On the Detection of Contemporaneous Relationships Among Multiple Time Series

A study is carried out of a sampling from a half-normal and exponential distributions to develop a test of hypothesis on the mean. Although these distributions are similar, the corresponding uniformly most paerful test statistics are different. The exact distributions of these statistics my be written in terms of the incomplete gamma function. If the experimental data my be fitted by either distributions, it is advisable to carryout the test based on the half-normal distribution as it is generally more powerful than the one based on the exponential one.     

Testing for departures from nominal dispersion in generalized nonlinear models with varying dispersion and/or additive random effects

This paper discusses the tests for departures from nominal dispersion in the framework of generalized nonlinear models with varying dispersion and/or additive random effects. We consider two classes of exponential family distributions. The first is discrete exponential family distributions, such as Poisson, binomial, and negative binomial distributions. The second is continuous exponential family distributions, such as normal, gamma, and inverse Gaussian distributions. Correspondingly, we develop a unifying approach and propose several tests for testing for departures from nominal dispersion in two classes of generalized nonlinear models. The score test statistics are constructed and expressed in simple, easy to use, matrix formulas, so that the tests can easily be implemented using existing statistical software. The properties of test statistics are investigated through Monte Carlo simulations.     

A Generalization of Ahmad's Class of Mann–Whitney–Wilcoxon Statistics

In the spirit of the recent work of Ahmad (1996) this paper introduces another class of Mann–Whitney–Wilcoxon test statistics. The test statistic compares the r th and s th powers of the tail probabilities of the underlying probability distributions. The choice of r + s = 4 improves the Pitman efficiency for uniform, exponential, lognormal and normal distributions and keeps the same efficiency as the Mann–Whitney–Wilcoxon test for logistic and double exponential distributions. The two-sample test is modified for the one-sample problem with symmetric underlying distribution.     

New Developments in Statistical Computing: Short Notices and Reviews of New Program Packages,other Sources of Information,and New Products

Graphs are presented on which the empirical distribution function can be plotted to test the assumption of normality by the Lilliefors test. A second set of graphs is presented for using the Lilliefors test on exponential distributions. The graphs allow for tests at the 10 percent, 5 percent, and 1 percent levels of significance. Use of these graphs makes it easy for students in a first course in statistics to test normal and exponential distributions without having to unravel the mystery associated with putting together a chi-squared goodness-of-fit test.     

A Diagnostic Test for Normality Within the Power Exponential Family

This article develops the locally uniformly most powerful unbiased Lagrange multiplier test of normality of regression disturbances within the family of power exponential distributions. The small sample power properties of the test are compared in a Monte Carlo study with 6 well-known tests across 12 alternative nonnormal distributions. In addition, the finite sample power properties for nonnormal alternatives within the power exponential family are summarized by estimating response surfaces. The results suggest that the proposed text is computationally convenient and possesses relatively attractive power properties even against alternatives outside the power exponential family.     

Power of a test for equality of k exponential location parameters

Kambo and Awad (1985) defined a test statistic based on doubly censored samples to test the equality of location parameters of K exponential distributions when their common scale parameter is unknown. The power function of the test is derived in this paper and some special cases are studied.     

Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

Randomized confidence intervals of a parameter for a family of discrete exponential type distributions

For a family of one-parameter discrete exponential type distributions, the higher order approximation of randomized confidence intervals derived from the optimum test is discussed. Indeed, it is shown that they can be asymptotically constructed by means of the Edgeworth expansion. The usefulness is seen from the numerical results in the case of Poisson and binomial distributions.     

A new estimator of Kullback–Leibler information and its application in goodness of fit tests*

We propose here a general statistic for the goodness of fit test of statistical distributions. The proposed statistic is constructed based on an estimate of Kullback–Leibler information. The proposed test is consistent and the limiting distribution of the test statistic is derived. Then, the established results are used to introduce goodness of fit tests for the normal, exponential, Laplace and Weibull distributions. A simulation study is carried out for examining the power of the proposed test and to compare it with those of some existing procedures. Finally, some illustrative examples are presented and analysed, and concluding comments are made.     

Likelihood ratio test for testing equality of location parameters of two exponential distributions from doubly censored samples

The Likelihood Ratio (LR) test for testing equality of two exponential distributions with common unknown scale parameter is obtained. Samples are assumed to be drawn under a type II doubly censored sampling scheme. Effects of left and right censoring on the power of the test are studied. Further, the performance of the LR test is compared with the Tiku(1981) test.     

Tests for generalized exponential laws based on the empirical Mellin transform

Recently, many standard families of distributions have been generalized by exponentiating their cumulative distribution function (CDF). In this paper, test statistics are constructed based on CDF–transformed observations and the corresponding moments of arbitrary positive order. Simulation results for generalized exponential distributions show that the proposed test compares well with standard methods based on the empirical distribution function.     

Attribute control chart for some popular distributions

The binomial distribution is often used to display attribute control data. In this paper, a statistical model is settled for attribute control chart under truncated life test. By Burr X & XII, inverse Gaussian (IG), and exponential lifetime-truncated distributions, a Shewhart-type attribute control chart is built to display the data. The performance of attributed control chart constructed on truncated life test is evaluated by average run length, which compares the performance of all distributions. Our study arranges that IG is better distribution among all.     

Testing exponentiality based on Type-I censored data

The problem of goodness-of-fit for the exponential distribution when the available data are subject to Type-I censoring is discussed here. A test procedure is proposed in this case that exhibits more power as compared to existing methods. The power of the proposed test is assessed for several alternative distributions by means of Monte Carlo simulations. Finally, the proposed test is illustrated with a real data set.     

On the choice of precedence tests

The best precedence test (BPT) is derived for testing the hypothesis that the lifetimes of two types of items on test have the same distribution. The test has maximum power in the class of the Lehmann type of alternatives F - 1 - (1-G) , A > 1, where F and G are probability distributions of the lifetimes of two types of items on test. This class includes exponential distributions, the Weibull distribution differing only in scale and distributions with proportional hazard rates. Exact power of the BPT is compared with other nonparametrie and parametric tests. The test may terminate before all the lifetimes of the items on test are recorded. In comparing with competing tests of equal size, the power functions are similar but a considerable number of items can be saved and the time on test can be reduced by using the BPT     

Testing exponentiality against L-distributions

In this paper we propose a test statistic for testing exponentiality versus L-class of life distributions. This test is based on an estimate of a functional of the cdf which discriminates between the exponential and L-family.     

ON TESTING THE EQUALITY OF LOCATION PARAMETERS IN k CENSORED EXPONENTIAL DISTRIBUTIONS

Kumar and Patel (1971) have considered the problem of testing the equality of location parameters of two exponential distributions on the basis of samples censored from above, when the scale parameters are the same and unknown. The test proposed by them is shown to be biased for n₁≠n₂, while for n₁=n₂ the test possesses the property of monotonicity and is equivalent to the likelihood ratio test, which is considered by Epstein and Tsao (1953) and Dubey (1963a, 1963b). Epstein and Tsao state that the test is unbiased. We may note that when the scale parameters of k exponential distributions are unknown the problem of testing the equality of location parameters is reducible to that of testing the equality of parameters in k rectangular populations for which a test and its power function were given by Khatri (1960, 1965); Jaiswal (1969) considered similar problems in his thesis. Here we extend the problem of testing the equality of k exponential distributions on the basis of samples censored from above when the scale parameters are equal and unknown, and we establish the likelihood ratio test (LET) and the union-intersection test (UIT) procedures. Using the results previously derived by Jaiswal (1969), we obtain the power function for the LET and for k= 2 show that the test possesses the property of monotonicity. The power function of the UIT is also given.     

Invariance properties of some classical tests for exponentiality

We examine some classical tests for the exponentiality of independent, identically distributed data. We show that a large number of these tests have the same distribution if the data follow certain multivariate Liouville distributions. These results highlight the role that the assumption of independence plays in the behavior of the classical test statistics. We use these results to derive some characterizations of the exponential distributions among the Liouville distributions.     

A new class of weighted exponential distributions

Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized exponential distributions.     

Goodness-of-fit test based on empirical characterisitc function

This paper discusses a goodness-of-fit test that uses the integral of the squared modulus of the difference between the empirical characteristic function of the sample data and the characteristic function of the hypothesized distribution. Monte Carlo procedures are employed to obtain the empirical percentage points for testing the fit of normal, logistic and exponential distributions with unknown location and scale parameters. Results of Monte Carlo power comparisons with other well-developed goodness-of-fit tests are summarized. Tne proposed test is shown to have superior power for testing the fit of the logistic distibotion (for moderate sample sizes) against a wide range of alternative distributions.     

Hypothesis testing on the common location parameter of several shifted exponential distributions: A note

This note deals with hypothesis testing on the common location parameter of several shifted exponential distributions with unknown and possibly unequal scale parameters. No exact test is available for the above mentioned problem; and one does not have the luxury of applying the asymptotic Chi-square test for the likelihood ratio test statistic since the distributions do not satisfy the usual regularity conditions. Therefore, we have proposed a few approximate tests based on the parametric bootstrap method which appear to work well even for small samples in terms of attaining the level. Powers of the proposed tests have been provided along with a recommendation of their usage.