34 s tat i s t i c a l i n f e r e n c e f o r e v e r yo n e
quantitatively resolve the level of uncertainty, and make valid infer- ences. It is in these cases that **statistical** **inference** is most useful.
**Statistical** **inference** refers to a field of study where we try to infer unknown properties of the world, given our observed data, in the face of uncertainty. It is a mathematical framework to quantify what our common sense says in many situations, but allows us to exceed our common sense in cases where common sense is not enough. Ig- norance of proper **statistical** **inference** leads to poor decisions and wasted money. As with ignorance in any other field, ignorance of sta- tistical **inference** can also allow others to manipulate you, convincing you of the truth of something that is false.

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D.CV. Lindley (1971). Making Decisions. Wiley Intersc., London.
A. Wald (1939). Annals Math. Stat., 10, 299–326.
Summary
This work was translated into English and published in the volume: Bruno De Finetti, Induction and Probability, Biblioteca di Statistica, eds. P. Monari, D. Cocchi, Clueb, Bologna, 1993. Bayesian **statistical** **Inference** is one of the last fundamental philosoph- ical papers in which we can find the essential De Finetti’s approach to the **statistical** **inference**.

Thus, the form of the asymptotic variance changes, in contrast to (6) it does no longer de- pend on g or x at all. The asymptotics in (14) could be used for constructing asymptotic conﬁdence intervals similarly as in (7), but with proper changes according to the form of the asymptotic variance, in particular, the estimate of g(x) is not needed. However, this is not recommended since convergence in (14) is quite slow, one should instead use bootstrap conﬁdence intervals. Another sensitive problem is the choice of the bandwidth in the super- smooth case, since this falls into the so-called bias-dominating case ([8]). This means that for optimal estimation, the bandwidth is chosen in such a way that the squared bias dominates the variance. However, for the construction of conﬁdence intervals one requires bandwidth which lead to variance-domination, but still to consistent estimation. This is still possible theoretically, but the construction of an adequate data-driven bandwidth choice is a hard, yet unsolved problem. Concerning uniform conﬁdence bands, no asymptotic results are avail- able for supersmooth deconvolution. Of course, one can simply bootstrap a supremum-type statistic, but no theoretical justiﬁcation is available here. Thus, there are still several open problems for **statistical** **inference** for supersmooth deconvolution.

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1. Introduction
The aim of this paper is to show the rich possibilities for asymptotically optimal **statistical** **inference** for “quantum i.i.d. models”. Despite the possibly exotic context, mathematical statistics has much to oﬀer, and much that we have learned—in particular through Jon Wellner’s work in semiparametric models and nonparametric maximum likelihood estimation—can be put to extremely good use. Exotic? In today’s quantum information engineering, measurement and estimation schemes are put to work to recover the state of a small number of quantum states, engineered by the physicist in his or her laboratory. New technologies are winking at us on the horizon. So far, the physicists are largely re-inventing **statistical** wheels themselves.

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CHAPTER 5. SUMMARY AND DISCUSSION
The three papers presented in this thesis established the validity of lineup protocol to use it as a tool for testing **statistical** hypothesis. Visual **statistical** **inference** is developed further by presenting the definitions of the terminologies. The methods of computing power of the visual test are proposed. Under some condition which is supported by experimental data, the power is obtained theoretically. A head to head comparison with the best available conventional test for regression slope is performed. The result suggests that visual test performs better when the effect size is large. For some super-visual individuals the performance is better even for small effect size. The influence of human factors on the visual test are examined and it is found that for some demographic and geographic factors the performance is better. But the practical impact of human factors is very negligible. Detailed procedures of human subject experiments are presented and the design of an web application to get lineups evaluated by human observer is provided. It offers various features that can be used by the researchers who intend to use lineup in decision making.

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ABSTRACT
HU, WENHAO. **Statistical** **Inference** for Model Selection. (Under the direction of Eric Laber and Leonard Stefanski.)
Penalized regression methods that perform simultaneous model selection and estimation are ubiquitous in **statistical** modeling. The use of such methods is often unavoidable as manual in- spection of all possible models quickly becomes intractable when there are more than a handful of predictors. However, such automated methods may fail to incorporate domain-knowledge, exploratory analyses, or other factors that might guide a more interactive model-building ap- proach. A hybrid approach is to use penalized regression to identify a set of candidate models and then to use interactive model-building to examine this candidate set more closely.

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We focus on the changes in Prolog syntax within SWI-Prolog that ac- commodate greater syntactic integration, enhanced user experience and improved features for web-services. We recount the full syntax and func- tionality of Real as well as presenting sister packages which include Pro- log code interfacing a number of common and useful tasks that can be delegated to R. We argue that Real is a powerful extension to logic programming, providing access to a popular **statistical** system that has complementary strengths in areas such as machine learning, **statistical** **inference** and visualisation. Furthermore, Real has a central role to play in the uptake of computational biology and bioinformatics as application areas for research in logic programming.

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You’ve already heard of a density since you’ve heard of the famous “bell curve”, or Gaussian density.
In this section you’ll learn exactly what the bell curve is and how to work with it.
Remember, everything we’re talking about up to at this point is a population quantity, not a statement about what occurs in our data. Think about the fact that 50% probability for head is a statement about the coin and how we’re flipping it, not a statement about the percentage of heads we obtained in a particular set of flips. This is an important distinction that we will emphasize over and over in this course. **Statistical** **inference** is about describing populations using data. Probability density functions are a way to mathematically characterize the population. In this course, we’ll assume that our sample is a random draw from the population.

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Various classical procedures, such as point estimates, hypotheses tests, and confidence intervals, have been pro- posed for **statistical** **inference**. For instance, estimations of unknown parameters, using methods such as the method of moment, the likelihood approach, least square methods, nonlinear regression estimators, robust estimation meth- ods, and Bayesian methods, have been applied to Weibull distribution (Bain & Engelhardt, 1980; Duffy, Starlinger,

So, our main contributions in the frontier analysis are to address a **statistical** **inference** which introduces the bivariate and the multivariate copula functions in the bootstrap procedure which serves this purpose. This copula density will model the association, if it exists, between the noise and the inefficiency terms of the frontier model. Indeed, some elliptical as the Gaussian copula, Archimedean as the Ali-Mikhail-Haq (AMH), Clayton and Frank copulas and other copula families as the Fairlie-Gumbel-Morgenstern (FGM) family are used. The prin- ciple is to model the inputs-output relationship under the dependence in the error term density to estimate efficiency and, in the bootstrap procedure of the confidence intervals for the chosen model and for each bootstrap replica- tion, draw dependently the two components using a bivariate copula density for the cross-sectional case and draw the noise variables independently each other but dependently with the inefficiency variable for the panel data. Besides, the existence of the association is performed with the nonparametric Kendall’s sta- tistical test related to the copula case.

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Unregularized M -estimation using SGD. Using SGD with a fixed step size, we demon- strate that the average of such SGD sequences can be used for **statistical** **inference**, after proper scaling. An intuitive analysis using the Ornstein-Uhlenbeck process suggests that such averages are asymptotically normal. From a practical perspective, our SGD-based **inference** procedure is a first order method, and is well-suited for large scale problems. To show its merits, we apply it to both synthetic and real datasets, and demonstrate that its accuracy is comparable to classical **statistical** methods, while requiring potentially far less computation.

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It is known that the boundary between sparse and dense cases is not always clear in prac- tice. Researchers may classify the same data set differently and therefore, a subjective choice between the sparse and dense cases might pose challenges for **statistical** **inference**. Hoover, et al. ( 1998 ), Wu and Chiang ( 2000 ), Chiang, et al. ( 2001 ), and Huang, et al. ( 2002 ) estab- lished some asymptotic bias and variance of their proposed estimates under some general conditions. However, the established limiting variances contain some unknown functions, which are not easy to estimate. Therefore, the bootstrap procedures were used to evalu- ate the variability of their proposed estimates. Li and Hsing ( 2010 ) established a uniform convergence rate for weighted local linear estimation of mean and variance functions for functional/longitudinal data. Nevertheless, Kim and Zhao ( 2013 ) showed that the conver- gence rates and limiting variances under sparse and dense assumptions are different. This motivated them to develop some unified nonparametric approaches that can be used to conduct longitudinal data analysis without deciding whether the data are dense or sparse. However, Kim and Zhao ( 2013 ) only considered estimating the mean response curve without the presence of covariates effect.

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We have given a short and selective review for causal **statistical** **inference** from obser- vational data. The proposed methodology (IDA Maathuis et al. 2010 ) is applicable to high-dimensional problems where the number of variables can greatly exceed sample size. Because some of the key assumptions for our (or any) modeling-based method are uncheckable in reality, there is an urgent need to validate the computational meth- ods and algorithms to better understand the limits and potential of causal **inference** machines. Of course, the validation should also provide new insights and further pri- oritization of future experiments in the field of the scientific study. We have pursued this route in Maathuis et al. ( 2010 ), Stekhoven et al. ( 2012 ).

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Emmanuel Grenier, Marc Hoffmann, Tony Leli` evre, Violaine Louvet, Cl´ ementine Prieur, Nabil Rachdi, Paul Vigneaux
To cite this version:
Emmanuel Grenier, Marc Hoffmann, Tony Leli` evre, Violaine Louvet, Cl´ ementine Prieur, et al.. **Statistical** **Inference** for Partial Differential Equations. SMAI 2013 - 6e Biennale Fran¸caise des Math´ ematiques Appliqu´ ees et Industrielles, May 2013, Seignosse, France. EDP Science, ESAM: ProcS, 45, pp.178-188, 2014, Congr` es SMAI 2013. <10.1051/proc/201445018>. <hal- 01102782>

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In this chapter we move from the framework of parametric **statistical** **inference** to nonparametric functional estimation, which aims to estimate a function without assuming any particular parametric form. The lack-of-fit problems of many para- metric models have made nonparametric curve estimation a very active research field in statistics. Here we will discuss the problems of drift estimation for the Brownian motion and of intensity estimation for the Cox process, extending Stein’s argument to an infinite dimensional setting using Malliavin Calculus. The first argument we present is a particular case of the theory treated in [PR08], while the second one is a slight generalization of [PR09], where the authors consider only a deterministic intensity. We aim at going further their results by proving that no unbiased estima- tor exist in H 1

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This paper focuses on conducting **statistical** **inference** for the QSR in a complex random sampling framework. However, the proposed method can be applied to other quantile share-based measures. Because the QSR is a non-linear function of the incomes, variance estimation is not straightforward and requires speciﬁc techniques. The variance estimators proposed here are based on the linearization approach by Deville (1999). **Inference** for the QSR using this approach has already been conducted by Osier (2006, 2009) and similar work has been done for the Gini index (Deville, 1996, 1999; Berger, 2008;

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A dot chart or dot plot is a **statistical** chart consisting of data points plotted on a
simple scale, typically using filled in circles. There are two common, yet very different, versions of the dot chart. The first is described by Wilkinson as a graph that has been used in hand-drawn (pre-computer era) graphs to depict distributions. The other version is described by Cleveland as an alternative to the bar chart, in which dots are used to depict the quantitative values (e.g. counts) associated with categorical

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be treated as constant as m varies. Indeed ν 0 might be approximately 1, so that
the prior expectation is that one of the null hypotheses is false. The dependence on m is thereby restored.
An important issue here is that to the extent that the **statistical** analysis is concerned with the relation between data and a hypothesis about that data, it might seem that the relation should be unaffected by how the hypothesis came to be considered. Indeed a different investigator who had focused on the particular hypothesis H
from the start would be entitled to use p
. But if simple significance tests are to be used as an aid to interpretation and discov- ery in somewhat exploratory situations, it is clear that some such precaution as the use of ( 5.20 ) is essential to ensure relevance to the analysis as imple- mented and to avoid the occurrence of systematically wrong answers. In fact, more broadly, ingenious investigators often have little difficulty in producing convincing after-the-event explanations of surprising conclusions that were unanticipated beforehand but which retrospectively may even have high prior probability; see Section 5.10 . Such ingenuity is certainly important but explan- ations produced by that route have, in the short term at least, different status from those put forward beforehand.

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Apart from psychological laws qua functional relationships between two or more variables, theories in psychology are qualitative explanatory theories. These explanatory theories are speculative statements about hypothetical mechanisms. Power-analysts have never shown how subtle conceptual differences in the qualitative theories may be faithfully represented by their limited range of ten or so 'reasonable' effect sizes. Furthermore, concerns about the **statistical** significance are ultimately concerns about data stability and the exclusion of chance influences as an explanation. These issues cannot be settled mechanically in the way depicted in power-analysis. The putative relationships among effect size, **statistical** power and sample size brings us to the putative dependence of **statistical** significance on sample size.

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Despite the tremendous attention on regression t-tests and F-tests, other methodology emerged in parallel as well. The earliest alternative is the permutation test, which justi- fies the significance of the test through the so-called “permutation distribution”. However, the early model to justify permutation tests is the “randomization model” in contrast to the “population model” that we considered in (3.2). The “randomization model” was introduced by Jerzy S. Neyman in his master thesis (Neyman 1923),which is also known as Neyman- Rubin model (Rubin 1974), or design-based **inference** (Särndal et al. (1978), in contrast to model-based **inference**), or “conditional-on-errors” model (Kennedy (1995), in contrast to “conditional-on-treatment” model), and the term was coined by Ronald A. Fisher in 1926 (Fisher 1926). The theoretical foundation of permutation test was laid by Edwin J. G. Pit- man in his three seminal papers (Pitman 1937a,b; Pitman 1938), where the last two were studied for regression problems, albeit under the “randomization model”. The early work view permutation tests as better devices in terms of the logical coherence and robustness to non-normality (e.g. Geary 1927; Eden and Yates 1933; Fisher 1935a). They found that the permutation distribution for “randomization models” mostly agree with the normality-based distribution for “population models”, until 1937 when Li B. Welch disproved the agreement for Latin-squares designs (Welch 1937). In the next half century, most of the work on per- mutation tests were established for “randomization models” without being justified under “population models”, except for rank-based tests which will be discussed later. We will skip the discussion for this period and refer to Berry et al. (2013) for a thorough literature review on this line of work, because our work focuses on the “population model” like (3.2).

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