# Top PDF Statistical Inference For Everyone ### Statistical Inference For Everyone

about the impending Russian satellite crash which said something like, “the scientists had studied the trajectory of the satellite, and determined that there was only a 25 % chance of it striking land, and even a much smaller chance striking a populated area.” The report was clearly designed to calm the public, and convince them that the scientists had a good handle on the situation. Unfortunately, given a little thought, one realizes that the Earth’s surface consists of about 25 % land and 75 % water, so if you knew nothing about the trajectory of the satellite, you would simply state that it had a 25 % chance of striking land. Instead of communicating knowledge of the situation, the news broadcast communicated (to those who knew basic statistical inference) that either the scientists were in complete ignorance of the trajectory or the reporter had misinterpreted a casual statement about probabilities and didn’t realize what was implied. Either way, the intent of the message and the content of the message (to those who understood basic probability) were in direct conflict. ### Stochastic gradients methods for statistical inference

Statistical inference, such as hypothesis testing and calculating a confidence interval, is an important tool for accessing uncertainty in machine learning and statistical problems, both for estimation and prediction purposes [FHT01, EH16]. E.g., in unregularized linear regression and high-dimensional LASSO settings [vdGBRD14, JM15, TWH15], we are interested in computing coordinate-wise confidence intervals and p-values of a p-dimensional variable, in order to infer which coordinates are active or not [Was13]. Traditionally, the inverse Fisher information matrix [Edg08] contains the answer to such inference questions; however it requires storing and computing a p × p matrix structure, often prohibitive for large-scale applications [TRVB06]. Alternatively, the Bootstrap [Efr82, ET94] method is a popular statistical inference algorithm, where we solve an optimization problem per dataset replicate, but can be expensive for large data sets [KTSJ14]. ### Statistical Inference For High-Dimensional Linear Models

Confidence sets play a fundamental role in statistical inference. Recently, confi- dence sets for high-dimensional linear models have been actively studied, where the focus is on the construction of confidence intervals for individual coordinates (Javan- mard & Montanari, 2014a; van de Geer et al., 2014) and the construction of confidence balls for the whole high-dimension vector β (Nickl & van de Geer, 2013). In addition, Gautier & Tsybakov (2011); Belloni et al. (2012); Fan & Liao (2014); Chernozhukov et al. (2015a) provide honest confidence intervals for a treatment effect in the frame- work of high-dimensional instrumental variable regression. However, compared to the estimation problem, there is still a paucity of methods and fundamental theoretical results on the inference problem for high-dimensional linear models. In this thesis, we will focus on the statistical inference problem in high-dimensional linear models. An outline of the thesis is presented in the next subsection. ### Statistical Inference for a Novel Health Inequality Index

choices to be made when measuring socioeconomic inequalities with rank-de- pendent inequality indices.  made an empirical comparison with several or- dinal and cardinal measures of health inequality.  proposed a new measure for ordinal health data to monitor income-related health differences between regions in Great Britain.  defined a new ratio-scale health status variable and developed positional stochastic dominance conditions that could be imple- mented in a context of multi-dimensionality categorical variables.  examined the measurement of social polarization with categorical and ordinal data.  introduced two approaches to measure social polarization in the case where the distance between groups is based on an ordinal variable, such as self-assessed health status. More examples on ordinal inequality measurements can be seen in ,  and so on. For statistical inference of these recent developed health inequality indices, some authors (e.g. , ) have derived standard errors for the inequality indices they have introduced.  presented a unified methodol- ogy for the estimation of inequality indices of the cumulative distribution func- tion. ### QInfer: Statistical inference software for quantum applications

In this work, we have presented QInfer, our open- source library for statistical inference in quantum in- formation processing. QInfer is useful for a range of different applications, and can be readily used for cus- tom problems due to its modular and extensible de- sign, addressing a pressing need in both quantum in- formation theory and in experimental practice. Impor- tantly, our library is also accessible, in part due to the extensive documentation that we provide (see ancil- lary files or docs.qinfer.org). In this way, QInfer sup- ports the goal of reproducible research by providing open-source tools for data analysis in a clear and un- derstandable manner. ### Statistical inference based on k-records

et al.  call this a Type 2 k-record sequence. For k = 1, note that the usual records are recovered. An analogous definition can be given for lower k-records as well. This sequence of k-records was introduced by Dziubdziela and Kopocinski  and it has found acceptance in the literature. Some work has been done on the statistical inference, based on k-records. See, for instance, Deheuvels and Nevzorov , Berred , Ali Mousa et al. , Malinowska and Szynal , Danielak and Raqab ,, Ahmadi et al. , Fashandi and Ahmadi  and references therein. ### Some aspects of statistical inference for econometrics

Research into statistical methodology for econometric modelling has tended to concentrate upon methods for estimation of the unknown parameters in theoretical specifications. In contrast, relatively little attention has been given to devising tools for discriminating between alternative specifications. The range of methods generally applied to the companion problem in statistical inference, that of testing hypotheses about the parameters, is somewhat limited compared with the degree of sophistication of estimators that are now available. Familiar "t-ratios" for testing whether a parameter can be set to zero and, in certain situations, the likelihood ratio criterion would be foremost in the econometrician's kit of testing tools. ### Variational methods for geometric statistical inference

The results of this thesis therefore concern the asymptotics for a selection of statistical inference problems. We construct our estimates as the minimizer of an appropriate functional and look at what happens in the large data limit. In each case we will show our estimates con- verge to a minimizer of a limiting functional. In certain cases we also add rates of convergence. The emphasis is on problems which contain a data association or classification compo- nent. More precisely we study a generalized version of the k-means method which is suitable for estimating multiple trajectories from unlabeled data which combines data association with spline smoothing. Another problem considered is a graphical approach to estimating the la- beling of data points. Our approach uses minimizers of the Ginzburg-Landau functional on a suitably defined graph. ### Making Neighborhoods Safer with Statistical Inference

As the crime rates shoot up year by year, in this era of data deluge, more and more data is getting piled up across the world. This study tries to find relationships between crime victims, arrests and neighbourhoods and dig out patterns that exist in the available data sets and help the police departments with statistically significant information. Using which the police can rearrange the man powers and fight crimes more effectively. The departments can use many such studies to keep the neighbourhoods safer. Keywords: Statistical Inference, Crime, Victims, Arrests, Neighbourhoods, Chi - squared test, Multinomial Logistic Regression ### Statistical Inference for Models with Intractable Normalizing Constants

In this dissertation, we have proposed two new algorithms for statistical inference for models with intractable normalizing constants: the Monte Carlo Metropolis-Hastings algorithm and the Bayesian Stochastic Approximation Monte Carlo algorithm. In addition, we has demonstrated how the SAMCMC method can be applied to estimate the parameters of ERGMs, which is one of the typical examples of statistical models with intractable normalizing constants, without the hinderance of model degeneracy. The MCMH algorithm is a Monte Carlo version of the Metropolis-Hastings al- gorithm. At each iteration, it replaces the unknown normalizing constant ratio by a Monte Carlo estimate. Although the algorithm violates the detailed balance con- dition, it still converges, as shown in the paper, to the desired target distribution under mild conditions. Unlike other auxiliary variable MCMC algorithms, such as the Møller and exchange algorithms, the MCMH algorithm avoids the requirement for perfect sampling, and thus can be applied to many statistical models for which perfect sampling is not available or very expensive. ### Statistical inference for spatial and spatio temporal processes

Before we move to the next two chapters th a t deal w ith some problems of statistical inference for processes on the regular d-dimensional lattice and before we propose various ways to solve them , we will need to summarize some basic definitions and results th a t have been given before, as well as to add some new results th a t will be extremely useful next. In Section 2.2, we recall the notion of unilateral ordering between any two locations v T, v T + j T e Z d, which was given by W hittle (1954) when d = 2 and by Guyon (1982) when d > 2. Section 2.3 defines the weakly and strictly stationary processes and states the Wold decomposition, which provides a link between (weakly) stationary processes and linear processes. In th a t same section, we prove some properties of processes, which are linear functions of independent and identically distributed random variables. A new definition of the so-called reverse strict stationarity might also be found there, which is an a tte m p t to extend the definition of strict stationarity in a way th a t does not favor any direction of each one of the d dimensions. Later in Proposition 2.5 and, consequently, in C hapter 3 and Sections 4.4 and 4.5, we have used conditions, which are satisfied if the process of interest is reverse strictly stationary. Thus, when we establish in the end of Section 2.3.2 th a t reverse strictly stationary processes exist, a t the same tim e we allow for some of our conditions used in C hapters 3 and for 4 to be more realistic. ### Statistical inference in a directed network model with covariates

Because of the form of the model and the independent assumption on the links, it appears that maximum likelihood estimation developed for logistic regression is all that is needed for inference. A major challenge of models of this kind is, however, that the number of parameters grows with the network size. In particular, the number of outgoingness and incomingness parameters needed by our model is already twice the size of the network, and the presence of the covariates poses additional challenges. See the literature review below. To a certain extent, our model can be seen as a special case of the exponential random graph model (ERGM) as discussed by Robins et, al. (2007a,b), as the sufficient statistics are the covariates and the bi-degree sequence. It is known, however, that fitting any nontrivial exponential random graph models is extremely challenging, not to mention developing valid procedures for their statistical inference (Goldenberg et al., 2009; Fienberg, 2012). Studying the asymptotic theory of the proposed directed network model is the main contribution of this paper. ### Statistical inference and spatial patterns in correlates of IQ

Controlling for autocorrelation may remove real biological patterns and this has been offered as an argument against controlling for both spatial (Legendre, 1993) and phylogenetic (Ricklefs & Starck, 1996) autocorrelation. However, any statistical analysis with an inherent spatial component should consider spatial autocorrelation, if only to demonstrate that its control is not necessary. Failure to account for this lack of independence in data violates statistical assumptions and renders statistical inference invalid. The initial dogmatism with which controls for spatial and phylogenetic autocorrelation were enforced has now given way to an acceptance that such controls are not always necessary. However, with the advent of numerous tools and techniques (such as those presented here) for assessing this need, we encourage researchers to at least give the topic due consideration as it can substantially influence results. ### Using Storm for scaleable sequential statistical inference

In sequential statistical inference, data arrive as a stream and inference is an iterative process that updates as new data are available. Numerous examples and applications exist, starting with the Kalman filter and its generalisations such as the dynamic state space model . Approaches to implement inference in this setting are the subject of much current work e.g. sequential Monte Carlo . The challenge is not only to work with data sources that require sophisticated analyses, but also for scaleable inference algorithms that can cope with increased data dimension and arrival rates. ### Students' understanding of statistical inference : implications for teaching

If students have pre-existing inappropriate or non-existent schemas about prob- ability they will not be able to understand statistical inference. Statistical infer- ence relies on the mathematics of probability from the selection of the sample to the drawing of the final conclusions. The literature shows, however, that a per- son‟s int uitive views of probability are often inappropriate or incomplete. These inappropriate or incomplete views may be difficult to detect because probability questions, using examples such as coin tosses, can be simple to answer. As people get older their intuitive perceptions of statistical phenomena, even though inap- propriate, may get stronger, and formal instruction may not correct these intuitive views (Moore, 1990). Furthermore, it has been found that students may use the formal views inside the classroom, but revert to their own intuitive views, whether correct or not, outside the classroom (Chance, delMas, & Garfield, 2004). ### On Asymptotic Quantum Statistical Inference

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. We think it is a pity statisticians are not more involved. If Jon is looking for some new challenges... ? ### V-Matrix Method of Solving Statistical Inference Problems

This paper presents direct settings and rigorous solutions of the main Statistical Inference problems. It shows that rigorous solutions require solving multidimensional Fredholm in- tegral equations of the first kind in the situation where not only the right-hand side of the equation is an approximation, but the operator in the equation is also defined ap- proximately. Using Stefanuyk-Vapnik theory for solving such ill-posed operator equations, constructive methods of empirical inference are introduced. These methods are based on a new concept called V -matrix. This matrix captures geometric properties of the observation data that are ignored by classical statistical methods. ### A Powerful Global Test Statistic for Functional Statistical Inference

We consider the problem of performing an association test between functional data and scalar variables in a varying co- efficient model setting. We propose a functional projection regression model and an associated global test statistic to ag- gregate relatively weak signals across the domain of functional data, while reducing the dimension. An optimal functional pro- jection direction is selected to maximize signal-to-noise ratio with ridge penalty. Theoretically, we systematically study the asymptotic distribution of the global test statistic and provide a strategy to adaptively select the optimal tuning parameter. We use simulations to show that the proposed test outperforms all existing state-of-the-art methods in functional statistical inference. Finally, we apply the proposed testing method to the genome-wide association analysis of imaging genetic data in UK Biobank dataset. ### Adaptive Sampling and Statistical Inference for Anomaly Detection

1. The techniques for volume anomaly detection are the methods that detect unusual volume changes in the univariate data. Many methods have been proposed to apply statistical tech- niques to detection volume anomalies. For example, Barford et al. treat anomalies as devi- ations in the overall traffic volume, and use the traffic variances at different time-frequency scales to distinguish predictable and anomalous traffic volume changes . It has been pro- posed to apply a variety of time series forecast models (e.g., ARIMA and Holt-Winters) on network tra ffi c, and look for tra ffi c flows with large forecast errors to detect tra ffi c anoma- lies [42, 43, 44, 45]. For example, the AnomalyDetection R package tool recently released by Twitter on GitHub  aims to detect anomalies in time series data. This tool assumes a smooth transition in normal data series such as activities of uploading photos on Twitter, and uses seasonal hybrid extreme studentized deviate (S-H-ESD) test to quantify deviation of a datapoint from its prediction. Datapoints that deviate from their predictions are flagged as anomalies in seasonal univariate time series. This tool is able to detect both global and local anomalies that do not necessarily appear to be extreme values. 