Semi-continuous Distributions Research Articles

Asymptotic results for recursive multivariate associated-kernel estimators of the probability density mass function of a data stream

. In this article, our central focus is to investigate the non parametric estimator of the probability density or mass function of a data stream by using the general family of kernels, also called multivariate associated kernels. Being able to estimate categorial, count, discrete, and (semi)continuous distributions, these multivariate associated kernel estimators unify several particular cases. Within this framework, we first introduce a recursive estimator for non classical associated kernels. Under reasonable assumptions, we subsequently exhibit their asymptotic results, such as the local uniform L q -consistency, the pointwise asymptotic normality and finally the strong global consistency. Illustrative examples of associated kernels are revisited.

The Zero-Adjusted Log-Symmetric Distributions: Point and Intervalar Estimation.

In this paper, a new class of semi-continuous distributions called zero-adjusted log-symmetric is introduced and studied. Some properties and parameters estimation by maximum likelihood method are derived and confidence intervals (CIs) are developed. A simulation study is conducted to evaluate properties of the maximum likelihood estimators in lighter/heavier-tailed distributions. Finally, an application in a real data set is presented to illustrate the flexibility of the proposed class of distributions.

Discriminating between and within (semi)continuous classes of both Tweedie and geometric Tweedie models

In both Tweedie and geometric Tweedie models, the common power parameter works as an automatic distribution selection. It separates two subclasses of semicontinuous (1<p<2) and positive continuous () distributions. Our paper centres around exploring diagnostic tools based on the maximum likelihood ratio test and minimum Kolmogorov-Smirnov distance methods to discriminate very close distributions within each subclass of these two models according to values of p. Grounded on the unique equality of variation indices, we also discriminate the gamma and geometric gamma distributions with p = 2 in Tweedie and geometric Tweedie families, respectively. Probabilities of correct selection for several parameters combination and sample sizes are examined by simulations. We thus perform a numerical comparison study to assess the discrimination procedures in these subclasses of two families. Finally, semicontinuous () distributions in the broad sense are significantly more distinguishable than the over-varied continuous (p>2) ones; and two datasets for illustration purposes are investigated.

A new two-part test based on density ratio model for zero-inflated continuous distributions

In this paper, we consider testing the hypothesis concerning the means of two independent semicontinuous distributions whose observations are zero-inflated, characterized by a sizable number of zeros and positive observations from a continuous distribution. The continuous parts of the two semicontinuous distributions are assumed to follow a density ratio model. A new two-part test is developed for this kind of data. The proposed test takes the sum of one test for equality of proportions of zero values and one conditional test for the continuous distribution. The test is proved to follow a χ2 distribution with two degrees of freedom. Simulation studies show that the proposed test controls the type I error rates at the desired level, and is competitive to, and most of the time more powerful than two popular tests. A real data example from a dietary intervention study is used to illustrate the usefulness of the proposed test.

On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance [Formula: see text] from a spacelike hypersurface [Formula: see text] is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure, compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.

Causality theory for closed cone structures with applications

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including: causal ladder, Fermat’s principle, notable singularity theorems in their causal formulation, Avez–Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, [Formula: see text]-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them, we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz–Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well-known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula à la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.

Mean and cell-mean imputation resulting in the use of a semicontinuous distribution and a mixture of semicontinuous distributions

ABSTRACTZero-inflated models are commonly used for modeling count and continuous data with extra zeros. Inflations at one point or two points apart from zero for modeling continuous data have been discussed less than that of zero inflation. In this article, inflation at an arbitrary point α as a semicontinuous distribution is presented and the mean imputation for a continuous response is discussed as a cause of having semicontinuous data. Also, inflation at two points and generally at k arbitrary points and their relation to cell-mean imputation in the mixture of continuous distributions are studied. To analyze the imputed data, a mixture of semicontinuous distributions is used. The effects of covariates on the dependent variable in a mixture of k semicontinuous distributions with inflation at k points are also investigated. In order to find the parameter estimates, the method of expectation–maximization (EM) algorithm is used. In a real data of Iranian Households Income and Expenditure Survey (IHIES), it is shown how to obtain a proper estimate of the population variance when continuous missing at random responses are mean imputed.

Treatment of Multivariate Outliers in Incomplete Business Survey Data

The distribution of multivariate quantitative survey data usually is not normal. Skewed and semi-continuous distributions occur often. In addition, missing values and non-response is common. All together this mix of problems makes multivariate outlier detection difficult. Examples of surveys where these problems occur are most business surveys and some household surveys like the Survey for the Statistics of Income and Living Condition (SILC) of the European Union. Several methods for multivariate outlier detection are collected in the R-package modi. This paper gives an overview of modi and its functions for outlier detection and corresponding imputation. The use of the methods is explained with a business survey dataset. The discussion covers pre- and post-processing to deal with skewness and zero-inflation, advantages and disadvantages of the methods and the choice of the parameters.