Non-continuous Distribution Research Articles

Development of pedestal-free large mode area fibers withTm3+ doped silica nanostructured core.

We present the pedestal-free thulium doped silica fiber with a large nanostructured core optimized for fiber lasers. The fiber is composed of over 6 thousand thulium doped silica nanorods with a diameter of 71 nm each which form a nanostructured step-index core. We study the influence of non-continuous distribution in nanoscale active areas on gain, beam quality, and fiber laser performance. The proof-of-concept fiber is effectively single mode for wavelength above 1.8 µm. We demonstrate the performance of the fiber in a laser setup pumped at 792 nm. Single mode laser emission with a slope efficiency of 29% at quasi-continuous output power of 4 W with M2 = 1.3 at the emission spectrum 1880-1925 nm is achieved.

Finite-sample generalized confidence distributions and sign-based robust estimators in median regressions with heterogeneous dependent errors

We study the problem of estimating the parameters of a linear median regression without any assumption on the shape of the error distribution – including no condition on the existence of moments – allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear and nonlinear). This is done through a reverse inference approach, based on a distribution-free sign-based testing theory, from which confidence sets and point estimators are subsequently generated. We propose point estimators, which have a natural association with confidence distributions. These estimators are based on maximizing test p-values and inherit robustness properties from the generating distribution-free tests. Both finite-sample and large-sample properties of the proposed estimators are established under weak regularity conditions. We show that they are median-unbiased (under symmetry and estimator unicity) and possess equivariance properties. Consistency and asymptotic normality are established without any moment existence assumption on the errors. A Monte Carlo study of bias and RMSE shows sign-based estimators perform better than LAD-type estimators in various heteroskedastic settings. We illustrate the use of sign-based estimators on an example of β-convergence of output levels across U.S. states.

Fast and Unbiased Estimation of Volume Under Ordered Three-Class ROC Surface (VUS) With Continuous or Discrete Measurements

Receiver Operating Characteristic (ROC) surfaces have been studied in the literature essentially during the last decade and are considered as a natural generalization of ROC curves in three-class problems. The volume under the surface (VUS) is useful for evaluating the performance of a trichotomous diagnostic system or a three-class classifier’s overall accuracy when the possible disease condition or sample belongs to one of three ordered categories. In the areas of medical studies and machine learning, the VUS of a new statistical model is typically estimated through a sample of ordinal and continuous measurements obtained by some suitable specimens. However, discrete scales of the prediction are also frequently encountered in practice. To deal with such scenario, in this paper, we proposed a unified and efficient algorithm of linearithmic order, based on dynamic programming, for unbiased estimation of the mean and variance of VUS with unidimensional samples drawn from continuous or non-continuous distributions. Monte Carlo simulations verify our theoretical findings and developed algorithms.

Denitrifying anaerobic methane oxidizing in global upland soil: Sporadic and non-continuous distribution with low influence

Both the distribution and contribution of denitrifying anaerobic methane oxidation (damo) have been extensively reported in enrichment culture and aquatic ecosystems; yet, the role of damo in upland soil remains unclear. In the present study, this role of damo bacteria in global upland soils was investigated via isotope tracing technique, quantitative PCR assay, and clone library construction, targeting both 16S rRNA and functional pmoA genes. Using samples from the surface (0–20 cm, n = 148) and deep soil profiles (from 0 to 70–6300 cm, n = 160), the damo bacterial distribution seemed to be sporadic in surface soils and non-continuous in soil profiles. The moisture content of the upland soils seems to be the key factor to influence damo occurrences. Among soil profiles with higher moisture content, the highest damo bacterial abundance of 4.62 × 106 copies g−1 soil was detected at 180–200 cm depth in Xingtai, China. Both a shift of damo bacterial community structure was observed with depth as well as strong temporal heterogeneity within surface soils. Damo activity (0.18 nmol C g−1 h−1) was only detected in the summer surface soil (0–20 cm) of Xingtai. The global distribution of damo in upland soils indicates that upland soils are not favorable environments for the occurrence of damo processes, and the potential role of damo in upland soils might have a limited contribution to the global CH4 cycle.

Preliminary Bioinformatic Analysis of Gallus gallus (Chicken) CD8a Gene

The experimental object of this study is Gallus gallus (chicken) CD8a gene. In this study, we performed sequence analysis of CD8a gene and analyzed the primary structure, secondary structure and tertiary structure of CD8a encoded protein by using bioinformatics. Through the analysis, we concluded that the sequence length is 648bp with various restriction sites identified, which is similar to CD8a of other species in Galliformes order. The sequence encodes a total of 216 amino acids. The amino acids have low hydrophobicity, with 12 phosphorylation sites and a transmembrane region, and no signal peptide. The protein secondary structure mainly consists ofαspiral, accounting for 22.69%; extension chain, accounting for 15.74%; random coil, accounting for 61.57 %, in non-continuous distribution. The relationship between CD8a and other species homologous sequences is basically consistent with the establishment of the system tree by the establishment of three trees. n the root tip cells of Vicia faba and increase in micronucleus rate. The results showed that the water extract of Solidago canadensis had different degree of inhibition and damage to silkworm root tip cells; it also had some genetic toxicity and rapid diffusion ability.

Generalized structural equations improve sexual-selection analyses.

Sexual selection is an intense evolutionary force, which operates through competition for the access to breeding resources. There are many cases where male copulatory success is highly asymmetric, and few males are able to sire most females. Two main hypotheses were proposed to explain this asymmetry: “female choice” and “male dominance”. The literature reports contrasting results. This variability may reflect actual differences among studied populations, but it may also be generated by methodological differences and statistical shortcomings in data analysis. A review of the statistical methods used so far in lek studies, shows a prevalence of Linear Models (LM) and Generalized Linear Models (GLM) which may be affected by problems in inferring cause-effect relationships; multi-collinearity among explanatory variables and erroneous handling of non-normal and non-continuous distributions of the response variable. In lek breeding, selective pressure is maximal, because large numbers of males and females congregate in small arenas. We used a dataset on lekking fallow deer (Dama dama), to contrast the methods and procedures employed so far, and we propose a novel approach based on Generalized Structural Equations Models (GSEMs). GSEMs combine the power and flexibility of both SEM and GLM in a unified modeling framework. We showed that LMs fail to identify several important predictors of male copulatory success and yields very imprecise parameter estimates. Minor variations in data transformation yield wide changes in results and the method appears unreliable. GLMs improved the analysis, but GSEMs provided better results, because the use of latent variables decreases the impact of measurement errors. Using GSEMs, we were able to test contrasting hypotheses and calculate both direct and indirect effects, and we reached a high precision of the estimates, which implies a high predictive ability. In synthesis, we recommend the use of GSEMs in studies on lekking behaviour, and we provide guidelines to implement these models.

Finite-Sample Generalized Confidence Distributions and Sign-Based Robust Estimators In Median Regressions with Heterogeneous Dependent Errors

We study the problem of estimating the parameters of a linear median regression without any assumption on the shape of the error distribution -- including no condition on the existence of moments -- allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear and nonlinear). This is done through a reverse inference approach, based on a distribution-free testing theory [Coudin and Dufour (2009, The Econometrics Journal)], from which confidence sets and point estimators are subsequently generated. The estimation problem is tackled in two complementary ways. First, we show how confidence distributions for model parameters can be applied in such a context. Such distributions -- which can be interpreted as a form of fiducial inference -- provide a frequency-based method for associating probabilities with subsets of the parameter space (like posterior distributions do in a Bayesian setup) without the introduction of prior distributions. We consider generalized confidence distributions applicable to multidimensional parameters, and we suggest the use of a projection technique for confidence inference on individual model parameters. Second, we propose point estimators, which have a natural association with confidence distributions. These estimators are based on maximizing test p-values and inherit robustness properties from the generating distribution-free tests. Both finite-sample and large-sample properties of the proposed estimators are established under weak regularity conditions. We show they are median unbiased (under symmetry and estimator unicity) and possess equivariance properties. Consistency and asymptotic normality are established without any moment existence assumption on the errors, allowing for noncontinuous distributions, heterogeneity and serial dependence of unknown form. These conditions are considerably weaker than those used to show corresponding results for LAD estimators. In a Monte Carlo study of bias and RMSE, we show sign-based estimators perform better than LAD-type estimators in heteroskedastic settings. We present two empirical applications, which involve financial and macroeconomic data, both affected by heavy tails (non-normality) and heteroskedasticity: a trend model for the S&P index, and an equation used to study β-convergence of output levels across U.S. States.

Effects of deep cryogenic treatment, cryogenic and annealing temperatures on mechanical properties and corrosion resistance of AA5083 aluminium alloy

Effects of deep cryogenic treatment (DCT), cryogenic and annealing temperatures on micro-structural evolution, mechanical properties and the intergranular corrosion (IGC) resistance of AA5083 aluminium alloy are investigated. Results showed that as the temperature of the annealing increased, its strength decreased while the IGC resistance improved. At cryogenic temperature, strength and ductility increased drastically. Failure modes observed were ductile particle dimples at 295 K and 213 K, and a mixture of intergranular and transgranular fractures at 110 K. Specimens annealed at 150°C and 200°C exhibited a higher vulnerability to IGC due to the presence of high dislocation densities and heterogeneous precipitation of β-phase (Al3Mg2) at the grain boundaries as the continuous phase. In contrast, specimens annealed from 250°C to 415°C exhibited excellent corrosion resistance due to the minimisation of dislocation densities and noncontinuous distribution of β-phase. DCT marginally decreased the strength and hardness of the alloy, while increasing the ductility and IGC resistance.

Effects of deep cryogenic treatment, cryogenic and annealing temperatures on mechanical properties and corrosion resistance of AA5083 aluminium alloy

Effects of deep cryogenic treatment (DCT), cryogenic and annealing temperatures on micro-structural evolution, mechanical properties and the intergranular corrosion (IGC) resistance of AA5083 aluminium alloy are investigated. Results showed that as the temperature of the annealing increased, its strength decreased while the IGC resistance improved. At cryogenic temperature, strength and ductility increased drastically. Failure modes observed were ductile particle dimples at 295 K and 213 K, and a mixture of intergranular and transgranular fractures at 110 K. Specimens annealed at 150°C and 200°C exhibited a higher vulnerability to IGC due to the presence of high dislocation densities and heterogeneous precipitation of β-phase (Al3Mg2) at the grain boundaries as the continuous phase. In contrast, specimens annealed from 250°C to 415°C exhibited excellent corrosion resistance due to the minimisation of dislocation densities and noncontinuous distribution of β-phase. DCT marginally decreased the strength and hardness of the alloy, while increasing the ductility and IGC resistance.

Beringian sub-refugia revealed in blackfish (Dallia): implications for understanding the effects of Pleistocene glaciations on Beringian taxa and other Arctic aquatic fauna.

BackgroundPleistocene climatic instability had profound and diverse effects on the distribution and abundance of Arctic organisms revealed by variation in phylogeographic patterns documented in extant Arctic populations. To better understand the effects of geography and paleoclimate on Beringian freshwater fishes, we examined genetic variability in the genus Dallia (blackfish: Esociformes: Esocidae). The genus Dallia groups between one and three nominal species of small, cold- and hypoxia-tolerant freshwater fishes restricted entirely in distribution to Beringia from the Yukon River basin near Fairbanks, Alaska westward including the Kuskokwim River basin and low-lying areas of Western Alaska to the Amguema River on the north side of the Chukotka Peninsula and Mechigmen Bay on the south side of the Chukotka Peninsula. The genus has a non-continuous distribution divided by the Bering Strait and the Brooks Range. We examined the distribution of genetic variation across this range to determine the number and location of potential sub-refugia within the greater Beringian refugium as well as the roles of the Bering land bridge, Brooks Range, and large rivers within Beringia in shaping the current distribution of populations of Dallia. Our analyses were based on DNA sequence data from two nuclear gene introns (S7 and RAG1) and two mitochondrial genome fragments from nineteen sampling locations. These data were examined under genetic clustering and coalescent frameworks to identify sub-refugia within the greater Beringia refugium and to infer the demographic history of different populations of Dallia.ResultsWe identified up to five distinct genetic clusters of Dallia. Four of these genetic clusters are present in Alaska: (1) Arctic Coastal Plain genetic cluster found north of the Brooks Range, (2) interior Alaska genetic cluster placed in upstream locations in the Kuskokwim and Yukon river basins, (3) a genetic cluster found on the Seward Peninsula, and (4) a coastal Alaska genetic cluster encompassing downstream Kuskokwim River and Yukon River basin sample locations and samples from Southwest Alaska not in either of these drainages. The Chukotka samples are assigned to their own genetic cluster (5) similar to the coastal Alaska genetic cluster. The clustering and ordination analyses implemented in Discriminant Analysis of Principal Components (DAPC) and STRUCTURE showed mostly concordant groupings and a high degree of differentiation among groups. The groups of sampling locations identified as genetic clusters correspond to geographic areas divided by likely biogeographic barriers including the Brooks Range and the Bering Strait. Estimates of sequence diversity (θ) are highest in the Yukon River and Kuskokwim River drainages near the Bering Sea. We also infer asymmetric migration rates between genetic clusters. The isolation of Dallia on the Arctic Coastal Plain of Alaska is associated with very low estimated migration rates between the coastal Alaska genetic cluster and the Arctic Coastal Plain genetic cluster.ConclusionsOur results support a scenario with multiple aquatic sub-refugia in Beringia during the Pleistocene and the preservation of that structure in extant populations of Dallia. An inferred historical presence of Dallia across the Bering land bridge explains the similarities in the genetic composition of Dallia in West Beringia and western coastal Alaska. In contrast, historic and contemporary isolation across the Brooks Range shaped the distinctiveness of present day Arctic Coastal Plain Dallia. Overall this study uncovered a high degree of genetic structuring among populations of Dallia supporting the idea of multiple Beringian sub-refugia during the Pleistocene and which appears to be maintained to the present due to the strictly freshwater nature and low dispersal ability of this genus.Electronic supplementary materialThe online version of this article (doi:10.1186/s12862-015-0413-2) contains supplementary material, which is available to authorized users.

The geological nature of dark material on Vesta and implications for the subsurface structure

Deposits of dark material appear on Vesta’s surface as features of relatively low-albedo in the visible wavelength range of Dawn’s camera and spectrometer. Mixed with the regolith and partially excavated by younger impacts, the material is exposed as individual layered outcrops in crater walls or ejecta patches, having been uncovered and broken up by the impact. Dark fans on crater walls and dark deposits on crater floors are the result of gravity-driven mass wasting triggered by steep slopes and impact seismicity. The fact that dark material is mixed with impact ejecta indicates that it has been processed together with the ejected material. Some small craters display continuous dark ejecta similar to lunar dark-halo impact craters, indicating that the impact excavated the material from beneath a higher-albedo surface. The asymmetric distribution of dark material in impact craters and ejecta suggests non-continuous distribution in the local subsurface. Some positive-relief dark edifices appear to be impact-sculpted hills with dark material distributed over the hill slopes. Dark features inside and outside of craters are in some places arranged as linear outcrops along scarps or as dark streaks perpendicular to the local topography. The spectral characteristics of the dark material resemble that of Vesta’s regolith. Dark material is distributed unevenly across Vesta’s surface with clusters of all types of dark material exposures. On a local scale, some craters expose or are associated with dark material, while others in the immediate vicinity do not show evidence for dark material. While the variety of surface exposures of dark material and their different geological correlations with surface features, as well as their uneven distribution, indicate a globally inhomogeneous distribution in the subsurface, the dark material seems to be correlated with the rim and ejecta of the older Veneneia south polar basin structure. The origin of the dark material is still being debated, however, the geological analysis suggests that it is exogenic, from carbon-rich low-velocity impactors, rather than endogenic, from freshly exposed mafic material or melt, exposed or created by impacts.

Structural modeling of 128× 128 InSb focal plane array detector

Higher fracture probability appearing in InSb infrared focal plane array (IRFPA) subjected to thermal shock test, restricts its final yield. In order to understand the fracture mechanism, in light of the proposed equivalent method, where a 32× 32 array is employed to replace the real 128× 128 array, to a three - dimensional structural model of IRFPA is developed by taking into account the temperature dependence of thermal expansion coefficient, anisotropic mechanical strength of InSb chip, damaging effects of the surface of the InSb chip, and a reduction of 90% the out-of-plane elastic modulus. Simulation results show that a maximum Von Mises stress appears in the N electrode zone in InSb chip, and the extremum values present a non-continuous distribution. This means that the cracks is most likely to emerge in the region of N electrode, besides, the number of crack tracks is more than one. These are well consistent with the 128× 128 InSb IRFPA fracture statistics results under thermal shock test. All these are beneficial to the further study of fracture inducing factors and structural reliability design of InSb IRFPA.

Mechanical parameters selection in InSb focal plane array detector normal direction

In order to learn the effects of front surface structural defects and back surface thinning process on the InSb chip deformation, its elastic modulus along normal direction is reduced in InSb structural modeling, and based on the typical strain character appearing under thermal shock, the mechanical parameter selection basis is deduced in this paper. Simulation results show that when the out-of-plane elastic modulus of InSb chip is set to be 30 percent Young's modulus, both the maximum Von Mises stress and Z component of strain appear in the N electrode zone, and the extremum values present non-continuous distribution. These are in good agreement with fracture origination zone and crack distribution in the fracture statistics results of 128 128 InSb infrared focal plane array under thermal shock. Besides, the region above the indium bump array is convex upward, and the domain above the isolation trough is concave downward, they are also identical with the scenario of Z component of strain in InSb chip under thermal shock. All these results indicate that the Z component of strain criterion can not only predict both crack origination zone and crack distribution, but also support both Z component of strain distribution in the central region and Z component of strain enhancement effect in the InSb chip N electrode zone.

Robust Sign-Based and Hodges-Lehmann Estimators in Linear Median Regressions with Heterogenous Serially Dependent Errors

We propose estimators for the parameters of a linear median regression without any assumption on the shape of the error distribution – including no condition on the existence of moments – allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear or nonlinear) including GARCH-type and stochastic volatility of unknown order. The estimators follow from a reverse inference approach, based on the class of distribution-free sign tests proposed in Coudin and Dufour (2009, Econometrics J.) under a median gale assumption. As a result, the estimators inherit strong robustness properties from their generating tests. Since the proposed estimators are based on maximizing a test statistic (or a p-value function) over different null hypotheses, they can be interpreted as Hodges-Lehmann-type (HL) estimators. It is easy to adapt the sign-based estimators to account for linear serial dependence. Both finite-sample and large-sample properties are established under weak regularity conditions. The proposed estimators are median unbiased (under symmetry and estimator unicity) and satisfy natural equivariance properties. Consistency and asymptotic normality are established without any condition on error moment existence, allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear or nonlinear). These conditions are considerably weaker than those used to show corresponding results for LAD estimators. In a Monte Carlo study on bias and mean square error, we find that sign-based estimators perform better than LAD-type estimators, especially in heteroskedastic settings. The proposed procedures are applied to a trend model of the Standard and Poor’s composite price index, where disturbances are affected by both heavy tails (non-normality) and heteroskedasticity.

Distribution and invasion genetics of the quagga mussel (Dreissena rostriformis bugensis) in German rivers

The quagga mussel Dreissena rostriformis bugensis, native to the Dnieper and the northern Black Sea, has become a major invasive species in both the Volga River and the North American Great Lakes since the early 1990s. Findings in the Netherlands (2006) and Germany (2007) mark the start of its establishment in Western Europe. We investigated the current distribution, time of first arrival and population structure of D. rostriformis bugensis from the rivers Rhine, Main and in the Main-Danube canal in Germany. Two putative sources of the German populations were analysed by genetically comparing these populations to older invasive populations from North America and the southeast Danube. Dreissena rostriformis bugensis was abundant in the Main and in three Rhine harbours, but rare in the actual Rhine river and absent south of the Main-Danube canal. Mussels found in the Rhine harbours were significantly smaller than in the Main. Population genetic analyses found no sign of founder effects and minimal differentiation between German, North American and southeast Danube populations. The genetic data suggest that these invasive populations derive from a common and rapidly expanding source. Based on the non-continuous distribution and shell size differences of Rhine harbour and Main populations, our results indicate that expansion in Germany involved at least two independent settling events, one of which happened before 2005, and most likely was caused by jump dispersal.

Influence of Rare Species on Electrofishing Distance When Estimating Species Richness of Stream and River Reaches

AbstractThe electrofishing distance needed to estimate fish species richness at the stream or river reach scale is an important question in fisheries science. This distance is governed by the shape of the species accumulation curve, which, in turn, is influenced by a combination of factors, including the number of species, their overall abundances, habitat associations, the efficiency of the sampling method, and the occurrence of rare species. In this study we document the influence of rare species on the species accumulation curves from stream and river sites in data sets from five dispersed regions of the USA. Spatial discontinuity (i.e., a noncontinuous distribution within reaches) was observed in four of the five data sets, and the four data sets contained numerically rare species represented by one or two individuals (termed singletons and doubletons, respectively). Numerically rare species were typically proportionately rare (i.e., <1% of the total number of individuals captured), but proportionately rare species were not always numerically rare and were dependent on the total number of fish captured. Species richness asymptotes were reached at shorter electrofishing distances when singletons and doubletons were removed. The number of singletons and doubletons in the samples remained relatively constant with increasing sampling effort (i.e., sampling distance and total abundance). Simulation modeling indicated that individual aggregation within species was not a plausible reason for spatially discontinuous species distributions. When accurately detecting the presence of species is a sampling goal, the presence and prevalence of numerically rare species may need to be considered in determining sampling protocols.

Microsatellite analyses reveal fine‐scale genetic structure in grey mouse lemurs (Microcebus murinus)

Information on genetic structure can be used to complement direct inferences on social systems and behaviour. We studied the genetic structure of the solitary grey mouse lemur (Microcebus murinus), a small, nocturnal primate endemic to western Madagascar, with the aim of getting further insight on its breeding structure. Tissue samples from 167 grey mouse lemurs in an area covering 12.3 km2 in Kirindy Forest were obtained from trapping. The capture data indicated a noncontinuous distribution of individuals in the study area. Using 10 microsatellite markers, significant genetic differentiation in the study area was demonstrated and dispersal was found to be significantly male biased. Furthermore, we observed an overall excess of homozygotes in the total population (F(IT) = 0.131), which we interpret as caused by fine-scale structure with breeding occurring in small units. Evidence for a clumped distribution of identical homozygotes was found, supporting the notion that dispersal distance for breeding was shorter than that for foraging, i.e. the breeding neighbourhood size is smaller than the foraging neighbourhood size. In conclusion, we found a more complex population structure than what has been previously reported in studies performed on smaller spatial scales. The noncontinuous distribution of individuals and the effects of social variables on the genetic structure have implications for the interpretation of social organization and the planning of conservation activities that may apply to other solitary and endangered mammals as well.

Fabric-bagging: Stress Distribution in Isotropic and Anisotropic Fabrics

The internal stress distribution of bagged fabric is studied theoretically and experimentally to investigate the influence of the stress distributions on fabric-residual-bagging deformation. A model is derived by using membrane theory to analyse the stress distributions for isotropic fabrics under different boundary conditions, including consideration of different bagging heights and the friction between the fabric sample and the steel ball deforming the fabric. The results indicate that a non-uniform distribution of meridian stress and a non-continuous distribution of hoop stress, along with bagging height, are important factors that may influence residual-bagging deformation and may cause localised damage of the fabric. To investigate the influence of fabric anisotropy on the bagging behaviour of a fabric, the stress distribution of an anisotropic fabric is studied on the basis of the measurements of tensile moduli in seven directions by using fabric strips and of the fabric strain calculated from the relation between the geometrical deformation and bagging height of a fabric. The analysis shows that a non-uniform stress distribution along the meridian direction and variation of the tensile angle θ may cause the difference in the yarn stresses between the warp and weft directions, resulting in different bagging shapes. Comparing predicted bagging forces with measured forces, it is found that the method is able to predict the trend of bagging force with some deviations.

A Distribution-Free Two-Sample Equivalence Test Allowing for Tied Observations

A new testing procedure is derived which enables to assess the equivalence of two arbitrary noncontinuous distribution functions from which unrelated samples are taken as the data to be analyzed. The equivalence region is defined to consist of all pairs (F,G) of distribution functions such that for independent X ∼ F, Y ∼ G the conditional probability of {X > Y} given {X ≠ Y} lies in some short interval around 1/2. The test rejects the null hypothesis of nonequivalence if and only if the standardized distance between the U-statistics estimator of P[X > Y ∣ X ≠ Y] and the center of the equivalence interval (1/2 — ε1, 1/2 + ε2) does not exceed a critical upper bound which has to be computed as the α-quantile of a χ2-distribution with one degree of freedom and a random noncentrality parameter proportional to the squared length of that interval. The test is shown to maintain the asymptotic significance level under very weak regularity conditions. Results of an extensive simulation study suggest that its level properties are very satisfactory in small samples as well. The power turns out to be inversely related to the rate P[X = Y] of ties between observations from different samples.

A Distribution-Free Two-Sample Equivalence Test Allowing for Tied Observations

A new testing procedure is derived which enables to assess the equivalence of two arbitrary noncontinuous distribution functions from which unrelated samples are taken as the data to be analyzed. The equivalence region is defined to consist of all pairs (F, G) of distribution functions such that for independent X ∼F, Y ∼G the conditional probability of {X > Y} given {X ¬= Y} lies in some short interval around 1/2. The test rejects the null hypothesis of nonequivalence if and only if the standardized distance between the U-statistics estimator of P|X > Y | X ¬= Y] and the center of the equivalence interval (1/2 - e 1 , 1/2 + e 2 ) does not exceed a critical upper bound which has to be computed as the α-quantile of a X 2 -distribution with one degree of freedom and a random noncentrality parameter proportional to the squared length of that interval. The test is shown to maintain the asymptotic significance level under very weak regularity conditions. Results of an extensive simulation study suggest that its level properties are very satisfactory in small samples as well. The power turns out to be inversely related to the rate P[X = Y] of ties between observations from different samples.