Avhad Ganesh Vishnu, Ananya Lahiri, Sudheesh K. Kattumannil
Heavy-tailed distributions, such as the Cauchy distribution, are acknowledged for providing more accurate models for financial returns, as the normal distribution is deemed insufficient for capturing the significant fluctuations observed in real-world assets. Data sets characterized by outlier sensitivity are critically important in diverse areas, including finance, economics, telecommunications, and signal processing. This article addresses a goodness-of-fit test for the Cauchy distribution. The proposed test utilizes empirical likelihood methods, including the jackknife empirical likelihood (JEL) and adjusted jackknife empirical likelihood (AJEL). Extensive Monte Carlo simulation studies are conducted to evaluate the finite sample performance of the proposed test. The application of the proposed test is illustrated through the analysing two real data sets.
{"title":"Jackknife Empirical Likelihood Ratio Test for Cauchy Distribution","authors":"Avhad Ganesh Vishnu, Ananya Lahiri, Sudheesh K. Kattumannil","doi":"arxiv-2409.05764","DOIUrl":"https://doi.org/arxiv-2409.05764","url":null,"abstract":"Heavy-tailed distributions, such as the Cauchy distribution, are acknowledged\u0000for providing more accurate models for financial returns, as the normal\u0000distribution is deemed insufficient for capturing the significant fluctuations\u0000observed in real-world assets. Data sets characterized by outlier sensitivity\u0000are critically important in diverse areas, including finance, economics,\u0000telecommunications, and signal processing. This article addresses a\u0000goodness-of-fit test for the Cauchy distribution. The proposed test utilizes\u0000empirical likelihood methods, including the jackknife empirical likelihood\u0000(JEL) and adjusted jackknife empirical likelihood (AJEL). Extensive Monte Carlo\u0000simulation studies are conducted to evaluate the finite sample performance of\u0000the proposed test. The application of the proposed test is illustrated through\u0000the analysing two real data sets.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this work is to estimate the drift coefficient of a fractional heat equation driven by an additive space-time noise using the Maximum likelihood estimator (MLE). In the first part of the paper, the first $N$ Fourier modes of the solution are observed continuously over a finite time interval $[0, T ]$. The explicit upper bounds for the Wasserstein distance for the central limit theorem of the MLE is provided when $N rightarrow infty$ and/or $T rightarrow infty$. While in the second part of the paper, the $N$ Fourier modes are observed at uniform time grid : $t_i = i frac{T}{M}$, $i=0,..,M,$ where $M$ is the number of time grid points. The consistency and asymptotic normality are studied when $T,M,N rightarrow + infty$ in addition to the rate of convergence in law in the CLT.
{"title":"Parameter estimation for fractional stochastic heat equations : Berry-Esséen bounds in CLTs","authors":"Soukaina Douissi, Fatimah Alshahrani","doi":"arxiv-2409.05416","DOIUrl":"https://doi.org/arxiv-2409.05416","url":null,"abstract":"The aim of this work is to estimate the drift coefficient of a fractional\u0000heat equation driven by an additive space-time noise using the Maximum\u0000likelihood estimator (MLE). In the first part of the paper, the first $N$\u0000Fourier modes of the solution are observed continuously over a finite time\u0000interval $[0, T ]$. The explicit upper bounds for the Wasserstein distance for\u0000the central limit theorem of the MLE is provided when $N rightarrow infty$\u0000and/or $T rightarrow infty$. While in the second part of the paper, the $N$\u0000Fourier modes are observed at uniform time grid : $t_i = i frac{T}{M}$,\u0000$i=0,..,M,$ where $M$ is the number of time grid points. The consistency and\u0000asymptotic normality are studied when $T,M,N rightarrow + infty$ in addition\u0000to the rate of convergence in law in the CLT.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the literature, derivations of exact null distributions of rank-sum statistics is often avoided in cases where one or more ties exist in the data. By deriving the null distribution in the no-ties case with the aid of classical $q$-series results of Euler and Rothe, we demonstrate how a natural generalization of the method may be employed to derive exact null distributions even when one or more ties are present in the data. It is suggested that this method could be implemented in a computer algebra system, or even a more primitive computer language, so that the normal approximation need not be employed in the case of small sample sizes, when it is less likely to be very accurate. Several algorithms for determining exact distributions of the rank-sum statistic (possibly with ties) have been given in the literature (see Streitberg and R"ohmel (1986) and Marx et al. (2016)), but none seem as simple as the procedure discussed here which amounts to multiplying out a certain polynomial, extracting coefficients, and finally dividing by a binomal coefficient.
通过借助欧拉和罗特的经典 q$ 系列结果推导无并列情况下的零分布，我们展示了如何利用该方法的自然概括来推导精确的零分布，即使数据中存在一个或多个并列。我们建议这种方法可以在计算机代数系统，甚至更简单的计算机语言中实现，这样在样本量较小的情况下就不必使用正态近似，因为正态近似不太可能非常精确。文献中已经给出了几种确定柄和统计量精确分布（可能有并列关系）的算法（见 Streitberg and R"ohmel (1986) and Marx et al. (2016)），但似乎都不如这里讨论的程序简单，它相当于乘出某个二项式，提取系数，最后除以二项式系数。
{"title":"On integer partitions and the Wilcoxon rank-sum statistic","authors":"Andrew V. Sills","doi":"arxiv-2409.05741","DOIUrl":"https://doi.org/arxiv-2409.05741","url":null,"abstract":"In the literature, derivations of exact null distributions of rank-sum\u0000statistics is often avoided in cases where one or more ties exist in the data.\u0000By deriving the null distribution in the no-ties case with the aid of classical\u0000$q$-series results of Euler and Rothe, we demonstrate how a natural\u0000generalization of the method may be employed to derive exact null distributions\u0000even when one or more ties are present in the data. It is suggested that this\u0000method could be implemented in a computer algebra system, or even a more\u0000primitive computer language, so that the normal approximation need not be\u0000employed in the case of small sample sizes, when it is less likely to be very\u0000accurate. Several algorithms for determining exact distributions of the\u0000rank-sum statistic (possibly with ties) have been given in the literature (see\u0000Streitberg and R\"ohmel (1986) and Marx et al. (2016)), but none seem as simple\u0000as the procedure discussed here which amounts to multiplying out a certain\u0000polynomial, extracting coefficients, and finally dividing by a binomal\u0000coefficient.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Existing concentration bounds for bounded vector-valued random variables include extensions of the scalar Hoeffding and Bernstein inequalities. While the latter is typically tighter, it requires knowing a bound on the variance of the random variables. We derive a new vector-valued empirical Bernstein inequality, which makes use of an empirical estimator of the variance instead of the true variance. The bound holds in 2-smooth separable Banach spaces, which include finite dimensional Euclidean spaces and separable Hilbert spaces. The resulting confidence sets are instantiated for both the batch setting (where the sample size is fixed) and the sequential setting (where the sample size is a stopping time). The confidence set width asymptotically exactly matches that achieved by Bernstein in the leading term. The method and supermartingale proof technique combine several tools of Pinelis (1994) and Waudby-Smith and Ramdas (2024).
{"title":"Empirical Bernstein in smooth Banach spaces","authors":"Diego Martinez-Taboada, Aaditya Ramdas","doi":"arxiv-2409.06060","DOIUrl":"https://doi.org/arxiv-2409.06060","url":null,"abstract":"Existing concentration bounds for bounded vector-valued random variables\u0000include extensions of the scalar Hoeffding and Bernstein inequalities. While\u0000the latter is typically tighter, it requires knowing a bound on the variance of\u0000the random variables. We derive a new vector-valued empirical Bernstein\u0000inequality, which makes use of an empirical estimator of the variance instead\u0000of the true variance. The bound holds in 2-smooth separable Banach spaces,\u0000which include finite dimensional Euclidean spaces and separable Hilbert spaces.\u0000The resulting confidence sets are instantiated for both the batch setting\u0000(where the sample size is fixed) and the sequential setting (where the sample\u0000size is a stopping time). The confidence set width asymptotically exactly\u0000matches that achieved by Bernstein in the leading term. The method and\u0000supermartingale proof technique combine several tools of Pinelis (1994) and\u0000Waudby-Smith and Ramdas (2024).","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shubhada Agrawal, Prashanth L. A., Siva Theja Maguluri
We consider the problem of estimating the asymptotic variance of a function defined on a Markov chain, an important step for statistical inference of the stationary mean. We design the first recursive estimator that requires $O(1)$ computation at each step, does not require storing any historical samples or any prior knowledge of run-length, and has optimal $O(frac{1}{n})$ rate of convergence for the mean-squared error (MSE) with provable finite sample guarantees. Here, $n$ refers to the total number of samples generated. The previously best-known rate of convergence in MSE was $O(frac{log n}{n})$, achieved by jackknifed estimators, which also do not enjoy these other desirable properties. Our estimator is based on linear stochastic approximation of an equivalent formulation of the asymptotic variance in terms of the solution of the Poisson equation. We generalize our estimator in several directions, including estimating the covariance matrix for vector-valued functions, estimating the stationary variance of a Markov chain, and approximately estimating the asymptotic variance in settings where the state space of the underlying Markov chain is large. We also show applications of our estimator in average reward reinforcement learning (RL), where we work with asymptotic variance as a risk measure to model safety-critical applications. We design a temporal-difference type algorithm tailored for policy evaluation in this context. We consider both the tabular and linear function approximation settings. Our work paves the way for developing actor-critic style algorithms for variance-constrained RL.
{"title":"Markov Chain Variance Estimation: A Stochastic Approximation Approach","authors":"Shubhada Agrawal, Prashanth L. A., Siva Theja Maguluri","doi":"arxiv-2409.05733","DOIUrl":"https://doi.org/arxiv-2409.05733","url":null,"abstract":"We consider the problem of estimating the asymptotic variance of a function\u0000defined on a Markov chain, an important step for statistical inference of the\u0000stationary mean. We design the first recursive estimator that requires $O(1)$\u0000computation at each step, does not require storing any historical samples or\u0000any prior knowledge of run-length, and has optimal $O(frac{1}{n})$ rate of\u0000convergence for the mean-squared error (MSE) with provable finite sample\u0000guarantees. Here, $n$ refers to the total number of samples generated. The\u0000previously best-known rate of convergence in MSE was $O(frac{log n}{n})$,\u0000achieved by jackknifed estimators, which also do not enjoy these other\u0000desirable properties. Our estimator is based on linear stochastic approximation\u0000of an equivalent formulation of the asymptotic variance in terms of the\u0000solution of the Poisson equation. We generalize our estimator in several directions, including estimating the\u0000covariance matrix for vector-valued functions, estimating the stationary\u0000variance of a Markov chain, and approximately estimating the asymptotic\u0000variance in settings where the state space of the underlying Markov chain is\u0000large. We also show applications of our estimator in average reward\u0000reinforcement learning (RL), where we work with asymptotic variance as a risk\u0000measure to model safety-critical applications. We design a temporal-difference\u0000type algorithm tailored for policy evaluation in this context. We consider both\u0000the tabular and linear function approximation settings. Our work paves the way\u0000for developing actor-critic style algorithms for variance-constrained RL.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces a new version of the smoothly trimmed mean with a more general version of weights, which can be used as an alternative to the classical trimmed mean. We derive its asymptotic variance and to further investigate its properties we establish the empirical likelihood for the new estimator. As expected from previous theoretical investigations we show in our simulations a clear advantage of the proposed estimator over the classical trimmed mean estimator. Moreover, the empirical likelihood method gives an additional advantage for data generated from contaminated models. For the classical trimmed mean it is generally recommended in practice to use symmetrical 10% or 20% trimming. However, if the trimming is done close to data gaps, it can even lead to spurious results, as known from the literature and verified by our simulations. Instead, for practical data examples, we choose the smoothing parameters by an optimality criterion that minimises the variance of the proposed estimators.
{"title":"Empirical likelihood for generalized smoothly trimmed mean","authors":"Elina Kresse, Emils Silins, Janis Valeinis","doi":"arxiv-2409.05631","DOIUrl":"https://doi.org/arxiv-2409.05631","url":null,"abstract":"This paper introduces a new version of the smoothly trimmed mean with a more\u0000general version of weights, which can be used as an alternative to the\u0000classical trimmed mean. We derive its asymptotic variance and to further\u0000investigate its properties we establish the empirical likelihood for the new\u0000estimator. As expected from previous theoretical investigations we show in our\u0000simulations a clear advantage of the proposed estimator over the classical\u0000trimmed mean estimator. Moreover, the empirical likelihood method gives an\u0000additional advantage for data generated from contaminated models. For the\u0000classical trimmed mean it is generally recommended in practice to use\u0000symmetrical 10% or 20% trimming. However, if the trimming is done close to\u0000data gaps, it can even lead to spurious results, as known from the literature\u0000and verified by our simulations. Instead, for practical data examples, we\u0000choose the smoothing parameters by an optimality criterion that minimises the\u0000variance of the proposed estimators.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the efficient estimation of a class of mean functionals in settings where a complete multivariate dataset is complemented by additional datasets recording subsets of the variables of interest. These datasets are allowed to have a general, in particular non-monotonic, structure. Our main contribution is to characterise the asymptotic minimal mean squared error for these problems and to introduce an estimator whose risk approximately matches this lower bound. We show that the efficient rescaled variance can be expressed as the minimal value of a quadratic optimisation problem over a function space, thus establishing a fundamental link between these estimation problems and the theory of generalised ANOVA decompositions. Our estimation procedure uses iterated nonparametric regression to mimic an approximate influence function derived through gradient descent. We prove that this estimator is approximately normally distributed, provide an estimator of its variance and thus develop confidence intervals of asymptotically minimal width. Finally we study a more direct estimator, which can be seen as a U-statistic with a data-dependent kernel, showing that it is also efficient under stronger regularity conditions.
我们研究的是在一个完整的多元数据集的基础上，通过记录相关变量子集的附加数据集，对一类均值函数进行有效估计的问题。允许这些数据集具有一般结构，特别是非单调结构。我们的主要贡献在于描述了这些问题的渐近最小均方误差，并引入了一种风险与该下限近似匹配的估计器。我们证明，有效的重标方差可以表示为函数空间上二次优化问题的最小值，从而在这些估计问题和广义方差分解理论之间建立了基本联系。我们的估算程序使用迭代非参数回归来模拟通过梯度下降得到的近似影响函数。我们证明了该估计值近似正态分布，提供了其方差的估计值，从而得出了渐近最小宽度的置信区间。最后，我们研究了一种更直接的估计器，它可以看作是具有数据依赖核的 U 统计量，并表明在更强的正则性条件下它也是有效的。
{"title":"Efficient estimation with incomplete data via generalised ANOVA decomposition","authors":"Thomas B. Berrett","doi":"arxiv-2409.05729","DOIUrl":"https://doi.org/arxiv-2409.05729","url":null,"abstract":"We study the efficient estimation of a class of mean functionals in settings\u0000where a complete multivariate dataset is complemented by additional datasets\u0000recording subsets of the variables of interest. These datasets are allowed to\u0000have a general, in particular non-monotonic, structure. Our main contribution\u0000is to characterise the asymptotic minimal mean squared error for these problems\u0000and to introduce an estimator whose risk approximately matches this lower\u0000bound. We show that the efficient rescaled variance can be expressed as the\u0000minimal value of a quadratic optimisation problem over a function space, thus\u0000establishing a fundamental link between these estimation problems and the\u0000theory of generalised ANOVA decompositions. Our estimation procedure uses\u0000iterated nonparametric regression to mimic an approximate influence function\u0000derived through gradient descent. We prove that this estimator is approximately\u0000normally distributed, provide an estimator of its variance and thus develop\u0000confidence intervals of asymptotically minimal width. Finally we study a more\u0000direct estimator, which can be seen as a U-statistic with a data-dependent\u0000kernel, showing that it is also efficient under stronger regularity conditions.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"396 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Min-max optimization problems, also known as saddle point problems, have attracted significant attention due to their applications in various fields, such as fair beamforming, generative adversarial networks (GANs), and adversarial learning. However, understanding the properties of these min-max problems has remained a substantial challenge. This study introduces a statistical mechanical formalism for analyzing the equilibrium values of min-max problems in the high-dimensional limit, while appropriately addressing the order of operations for min and max. As a first step, we apply this formalism to bilinear min-max games and simple GANs, deriving the relationship between the amount of training data and generalization error and indicating the optimal ratio of fake to real data for effective learning. This formalism provides a groundwork for a deeper theoretical analysis of the equilibrium properties in various machine learning methods based on min-max problems and encourages the development of new algorithms and architectures.
{"title":"Statistical Mechanics of Min-Max Problems","authors":"Yuma Ichikawa, Koji Hukushima","doi":"arxiv-2409.06053","DOIUrl":"https://doi.org/arxiv-2409.06053","url":null,"abstract":"Min-max optimization problems, also known as saddle point problems, have\u0000attracted significant attention due to their applications in various fields,\u0000such as fair beamforming, generative adversarial networks (GANs), and\u0000adversarial learning. However, understanding the properties of these min-max\u0000problems has remained a substantial challenge. This study introduces a\u0000statistical mechanical formalism for analyzing the equilibrium values of\u0000min-max problems in the high-dimensional limit, while appropriately addressing\u0000the order of operations for min and max. As a first step, we apply this\u0000formalism to bilinear min-max games and simple GANs, deriving the relationship\u0000between the amount of training data and generalization error and indicating the\u0000optimal ratio of fake to real data for effective learning. This formalism\u0000provides a groundwork for a deeper theoretical analysis of the equilibrium\u0000properties in various machine learning methods based on min-max problems and\u0000encourages the development of new algorithms and architectures.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the incorporation of any new piece of information allows for improved decision making in the sense that the expected costs of an optimal decision decrease (or, in boundary cases where no or not enough new information is incorporated, stays the same) whenever this is done by the appropriate update of the probabilities of the hypotheses. Versions of this result have been stated before. However, previous proofs rely on auxiliary constructions with proper scoring rules. We, instead, offer a direct and completely general proof by considering elementary properties of likelihood ratios only. We do point out the relation to proper scoring rules. We apply our results to make a contribution to the debates about the use of score based/feature based and common/specific source likelihood ratios. In the literature these are often presented as different ``LR-systems''. We argue that deciding which LR to compute is simply a matter of the available information. There is no such thing as different ``LR-systems'', there are only differences in the available information. In particular, despite claims to the contrary, scores can very well be used in forensic practice and we illustrate this with an extensive example in DNA kinship context.
我们证明，只要适当更新假设的概率，纳入任何新信息都能改善决策，即最优决策的预期成本会降低（或者，在没有或没有纳入足够新信息的边界情况下，预期成本保持不变）。这一结果的不同版本以前也曾提出过。然而，以前的证明依赖于具有适当评分规则的辅助构造。而我们只考虑似然比的基本性质，就提供了一个直接的、完全通用的证明。我们指出了与适当评分规则的关系。我们运用我们的结果，为关于使用基于分数/基于特征和基于共同/特定来源的似然比的争论做出了贡献。在文献中，这些通常被表述为不同的 "LR 系统"。我们认为，决定计算哪种 LR 只是可用信息的问题。并不存在不同的 "LR 系统"，只有可用信息的差异。特别是，尽管有相反的说法，分数在法医实践中还是可以很好地使用，我们以 DNA 亲缘关系中的大量实例来说明这一点。
{"title":"Common or specific source, features or scores; it is all a matter of information","authors":"Aafko Boonstra, Ronald Meester, Klaas Slooten","doi":"arxiv-2409.05403","DOIUrl":"https://doi.org/arxiv-2409.05403","url":null,"abstract":"We show that the incorporation of any new piece of information allows for\u0000improved decision making in the sense that the expected costs of an optimal\u0000decision decrease (or, in boundary cases where no or not enough new information\u0000is incorporated, stays the same) whenever this is done by the appropriate\u0000update of the probabilities of the hypotheses. Versions of this result have\u0000been stated before. However, previous proofs rely on auxiliary constructions\u0000with proper scoring rules. We, instead, offer a direct and completely general\u0000proof by considering elementary properties of likelihood ratios only. We do\u0000point out the relation to proper scoring rules. We apply our results to make a\u0000contribution to the debates about the use of score based/feature based and\u0000common/specific source likelihood ratios. In the literature these are often\u0000presented as different ``LR-systems''. We argue that deciding which LR to\u0000compute is simply a matter of the available information. There is no such thing\u0000as different ``LR-systems'', there are only differences in the available\u0000information. In particular, despite claims to the contrary, scores can very\u0000well be used in forensic practice and we illustrate this with an extensive\u0000example in DNA kinship context.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Testing has developed into the fundamental statistical framework for falsifying hypotheses. Unfortunately, tests are binary in nature: a test either rejects a hypothesis or not. Such binary decisions do not reflect the reality of many scientific studies, which often aim to present the evidence against a hypothesis and do not necessarily intend to establish a definitive conclusion. To solve this, we propose the continuous generalization of a test, which we use to measure the evidence against a hypothesis. Such a continuous test can be interpreted as a non-randomized interpretation of the classical 'randomized test'. This offers the benefits of a randomized test, without the downsides of external randomization. Another interpretation is as a literal measure, which measures the amount of binary tests that reject the hypothesis. Our work also offers a new perspective on the $e$-value: the $e$-value is recovered as a continuous test with $alpha to 0$, or as an unbounded measure of the amount of rejections.
检验已发展成为证伪假设的基本统计框架。遗憾的是，检验在本质上是二元对立的：检验要么否定假设，要么不否定假设。为了解决这个问题，我们提出了检验的连续泛化，我们用它来衡量反对假设的证据。这种连续检验可以解释为经典 "随机检验 "的非随机化解释。它既有随机试验的优点，又没有外部随机化的缺点。另一种解释是字面测量，即测量拒绝假设的二元检验的数量。我们的工作还为e$值提供了一个新的视角：e$值被恢复为$alpha to 0$的连续检验，或作为拒绝量的无界度量。
{"title":"A Continuous Generalization of Hypothesis Testing","authors":"Nick W. Koning","doi":"arxiv-2409.05654","DOIUrl":"https://doi.org/arxiv-2409.05654","url":null,"abstract":"Testing has developed into the fundamental statistical framework for\u0000falsifying hypotheses. Unfortunately, tests are binary in nature: a test either\u0000rejects a hypothesis or not. Such binary decisions do not reflect the reality\u0000of many scientific studies, which often aim to present the evidence against a\u0000hypothesis and do not necessarily intend to establish a definitive conclusion.\u0000To solve this, we propose the continuous generalization of a test, which we use\u0000to measure the evidence against a hypothesis. Such a continuous test can be\u0000interpreted as a non-randomized interpretation of the classical 'randomized\u0000test'. This offers the benefits of a randomized test, without the downsides of\u0000external randomization. Another interpretation is as a literal measure, which\u0000measures the amount of binary tests that reject the hypothesis. Our work also\u0000offers a new perspective on the $e$-value: the $e$-value is recovered as a\u0000continuous test with $alpha to 0$, or as an unbounded measure of the amount\u0000of rejections.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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