Tests for symmetry of two-piece normal distribution

In this paper, classical optimum tests for symmetry of two-piece normal distribution is derived. Uniformly most powerful one-sided test for the skewness parameter is obtained when the location and scale parameters are known and is compared with sequential probability ratio test. An ad-hoc test for symmetry and likelihood ratio test for symmetry for large samples, can be found in literature for this distribution. But in this paper, we derive exact likelihood ratio test for symmetry, when location parameter is known. The exact power of the test is evaluated for different sample sizes.     

Modified cramer-von mises and anderson-darling tests for weibull distributions with unknown location and scale parameters

The standard Cramer-von Mises and Anderson-Darling goodness-of-fit tests require continuous underlying distributions with known parameters. In this paper, tables of critical values are generated for both tests for Weibull distributions with unknown location and scale parameters and known shape parameters. The powers of the Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Chi-Square tests for this situation are investigated. The Cramer-von Mises test has most power when the shape is 1.0 and the Anderson-Darling test has most power when the shape is 3.5. Finally, a relation between critical value and inverse shape parameter is presented.     

Exponentiated Sinh Cauchy Distribution with Applications

The exponentiated sinh Cauchy distribution is characterized by four parameters: location, scale, symmetry, and asymmetry. The symmetry parameter preserves the symmetry of the distribution by producing both bimodal and unimodal densities having coefficient of kurtosis values ranging from one to positive infinity. The asymmetry parameter changes the symmetry of the distribution by producing both positively and negatively skewed densities having coefficient of skewness values ranging from negative infinity to positive infinity. Bimodality, skewness, and kurtosis properties of this regular distribution are presented. In addition, relations to some well-known distributions are examined in terms of skewness and kurtosis by constructing aliases of the proposed distribution on the symmetry and asymmetry parameter plane. The maximum likelihood parameter estimation technique is discussed, and examples are provided and analyzed based on data from astronomy and medical sciences to illustrate the flexibility of the distribution for modeling bimodal and unimodal data.     

Default Bayesian goodness-of-fit tests for the skew-normal model

In this paper we propose a series of goodness-of-fit tests for the family of skew-normal models when all parameters are unknown. As the null distributions of the considered test statistics depend only on asymmetry parameter, we used a default and proper prior on skewness parameter leading to the prior predictive p-value advocated by G. Box. Goodness-of-fit tests, here proposed, depend only on sample size and exhibit full agreement between nominal and actual size. They also have good power against local alternative models which also account for asymmetry in the data.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

Exact conditional inference for two uniform populations

We discuss the nature of ancillary information in the context of the continuous uniform distribution. In the one-sample problem, the existence of sufficient statistics mitigates conditioning on the ancillary configuration. In the two-sample problem, additional ancillary information becomes available when the ratio of scale parameters is known. We give exact results for conditional inferences about the common scale parameter and for the difference in location parameters of two uniform distributions. The ancillary information affects the precision of the latter through a comparison of the sample value of the ratio of scale parameters with the known population value. A limited conditional simulation compares the Type I errors and power of these exact results with approximate results using the robust pooled t-statistic.     

Kolmogorov-Smirnov Type Test-Statistics For The Gamma,Erlang-2 And The Inverse Gaussian Distributions When The Parameters Are Unknown.

Critical values are presented for the Kolmogorov-Smirnov type test statistics for the following three cases: (i) the gamma distribution when both the scale and the shape parameters are not known, (ii) the scale parameter of the gamma distribution is not known and (iii) the inverse Gaussian distribution when both the parameters are unknown. This study was motivated by the necessity to fit the gamma, the Erlang-2 and the inverse Gaussian distributions to the interpurchase times of individuals for coffee in marketing research.     

Power studies of tests for uniformity,II

Results from a power study of six statistics for testing that a sample is from a uniform distribution on the unit interval (0,1) are reported. The test statistics are all well-known and each of them was originally proposed because they should have high power against some alternative distributions. The tests considered are the Pearson probability product test, the Neyman smooth test, the Sukhatme test, the Durbin-Kolmogorov test, the Kuiper test, and the Sherman test. Results are given for each of these tests against each of four classes of alternatives. Also, the most powerful test against each member of the first three alternatives is obtained, and the powers of these tests are given for the same sample sizes as for the six general "omnibus" test statistics. These values constitute a "power envelope" against which all tests can be compared. The Neyman smooth tests with 2nd and 4th degree polynomials are found to have good power and are recommended as general tests for uniformity.     

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.     

CONDITIONAL INFERENCE PROCEDURES FOR THE PARETO AND POWER FUNCTION DISTRIBUTIONS BASED ON TYPE-II RIGHT CENSORED SAMPLES

In this paper we consider conditional inference procedures for the Pareto and power function distributions. We develop procedures for obtaining confidence intervals for the location and scale parameters as well as upper and lower n probability tolerance intervals for a proportion g, given a Type-II right censored sample from the corresponding distribution. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since, for each distribution, the procedures assume that a shape parameter x is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in x.     

Testing overdispersion in the zero-inflated Poisson model

The zero-inflated negative binomial (ZINB) model is used to account for commonly occurring overdispersion detected in data that are initially analyzed under the zero-inflated Poisson (ZIP) model. Tests for overdispersion (Wald test, likelihood ratio test [LRT], and score test) based on ZINB model for use in ZIP regression models have been developed. Due to similarity to the ZINB model, we consider the zero-inflated generalized Poisson (ZIGP) model as an alternate model for overdispersed zero-inflated count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes score tests for overdispersion based on the ZIGP model and illustrates that the derived score statistics are exactly the same as the score statistics under the ZINB model. A simulation study indicates the proposed score statistics are preferred to other tests for higher empirical power. In practice, based on the approximate mean–variance relationship in the data, the ZINB or ZIGP model can be considered, and a formal score test based on asymptotic standard normal distribution can be employed for assessing overdispersion in the ZIP model. We provide an example to illustrate the procedures for data analysis.     

Locally optimal test of normality against skew-normality

Efficient Score testing procedures for testing for the skewness parameter of the one-parameter and three-parameter skew-normal distributions are investigated. Approximate percentage points for the test statistics are obtained via the Monte Carlo method. Salvan's locally optimal test is extended and simulation is used to show that the power of the extended test is higher than that of the skewness test.     

Tests for Equality of Parameters of Two Exponential Distributions with Common Known Coefficient of Variation

ABSTRACT

This article considers the problem of testing equality of parameters of two exponential distributions having common known coefficient of variation, both under unconditional and conditional setup. Unconditional tests based on BLUE'S and LRT are considered. Using the Conditionality Principle of Fisher, an UMP conditional test for one-sided alternative is derived by conditioning on an ancillary. This test is seen to be uniformly more powerful than unconditional tests in certain given ranges of ancillary. Simulation studies on the power functions of the tests are done for this purpose.     

Tests for symmetry with right censoring

Permutation tests for symmetry are suggested using data that are subject to right censoring. Such tests are directly relevant to the assumptions that underlie the generalized Wilcoxon test since the symmetric logistic distribution for log-errors has been used to motivate Wilcoxon scores in the censored accelerated failure time model. Its principal competitor is the log-rank (LGR) test motivated by an extreme value error distribution that is positively skewed. The proposed one-sided tests for symmetry against the alternative of positive skewness are directly relevant to the choice between usage of these two tests.

The permutation tests use statistics from the weighted LGR class normally used for making two-sample comparisons. From this class, the test using LGR weights (all weights equal) showed the greatest discriminatory power in simulations that compared the possibility of logistic errors versus extreme value errors.

In the test construction, a median estimate, determined by inverting the Kaplan–Meier estimator, is used to divide the data into a “control” group to its left that is compared with a “treatment” group to its right. As an unavoidable consequence of testing symmetry, data in the control group that have been censored become uninformative in performing this two-sample test. Thus, early heavy censoring of data can reduce the effective sample size of the control group and result in diminished power for discriminating symmetry in the population distribution.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

An alternative skew exponential power distribution formulation

In this note we propose a newly formulated skew exponential power distribution that behaves substantially better than previously defined versions. This new model performs very well in terms of the large sample behavior of the maximum likelihood estimation procedure when compared to the classically defined four parameter model defined by Azzalini. More recently, approaches to defining a skew exponential power distribution have used five or more parameters. Our approach improves upon previous attempts to extend the symmetric power exponential family to include skew alternatives by maintaining a minimum set of four parameters corresponding directly to location, scale, skewness and kurtosis. We illustrate the utility of our proposed model using translational and clinical data sets.     

Hypothesis testing for quantitative trait locus effects in both location and scale in genetic backcross studies

Testing the existence of a quantitative trait locus (QTL) effect is an important task in QTL mapping studies. Most studies concentrate on the case where the phenotype distributions of different QTL groups follow normal distributions with the same unknown variance. In this paper we make a more general assumption that the phenotype distributions come from a location-scale distribution family. We derive the limiting distribution of the likelihood ratio test (LRT) for the existence of the QTL effect in both location and scale in genetic backcross studies. We further identify an explicit representation for this limiting distribution. As a complement, we study the limiting distribution of the LRT and its explicit representation for the existence of the QTL effect in the location only. The asymptotic properties of the LRTs under a local alternative are also investigated. Simulation studies are used to evaluate the asymptotic results, and a real-data example is included for illustration.     

A new test for two-sample location problem based on empirical distribution function

We propose a new test for testing the equality of location parameter of two populations based on empirical distribution function (ECDF). The test statistics is obtained as a power divergence between two ECDFs. The test is shown to be distribution free, and its null distribution is obtained. We conducted empirical power comparison of the proposed test with several other available tests in the literature. We found that the proposed test performs better than its competitors considered here under several population structures. We also used two real datasets to illustrate the procedure.     

Goodness-of-Fit Tests for the Skew-Normal Distribution When the Parameters Are Estimated from the Data

In this article, tests are developed which can be used to investigate the goodness-of-fit of the skew-normal distribution in the context most relevant to the data analyst, namely that in which the parameter values are unknown and are estimated from the data. We consider five test statistics chosen from the broad Cramér–von Mises and Kolmogorov–Smirnov families, based on measures of disparity between the distribution function of a fitted skew-normal population and the empirical distribution function. The sampling distributions of the proposed test statistics are approximated using Monte Carlo techniques and summarized in easy to use tabular form. We also present results obtained from simulation studies designed to explore the true size of the tests and their power against various asymmetric alternative distributions.