A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

On a modified test of equality of scale parameters of exponential distributions

In this paper, an exact distribution of a modifier likelihood ratio criterion for testing the equality of scale parameters of several two parameter exponential distributions is obtained for the case of unequal sample size in a computational form. A short table of critical values of the proposed statistic is also presented.     

Changepoint Estimation in a Segmented Linear Regression via Empirical Likelihood

For a segmented regression system with an unknown changepoint over two domains of a predictor, a new empirical likelihood ratio statistic is proposed to test the null hypothesis of no change. Under the null hypothesis of no change, the proposed test statistic is shown empirically to be Gumbel distributed with robust location and scale estimators against various parameter settings and error distributions. A power analysis is conducted to illustrate the performance of the test. Under the alternative hypothesis with a changepoint, the test statistic is utilized to estimate the changepoint between the two domains. A comparison of the frequency distributions between the proposed estimator and two parametric methods indicates that the proposed method is effective in capturing the true changepoint.     

Detection of a changepoint,a mean-shift accompanied with a trend change,in short time-series with autocorrelation

In this study, a changepoint model, which can detect either a mean shift or a trend change when accounting for autocorrelation in short time-series, was investigated with simulations and a new method is proposed. The changepoint hypotheses were tested using a likelihood ratio test. The test statistic does not follow a known distribution and depends on the length of the time-series and the autocorrelation. The results imply that it is not possible to detect autocorrelation and that the estimate of the autocorrelation parameter is biased. It is therefore recommended to use critical values from the empirical distribution for a fixed autocorrelation.     

SHRINKAGE PREDICTION IN THE EXPONENTIAL DISTRIBUTION WITH A PRIOR INTERVAL FOR THE SCALE PARAMETER

The author proposes the best shrinkage predictor of a preassigned dominance level for a future order statistic of an exponential distribution, assuming a prior estimate of the scale parameter is distributed over an interval according to an arbitrary distribution with known mean. Based on a Type II censored sample from this distribution, we predict the future order statistic in another independent sample from the same distribution. The predictor is constructed by incorporating a preliminary confidence interval for the scale parameter and a class of shrinkage predictors constructed here. It improves considerably classical predictors for all values of the scale parameter within its dominance interval containing the confidence interval of a preassigned level.     

A two-sample distribution-free scale test of the smirnov type

The two-sample, distribution-free statistics of Smirnov (1939) are used to define a new statistic. While the Smirnov statistics are used as a general goodness-of-fit test, a distribution-free scale test based on this new statistic is developed. It is shown that this new test has higher power than the two-sided Smirnov statistic in detecting differences in scale for some symmetric distributions with equal means/medians. The critical values of the proposed test statistic and its limiting distribution are given     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

On a life testing problem involving the two parameter exponential distribution

For a hypothesis testing problem involving the location and scale parameters of an exponential distribution, Perng (1977) proposed a test procedure based on the first r out of n observed failure times. In this paper the likelihood ratio test is determined, critical values are provided and the asymptotic null distribution is determined. An alternate test based on an F statistic is also proposed and the critical regions and power functions of the procedures are compared.     

A Goodness-of-fit Test for Exponentiality Based on the Memoryless Property

In this investigation a test of goodness of fit for exponentiality is proposed. This procedure applies equally whether the scale and/or the location parameters of the distribution are known or not. The limiting null and non-null distributions of the test statistic are normal under minimal conditions. Monte Carlo critical values for small sample sizes are given and the power of the test is calculated for various alternatives showing that it compares favourably relatively to other more complicated published procedures.     

Tests of fit for the Gumbel distribution: EDF-based tests against entropy-based tests

In this article, we propose some tests of fit based on sample entropy for the composite Gumbel (Extreme Value) hypothesis. The proposed test statistics are constructed using different entropy estimates. Through a Monte Carlo simulation, critical values of the test statistics for various sample sizes are obtained. Since the tests based on the empirical distribution function (EDF) are commonly used in practice, the power values of the entropy-based tests with those of the EDF tests are compared against various alternatives and different sample sizes. Finally, two real data sets are modeled by the Gumbel distribution.KEYWORDS: Entropy estimator, Gumbel distribution, Monte Carlo simulation, test power     

Goodness of fit test for the generalized Rayleigh distribution with unknown parameters

The use of goodness-of-fit test based on Anderson–Darling (AD) statistic is discussed, with reference to the composite hypothesis that a sample of observations comes from a generalized Rayleigh distribution whose parameters are unspecified. Monte Carlo simulation studies were performed to calculate the critical values for AD test. These critical values are then used for testing whether a set of observations follows a generalized Rayleigh distribution when the scale and shape parameters are unspecified and are estimated from the sample. Functional relationship between the critical values of AD is also examined for each shape parameter (α), sample size (n) and significance level (γ). The power study is performed with the hypothesized generalized Rayleigh against alternate distributions.     

Maximum log-likelihood ratio test for a change in three parameter Weibull distribution

Asymptotic behavior of a log-likelihood ratio statistic for testing a change in a three parameter Weibull distribution is studied. It is shown that if a shape parameter α>2$\alpha >2$ the law of iterated logarithm for maximum-likelihood estimators is still valid and the log-likelihood testing statistic is asymptotically distributed (after an appropriate normalization) according to a Gumbel distribution.     

Change-point analysis with bathtub shape for the exponential distribution

Likelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.     

Improving on truncated linear estimates of exponential and gamma scale parameters

We consider the problem of estimating the scale parameter of an exponential or a gamma distribution under squared error loss when the scale parameter θ is known to be greater than some fixed value θ₀. Natural estimators in this setting include truncated linear functions of the sufficient statistic. Such estimators are typically inadmissible, but explicit improvements seem difficult to find. Some are presented here. A particularly interesting finding is that estimators which are admissible in the untruncated problem which take values only in the interior of the truncated parameter space are found to be inadmissible for the truncated problem.     

An entropy-based test for goodness of fit of the von mises distribution

The maximum entropy characterization of the von Mises distribution on the circle is exploited to derive a consistent goodness of fit test for the von Mises distribution. Monte Carlo simulation results are tabulated giving critical values of the test statistic for various sample sizes and values of the concentration parameter. A power analysis is presented for various alternative hypotheses, comparing this entropy statistic to two other competing goodness of fit statistics. The entropy statistic is shown to compare favorably and may be more attractive, especially considering its ease of computation.     

Detection and estimation of structural change in heavy-tailed sequence

In this paper, bootstrap detection and ratio estimation are proposed to analysis mean change in heavy-tailed distribution. First, the test statistic is constructed into a ratio form on the CUSUM process. Then, the asymptotic distribution of test statistic is obtained and the consistency of the test is proved. To solve the problem that the null distribution of the test statistic contains unknown tail index, we present a bootstrap approximation method to determine the critical values of the null distribution. We also discuss how to estimate change point based on ratio method. The consistency and rate of convergence for the change-point estimator are established. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real data sets. Especially the simulation results of bootstrap test are better than those of another existing method.     

A confidence interval for the scale parameter based on sukhtmes two-sample statistic

Sukhatme (1957) introduced a statistic which may be used to test the equality of variances in two independent samples from continuous distributions, centered at zero. It has become evident that this statistic cound be used to construct, analytically, a confidence interval for the scale parameter relating the two random variables(of.Laubscher (1968) and Noether (1967, pp. 66-69 and 1972)). In view of this additional use of the statistic, and since no tables of critical values exist, we provide such a table which makes the application of the statistic a practical proposition. In addition, a normal approximation is suggested for the use outside the range of Table III and the accuracy of this approximation is investigated. The Sukhatme test applied to sample values reduced by t n e i r medians i s studied in a small simulation exercise. It appears that this test when used in conjunction with the critical values of Sukhatme's statistic, is a very conservative one and that it is probably fairly robust with respect to the underlying population distribution.     

A monte carlo investigation of a statistic for a bivariate missing data problem

Testing the equal means hypothesis of a bivariate normal distribution with homoscedastic varlates when the data are incomplete is considered. If the correlational parameter, ρ, is known, the well-known theory of the general linear model is easily employed to construct the likelihood ratio test for the two sided alternative. A statistic, T, for the case of ρ unknown is proposed by direct analogy to the likelihood ratio statistic when ρ is known. The null and nonnull distribution of T is investigated by Monte Carlo techniques. It is concluded that T may be compared to the conventional t distribution for testing the null hypothesis and that this procedure results in a substantial increase in power-efficiency over the procedure based on the paired t test which ignores the incomplete data. A Monte Carlo comparison to two statistics proposed by Lin and Stivers (1974) suggests that the test based on T is more conservative than either of their statistics.     

Modified Baumgartner statistics for the two-sample and multisample problems: a numerical comparison

Various non-parametric rank tests based on the Baumgartner statistic have been proposed for testing the location, scale and location–scale parameters. The modified Baumgartner statistics are not suitable for the scale shifts for a two-sample problem. Two modified Baumgartner statistics are proposed by changing the weight function. The suggested statistics are extended to the multisample problem. Some exact critical values of the suggested test statistics are evaluated. Simulations are used to investigate the power of the modified Baumgartner statistics.     

The generalized Cucconi test statistic for the two-sample problem

When testing hypotheses in two-sample problem, the Lepage test statistic is often used to jointly test the location and scale parameters, and this test statistic has been discussed by many authors over the years. Since two-sample nonparametric testing plays an important role in biometry, the Cucconi test statistic is generalized to the location, scale, and location–scale parameters in two-sample problem. The limiting distribution of the suggested test statistic is derived under the hypotheses. Deriving the exact critical value of the test statistic is difficult when the sample sizes are increased. A gamma approximation is used to evaluate the upper tail probability for the proposed test statistic given finite sample sizes. The asymptotic efficiencies of the proposed test statistic are determined for various distributions. The consistency of the original Cucconi test statistic is shown on the specific cases. Finally, the original Cucconi statistic is discussed in the theory of ties.