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Journal ArticleDOI

Tests and confidence bands for bivariate cumulative distribution functions

01 Jan 1990-Communications in Statistics - Simulation and Computation (Marcel Dekker, Inc.)-Vol. 19, Iss: 1, pp 25-36

AboutThis article is published in Communications in Statistics - Simulation and Computation.The article was published on 1990-01-01 and is currently open access. It has received 2 citation(s) till now. The article focuses on the topic(s): Kolmogorov–Smirnov test & Confidence and prediction bands. more

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University of Pennsylvania University of Pennsylvania
ScholarlyCommons ScholarlyCommons
Statistics Papers Wharton Faculty Research
Tests and Con<dence Bands for Bivariate Cumulative Distribution Tests and Con<dence Bands for Bivariate Cumulative Distribution
Functions Functions
Robert J. Adler
Lawrence D. Brown
University of Pennsylvania
Kun-Liang Lu
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Recommended Citation Recommended Citation
Adler, R. J., Brown, L. D., & Lu, K. (1990). Tests and Con<dence Bands for Bivariate Cumulative Distribution
Communications in Statistics - Simulation and Computation,
(1), 25-36.
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Tests and Con<dence Bands for Bivariate Cumulative Distribution Functions Tests and Con<dence Bands for Bivariate Cumulative Distribution Functions
Keywords Keywords
cumulative distribution functions, non-parametric tests, con<dence bands, Kolmogrorov-Smirnov statistic
Disciplines Disciplines
Statistics and Probability
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Abstract: Detrital zircon from unconsolidated, Cenozoic sediments from eastern South Africa has been analysed for U–Pb and Lu–Hf isotopes by laser ablation inductively coupled plasma mass spectrometry. Identifiable bedrock sources have made local contributions to the detrital zircon populations, but the dominant zircon components are of regional distribution: late Mesoproterozoic (e Hf = –5 to +10), Neoproterozoic to early Palaeozoic (e Hf = –10 to +10), and minor late Palaeozoic (e Hf ≈ 0). Archaean zircons are scarce even in sediments deposited on exposed Archaean basement or by rivers eroding it. The dominant components cannot be tied to specific first-generation sources in southern Africa or its former Gondwana neighbours. Instead, we see the effect of mixing and remobilization of debris from large parts of the supercontinent in the early Phanerozoic, which was stored in the Karoo basin and other continental cover sequences and shed from there to the present site of deposition. Therefore, data from detrital zircon in these deposits tell us less about the path of detritus from source to sink in a recent sedimentary system than about processes in much earlier erosion–transport–deposition cycles. To facilitate comparison of detrital zircon age distribution patterns, a simple and intuitive method that takes sampling uncertainty explicitly into account is proposed. Supplementary materials: U–Pb and Lu–Hf data, and concordia diagrams and discussion of effects of discordance are available at

34 citations

Cites methods from "Tests and confidence bands for biva..."

  • ...Methods to assign confidence limits to bivariate distributions have been described in the literature (e.g. Adler et al. 1990), but because sorting of the bivariate data is required, results may not be unique and the geologically important relationship between age and Hf isotope composition will be…...


01 Jan 2007
Abstract: Let Xi,X2,. • • ,Xn be a random sample of size n from a continuous distribution F and Y1}Y2,..., Ym be a random sample of size m from a continuous distribution G. One of the ways to test the hypothesis of equality of F and G against the alternative that F < G when both distributions are univariate is to perform a precedence test -a test that not only requires only a portion of the samples, but which is distribution-free under the null hypothesis. The initial purpose of this thesis was to extend the notion of a precedence test to higher dimensions. In doing so, we found two different tests that are appropriate for both partial and complete data sets. These tests are based on two different extensions of the usual definition of a procentile-procentile plot -which is closely related to the precedence test statistic on the lineto the plane. The first of the above mentioned extensions involves the contours formed by the distribution function F; the second of our tests uses the marginal quantiles of F. For both extensions of the empirical p — p plot, we have proven a Glivenko-Cantelli type of result. Also, we have developed their asymptotic convergence to Gaussian limits. The choice between tests based on these two plots depends on the kind of information that the data of our experiment generates. All the results presented here, although mostly presented for !ft, are valid for 3?-valued data.

2 citations

Cites background or methods from "Tests and confidence bands for biva..."

  • ...Again, simulated values for JT+X can be found in [2] and the resulting critical values are in table 5....


  • ...We can find in [2] simulated values of dt and -£ for various values of n and m; the values for our test for different values of n and m are shown in Table 5....


  • ...In view of this two results, Adler, Brown and Lu [2] found, using simulation techniques, several critical values which we will make use of next....


  • ...We define the following notation (as in [2]): \F — G\ = sup(F(a;) — G(x)), \F - G\~ = sup(G(x) - F(x)), and \F - G\ = max( |F - G\, \F - G\~) for x e 3?(2)...


  • ...Again, we can use [2] to find d\ and - 1 ; these values for our test for several levels of significance and combinations of n and m are shown in Table 5....


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Abstract: The authors study the problem of testing whether the distribution function (d.f.) of the observed independent chance variables $x_1, \cdots, x_n$ is a member of a given class. A classical problem is concerned with the case where this class is the class of all normal d.f.'s. For any two d.f.'s $F(y)$ and $G(y)$, let $\delta(F, G) = \sup_y | F(y) - G(y) |$. Let $N(y \mid \mu, \sigma^2)$ be the normal d.f. with mean $\mu$ and variance $\sigma^2$. Let $G^\ast_n(y)$ be the empiric d.f. of $x_1, \cdots, x_n$. The authors consider, inter alia, tests of normality based on $ u_n = \delta(G^\ast_n(y), N(y \mid \bar{x}, s^2))$ and on $w_n = \int (G^\ast_n(y) - N(y \mid \bar{x}, s^2))^2 d_yN (y \mid \bar{x}, s^2)$. It is shown that the asymptotic power of these tests is considerably greater than that of the optimum $\chi^2$ test. The covariance function of a certain Gaussian process $Z(t), 0 \leqq t \leqq 1$, is found. It is shown that the sample functions of $Z(t)$ are continuous with probability one, and that $\underset{n\rightarrow\infty}\lim P\{nw_n < a\} = P\{W < a\}, \text{where} W = \int^1_0 \lbrack Z(t)\rbrack^2 dt$. Tables of the distribution of $W$ and of the limiting distribution of $\sqrt{n} u_n$ are given. The role of various metrics is discussed.

189 citations

Journal ArticleDOI

110 citations

Journal ArticleDOI
Abstract: The confidence sets for a $q$-dimensional distribution studied in this paper have several attractive features: affine invariance, correct asymptotic level whatever the actual distribution may be, numerical feasibility, and a local asymptotic minimax optimality property. When dimension $q$ equals one, the confidence sets reduce to the usual Kolmogorov-Smirnov confidence bands, except that critical values are determined by bootstrapping.

68 citations

Journal ArticleDOI
Abstract: We consider multivariate empirical processes $X_n(t) := \sqrt n (F_n(t) - F(t))$, where $F_n$ is an empirical distribution function based on i.i.d. variables with distribution function $F$ and $t \in \mathbb{R}^k$. For $X_F$ the weak limit of $X_n$, it is shown that $c(F, k)\lambda^{2(k-1)}e^{-2\lambda^2} \leq P\big\{\sup_t X_F(t) > \lambda\big\} \leq C(k)\lambda^{2(k-1)}e^{-2\lambda^2}$ for large $\lambda$ and appropriate constants $c, C$. When $k = 2$ these constants can be identified, thus permitting the development of Kolmogorov--Smirnov tests for bivariate problems. For general $k$ the bound can be used to obtain sharp upper-lower class results for the growth of $\sup_tX_n(t)$ with $n$.

32 citations

Journal ArticleDOI
Abstract: We describe a general class of multivariate infinitely divisible distributions and their related stochastic processes. Then we prove inequalities which are the analogs of Slepian's inequality for these distributions. These inequalities are applied to the distributions of $M/G/\infty$ queues and of sample cumulative distribution functions for independent multivariate random variables.

10 citations