-form expression, all the derivations are cast in a form that is particularly suitable for machine computation. Specifically, results are presented for the probability density function (pdf) of the envelope and the instantaneous value, the moments of these distributions, and the relative cumulative density function (cdf). The analysis hinges on expanding the functions of interest in a way that allows computation by means of recursive relations. Specifically, all the expansions are expressed in terms of sums of products of Gaussian hypergeometric functions and Laguerre polynomials. Computer results obtained on a CDC 6600 are presented. If a and b are the amplitudes of the two sine waves, normalized to the rms noise level, the expansions presented are useful up to values of a,b of about 17 dB, in double precision on the CDC 6600. A different approximation is also given for the case of very high SNR.

On validation of simulation models

Abstract: Before an investigator can claim that his simulation model is a useful tool for studying behavior under new hypothetical conditions, he is well advised to check its consistency with the true system, as it exists before any change is made. The success of this validation establishes a basis for confidence in results that the model generates under new conditions. After all, if a model cannot reproduce system behavior without change, then we hardly expect it to produce truly representative results with change.The problem of how to validate a simulation model arises in every simulation study in which some semblance of a system exists. The space devoted to validation in Naylor's book Computer Simulation Experiments with Models of Economic Systems indicates both the relative importance of the topic and the difficulty of establishing universally applicable criteria for accepting a simulation model as a valid representation.One way to approach the validation of a simulation model is through its three essential components; input, structural representation and output. For example, the input consist of exogenous stimuli that drive the model during a run. Consequently one would like to assure himself that the probability distributions and time series representations used to characterize input variables are consistent with available data. With regard to structural representation one would like to test whether or not the mathematical and logical representations do not conflict with the true system's behavior. With regard to output one could feel comfortable with a simulation model if it behaved similarly to the true system when exposed to the same input.Interestingly enough, the greatest effort in model validation of large econometric models has concentrated on structural representation. No doubt this is due to the fact that regression methods, whether it be the simple leastsquares method or a more comprehensive simultaneous equations techniques, in addition to providing procedures for parameter estimation, facilitate hypothesis testing regarding structural representation. Because of the availability of these regression methods, it seems hard to believe that at least some part of a model's structural representation cannot be validated. Lamentably, some researchers choose to discount and avoid the use of available test procedures.With regard to input analysis, techniques exist for determining the temporal and probabilistic characteristics of exogeneous variables. For example the autoregressive---moving average schemes described in Box and Jenkins' book, Time Series Analysis: Forecasting and Control, are available today in canned statistical computer programs. Maximum likelihood estimation procedures are available for most common probability distribution and tables based on sufficient statistics have begun to appear in the literature. Regardless of how little data is available, a model's use would benefit from a conscientious effort to characterize the mechanism that produced those data.As mentioned earlier a check of consistency between model and system output in response to the same input would be an appropriate step in validation. A natural question that arises is: What form should the consistency check take? One approach might go as follows: Let X1, ..., Xn be the model's output in n consecutive time intervals and let Y1, ..., Yn be the system's output for n consecutive time intervals in response to the same stimuli. Test the hypothesis that the joint probability distribution of X1, ..., Xn is identical with that of Y1, ..., Yn.My own feeling is that the above test is too stringent and creates a misplaced emphasis on statistical exactness. I would prefer to frame output validation in more of a decision making context. In particular, one question that seems useful to answer is: In response to the same input, does the model's output lead decision makers to take the same action that they would take in response to the true system's output? While less stringent than the test first described, its implementation requires access to decision makers. This seems to me to be a desirable requirement for only through continual interaction with decision makers can an investigator hope to gauge the sensitive issues to which his model should be responsive and the degree of accuracy that these sensitivities require.

First-order Probability Densities of Laser Speckle Patterns Observed through Finite-size Scanning Apertures

Abstract: The first-order probability density function of laser speckle patterns observed through finite size apertures is theoretically studied. An exact solution is obtained through the use of a two-dimens...

Polymer conformational statistics. I. Probability distribution

Abstract: The probability distribution P(R) of the end‐to‐end distance is studied for the rotational‐isomeric model of polymer chains. A Monte Carlo investigation provided reliable numerical data for P(R), which was then compared with results from two relatively analytic studies. The first of these, and by far the more complicated, was a steepest descent inversion of the characteristic function. This method was found to be more satisfactory, particularly for large R, than Hermite function expansions that have been used in the past. However for chains of N bonds the method fails, for N<40, to show the stiffness evident for small R. A second quite simple method was to maximize the entropy functional of P(R) subject to assigned 〈 R2〉 and 〈 R4〉. The results for P(R) were in good agreement with Monte Carlo results down to N=12, and for all R.

Estimating the Parameters of a Bivariate Exponential Distribution in Several Sampling Situations.

Abstract: : Estimation for the three parameter, bivariate exponential distribution of Marshall and Olkin (BVE) is investigated for several sampling plans. Sample data arising from a set of two-component systems on life test may be of three basic types depending on whether the systems on test are series, parallel, or parallel with observable causes of individual component failure. Series and parallel BVE sampling have been investigated previously, but here the theory is unified by characterizing the accrual of information, i.e., by specifying the densities of the additional random variables that become observable, in going from series to parallel to cause identifiable sampling. Using these results, a likelihood is computed for samples that may contain any combination of series, parallel, or cause identifiable subsamples. By slightly modifying the densities involved in the information accrual characterization, the likelihoods of time truncated parallel and cause identifiable samples are obtained. (Modified author abstract)

Limit Laws for Sequences of Normed Sums Satisfying Some Stability Conditions

Abstract: This chapter discusses the limit laws arising from normed sums of independent random variables satisfying some stability conditions. These are sequences for which the limit properties of suitably normed sums are similar to those for sequences of identically distributed random variables. The chapter presents a result that is an analogue of the Levy–Khinchin representation of infinitely divisible laws. A probability measure is said to be a limit distribution of the triangular array { X kn } if it is the weak limit of the sequence of probability distributions of normed sums (1/ n ) Σ k=1 n X kn − a n for suitably chosen constants a n . The limit distribution of { X kn } is defined uniquely up to a shift transformation.

The return of the linear search problem

Abstract: The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes.

A Semi-Markovian Population Model with Application to Hospital Planning

Abstract: A semi-Markovian population model can be used to describe patient movements in a health-care delivery system. The recovery progress of a patient at any time is defined by one of a finite number of "states." The movement of a patient among the states is governed by a discrete-time semi-Markov process which is called a " path." The inputs to the system are various patient "groups." Each group is characterized by a distinct input distribution with known mean and variance. Assuming that the input distributions are relatively stable in the long run, estimates of the means and variances of steady-state census mix are derived. On a short-term basis, the model will accommodate daily fluctuations in input distributions and use an information vector, which summarizes the state of the system at the time of prediction, to improve the estimation. An illustrative example using field data is presented.

The Use of Distribution Functions to Represent Utility Functions

Abstract: This paper considers the decision maker whose evaluation and consequent choice of actions is accomplished through the use of the expected utility hypothesis. In cases where the utility function is increasing with upper and lower bounds then the utility function can be characterized by a distribution function, and we can take advantage of the various properties of such functions as well as existing results with respect to such functions. Using these properties and results we can determine the certainty equivalents as a function of the parameters of the distribution function utility function and the parameters of the probability distribution on the uncertain payoff. The following cases are considered: 1 Gaussian distribution function and Gaussian probability distribution, 2 Exponential distribution function and exponential distribution and 3 Exponential distribution function and Gaussian probability distribution.

Bayesian Models for Forecasting Future Security Prices

Abstract: The field of investment analysis provides an example of a situation in which individuals or corporations make inferences and decisions in the face of uncertainty about future events. The uncertainty concerns future security prices and related variables, and it is necessary to take account of this uncertainty when modeling inferential or decision-making problems relating to investment analysis. Since probability can be thought of as the mathematical language of uncertainty, formal models for decision making under uncertainty require probabilistic inputs. In financial decision making, this is illustrated by the models that have been developed for the portfolio selection problem; such models generally require the assessment of probability distributions (or at least some summary measures of probability distributions) for future prices or returns of the securities that are being considered for inclusion in the portfolio (e.g., see Markowitz [11] and Sharpe [19]).

Generation of a pseudo-random set with desired correlation and probability distribution

Abstract: Pseudo-random sequences can be computed to approximate a distribution or a sample correlation function. How ever, conventional techniques do not allow the control of both distribution and correlati...