On a Class of Tests of Exponentiality

The class of statistics T, = (Σ^α _i=1, X ^α _i /n)/ ^x , where α > 0 and ≠ 1, have been considered in the literature for testing exponentiality versus omnibus alternatives. These tests have a twosided rejection region. It is shown here that tests based on these statistics are not consistent for certain alternatives. However, it is shown that one-sided tests based on these statistics are consistent for IFRA (DFRA) distributions. A Monte Carlo power study suggests that lower tail tests based on T _½ and T ₂ are very competitive for DFRA and IFRA alternatives, respectively. The null distributions of T _½ and T ₂ are approximated by members of the Johnson families of distributions.     

The Percentile Points of Distributions Having Known Cumulants

In an earlier study of the uses of moments and cumulants in the specification of statistical distributions, the authors developed explicit asymptotic expansions, expressing any desired percentile point of such distributions in terms of known cumulants.

The general formulae are now presented as far as the sixth adjustment, based on the eighth cumulant, and also numerical tables showing the coefficients of all terms for ten chosen levels of significance over the range 0.5 to 0.0005 (single tail), together with the first five Ilermite polynomials and tables for the common tests of significance, χ², t and z. at the same levels.     

ε-Contaminated priors in contingency tables

An r×s table is used for different approaches to statistical inference. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. We consider testing equality of proportions of independent multinomial distributions when the common proportions are known. A lower bound of the posterior probabilities of the null hypothesis is calculated with respect to a mixture of a point prior on the null and an ε-contaminated prior on the proportions under the alternative. The resulting Bayes tests are compared numerically to Pearson’s χ ² in a number of examples. For the examined examples the lower bound and the p-value can be made close. The obtained results are generalized when the common proportions vector under the null is unknown or has a known functional form.      

The Performance of Some Rough Tests for Bivariate Normality Before and After Coordinate Transformations to Normality

Some rough tests for bivariate normality are employed in an attempt to quantify the intuitive notion that coordinate transformations to normality produce distributions which are “more bivariate normal” than the original variables. These tests are not rigorous procedures but are intuitively satisfying, based on natural statistics, and provide numerical measures of the “distance” of a bivariate distribution from the normal model. It is shown that, for a wide class of non-normal (X, Y) distributions, coordinate transformations to normality decrease this distance as measured by these tests. It is indicated how one may estimate the coordinate transformations and applications to correlation theory are explored.     

Proof testing of ceramics

Theoretical estimates are made of strength distributions after proof testing. Assuming that the crack velocity can be expressed as a power function of the stress intensity factor,v=AK _I ⁿ , an analysis of the amount of strength loss during a load cycle is presented for single-region crack propagation. For multi-region crack propagation, a numerical analysis is used to describe strength loss. In both analyses, the effects of environment and loading rate are studied. For single region crack propagation, the strength after proof testing can be represented by two Weibull curves: one with a slope ofm at high cumulative failure probability levels; the other with a slope ofn–2 at low failure probability levels. Truncation of the strength distribution always occurs as the result of proof testing; the truncation strength depends on the rate of unloading. Multi-region crack propagation results in a more complicated strength distribution after proof testing. Bimodal strength distributions occur as a consequence of region II type crack growth (i.e.n=0). Theoretical results confirm experimental findings that proof tests must be conducted at rapid unloading rates and with good environmental control to be effective.     

Families of distributions arising from distributions of order statistics

Consider starting from a symmetric distributionF on ℜ and generating a family of distributions from it by employing two parameters whose role is to introduce skewness and to vary tail weight. The proposal in this paper is a simple generalisation of the use of the collection of order statistic distributions associated withF for this purpose; an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution. General properties of the proposed family of distributions are explored. It is argued that two particular special cases are especially attractive because they appear to provide the most tractable instances of families with power and exponential tails; these are the skewt distribution and the logF distribution, respectively. Limited experience with fitting the distributions to data in their four-parameter form, with location and scale parameters added, is described, and hopes for their incorporation into complex modelling situations expressed. Extensions to the multivariate case and to ℜ⁺ are discussed, and links are forged between the distributions underlying the skewt and logF distributions and Tadikamalla and Johnson'sL _U family.     

A Semiparametric Approach to the One-Way Layout

We consider m distributions in which the first m ? 1 are obtained by multiplicative exponential distortions of the mth distribution, which is a reference. The combined data fromm samples, one from each distribution, are used in the semiparametric large-sample problem of estimating each distortion and the reference distribution and testing the hypothesis that the distributions are identical. The approach generalizes the classical normal-based one-way analysis of variance in the sense that it obviates the need for a completely specified parametric model. An advantage is that the probability density of the reference distribution is estimated from the combined data and not only from the mth sample. A power comparison with the t and F tests and with two nonparametric tests, obtained by means of a simulation, points to the merit of the present approach. The method is applied to rain-rate data from meteorological instruments.     

Genesis of Bimodal Distributions

Conditions for which the density function of a mixture of two normal distributions is bimodal are investigated. For fixed values of the variances σ₁ ² and σ₂ ² of the normal distributions if the difference between the means is sufficiently small, the distribution of the mixture will be unimodal, independent of the proportions p and 1 – p, 0 < p < 1. If the difference exceeds a critical value which depends on σ₁ ² and σ₂ ² the bimodality property then depends on p. Values of p sufficiently close to zero and one always exist for which the distribution is unimodal.     

Statistical discrimination of climate modification by urbanization

The statistical significance of observed changes in precipitation characteristics brought about by urbanization is investigated by using several statistical tests, including bivariatex ²,T ², likelihood ratio and conditional-t tests. Annual precipitation data from a number of stations in the LaPorte, St. Louis, Tulsa and Kansas City areas are analysed and the results are presented. The nature and limitations of these tests are discussed from the view points of assumptions made in the tests and the number of observations needed.     

Conversion of Ca/K projectile fragments into thermalized ion beams

The results of tests for stopping and extraction of energetic, 92 MeV/u, short-lived ³⁸Ca and ³⁷K fragments of 0.5% and 0.1% full-width of momentum spread with the first version of the National Superconducting Cyclotron Laboratory (NSCL) gas cell are reported. The projectile fragments were thermalized in 51 cm of helium at 1 bar and were transported by electric fields and gas flow into an expansion chamber through a supersonic nozzle, guided by an radio frequency quadrupole (RFQ) guide and collected on a metallic wire. The extraction efficiency was measured for different implantation rates, different electrical field distributions inside the gas cell, and different values of beam degrader thicknesses. The extraction measurements were compared with the experimental stopping efficiency and measurement of ionization induced in the helium gas. Mechanisms responsible for ion losses were identified and possible improvements of the gas cell performance were discussed.     

Tests for Exponentiality When Origin and Scale Parameters Are Unknown

Several methods are examined for testing for the exponential distribution, using complete samples, when both the scale and location parameters are unknown. Percentage points are given for empirical distribution function (EDF) tests in which both parameters are estimated from the data, and power studies are reported to compare these tests with other goodness-offit techniques for the exponential distribution, particularly those based on regression methods. It is found that the EDF statistics W ² and A ² have overall good power properties, with A ² slightly superior.     

On Bayesian interval prediction of future records

Based on the one-sample scheme, Bayesian prediction bounds for the s ^th future record value are obtained. All of the informative and future observations are assumed to be obtained from a general class of distributions which includes the Weibull, compound Weibull, Pareto, beta, Gompertz, compound Gompertz among other distributions. The prior belief of the experimenter is measured by a proper general conjugate prior which was suggested by AL-Hussaini (1999b).     

Above threshold dissociation in HD+ using frequency chirped laser pulses

We have theoretically studied the dynamics of above threshold dissociation (ATD) in molecular ions HD⁺ using frequency chirped femtosecond laser pulses from numerical solutions of the time-dependent Schrödinger equation by using the three-dimensional time-dependent quantum wave packet method. Energy-dependent distributions of ATD fragments are analyzed by an asymptotic-flow expression in momentum space. Linearly positive and negative frequency chirped laser pulses are adopted. It is found that varying frequency chirped parameters can change branching ratios of the 1sσ _g and 2pσ _u dissociations channels. The concept of a light-induced potential is used to interpret the ATD process. The angular resolved energy distributions of the photofragments are also illustrated.     

Linear Hypothesis Testing With Functional Data

In real data analysis, it is often interesting to consider a general linear hypothesis testing (GLHT) problem for functional data, which includes the one-way ANOVA, post hoc, or contrast analysis as special cases. Existing tests for this GLHT problem include an L²-norm-based test and an F-type test but their theoretical properties have not been investigated. In addition, for functional one-way ANOVA, simulation studies in the literature indicate that they are less powerful than the globalizing pointwise F (GPF) test and the F_max?-test. The GPF and F_max?-test enjoy several other good properties. They are scale-invariant in the sense that their test statistics do not change if we multiply each of functional curves with a nonzero function of the observed locations. In this article, the GPF and F_max?-test are adapted to the above GLHT problem. Their theoretical properties, for example, root-n consistency as well as those of the L²-norm-based and F-type tests are established. Intensive simulation studies are carried out to compare the finite-sample behavior of the tests under consideration in scenarios reflecting various practical characteristics of functional data. Simulation results indicate that the GPF test has higher power than other tests when the functional data are less correlated, and the F_max?-test has higher power than other tests when the functional data are moderately or highly correlated. These results are also confirmed by application of the GPF and F_max? tests to the corneal surface data coming from medical industry. This application suggests the new methods may help to make more clear and sure decisions in practice. For a convenient application of the considered testing procedures, their implementation is developed in the R programming language. Supplementary materials for the article are available online.     

Prediction Limits for the Inverse Gaussian Distribution

Prediction intervals for the inverse Gaussian are obtained from both a frequentist and a Bayesian viewpoint. The frequentist intervals are obtained by constructing pivotals that have the x ² and F distributions. The method involves inversion of probability statements, which results in two-sided prediction intervals. A Bayesian predictive density is obtained using a vague prior, from which one- or two-sided Bayesian prediction intervals can be determined. An example for which Bayesian prediction limits are narrower than the frequentist is given.     

Some Properties of Fresnel Images of a Square Wave Amplitude Grating

The self-imaging phenomenon of a square wave binary amplitude grating is studied. The intensity distributions in the Fresnel diffraction image planes lying at distances z = (M -1/4)d ²/u from the grating illuminated by a plane wave are calculated. It is found that these images are very similar to the object structure. As a consequence, some potential applications are proposed.     

A Bivariate Semiparametric Control Chart Based on Order Statistics

In this article, a new bivariate semiparametric Shewhart‐type control chart is presented. The proposed chart is based on the bivariate statistic (X_(r), Y_(s)), where X_(r) and Y_(s) are the order statistics of the respective X and Y test samples. It is created by considering a straightforward generalization of the well‐known univariate median control chart and can be easily applied because it calls for the computation of two single order statistics. The false alarm rate and the in‐control run length are not affected by the marginal distributions of the monitored characteristics. However, its performance is typically affected by the dependence structure of the bivariate observations under study; therefore, the suggested chart may be characterized as a semiparametric control chart. An explicit expression for the operating characteristic function of the new control chart is obtained. Moreover, exact formulae are provided for the calculation of the alarm rate given that the characteristics under study follow specific bivariate distributions. In addition, tables and graphs are given for the implementation of the chart for some typical average run length values and false alarm rates. The performance of the suggested chart is compared with that of the traditional χ² chart as well as to the nonparametric SN² and SR² charts that are based on the multivariate form of the sign test and the Wilcoxon signed‐rank test, respectively. Finally, in order to demonstrate the applicability of our chart, a case study regarding a real‐world problem related to winery production is presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal testing policy for a computer system with intermittent faults

A computer system with intermittent faults fails with probability p when it is used in hidden faults. Periodic tests are scheduled at times kT(k = 1, 2,…) to detect a hidden fault. The mean time, the expected number of tests and the expected cost until detection of a fault or system failure are derived, using the theory of Markov renewal processes. An optimal testing time T^* to minimize the expected cost is discussed. A finite T^* is given by a unique solution of an equation.     

Nonequilibrium distributions in superconductors

The nonequilibrium distribution functions of quasiparticles and phonons in superconductors are calculated for various cases. The conditions at which the nonequilibrium distributions exist are found. The dependences of the temperature and concentration of excitations on the pumping intensity, the sample thickness, and other parameters of the superconductor are calculated. In the current state the dependences of these quantities on the superfluid velocity and the current are investigated, and it is found that the dependenceT(v _s ) has a minimum, and the current for v_s>v _so ⁽¹⁾ becomes negative. It is also shown that in the nonequiibrium superconductor the state withv _s ^* 0 andJ(v _s ^* )=0 may exist. We have determined the nonequilibrium distribution function for a tunnel junction and investigated theV-A characteristic at some conditions for which an absolute negative resistance may exist.