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Standard nonparametric prediction intervals for a future order statistic are obtained by taking the interval between two order statistics of the initial sample. We obtain improved prediction intervals by taking the shortest of two or more standard intervals.

In this paper, a new approximation of the marginal posterior distribution function is obtained. Moreover, for the location-scale model, by applying the shrinkage argument, a new approximation of the conditional distribution function of the signed likelihood ratio statistic given an ancillary statistic is derived from the approximated marginal posterior distribution.

We show that the fact that any uniformly minimum variance unbiased estimator (UMVUE) is uncorrelated with all unbiased estimators of zero implies certain conclusions of the Rao–Blackwell and Lehmann–Scheffe theorems. We also show that the conditional mean and variance of any UMVUE, given any ancillary statistic, must be independent of the ancillary. We discuss some implications of these results when a minimal sufficient statistic is incomplete and contains ancillaries.

A simple connection between the Bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant of the p ∗ formula–an approximate conditional density for the maximum likelihood estimate given an exact or an approximate ancillary statistic–was established in Barndorff-Nielsen and Cox (1984) . In this paper, the explicit form of the normalizing constant of the p ∗ formula for the scalar parameter model is derived. By change of variables, the mean and variance of the signed root log likelihood ratio statistic are obtained explicitly, and, hence, tail probabilities can be calculated from the standardized signed root log likelihood ratio statistic. Examples are used to illustrate the implementation and accuracy of the proposed method.

This paper considers a multivariate isotonic regression model in which the columns of the mean matrix are constrained by a quasi-ordering. For the test on the homogeneity of the means an ad hoc test statistic is derived. When the order restriction is confined to the elements of only one row, another statistic is obtained. A sufficient condition is established under which the p-value of the test is expressed through the distribution of the second statistic at a specified parameter point. This result not only simplifies the expression of the p-values but also makes the computation for the p-values by the Monte Carlo method possible.

The G statistic and its local version have been used extensively in spatial data analysis. The paper proves the asymptotic normality of the G statistic. Theorems in this paper imply that the regular permutation test for the G statistic is valid.

In this article we introduce a maximum scan score-type statistic for testing the null hypothesis that the observations are iid according to a specified distribution, against an alternative that the observations cluster within a window of unknown length. This statistic is a variable window scan statistic, based on a finite number of standardized fixed window scan statistics. Approximations for the significance level of this statistic are derived for 0–1 iid Bernoulli trials and for iid uniform observations on the interval [0,1). The advantage in using a maximum scan score-type statistic, rather than a single fixed window scan statistic, is that it is more effective in detecting window-type clustering of observations.

In the area of exact simultaneous confidence bands for all contrasts of three or more regression lines one quickly encounters a certain statistic that aids in the discovery of critical values for developing the confidence bands. In this paper, the exact distribution of this statistic is derived.

In this paper, we consider the problem of testing for a change of the marginal density of a strong mixing process. The test statistic is constructed based on the sequential kernel estimate. In order to derive the asymptotic distribution of the test statistic, we first show that a functional central limit theorem holds for the sequential density estimator under some regularity conditions. Based on the result, we show that the limiting distribution of the test statistic is a function of independent Brownian bridges.

The Anderson–Darling statistic is basically a weighted average of Pearson statistics. In this paper, we first propose to use other weights and next we generalize the Anderson–Darling statistic by inserting the Cressie-and-Read family of statistics into the Anderson–Darling statistic.

The similar test for an affine hypothesis in the exponential family is approximated by a score statistic, derived from the Edgeworth expansion of the conditional distribution given the minimal sufficient statistic under the null hypothesis.

This paper obtains a complete characterization of the previously unknown exact finite-sample distribution of a routine statistic for testing a linear regression function against a non-nested alternative. Simple approximations to the exact distribution are investigated.

Lachenbruch suggested the use of the hold-out method for assessing the chance of misclassification when a new observation is to be classified into one of two normal populations, using Anderson's classification statistic. The successive values of the Anderson statistic, for this method, are then correlated. This correlation in large samples is derived in this paper and its effect on the confidence interval for the true chance of misclassification is investigated.

A new family of goodness-of-fit statistics for discrete multivariate data is being introduced, which has the characteristic of linking Pearson's Chi-square statistic with the log-likelihood ratio statistic, thus leading to new compromises between these two classical test statistics. Preliminary simulation studies indicate that this family has further attractive elements, which are not “close” to standard statistics and which prove to have substantially higher power in specific situations. The new test procedure seems especially helpful with a null-hypothesis, where some cells have low probabilities of entry and an alternative hypothesis, which is “one-sided” with respect to those low-entry-cells: either postulating uniformly even lower or postulating uniformly higher probabilities of entry.

The modified likelihood ratio (MLR) test statistic is frequently used to detect unequal covariance matrices. We are concerned with examining this statistic with respect to departures from the usual i.i.d. assumptions on the sample data. In particular we characterize the joint covariance structure of two groups of multivariate normal observations so that the distribution of this MLR test statistic is identical to that under the usual assumption of independent identically distributed observations.

For testing the equality of normal variances with an increasing alternative, under the null hypothesis the likelihood ratio test statistic is asymptotically distributed as a mixture of chi-squared distributions. In this paper a Bartlett-type adjustment is proposed to improve the approximation of the null distribution of the likelihood ratio test statistic with an ordered alternative.

The bivariate Cramer—von Mises statistic has been proposed as a basis for goodness-of-fit testing for continuous bivariate distributions. This statistic can be expressed as the integral, over the unit square I2, of the square of the bivariate uniform empirical process. For some time it apparently went unnoticed that this statistic's value is not ‘origin-invariant’, i.e., its value depends on which corner of I2 is identified as the origin. Zimmerman (1993) proposed a modification of the bivariate Cramer—von Mises statistic that does not suffer from this defect. In this paper this statistic's limiting null distribution is derived and tabulated using the principal component decomposition method.

This note provides simple sufficient conditions for the non-existence of non-trivial ancillary statistics. The results show that in random sampling from some commonly-used families of discrete distributions there are no non-trivial ancillary statistics, and hence the statistical implication, under this setting, is there can be no exact distribution-free procedures.

Recent research of asymptotic robustness shows that the likelihood ratio (LR) test statistic for test-of-independence based on normal theory remains valid for a general case where only independence is assumed. In contrast, under elliptical populations the LR statistic is correct if a kurtosis adjustment is made. Thus, the LR statistic itself is available for the first case, whereas a certain correction is needed for the second framework, which is seriously inconvenient for practitioners. In this article, we propose an alternative adjustment to the LR statistic which can be utilized for both of the distribution families. The theory is derived in the context of general linear latent variate models.

Based on the concept of a logarithmic mean in two arguments, Pearson's Chi-square statistic is reformulated. Generalizations of the logarithmic mean are discussed, which immediately lead to new families of goodness-of-fit statistics. Thus it is possible to bridge the gap between Pearson's Chi-square statistic and the log-likelihood ratio statistic in a new way.

The asymptotic null distribution of a test statistic for stability over time of a correlation coefficient in a bivariate sequence is derived. The statistic is an adaptation of the classical Fisher's z.

This note gives a simple condition for the nonexistence of nontrivial ancillary statistics. The result shows that in sampling from a discrete exponential family with a regression model, usually there is no nontrivial ancillary statistic.

A mathc statistic considered by Khidr (1981) is interpreted in terms of crossings of the empirical and true distribution functions and a simpler alternate derivation of its distribution provided. This approach can also be used to obtain the distribution of a two-sample match statistic, considered earlier by Takacs (1971).

The expected value is computed for a statistic which is used to test that a specified unit vector is actually the first principal component vector. This suggests an adjusted statistic with a better chi-squared approximation.

A discrete time process on [0,1] considered in this paper is related to various problems involving two independent samples. In particular one may suggest a simple matching rule for the case of continuously generated samples and a goodness-of-fit test based on the number of unmatched elements. A recurrence formula for computing the exact distribution of this statistic and the asymptotic behavior of its expectation are found.