A saddlepoint approximation of the Student's t-statistic was derived by Daniels and Young [Biometrika 78 (1991) 169-179] under the very stringent exponential moment condition that requires that the underlying density function go down at least as fast as a Normal density in the tails. This is a severe restriction on the approximation's applicability. In this paper we show that this strong exponential moment restriction can be completely dispensed with, that is, saddlepoint approximation of the Student's t-statistic remains valid without any moment condition. This confirms the folklore that the Student's t-statistic is robust against outliers. The saddlepoint approximation not only provides a very accurate approximation for the Student's t-statistic, but it also can be applied much more widely in statistical inference. As a result, saddlepoint approximations should always be used whenever possible. Some numerical work will be given to illustrate these points.
We define a moment-based estimator that maximizes the empirical saddlepoint (ESP) approximation of the distribution of solutions to empirical moment conditions. We call it the ESP estimator. We prove its existence, consistency and asymptotic normality, and we propose novel test statistics. We also show that the ESP estimator corresponds to the MM (method of moments) estimator shrunk toward parameter values with lower estimated variance, so it reduces the documented instability of existing moment-based estimators. In the case of just-identified moment conditions, which is the case we focus on, …
An Extended Empirical Saddlepoint Approximation for Intractable Likelihoods
计算生态学、生物学和遗传学中使用的复杂随机模型所带来的挑战刺激了统计推断近似方法的发展。在这里，我们专注于合成似然函数（SL），一个程序，减少了观察到的和模拟的数据，一组汇总统计量，并通过一个合成的似然函数量化它们之间的差异。SL几乎不需要调整，但它依赖于汇总统计量的近似正态性。我们放松这个假设，提出了一种新的，更灵活的，密度估计：扩展经验鞍点近似。除了证明SL的一致性，无论是新的或高斯密度估计，我们用两个例子来说明该方法。其中之一是一个复杂的基于个体的森林模型，SL为统计推断提供了为数不多的实际可能性之一。例子表明，新的密度估计是能够捕捉大偏离正态分布，同时可扩展到高维，这反过来又导致更准确的参数估计，相对于高斯替代。新的密度估计器由esaddle R包实现，该包可以在Comprehensive R Archive Network（CRAN）上找到。
For certain types of statistical models, the characteristic function (Fourier transform) is available in closed form, whereas the probability density function has an intractable form, typically as an infinite sum of probability weighted densities. Important such examples include solutions of stochastic differential equations with jumps, the Tweedie model, and Poisson mixture models. We propose a novel and general numerical method for retrieving the probability density function from the characteristic function, conditioned on the existence of the moment generating function. Unlike methods based…
Consistency of the jackknife-after-bootstrap variance estimator for the bootstrap quantiles of a studentized statistic
埃夫隆[J.罗伊。中央集权主义者Soc.Ser.B54（1992）83- 111]提出了一种计算效率高的方法，称为自举后折刀法，用于估计独立数据的自举估计量的方差。对于相依数据，Lahiri最近提出了一种自举后折刀法[Econometric Theory 18（2002）79- 98]。本文证明了对于相依和独立数据，Bootstrap分位数方差的Jackknife-after-Bootstrap估计是相合的。模拟研究的结果也被提出。
Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject to be error prone because of statistical estimation error. Lo (2002), Mertens (2002) derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory un…
Drawing inference from data is an important skill for students to understand their everyday life, so that the sampling distribution as a central topic in statistical inference is necessary to be learned by the students. However, little is known about how to teach the topic for high school students, especially in Indonesian context. Therefore, the present study provides a teaching experiment to support the students' informal inferential reasoning in understanding the sampling distribution, as well as the students' perceptions toward the teaching experiment. The subjects in the present study wer…
Entropy and its various generalizations are important in many fields, including mathematical statistics, communication theory, physics and computer science, for characterizing the amount of information associated with a probability distribution. In this paper we propose goodness-of-fit statistics for the multivariate Student and multivariate Pearson type II distributions, based on the maximum entropy principle and a class of estimators for Rényi entropy based on nearest neighbour distances. We prove the L^2-consistency of these statistics using results on the subadditivity of Euclidean functio…
