Marc Noe

Bio: Marc Noe is an academic researcher. The author has contributed to research in topics: Chi distribution & Distribution function. The author has an hindex of 2, co-authored 3 publications receiving 105 citations.

The Calculation of Distributions of Two-Sided Kolmogorov-Smirnov Type Statistics

Abstract: Let $X_1^n \leqq X_2^n \leqq \cdots \leqq X_n^n$ be the order statistics of a size $n$ sample from any distribution function $F$ not necessarily continuous. Let $\alpha_j, \beta_j, (j = 1,2, \cdots, n)$ be any numbers. Let $P_n = P(\alpha_j < X_j^n \leqq \beta_j, j = 1,2, \cdots, n)$. A recursion is given which calculates $P_n$ for any $F$ and any $\alpha_j, \beta_j$. Suppose now that $F$ is continuous. A two-sided statistic of Kolmogorov-Smirnov type has the distribution function $P_{\mathrm{KS}} = P\lbrack\sup n^{\frac{1}{2}}\psi(F) \cdot |F^n - F| \leqq \lambda\rbrack$, where $F^n$ is the empirical distribution function of the sample and $\psi(x)$ is any nonnegative weight function. As $P_{\mathrm{KS}}$ has the form $P_n$, its calculation as a function of $\lambda$ can be carried out by means of the recursion. This has been done for the case $\psi(x) = \lbrack x(1 - x)\rbrack^{-\frac{1}{2}}$. Curves are given which represent $\lambda$ versus $1 - P_{\mathrm{KS}}$ for $n = 1,2, 10, 100$. From additional computations, the precision of a truncated development of $1 - P_{\mathrm{KS}}$ in powers of $\lambda^{-2}$ has been determined.

Confidence regions and tests for a change-point in a sequence of exponential family random variables

Abstract: SUMMARY Maximum likelihood methods are used to test for a change in a sequence of independent exponential family random variables, with particular emphasis on the exponential distribution. The exact null and alternative distributions of the test statistics are found, and the power is compared with a test based on a linear trend statistic. Exact and approximate confidence regions for the change-point are based on the values accepted by a level x likelihood ratio test and a modification of the method proposed by Cox & Spj0tvoll (1982). The methods are applied to a classical data set on the time intervals between coal mine explosions, and the change in variation of stock market returns. In both cases the confidence regions for the change-point cover historical events that may have caused the changes.

Boundary-crossing probabilities for the Brownian motion and poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test

Abstract: Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t 1)w(t 2)}= min (t 1, t 2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.

Confidence Bands for Survival Functions With Censored Data: A Comparative Study

Abstract: Consider the problem of obtaining simultaneous confidence bands for the survival function S(x) when the data are arbitrarily right censored. The usual pointwise confidence intervals based on Greenwood's variance formula can be adapted to yield a large-sample confidence band. This band has, in a certain sense, equal precision at each point of S(x). It is compared with the censored versions of the Kolmogorov band and the Renyi band. The comparisons are made in terms of the widths and the adequacy of large-sample approximations and are carried out under various censoring models and degrees of censoring. The bands are illustrated by applying them to data from a mechanical-switch life test.

The stabilized probability plot

Abstract: SUMMARY The stabilized probability plot is introduced. An attractive feature of the plot that enhances its interpretability is that the variances of the plotted points are approximately equal. This prompts the definition of a new and powerful goodness-of-fit statistic Dsp which, analogous to the standard Kolmogorov-Smirnov statistic D, is defined to be the maximum deviation of the plotted points from their theoretical values. Using either D or Dsp it is shown how to construct acceptance regions for QQ,, Pp and the new plots. Acceptance regions can help remove much of the subjectivity from the interpretation of these probability plots.

Goodness-of-fit tests via phi-divergences

Abstract: A unified family of goodness-of-fit tests based on $\phi$-divergences is introduced and studied. The new family of test statistics $S_n(s)$ includes both the supremum version of the Anderson--Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47--59] as special cases ($s=2$ and $s=1$, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47--59] and Wellner and Koltchinskii [High Dimensional Probability III (2003) 321--332. Birkhauser, Basel] for the Berk--Jones statistic applies to the whole family of statistics $S_n(s)$ with $s\in[-1,2]$. On the side of power behavior, we study the test statistics under fixed alternatives and give extensions of the ``Poisson boundary'' phenomena noted by Berk and Jones for their statistic. We also extend the results of Donoho and Jin [Ann. Statist. 32 (2004) 962--994] by showing that all our new tests for $s\in[-1,2]$ have the same ``optimal detection boundary'' for normal shift mixture alternatives as Tukey's ``higher-criticism'' statistic and the Berk--Jones statistic.