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Showing papers on "Cumulative distribution function published in 1969"


Journal ArticleDOI

657 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that for a general class of kernels, the assumption that the distribution of the kernel is uniformly continuous is necessary and sufficient for convergence to the real line.
Abstract: Let $X_1, X_2, \cdots$ be independent identically distributed random variables having a common probability density function $f$. After a so-called kernel class of estimates $f_n$ of $f$ based on $X_1, \cdots, X_n$ was introduced by Rosenblatt [7], various convergence properties of these estimates have been studied. The strongest result in this direction was due to Nadaraya [5] who proved that if $f$ is uniformly continuous then for a large class of kernels the estimates $f_n$ converges uniformly on the real line to $f$ with probability one. For a very general class of kernels, we will show that the above assumptions on $f$ are necessary for this type of convergence. That is, if $f_n$ converges uniformly to a function $g$ with probability one, then $g$ must be uniformly continuous and the distribution $F$ from which we are sampling must be absolutely continuous with $F' (x) = g(x)$ everywhere. When in addition to the conditions mentioned above, it is assumed that $f$ and its first $r + 1$ derivatives are bounded, we are able to show how to construct estimates $f_n$ such that $f^{(s)}_n$ converges uniformly to $f^{(s)}$ at a given rate with probability one for $s = 0, 1, \cdots, r$.

196 citations


Journal ArticleDOI
J. S. Smart1
TL;DR: In this paper, an intermediate method of classification, called "ambilateral", is proposed, where two topologically distinct channel networks (TDCN) belong to the same class if one can be converted into the other by reversal of the right-left order at one or more junctions.
Abstract: There are two common ways of describing the topological properties of channel networks; one is to provide an exact description that gives all junctions and sources in sequence, and the other is to specify the Strahler stream numbers. For many purposes, the first method is too detailed and the second too broad. An intermediate method of classification, called “ambilateral,” is proposed. The basic rule of this classification scheme is that two topologically distinct channel networks (TDCN) belong to the same class if one can be converted into the other by reversal of the right-left order at one or more junctions. Calculations based on an idealized model suggest that the ambilateral classification is more closely correlated with geomorphic and hydrologic properties than is the usual stream-number classification. Shreve (1966) has proposed that natural channel networks developed in the absence of geological controls are topologically random. This hypothesis can be tested by various statistical studies of the topological properties of a random sample of natural networks, including (1) distribution of individual TDCN for given N 1 , (2) distribution of ambilateral classes for given N 1 , (3) distribution of N 2 values for given N 1 , (4) distribution of stream numbers in cumulative probability ranks for a given range of N 1 values (a method employed by Shreve), and (5) distribution of the junction points of excess tributaries. Most of the tests listed above require that the sample of networks be selected without regard to order, a condition that is not satisfied by much of the published data. In order to obtain a suitable sample, 86 networks ranging in magnitude from 23 to 1172 were determined from stream systems in the United States. The statistical analyses generally support Shreve9s hypothesis of topological randomness. One exception to this general remark is that the stream-number data for the western United States shows too many large (>4) bifurcation ratios; the discrepancy becomes more pronounced as N 1 increases. Procedures are suggested for using observed deviations from the topologically random model to draw inferences about the presence of geological controls.

61 citations


ReportDOI
01 Oct 1969
TL;DR: In this paper, confidence limits for the expected value EX were found for all continuous distribution functions with (a, b) for known finite numbers a and b (a < b).
Abstract: : Consider a random variable X with a continuous cumulative distribution function F(x) such that F(a) = 0 and F(b) = 1 for known finite numbers a and b (a < b) The distribution function F(x) is unknown A sample of size n is drawn from this distribution Confidence limits for the expected value EX are to be found that hold for all continuous distribution functions with (a, b)

35 citations


Journal ArticleDOI
11 Aug 1969
TL;DR: In this paper, a method for direct numerical evaluation of the cumulative probability distribution function from the characteristic function in terms of a single integral is presented, and no moment evaluation or series expansions are required.
Abstract: A method for direct numerical evaluation of the cumulative probability distribution function from the characteristic function in terms of a single integral is presented. No moment evaluation or series expansions are required. Intermediate evaluation of the probability density function is circumvented. The method takes on a special form when the random variables are discrete.

26 citations


Journal ArticleDOI
William C. Y. Lee1
TL;DR: In this article, a comparison of the statistical properties of energy density reception of mobile radio signals with those of multichannel predetection combining diversity reception using an array of whip antennas lined up transverse to the direction of motion was made.
Abstract: A comparison is made of the statistical properties of energy density reception of mobile radio signals with those of multichannel predetection combining diversity reception using an array of whip antennas lined up transverse to the direction of motion. Theoretically, the energy density system is about as effective as a 3-channel diversity system in reducing fading. This conclusion was obtained from cumulative distribution function (cdf) curves, level crossing rate curves, and power spectra density function curves. Experimentally, the probability distribution curves show that the variation of the signal amplitude of a 4-channel diversity system is less than that of an energy density signal, but the fading rates of the two signals are almost the same. The experimental power spectra curves verify that the shape of the output spectrum for the diversity receiver is independent of the number of channels, but the level is reduced in magnitude with more channels. The output spectrum for the energy density system is more concentrated at lower frequencies than the diversity receiver.

19 citations


Journal ArticleDOI
TL;DR: In this article, the cumulative bivariate t-distribution associated with random variables T1 = X1/(S/k)112, T2 = X2/(S /k)"12 is considered where X1, X2 are bivariate normal with correlation coefficient p and S is an independent x2 random variable with k degrees of freedom.
Abstract: The cumulative bivariate t-distribution associated with random variables T1 = X1/(S/k)112, T2 = X2/(S/k)"12 is considered where X1, X2 are bivariate normal with correlation coefficient p and S is an independent x2 random variable with k degrees of freedom. Representations in terms of series and simple, one-dimensional quadratures are presented together with efficient computational procedures for the special functions used in numerical evaluation. U Preliminary Representations. The bivariate t-distribution derived below has been of interest to many authors (1), (3), (8), (15). The work of Dunnett and Sobel (2) on the cumulative distribution in terms of incomplete beta functions stands out for computational convenience. These results, coupled with the more recent work of Gautschi (7) on efficient computational procedures for many of the special functions, makes these results even more accessible. The need for other computational formulae stems from possible losses of significance by subtraction in numerical evaluation. A simple quadrature derived below overcomes this difficulty and certain series representations offer computational advantages for large degrees of freedom. The usual procedure for deriving this t-distribution starts with the bivariate normal with correlation matrix 2 associated with the random variables X1, X2 and a X2-distribution with k degrees of freedom associated with an independent random variable A,

14 citations



ReportDOI
23 Dec 1969
TL;DR: In this paper, the basic equation for pulse-radar maximum-range calculation is presented in a form convenient for numerical computation, including graphs for the required signal-to-noise ratio as a function of probability of detection, false-alarm probability, and number of pulses integrated.
Abstract: : The basic equation for pulse-radar maximum-range calculation is presented in a form convenient for numerical computation. Charts, graphs, tables, and auxiliary equations are presented for evaluation of the various factors in the range equation. Included are graphs for the required signal-to- noise ratio as a function of probability of detection, false-alarm probability, and number of pulses integrated, for both nonfluctuating and fluctuating (Swerling Cases 1 and 3) echoes. Also treated are the effects of receiver bandwidth, antenna and receiver noise, sea-reflection interference, refraction and absorption by the atmosphere, and various system losses. Standard definitions of range-equation quantities are given. The effects of jamming and clutter echoes are treated briefly, as are also cumulative probability of detection and accuracy of radar range prediction. A systematic procedure for range calculation, employing a work sheet, is presented.

11 citations


Journal ArticleDOI
TL;DR: In this paper, the authors illustrate a "give and take" type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science on the other.
Abstract: The purpose of the paper is to illustrate a “give and take” type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science, on the other. Invariably, research in science involves some observations x and the unknown “true” state of nature σ. Quite often, replication of the experiments reveals considerable variation in x, indicating that no mathematical treatment of the problem is possible without the assumption that x is a sample value of a random variable X, treated in terms of frequentist theory of probability. As to the true state of nature, σ, situations vary. Indeed, there are cases where it appears natural to consider that σ is selected at random out of a certain known set Σ, with either known or unknown probability law. Section 2 lists three typical examples in which the frequentist theory of probability can “give” something to science. In each example, however, the assumption of the randomness of σ appears extraneous. Section 3 de...

10 citations


Journal ArticleDOI
TL;DR: In this article, the distribution of the maximum and the minimum distributions of the Jth-order statistics is investigated and the distribution-free subset selection rules using the percentage points of these order statistics are investigated.
Abstract: : Let X sub i(i = 0,1,...,p) be (p + 1) independent and identically distributed nonnegative random variables each representing the jth order statistic in a random sample of size n from a continuous distribution G(x) of a nonnegative random variable. Let H sub (j,n) (x) be the cumulative distribution function of X sub i(i = 0,1,...,p). Consider the ratios Y sub i = X sub i/X sub o (i = 1,2,...,p). The random variables Y sub i (i = 1,2,...,p) are correlated and the distribution of the maximum and the minimum is of interest in problems of selection and ranking for restricted families of distribution. The distribution-free subset selection rules using the percentage points of these order statistics are investigated in a companion paper by Barlow and Gupta (1967). In the present paper, we discuss the distribution of these statistics, in general, for any G(x) and then derive specific results for G(x) = 1-(e to the power (-x/theta)), x > 0, theta > 0. Section 2 deals with the distribution of the maximum while Section 3 discusses the distribution of the minimum. Section 4 describes the tables of the percentage points of the two statistics. (Author)

Journal ArticleDOI
TL;DR: In this article, the limiting distributions of these statistics for sample size n ∞ are discussed, and sample sizes are indicated for which these limiting distributions can be used instead of the exact distributions.
Abstract: Let X(1) ≤ X (2) ≤ … ≤ X (n) be an ordered sample of a random variable X which has continuous probability distribution function F (x), and let Fn (x) be the corresponding empirical distribution function. The following three statistics, introduced by A. Renyi, are considered: The paper presents table of exact probabilities for these statistics for finite sample sizes. The limiting distributions of these statistics for sample size n ∞ are discussed, and sample sizes are indicated for which these limiting distributions can be used instead of the exact distributions. Numerical examples for the use of the tables are presented, as well as applications to testing hypotheses on life distributions and to one-sided estimation of probability distribution functions from censored data.

Journal ArticleDOI
01 Mar 1969
TL;DR: In this paper, a numerical method for calculating the cumulative distribution of a positive random variable from its moment-generating function is presented, which involves an expansion of the rectangular function in Laguerre functions.
Abstract: A numerical method is presented for calculating the cumulative distribution of a positive random variable from its moment-generating function. It involves an expansion of the rectangular function in Laguerre functions. As examples, the cumulative exponential and cumulative Poisson probability functions are approximated.


Journal ArticleDOI
TL;DR: Based upon the known theoretic distribution of ordered data for any continuous probability distribution, a graphic procedure for setting confidence limits can be used as discussed by the authors, which requires foreknowledge or an assumption of the population parameters.
Abstract: Based upon the known theoretic distribution of ordered data for any continuous probability distribution, a graphic procedure for setting confidence limits can be used. Such a procedure requires foreknowledge or an assumption of the population parameters. Although the procedure does not define confidence limits beyond the maximum return period of the series, a reasonable extrapolation based upon families of curves of various lengths n indicates a rapid deterioration of confidence in any extensions of the cumulative probability curve.

Journal ArticleDOI
TL;DR: In this article, a method for deriving an estimation procedure from the recurrence relationship between probabilities in a discrete distribution is proposed, which is limited to cases in which only one parameter occurs in the Recurrence relationship.
Abstract: A method is proposed for deriving an estimation procedure from the recurrence relationship between probabilities in a discrete distribution. It is limited to cases in which only one parameter occurs in the recurrence relationship. To find a confidence interval for the parameter we consider the estimating function obtained and this may be shown to be dependent on differences between the cumulative probability function and its estimated value.