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.

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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.

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choices to be made when measuring socioeconomic inequalities with rank-de- pendent inequality indices. [9] made an empirical comparison with several or- dinal and cardinal measures of health inequality. [10] proposed a new measure for ordinal health data to monitor income-related health differences between regions in Great Britain. [11] 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. [12] examined the measurement of social polarization with categorical and ordinal data. [13] 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 [14], [15] and so on. For **statistical** **inference** of these recent developed health inequality indices, some authors (e.g. [4], [16]) have derived standard errors for the inequality indices they have introduced. [17] presented a unified methodol- ogy for the estimation of inequality indices of the cumulative distribution func- tion.

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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.

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et al. [5] 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 [12] 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 [11], Berred [7], Ali Mousa et al. [4], Malinowska and Szynal [17], Danielak and Raqab [9],[10], Ahmadi et al. [2], Fashandi and Ahmadi [13] and references therein.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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 [4]. Approaches to implement **inference** in this setting are the subject of much current work e.g. sequential Monte Carlo [3]. 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.

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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).

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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... ?

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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.

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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.

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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 [41]. 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 [45] 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.

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Hence, in this contribution we complement the existing results in the literature by developing a full- ‡edged asymptotic theory for the WLS estimator of ' in (1.1). The WLS estimator avoids the issues described above, and it has been shown, in a time series context, to work well in comparison with the maximum likelihood approach (Schick, 1996; Koul and Schick, 1996). We show that in general, thanks to the self-normalised nature of the WLS estimator, there is no “unit root problem”in case of the RCA model de…ned by (1.1). Further, we study an estimator of the asymptotic variance of the WLS estimator which is consistent under any degree of cross sectional dependence; hence, it is possible to normalise the estimated ' and recover standard normal **inference**, with no need for any prior **inference** on the possible presence of a factor structure. We also show that, contrary to the time series case, the suitably normalised WLS estimator converges to a normal distribution even when the autoregressive root is not random, i.e. when b i;t = 0 for all 1 i N and 1 t T ; the only case in which asymptotic normality

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