Search

This article proposes new tests of randomness for innovations in a large class of time series models. These tests are based on functionals of empirical processes constructed from either the model residuals or their associated ranks. The asymptotic behavior of these processes is determined under the null hypothesis of randomness. The limiting distributions are seen to be independent of estimation errors under appropriate regularity conditions. Several test statistics are derived from these processes; the classical Brock, Dechert, and Scheinkman statistic and a rank-based analog are included as special cases. Because the limiting distributions of the rank-based test statistics are margin-free, their finite-sample p values can be easily calculated by simulation. Monte Carlo experiments show that these statistics are quite powerful against several classes of alternatives.

In many applications, it is of interest to test whether a spatial point pattern is randomly generated after adjusting for an underlying spatial inhomogeneity. A great variety of different test statistics have been proposed for this purpose by scientists in different fields; these are reviewed in this article. Despite apparent dissimilarities in terms of their original formulations, most of these statistics can be placed into one general framework of which they are special cases. This makes it easier to see exactly how they relate to one another and also to determine which test to use for a particular application. The general framework can also be used for proposing new tests by combining properties of existing tests, for developing theoretical foundations for these types of test statistics, for doing structured comparative evaluations, and for software development.

In medical applications there is interest in whether deaths or other events occur randomly throughout the day, as opposed to in a definite circadian pattern. At times, knowledge of the circadian phenomenon will suggest that a single cluster of events should occur during a specific period during the day, such as mid-morning. We propose a test that incorporates such knowledge. The test can be viewed as the likelihood ratio test for a restricted alternative for data from the von Mises (i.e., circular normal) distribution. The test's power is compared to a commonly used test of randomness that ignores when a peak is likely to occur. Methods for dealing with interval censored data are discussed, and an example is given. A Bayesian alternative is also explored.

Psychologists often want to detect category structure in subjects' free recall protocols. Although runs tests based on the binomial distribution are commonly used to detect nonrandomness within a sequence, many research situations require tests based on the multinomial distribution. We propose a test of randomness versus clustering based on the number of runs in multinomial data. We illustrate its use with data from a mass communication experiment using a constrained free recall procedure.

One notion of language development suggests that infants' and toddlers' vocalizations are produced at random with no particular patterns or organization. In this article two tests (the pairs test and the gap test) that are used to check the randomness properties of computer-generated pseudorandom numbers are modified to study the sounds produced by infants 13 months of age and toddlers 24 months of age. Deviations from the expected patterns that would be observed if the children did produce sound segments at random suggest features of the development of language skills. The information provided by the lack of fit of the model corresponding to the pairs test is used to develop models that capture the common patterns in transitions among sound classifications in children's early speech.

Randomness, mathematics, and computers. Fundamental concepts of probability. Random variables and distribution. The multivariate environment. Concepts in sampling. Analysis of sample data: estimation methods. Analysis of sample data: making decisions. Analysis of discrete event stochastic processes.

A test of independence in a (r × c) contingency table using a specified alternative hypothesis is proposed. The test is created using Neyman's Cα procedure and the approximate power computed. This test is useful when alternatives of a specific “slippage” type can be hypothesized. The usual χ2 test tends to detect all departures from randomness indiscriminately and thus not employ the added directional a priori information.

The concept of power for monotone invariant clustering procedures is developed via the possible partitions of objects at each iteration level in the obtained hierarchy. At a given level, the probability of rejecting the randomness hypothesis is obtained empirically for the possible types of partitions of the n objects employed. The results indicate that the power of a particular hierarchical clustering procedure is a function of the type of partition. The additional problem of estimating a “true” partition at a certain level of a hierarchy is discussed briefly.

The method presented here assumes that the variable under consideration is a weighted moving average of annual values, with weights to be estimated by means of quadratic programming. This multivariate (MV) method is used to estimate quarterly income figures on the basis of annual income data in quarterly consumption functions for the U.S. and the Netherlands. The MV method appears to perform slightly better than the competing single-variate (SV) smoothing methods of Feibes, Boot and Lisman with respect to parameter estimates, and smallness and randomness of the differences between calculated and observed values of consumption.

Statistics based on a theory of random graphs have been proposed as an analytic aid to assess the randomness of a clustered structure. Probability tables for two such statistics are tabulated. Exact values of Pn,v , the cumulative probabilities of the minimum number of edges needed to connect a random graph, are tabulated for n = 10(1)30(5)80-(10)100. Exact and approximate values of En,v, the expected number of components in a random graph with n vertices and v edges, are tabulated for n = 10(1)30(5)100.

Optimal rank-based procedures were derived in Hallin, Ingenbleek, and Puri (1985, 1987) and Hallin and Puri (1988) for some fundamental testing problems arising in time series analysis. The optimality properties of these procedures are of an asymptotic nature, however, whereas much of the attractiveness of rank-based methods lies in their small-sample applicability and robustness features. Accordingly, the objective of this article is twofold: (a) a study of the finite-sample behavior of the asymptotically optimal tests for randomness against first-order autoregressive moving average dependence proposed in Hallin et al. (1985), both under the null hypothesis (tables of critical values) and under alternatives of serial dependence (evaluation of the power function), and (b) an (heuristic) investigation of the robustness properties of the proposed procedures (with emphasis on the identification problem in the presence of “outliers”). We begin (Sec. 2) with a brief description of the rank-based measu...

Most traditional applications of runs tests have entailed bivariate runs distributions. Extensions to more general runs tests for randomness have been hindered by the lack of exact critical values for distributions with more than two kinds of elements. Combinatorial recurrences that can be used to compute exact significance levels for small sample sizes are presented for runs distributions for ν ≥ 2 kinds of elements. Critical values for small sample sizes are tabled for ν = 2(1)6. More extensive tables are available from the author.

Although rank tests for randomness were proposed in the literature as early as 1943, no such test has gained wide acceptance comparable to, say, Spearman's rho test. This may be due to the lack of small-sample theory and of tables of critical values to enable such a test to be carried out on small samples. In this article, we consider the rank version of the von Neumann ratio statistic and we obtain the critical values of this statistic under the randomization hypothesis. In a Monte Carlo experiment we then show that the resulting nonparametric test for randomness has far greater power than the test based on the number of runs up and down. Moreover, under normality, its power vis-a-vis the normal theory von Neumann ratio test is also very good. It is therefore suggested that with the tables presented in this article, the rank von Neumann ratio test for randomness provides an easy and powerful alternative to nonparametric tests now in common use.

There exist several tests for autocorrelation and randomness in multiple time series. Unfortunately, the exact distributions of these statistics are unknown and the asymptotic distributions that are known do not provide adequate approximation to the exact ones in small samples. In this article, the test statistics are modified. Asymptotically, these modified statistics are equivalent to their original counterparts; however, it is found that the asymptotic distributions of these statistics provide adequate approximation to the exact ones in relatively small samples and possibly when the time series are nonnormal. The adequacy of the approximations is examined by simulation experiments. The original test statistics are based on sample lag cross-covariances, autocovariances, cross-correlations, and autocorrelations standardized by their asymptotic means and covariances. The modified statistics are obtained when the asymptotic means and covariances in the standardization are replaced by the exact mea...

The sample Lorenz curve provides a powerful, easily computed goodness-of-fit test for exponentiality which does not depend on the unknown scale parameter. Exact critical values and asymptotic results are given, which can also be used to test the related hypothesis of randomness on an interval. General limit theorems show that the sample Lorenz curve converges almost surely to the population Lorenz curve and that the standardized Lorenz statistic converges to normality provided the underlying random variable has finite variance. Asymptotic relative efficiencies and Monte Carlo power estimates are also given.

This article explores the identical distribution hypothesis for stock-price changes through a set of optimum non-parametric tests for randomness against location- and scale-shift alternatives. An application of these tests to the daily, weekly, monthly, and quarterly rates of return from 1966 through 1976 for each of the first 15 stocks in the Dow Jones Industrial Average reveals that these series may have constant location parameters throughout the sample period, but the stability of their scale parameters is questionable.

It is sometimes important to test the hypothesis that a given set of N unit vectors, representing the directions of a physical quantity, comes from a population of vectors which are randomly oriented in two or three dimensions, as opposed to a population in which the vectors are clustered to some degree. For three dimensions, Watson has proposed the length of the sample resultant as a suitable statistic, and given the appropriate tests. An improved table is given for the application of Watson's tests. The power of the test against a specific alternative distribution, used in geophysical problems, is used to give a table of sample sizes necessary to detect a degree of clustering.

In the case of moving average schemes or what are termed the m-dependent stochastic processes, the successive observations in the series are not stochastically independent. As a result, the usual sign test, the test based on the rank correlation τ (for randomness against trend alternatives), and the test proposed by Moore and Wallis [9] and further developed by Goodman and Grunfeld [2] (for the comovement between two time series) are not valid and require some modifications. Accordingly, the modified forms of these non-parametric tests are studied here in detail, and their applications considered.

Two procedures are suggested for testing the hypothesis of randomness against alternatives of a linear trend and a cyclical trend, respectively. Both procedures reduce to tests of the hypothesis that the parameter of a binomial distribution is p 0 against the alternative that it is p 1 and can, therefore, be easily sequentialized. The expected number of observations required by the sequential test against linear trend is smaller than the corresponding number of observations of other tests in current use. * This research was supported by the United States Air Force under Contract AF 18(600)-778, monitored by the Office of Scientific Research, Air Research and Development Command. A more theoretical discussion of the two tests—as well as others—is given in the author's report to the Office of Scientific Research, “Sequential Tests of Randomness,” dated December, 1953.

Speculative markets have often been characterized as exhibiting random price behavior where prices tend to move in a series of unconnected steps. The hypothesis of random behavior in commodity markets is investigated by three tests of randomness for a total of 167 futures contracts representing a variety of commodities. The autocorrelation function, spectral density function, and integrated period-ogram are estimated for each futures contract. The empirical results indicate that futures price behavior cannot generally be characterized by a simple random walk model, though several commodities appear to yield random behavior.

Some random normal number generators that were satisfactory for 36-bit-word computers are inadequate for equipment such as the IBM System/360, using a 32-bit-word. A generator is presented and shown to be satisfactory relative to several criteria for testing normality and randomness.

Tests for randomness are described, when the data consists of points on the circumference of a unit circle, and the alternative to the uniform distribution (randomness) is either a unimodal or a bimodal distribution: the von Mises distribution or an adaptation. Tables of significance points are given for the test statistics, and the power is used to give a table of sample sizes needed to detect a given degree of clustering, measured by the parameters of the distributions. This paper is a close parallel, for two dimensions, to Stephens [15], which gave corresponding tests and tables for three dimensions.

S EVERAL writers, notably Hotelling and Pabst [5], have explicitly assumed that the relative efficiency of two test statistics is to be measured by their estimating efficiencies. While this seems reasonable, it is by no means obvious, since if the two tests are consistent, the ratio of their powers against any fixed alternative hypothesis must tend to unity with increasing sample size n, and it may easily be shown that for any n, the less efficient estimator may provide a more powerful test (Sundrum [14]). Pitman [11] has proposed a measure of the asymptotic relative efficiency of consistent tests. Given that the two statistics, t1 and t2, have normal limit distributions with variances of order n-1, and that certain general regularity conditions are satisfied, he considered a limiting process in which the alternative hypothesis Hi differs from the null hypothesis Ho by a quantity of order n-12, so that as n increases, Hi tends to Ho. Under these conditions, he showed that the reciprocal of the ratio of sample sizes required to attain equal power against the same alternative was, in the limit

We point out that autocorrelated error terms require modification of the usual methods of estimation and prediction; and we present evidence showing that the error terms involved in most current formulations of economic relations are highly positively autocorrelated. In doing this we demonstrate that when estimates of autoregressive properties of error terms are based on calculated residuals there is a large bias towards randomness. We demonstrate how much efficiency may be lost by current methods of estimation and prediction; and we give a tentative method of procedure for regaining the lost efficiency. * We wish to express our thanks for the considerable assistance we have received from Richard Stone.

Recently, Marsaglia (1984, Proceedings of the Sixteenth Symposium on the Interface) noted a trend toward increasingly more demanding use of random number generators. After observing that, for current uses, generators must pass increasingly more sophisticated tests for randomness in order to provide acceptable results, Marsaglia described a class of “combination” generators and recommended two specific combination generators that passed an extensive battery of stringent tests for randomness. The present article describes an alternative class of generators, which produce random values by compounding, or intermixing, several different sequences of random values. These “compound” generators are easier to implement, especially on smaller computers, than are Marsaglia's recommended generators. Five specific compound generators, two standard generators, and Marsaglia's two combination generators were subjected to several tests for randomness, including Marsaglia's stringent tests. The standard generator...

A statistical hypothesis testing procedure is maximin if the procedure maximizes the minimum power over the space of the alternative hypothesis separated from the space of the null hypothesis by an indifference zone. Given a multinomial distribution with k events, maximin tests of the homogeneity of the probabilities of the k multinomial events against ordered alternatives are developed.

Let {Xj } be a sequence of independently distributed real valued random variables, and let Rj denote the rank of Xj among X 1, ···, Xj for j = 1, ···, n. The Rj 's are called “sequential ranks.” A class of unbiased tests, based on the sequential ranks, is given for testing the hypothesis that the random variables are identically distributed against the alternative hypothesis that they are stochastically ordered.