Showing papers on "Cumulative distribution function published in 1971"

Nonparametric roughness penalties for probability densities

Abstract: ONE of the most fundamental problems in statistics is the estimation of a probability density function from a sample, the smoothing of a histogram being the usual non-parametric method. This method requires a large sample and even so it is difficult to decide whether “bumps” are genuinely in the population. A method is presented here that should help to overcome this difficulty.

Probabilistic methods of signal and system analysis

Abstract: Preface 1. Introduction To Probability 1-1 Engineering Applications Of Probability 1-2 Random Experiments And Events 1-3 Definitions Of Probability 1-4 The Relative-Frequency Approach 1-5 Elementary Set Theory 1-6 The Axiomatic Approach 1-7 Conditional Probability 1-8 Independence 1-9 Combined Experiments 1-10 Bemoulli Trials 1-11 Applications Of Bemoulli Trials 2. Random Variables 2-1 Concept Of A Random Variable 2-2 Distribution Functions 2-3 Density Functions 2-4 Mean Values And Moments 2-5 The Gaussian Random Variable 2-6 Density Functions Related To Gaussian 2-7 Other Probability Density Functions 2-8 Conditional Probability Distribution And Density Functions 2-9 Examples And Applications 3. Several Random Variables 3-1 Two Random Variables 3-2 Conditional Probability-Revisited 3-3 Statistical Independence 3-4 Correlation Between Random Variables 3-5 Density Function Of The Sum Of Two Random Variables 3-6 Probability Density Function Of A Function Of Two Random Variables 3-7 The Characteristic Function 4. Elements oOf Statistics 4-1 Introduction 4-2 Sampling Theory- The Sample Mean 4-3 Sampling Theory- The Sample Variance 4-4 Sampling Distributions And Confidence Intervals 4-5 Hypothesis Testing 4-6 Curve Fitting And Linear Regression 4-7 Correlation Between Two Sets of Data 5. Random Processes 5-1 Introduction 5-2 Continuous And Discrete Random Processes 5-3 Deterministic And Nondeterministic Random Processes 5-4 Stationary and Nonstationary Random Processes 5-5 Ergodic And Nonergodic Random Processes 5-6 Measurement Of Process Parameters 5-7 Smoothing Data With A Moving Window Average 6. Correlation Functions 6-1 Introduction 6-2 Example:Autocorrelation Function Of A Binary Profess 6-3 Properties Of Autocorrelation Functions 6-4 Measurement Of Autocorrelation Functions 6-5 Examples Of Autocorrelation Functions 6-6 Crosscorrelation Functions 6-7 Properties Of Crosscorrelation Functions 6-8 Examples And Applications Of Crosscorrelation Functions 6-9 Correlation Matrices For Sampled Functions 7. Spectral Density 7-1 Introduction 7-3 Properties Of Spectral Density 7-4 Spectral Density And The Complex Frequency Plane 7-5 Mean-Square Values From Spectral Density 7-6 Relation Of Spectral Density To The Autocorrelation Function 7-7 White Noise 7-8 Cross-Spectral Density 7-9 Measurement Of Spectral Density 7-10 Periodogram Estimate Of Spectral Density 7-11 Examples And Applications Of Spectral Density 8. Repines Of Linear Systems To Random Inputs 8-1 Introduction 8-2 Analysis In The Time Domain 8-3 Mean And Mean-Swquare Value Of System Output 8-4 Autocorrelation Function Of System Output 8-5 Crosscorrelation Between Input And Output 8-6 Example Of Time-Domain Analysis 8-7 Analysis In The Frequency Domain 8-8 Spectral Density At The System Output 8-9 Cross-Spectral Densities Between Input And Output 8-10 Examples Of Frequency-Domain Analysis 8-11 Numerical Computation Of System Output 9. Optimum Linear Systems 9-1 Introduction 9-2 Criteria Of Optimaility 9-3 Restrictions On The Optimum System 9-4 Optimization By Parameter Adjustment 9-6 Systems That Minimize Mean-Square Error Appendices Appendix A: Mathematical Tables Appendix B: Frequently Encountered Probability Distributions Appendix C: Binomial Coefficients Appendix D: Normal Probability Distribution Function Appendix E: The Q-Function Appendix F: Student's T-Distribution Function Appendix G: Computer Computations Appendix H: Table Of Correlation Function-Spectral Density Pairs Appendix I: Contour Integration

A general approximation to the distribution of instrumental variables estimates

Abstract: This paper develops approximations of the Gram-Charlier type to the cumulative distribution function of the instrumental variables estimator on classical assumptions. In the special case where there are only two endogenous variables in the estimated equation, exact values of the cumulative distribution function are computed by numerical integration and compared with the approximations. Although the error in the approximation depends critically on the parameters of the stochastic model, the approximation is good for the special case even for small sample size over a wide range of values of the parameters. THIS PAPER was originally conceived as a study of the finite sample distribution of two stage least squares estimates. Since it was found that the distribution of a more general class of instrumental variables estimates can be discussed in the same way with a trifling complication of the algebra, the paper was modified to cover these estimates. The basic approach is somewhat similar to that of Nagar [15], since it involves expanding the formulae for the estimator as a series of terms of 0(1), O(T-+), O(T- 1), O(T- 1+), etc., and from this a similar expansion is found for the cumulative probability of the form

On Estimating the Reliability of a Component Subject to Several Different Stresses

Abstract: : A great deal has been written concerning the estimation of the probability and testing of whether one of two random variables is stochastically larger than the other and its relationship to the estimation of reliability for stress-strength relationships. A more general problem is the estimation and testing of whether one of N + 1 random variables is simultaneously stochastically larger (smaller) than the others. An initial paper which deals with this problem for the special case N = 2 is that of D. R. Whitney (1951), A Bivariate Extension of the U Statistic, where he provides a test function and discusses the asymptotic normality of the statistics proposed under the null hypothesis that all the random variables have the same distribution function. In the report, the problem of estimation of the probability of whether one of N + 1 mutually independent random variables, each having a continuous cumulative distribution function, is simultaneously stochastically larger (smaller) than the others has been considered.

On Unbiased Estimation of Density Functions

Abstract: Let $X^{(n)} = (X_1, \cdots, X_n)$ be a random sample of size $n$ from the distribution of a real-valued random variable $X$ with an absolutely continuous distribution function $F$ and a density function $f$. Rosenblatt (1956) showed that in this setting there exists no unbiased estimator of $f$ based on the order statistics. His result follows from the fact that the empirical distribution function is not absolutely continuous. He also assumed that $f$ is continuous, but this condition is unnecessary. Rosenblatt's result also arises as a consequence of general results by Bickel and Lehmann (1969) on unbiased estimation in convex families, such as the family of all such $F$ (above). A number of writers (Kolmogorov (1950), Schmetterer (1960), Ghurye and Olkin (1969)) have obtained unbiased estimators of particular normal-related families as well as for other estimable functions. Washio, Morimoto and Ikeda (1956) considered related questions for the Koopman-Pitman family of densities, and Tate (1959) confined his attention to functions of scale and location parameters. A question arises as to exactly when unbiased--uniform minimum variance unbiased (UMVU)--estimators of density functions exist and when they do not. In a recent publication, Lumel'skii and Sapozhnikov (1969) considered such a question in relation to estimating the density function at a point, whereas, in this paper our definition of unbiasedness requires the estimator to be unbiased at every point. The so-called "Bayesian" methods they employ yield estimators for most of the well-known families of distributions as well as for several types of $p$-dimensional discrete distributions. In Section 2 we formulate the problem in a fairly general setting and obtain results in terms of unbiased estimators of probability measures (or distribution functions) which always exist. In Section 3 we consider examples to illustrate the theory of the preceding section and in Section 4 give a theorem which generalizes a lemma stated by Ghurye and Olkin (1969) which formalizes the approach used by Schmetterer (1960) for obtaining unbiased estimators of certain types of parametric functions.

Bayesian Confidence Limits for the Reliability of Mixed Exponential and Distribution-Free Cascade Subsystems

Abstract: The problem treated here is the theoretical one of deriving exact Bayesian confidence intervals for the reliability of a system consisting of some independent cascade subsystems with exponential failure probability density functions (pdf) mixed with other independent cascade subsystems whose failure pdf's are unknown. The Mellin integral transform is used to derive the posterior pdf of the system reliability. The posterior cumulative distribution function (cdf) 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 derived pdf. A computer program written in Fortran IV is operational. It utilizes multiprecision to obtain the posterior pdf to any desired degree of accuracy in both functional and tabular form. The posterior cdf is tabulated at any desired increments to any required degree of accuracy.

Statistical Estimation of the Derivatives of Density Functions

Abstract: THE need to estimate the slope (or higher derivatives) of a probability density function from a sample of independent observations has suggested an extension of a technique previously suggested1 for estimating density functions. The proposed extension is straightforward and is summarized here for the first derivative. Let the sample cumulative distribution function be Fs(u) and let the true density function be f(u), with successive derivatives f1(u),f2(u), …. Let the estimate of f1(u) at x be

Probability Distribution of Number of Networks in Topologically Random Network Patterns

Abstract: The probability distribution of the number of channel networks for Werner's model of topologically random network patterns is shown to obey a shifted negative or inverse hypergeometric probability law and, when the number of exterior links is large, to be approximated by a shifted negative binomial probability law.

A technique for obtaining probabilities of correct selection in a two-stage selection problem

Abstract: SUMMARY Suppose one has several normal populations, identically distributed except for their means. At stage one a sample of size n1 is taken from each population. At stage two, a sample of size n2 is taken from the two populations producing the largest means in stage one and the population having the largest cumulative mean selected as best. For the least favourable configuration of means an algorithm is developed for calculating po' the probability of 'correct' selection. The technique involves using a finite representation of the standard normal distribution, counting methods and the use of a high-speed computer both for enumeration and for later smoothing and filtering. A two-stage procedure for the selection of the population with the largest mean from a set of normal populations with unknown means and a common known variance was proposed by Somerville (1954) and extended by Fairweather (1968). The procedure eliminates a predetermined number of populations after the first stage and, from the survivors, selects after the second stage the one with the largest cumulative mean. Costs from incorrect selection and from sampling are assumed known. For a given ratio of first-stage to second-stage sample sizes, the total sample size which would minimize the maximum expected loss was derived. The maximum was taken over all possible configurations of the true population means. For a wide range of losses due to incorrect selection and for a wide range of procedures, one-stage, two-stage, etc., for selecting the 'best' population, viz. the one with the largest mean, it was shown that the maximum expected loss occurs when the means of all the populations except the best are equal. We describe this as the 'least favourable configuration.' Under this configuration of means, a considerable portion of the effort in Somerville (1954) and Fairweather (1968) was devoted to the problem of the evaluation of po, defined to be the probability of a correct selection, selecting the best population. For example, Fairweather showed that the determination of po involved the evaluation of the cumulative distribution function of a multivariate normal variable with at least five different values in the variance-covariance matrix. The dimension of the multivariate integral is 2k - t, where k + 1 is the number of populations and t is the number eliminated after the first stage. The amount of work involved in the computation of po even for small values of k is seen to be very large and increases exponentially with k. Fairweather's computations were thus limited to the case of four populations. Curnow & Dunnett (1962) have shown that any n-variate normal cumulative distribution function can always be written as a single integral with an (n - 1)-variate normal cumulative distribution function in the integrand, with the integration extending over a singly infinite

Asymptotic Distribution of the Sample Size for a Sequential Probability Ratio Test

Abstract: The derivation is presented of the asymptotic distribution of the number N of observations required to terminate the classical sequential probability ratio test (SPRT) with bounds a, b (a > 0 > b) of a simple parametric hypothesis H: θ = θ1 against a simple parametric alternative K:θ=θ1+Δ(Δ>0) when Δ tends to zero. It is shown that if the common probability density function (p.d.f.) of the basic random variables belongs to the Darmois-Koopman exponential class and if the true value of θ0 of θ is different from θ1 then as Δ→0, NΔ tends in probability to a constant A > 0 which is a specified multiple of a if θ0 > θ1 and of - b if θ0 0 which is finite and depends on a if θ0 < θ1 and on − b if θ0 < θ1 such that tends to N(0, 1) as Δ→0. Finally, a few numerical examples are worked to illustrate the magnitudes of A and B.

Broken-line and complex frequency distributions

Abstract: Broken straight lines on probability paper have been proposed as representing the cumulative probability distributions of SiO2, Na, and Cl in certain plutons. This corresponds to pairs of complementarily truncated normal or lognormal distributions. It is shown that alternative representations, such as pairs of overlapping complete normal or lognormal distributions, yield fits equally acceptable statistically if the sample selection is random. The method also yields a good description of distributions described as “complex.” Computer methods of optimizing the free parameters are used; significance testing is discussed in some detail. It is stressed that although significance testing is helpful, it can neither relieve the geochemist of the burden nor take from him the privilege of being independent, and forming his opinion on the total evidence.

A Statistical Investigation into the Development of Energy Absorber Design Criteria.

Abstract: : A Statistical technique for evaluating the injury reduction capability of energy absorbers was developed. Existing statistical data on crash accelerations and man-weight distributions were used as inputs to a MIMIC computer program. A Dynamic Response Index (D.R.I) and stroke length were computed for each acceleration - man-weight combination. The joint probability for each acceleration, man-weight, DRI combination was figured. The summation of all such combinations produced a cumulative probability of injury for the specific energy absorber. The waveform of the force deflection curve of the energy absorber was varied to determine the effect on cumulative probability of injury and required stroke length. The results of the program can be used as a guideline in selecting candidate energy absorbers and the technique developed is appropriate for evaluating the absorbers so selected. (Author)