We consider the minimax estimation problem of a discrete distribution with support size <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> under privacy constraints. A ...privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula> measures the privacy level of a privatization scheme. For a given <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula>, we consider the problem of constructing optimal privatization schemes with <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula>-privacy level, i.e., schemes that minimize the expected estimation loss for the worst-case distribution. Two schemes known in the literature provide order optimal performance in the high privacy regime where <inline-formula> <tex-math notation="LaTeX">\epsilon </tex-math></inline-formula> is very close to 0, and in the low privacy regime where <inline-formula> <tex-math notation="LaTeX">e^{\epsilon }\approx k </tex-math></inline-formula>, respectively. In this paper, we propose a new family of schemes which substantially improve the performance of the existing schemes in the medium privacy regime when <inline-formula> <tex-math notation="LaTeX">1\ll e^{\epsilon } \ll k </tex-math></inline-formula>. More concretely, we prove that when <inline-formula> <tex-math notation="LaTeX">3.8 < \epsilon <\ln (k/9) </tex-math></inline-formula>, our schemes reduce the expected estimation loss by 50% under <inline-formula> <tex-math notation="LaTeX">\ell _{2}^{2} </tex-math></inline-formula> metric and by 30% under <inline-formula> <tex-math notation="LaTeX">\ell _{1} </tex-math></inline-formula> metric over the existing schemes. We also prove a lower bound for the region <inline-formula> <tex-math notation="LaTeX">e^{\epsilon } \ll k </tex-math></inline-formula>, which implies that our schemes are order optimal in this regime.
In a variety of research areas, the weighted bag of vectors and the histogram are widely used descriptors for complex objects. Both can be expressed as discrete distributions. D2-clustering pursues ...the minimum total within-cluster variation for a set of discrete distributions subject to the Kantorovich-Wasserstein metric. D2-clustering has a severe scalability issue, the bottleneck being the computation of a centroid distribution, called Wasserstein barycenter, that minimizes its sum of squared distances to the cluster members. In this paper, we develop a modified Bregman ADMM approach for computing the approximate discrete Wasserstein barycenter of large clusters. In the case when the support points of the barycenters are unknown and have low cardinality, our method achieves high accuracy empirically at a much reduced computational cost. The strengths and weaknesses of our method and its alternatives are examined through experiments, and we recommend scenarios for their respective usage. Moreover, we develop both serial and parallelized versions of the algorithm. By experimenting with large-scale data, we demonstrate the computational efficiency of the new methods and investigate their convergence properties and numerical stability. The clustering results obtained on several datasets in different domains are highly competitive in comparison with some widely used methods in the corresponding areas.
The paper is concerned with an age based preventive replacement policy for an arbitrary coherent system that consists of components that are independent and have common discrete lifetime ...distribution. The system having an arbitrary structure is replaced preventively after a specific number of cycles or correctively at its failure time. The necessary conditions for the unique and finite replacement cycle that minimize the expected cost per unit of time are obtained. The policy is studied for some particular system models including the well-known k-out-of-n structure. The findings of the paper extend the results in the literature from single unit and parallel systems to an arbitrary coherent system. Numerical results are presented for particular discrete component lifetime distributions.
•Discrete time age replacement policy is considered.•The extension from parallel system to a coherent system.•The components follow discrete lifetime distribution.•The necessary conditions for the optimal replacement cycle.
In this study, a new one-parameter discrete distribution obtained by compounding the Poisson and xgamma distributions is proposed. Some statistical properties of the new distribution are obtained ...including moments and probability and moment generating functions. Two methods are used for the estimation of the unknown parameter: the maximum likelihood method and the method of moments. Additionally, the count regression model and integer-valued autoregressive process of the proposed distribution are introduced. Some possible applications of the introduced models are considered and discussed.
Dairy cattle of different ages experience different living conditions and varied frequency of antibiotic administration that likely influence the distribution of microbiome and resistome in ways that ...reflect different risks of microbial transmission. To assess the degree of variance in these distributions, fecal and soil samples were collected from six distinct housing areas on commercial dairy farms (n = 7) in Washington State. 16S rRNA gene sequencing indicated that the microbiota differed between different on-farm locations in feces and soil, and in both cases, the microbiota of dairy calves was often distinct from others (P < 0.05). Thirty-two specific antibiotic resistance genes (ARGs) were widely distributed on dairies, of which several clinically relevant ARGs (including cfr, cfrB, and optrA) were identified for the first time at U.S. dairies. Overall, ARGs were observed more frequently in feces and soil from dairy calves and heifers than from hospital, fresh, lactation and dry pens. Droplet-digital PCR demonstrated that the absolute abundance of floR varied greatly across housing areas and this gene was enriched the most in calves and heifers. Furthermore, in an extended analysis with 14 dairies, environmental soils in calf pens had the most antibiotic-resistant Escherichia coli followed by heifer and hospital pens. All soil E. coli isolates (n = 1,905) are resistant to at least 4 different antibiotics, and the PFGE analysis indicated that florfenicol-resistant E. coli is probably shared across geographically-separated farms. This study identified a discrete but predictable distribution of antibiotic resistance genes and organisms, which is important for designing mitigation for higher risk areas on dairy farms.
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•Fecal and soil microbiota vary significantly across housing areas on dairy farms.•Feces and soil from young animals have higher prevalence of ARGs than other groups.•Environmental soils in calf pen have the most antibiotic-resistant E. coli.•Florfenicol-resistant E. coli likely migrates across geographically separated farms.
In this article we present a stochastic ordering verification algorithm between multivariate discrete distributions implemented in the C++ programming language. This algorithm is essential in ...problems of finding the optimal portfolio when dealing with discrete distributions.
In the literature of distribution theory, a vast proportion is acquired by discrete distributions and their applications in real-world phenomena. However, in a rapidly changing technological era, the ...data generated is becoming increasingly complex day by day, making it difficult for us to capture various aspects of this real data through existing discrete models. In view of this, we propose a new flexible discrete distribution with one parameter. Some statistical and reliability are derived. These properties can be expressed as closed-forms. One of the important virtues of this newly evolved model is that it can model not only over-dispersed, positively skewed and leptokurtic data sets, but it can also be utilized for modeling increasing, decreasing and unimodal failure rate. Various estimation approaches are utilized to estimate the model parameter. A simulation study is carried out to examine the performance of the estimators for different sample size. The flexibility of the new model for analyzing different types of data is explained by utilizing four real data sets in different fields. Finally, the proposed model can serve as an alternative model to other distributions in the existing literature for modeling positive real data in several areas.
In this paper, we introduce a new distribution defined on
$ \mathbb {Z} $
Z
, called the
$ {{ZPMOIEP}} $
ZPMOIEP
distribution, which can be viewed as a natural extension of the zero-and-one-inflated ...Poisson (
$ {ZOIP} $
ZOIP
) distribution. It is designed to fit the count data with potentially excess zeros and/or ones, and/or minus ones. We explore its various properties and investigate the estimation of the unknown parameters. Moreover, simulation experiments are carried out to attest to the performance of the estimation. Through the use of a useful data set on football scores, the applicability of the proposed distribution is examined.
This paper studies the matching problem of cross-modality video data from a discrete distribution alignment view. Central to this discussion is the visible-infrared person re-identification ...(VI-reID), a crucial feature that bolsters surveillance systems’ efficacy in monitoring individuals across diverse lighting conditions. Going beyond traditional image-to-image matching paradigms, a recent study shows that temporal information can bring richer cues to encode the pedestrian representation, improving the representation power of deep neural networks. However, this integration further complicates cross-modality data matching due to the joint processing of spatial and temporal information. This paper formulates the video data as a discrete distribution and aligns the cross-modality video representation by reducing the matching cost between the two distributions. To this end, a natural idea for aligning the videos is to reduce the divergence of distributions. Moreover, the powerful optimal transport (OT) scheme, which generates the optimal matching flows and establishes the relevance of two distributions, is also employed as a way to measure the distance of distributions. Nevertheless, we observe that endowing the OT in the advanced VI-reID feature extractor leads to a non-symmetric measurement. To mitigate this, the paper introduces a new metric, namely symmetric optimal transport (SOT), reformulating OT into a symmetric form. Thorough analyses and empirical studies affirm the superiority of the proposed SOT, which significantly outperforms the current state-of-the-art methods according to standard benchmarking evaluations.
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•This paper formally formulates the video sequences into a discrete distribution.•This paper further provides a symmetric optimal transport (SOT) scheme.•Thorough experiments are conducted to evaluate the proposed SOT.
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