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Register a new userConfidence Distribution and Distribution Estimation for Modern Statistical Inference
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Added by
Yifan Cui
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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This paper introduces to readers the new concept and methodology of confidence distribution and the modern-day distributional inference in statistics. This discussion should be of interest to people who would like to go into the depth of the statistical inference methodology and to utilize distribution estimators in practice. We also include in the discussion the topic of generalized fiducial inference, a special type of modern distributional inference, and relate it to the concept of confidence distribution. Several real data examples are also provided for practitioners. We hope that the selected content covers the greater part of the developments on this subject.
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Distribution function is essential in statistical inference, and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces and no longer convenient to use in rapidly evolving data objects of complex nature. It is imperative to develop the concept of distribution function in a more general space to meet emerging needs. Note that the linearity allows us to use hypercubes to define the distribution function in a Euclidean space, but without the linearity in a metric space, we must work with the metric to investigate the probability measure. We introduce a class of metric distribution functions through the metric between random objects and a fixed location in metric spaces. We overcome this challenging step by proving the correspondence theorem and the Glivenko-Cantelli theorem for metric distribution functions in metric spaces that lie the foundation for conducting rational statistical inference for metric space-valued data. Then, we develop homogeneity test and mutual independence test for non-Euclidean random objects, and present comprehensive empirical evidence to support the performance of our proposed methods.
This paper proposed a new probability distribution named as inverse xgamma distribution (IXGD). Different mathematical and statistical properties,viz., reliability characteristics, moments, inverse moments, stochastic ordering and order statistics of the proposed distribution have been derived and discussed. The estimation of the parameter of IXGD has been approached by different methods of estimation, namely, maximum likelihood method of estimation (MLE), Least square method of estimation (LSE), Weighted least square method of estimation (WLSE), Cramer-von-Mises method of estimation (CME) and maximum product spacing method of estimation (MPSE). Asymptotic confidence interval (ACI) of the parameter is also obtained. A simulation study has been carried out to compare the performance of the obtained estimators and corresponding ACI in terms of average widths and corresponding coverage probabilities. Finally, two real data sets have been used to demonstrate the applicability of IXGD in real life situations.
In this article, we proposed a new probability distribution named as power Maxwell distribution (PMaD). It is another extension of Maxwell distribution (MaD) which would lead more flexibility to analyze the data with non-monotone failure rate. Different statistical properties such as reliability characteristics, moments, quantiles, mean deviation, generating function, conditional moments, stochastic ordering, residual lifetime function and various entropy measures have been derived. The estimation of the parameters for the proposed probability distribution has been addressed by maximum likelihood estimation method and Bayes estimation method. The Bayes estimates are obtained under gamma prior using squared error loss function. Lastly, real-life application for the proposed distribution has been illustrated through different lifetime data.
Understanding causal relationships is one of the most important goals of modern science. So far, the causal inference literature has focused almost exclusively on outcomes coming from a linear space, most commonly the Euclidean space. However, it is increasingly common that complex datasets collected through electronic sources, such as wearable devices and medical imaging, cannot be represented as data points from linear spaces. In this paper, we present a formal definition of causal effects for outcomes from non-linear spaces, with a focus on the Wasserstein space of cumulative distribution functions. We develop doubly robust estimators and associated asymptotic theory for these causal effects. Our framework extends to outcomes from certain Riemannian manifolds. As an illustration, we use our framework to quantify the causal effect of marriage on physical activity patterns using wearable device data collected through the National Health and Nutrition Examination Survey.
Generative Adversarial Networks (GANs) have achieved great success in unsupervised learning. Despite the remarkable empirical performance, there are limited theoretical understandings on the statistical properties of GANs. This paper provides statistical guarantees of GANs for the estimation of data distributions which have densities in a H{o}lder space. Our main result shows that, if the generator and discriminator network architectures are properly chosen (universally for all distributions with H{o}lder densities), GANs are consistent estimators of the data distributions under strong discrepancy metrics, such as the Wasserstein distance. To our best knowledge, this is the first statistical theory of GANs for H{o}lder densities. In comparison with existing works, our theory requires minimum assumptions on data distributions. Our generator and discriminator networks utilize general weight matrices and the non-invertible ReLU activation function, while many existing works only apply to invertible weight matrices and invertible activation functions. In our analysis, we decompose the error into a statistical error and an approximation error by a new oracle inequality, which may be of independent interest.
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