Kuiper Statistic Research Articles

Percentage Points of the Asymptotic Distributions of One and Two Sample Kuiper Statistics for Truncated or Censored Data

A table of percentage points of the asymptotic distribution of the one sample truncated K uiper statistic is provided. The table is useful for goodness of fit problems involving truncated or censored data, and in two sample testing problems.

Percentage Points of the Asymptotic Distributions of One and Two Sample Kuiper Statistics for Truncated or Censored Data

Percentage Points of the Asymptotic Distributions of One and Two Sample Kuiper Statistics for Truncated or Censored Data

On some properties of the scan statistic on the circle and the line

The scan statistic is defined as the supremum of a particular continuous-time stochastic process, and is used as a test statistic for testing uniformity against a simple clustering type of alternative. Its distribution under the null hypothesis is investigated and weak convergence of the stochastic process to the appropriate Gaussian process is proved. An interesting link is forged between the circular scan statistic and Kuiper's statistic, which rids us of the trouble of estimating a nuisance parameter. Distributions under the alternative are then derived, and asymptotic power comparisons are made.

On some properties of the scan statistic on the circle and the line

The scan statistic is defined as the supremum of a particular continuous-time stochastic process, and is used as a test statistic for testing uniformity against a simple clustering type of alternative. Its distribution under the null hypothesis is investigated and weak convergence of the stochastic process to the appropriate Gaussian process is proved. An interesting link is forged between the circular scan statistic and Kuiper's statistic, which rids us of the trouble of estimating a nuisance parameter. Distributions under the alternative are then derived, and asymptotic power comparisons are made.

Limiting Distributions of Kolmogorov-Smirnov Type Statistics Under the Alternative

Let $X_1, X_2, \cdots$ be a sequence of independent and identically distributed random variables with the common distribution being uniform on [0, 1]. Let $Y_1, Y_2, \cdots$ be a sequence of i.i.d. variables with continuous $\operatorname{cdf}F(t)$ and with [0, 1] support. Let $F_n(t, \omega)$ denote the empirical distribution function based on $Y_1(\omega), \cdots, Y_n(\omega)$ and let $G_m(t, \omega)$ the empirical $\operatorname{cdf}$ pertaining to $X_1(\omega), \cdots, X_m(\omega)$. Let $\sup_{0\leqq t \leqq 1}|F(t) - t| = \lambda$ and $D_n = \sup_{0 \leqq t \leqq 1}|F_n(t, \omega) - t|$. The limiting distribution of $n^{\frac{1}{2}}(D_n - \lambda)$ is obtained in this paper. The limiting distributions under the alternative of the corresponding one-sided statistic in the one-sample case and the corresponding Smirnov statistics in the two-sample case are also derived. The asymptotic distributions under the alternative of Kuiper's statistic are also obtained.

Parameter estimation in the exponential distribution, confidence intervals and a monte carlo study for a goodness of fit test

For the one and two parameter exponential distribution minimum variance unbiased estimators and shortest confidence intervals are calculated. For testing goodness of fit we have used the Kuiper statistic. Critical values and power results of the test statistic are derived using Monte Carlo methods.

On the Kuiper test for normality with mean and variance unknown

Summary A modified form of the Kuiper statistic Vn is developed for testing the composite hypothesis that a sample of size n comes from a normal population with unspecified mean and variance. Its distribution is derived using Monte Carlo methods. Power comparison with the adjusted Kuiper test proposed by Louter and Koerts [6] indicates that our test is superior with respect to certain alternatives.

On the Kuiper test for normality with mean and variance unknown

Summary If one wants to test the hypothesis as to whether a set of observations comes from a completely specified continuous distribution or not, one can use the Kuiper test. But if one or more parameters have to be estimated, the standard tables for the Kuiper test are no longer valid. This paper presents a table to use with the Kuiper statistic for testing whether a sample comes from a normal distribution when the mean and variance are to be estimated from the sample. The critical points are obtained by means of Monte‐Carlo calculation; the power of the test is estimated by simulation; and the results of the powers for several alternative distributions are compared with the estimated powers of the Kolmogorov‐Smirnov test.

Exact Bahadur Efficiencies for the Kolmogorov-Smirnov and Kuiper One- and Two-Sample Statistics

1. Introduction. In 1960, Bahadur [3] proposed two measures of the asymptotic performance of tests: one approximate measure, based on the limiting distribution of the test statistic; and one exact, based on the limiting form of the probability of a large deviation of the statistic from its asymptotic mean. Although knowledge of the exact measure is more desirable, it is often difficult to compute, whereas the approximate measure is usually trivially available. It is to be hoped that the approximate measure will nearly always be a good approximation to the exact measure, and there is considerable evidence in support of this conjecture, but counterexamples do exist (see e.g. [1], p. 20). This paper explores the question for the Kolmogorov-Smirnov (K-S) and Kuiper statistics-two closely related measures of goodness-of-fit based on deviations of the sample distribution functions from the null case-in one- and two-sample situations. In the case of the weighted one-sample K-S statistic and the two-sample Kuiper statistic, it is possible to obtain exact measures although the corresponding approximate measures are not available. The relative efficiency (in a sense defined in Section 2) of the weighted one-sample K-S statistic to the unweighted K-S statistic is computed in a number of cases, and the relative efficiency of the Kuiper statistic to the unweighted K-S statistic is also examined, from which it appears that the Kuiper statistic is always at least as good (and often much better) than the K-S statistic. In the cause of brevity, a great deal of the theory and all the computational detail (generally very tedious) has been omitted. Much of it is available in [1]. However, a short resum6 of the theory of the Bahadur measures of performance and their properties is given in Section 2 for the sake of completeness, although the details here are also omitted.