European Mathematical Society - 62 Statistics
https://euro-math-soc.eu/msc/62-statistics
enDark Data
https://euro-math-soc.eu/review/dark-data
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Dark matter and dark energy in cosmology is the matter and energy that cannot be directly observed with the techniques currently available, but we know that it must be there since it is the only explanation for the data that we observe and that cannot be explained by all the measurable matter and energy. This may be our first insight into the existence of dark data. Currently we live in a period of big data thanks to computers and the World Wide Web (part of if is called dark as well) and there are many different ways to collect and process these data. However there are also many different ways in which our analysis may lead to the wrong conclusions because part of the data are missing or wrong, or "dark" as Hand calls it. These dark data can exist for many different reasons. This book is taking a closer look at the phenomenon. Countless examples are described in the book (mostly for UK data). What type of obfuscations can darken our data? Why are some datasets dark? What are the consequences? What could possibly be done to remedy the situation?</p>
<p>Hand starts with a kind of taxonomy of dark data. In what kind of situations are we dealing with dark data? He describes, with many examples, fifteen different phenomena that can lead to dark data. Some are quite obvious like missing data that we know are missing (known unknowns), but there might also be data missing that we are not aware of (unknown unknowns). Some data are intentionally wrong (falsification) or unintentionally (over-simplification or rounding). Conclusions may be obtained for a whole population or over a larger period of time based on data that were only collected for part of the population or were only valid at a certain moment in time (extrapolation), etc. Quite often, the data are wrong or misused for more than one reason.</p>
<p>There is not a formal definition in this book, but nevertheless, using his examples, Hand explains what the different types of dark data are and how they come about, and identifies some of the concepts that he uses throughout the book. For example dark data caused by "self-selection" refers to the fact that data are corrupted because some participants, invited for and online poll, decide not to participate, or prefer not to answer some of the questions. There are problems of designing the sample (even a sample can be big data, in any case the data collected should represent the whole population for which the conclusion is supposed to hold), one has to be careful not to miss what really matters (like causality between data used and the conclusion derived), data can be corrupted by human errors, by summarising or simplifying or rounding the data. People can manipulate data in a creative way (like tax evasion) or corrupt data by deliberately feeding false data (criminal activity, insurance fraud).</p>
<p>Hand also has a chapter on science and dark data, not only were scientists in the course of history tricked in their conclusions by dark data, some also contributed by falsifying published data intentionally, or they may have been biassed by a general belief or intuition. John Ioannidis threw the cat among the pigeons with his 2005 paper <a target="_blank" href="https://doi.org/10.1371/journal.pmed.0020124"><em>Why Most Published Research Findings Are False</em></a>. Reproducibility was recognised as a major problem and research institutions are demanding data management plans in funding applications. Data management has grown into big business interfering with problems of privacy and GDPR. Distinguishing truth from reality, has become increasingly difficult in our digital world. Encryption, verification, identification, authentication, etc. can hardly keep up chasing creative fictionalisation. Artificial Intelligence algorithms based on machine-learning try to analyse the data that are too massive for humans to deal with, but even these machines can be led astray by dark data.</p>
<p>Thus it has become a major problem to recognise dark data and to know how to deal with it and avoid wrong conclusions. This is what Hand is discussing in the first two chapters of part II of the book (part II has a third chapter that is also the last chapter of the book which is summarising the taxonomy that was described in the early chapters). First we need identify why some data are missing. Here Hand considers three different types: it can be a random phenomenon but related to the missing data (UDD = Unseen Data Dependent) like some may be reluctant to give their BMI when it is high. Or missing data may depends on the data previously observed (SDD = Seen Data Dependent) like a BMI not given because it has increased since the last registered observation. Finally, data are missing but that does not depend in any way on the data observed (NDD = Not Data Dependent). Recognising the mechanism behind the missing data is important because it defines how one should deal with the data, for example on whether and how to complete the missing data or not. Effects of NDD and SDD can be be cured, but UDD is more difficult to deal with.</p>
<p>Dark data can also be beneficial if it is detected and if that leads to a reformulation of the question that we want to answer, or it may lead to strategic elimination of some data that would bias the result. To avoid dark data, it helps to randomise the sample and even hide data from the researcher (like not revealing who did and who did not get the placebo). An obvious way to fill up missing data is to use averages, but a somewhat strange advise is to fill up these data by simulation. It is a valid way to generate data in case of a simple model with known probability distribution, but when it involves a complicated model, then these models are simplifications of reality built upon observations that may involve dark data. Similarly machine-learning techniques involve massive analysis of data that may be corrupted. In these cases it can only be hoped that over the repeated simulations or over the whole learning process the wrong data are not systematic and are averaged out over the iteration. Or one may apply techniques such as boosting and bootstrapping to reduce bias. Bayesian statistics helps to test hypotheses and thus confirm or refute intuitive assumptions. Cryptography can help to make data anonymous so that people are more willing to provide correct data or to prevent the introduction of false data by fake persons. It may even help to make some data deliberately dark, not making them available to users but still using them in computations.</p>
<p>As mentioned above, the description is an exploration of a major problem in data analysis with an attempt of classification, analysing causes, mechanisms, and to some extent also suggest mitigations. However most of the book consists of examples and particular cases to clearly explain the ideas. Nowhere, however is there a concrete or general well defined statistical, mathematical, or algorithmic solution given. This is mainly a wake-up call that clearly points at a major problem that any scientist has to be aware of and that he or she should think about how to deal with. Certainly statisticians, applied mathematicians, computer scientists, but in fact anyone dealing with data (big or not) should be well aware of the "darkness" of their data.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a description of how important it can be that in our treatment of data, some of them are missing, or fake. Conclusions derived from these corrupted data can be biassed or wrong. How do we recognize the dark data? How can we deal with the phenomenon? These are answers that Hand deals with in this book. The approach is mainly descriptive with an abundance of examples, mainly from data related to the UK situation. Suggestions are given, but no concrete precise or detailed mathematical or statistical analysis or algorithms is discussed in detail.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-j-hand" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David J. Hand</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691182377 (hbk), 9780691198859 (ebk), 9780691199184 (abk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 26.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">344</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691182377/dark-data" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691182377/dark-data</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62d99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62D99</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/68p99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68P99</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68t09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68T09</a></li></ul></span>Mon, 03 Feb 2020 12:29:51 +0000Adhemar Bultheel50375 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/dark-data#comments Benford's Law: Theory and Applications
https://euro-math-soc.eu/review/benfords-law-theory-and-applications
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is an excellent companion to the book by Arno Berger and Theodore P. Hill <em>An Introduction to Benford's Law </em> (Princeton University Press, 2015) that gives a sound mathematical introduction to the law and that is also reviewed <a href="/review/introduction-benfords-law" target="_blank">here</a>. More info on the nature of Benford's law can be found there. A 40-page summary of that introduction can also be found as a chapter in the current volume. Miller has also an introductory chapter where he gives arguments based on Fourier analysis to show that most sequences will (almost) satisfy Benford's law.</p>
<p>
Many other specialists on the topic contribute the 16 remaining papers. Some of them deal with further theoretical issues such as convergence. However, most of them discuss applications in many different fields such as accounting and voting systems, in economics and finance, psychology, games, clinical data, and medical images. An additional chapter provides exercises for all the previous chapters. The editor has a special site devoted to the book. For example all the exercises that can be associated with each of the chapters are downloadable in pdf format. It is advisable to also look up the editor's <a href="http://web.williams.edu/Mathematics/sjmiller/public_html/benford/" target="_blank">website</a> for supplementary material. This can include references for additional reading, software, homework assignments, and occasionally even a video.</p>
<p>
Note that although Benford's law is rather popular among mathematical hobbyists, it requires a good mathematical training to embark on this book and a sound introduction is given by the introductory book by berger and Hill mentioned above.</p>
<p>
Both the present book and the introductory book by Berger and Hill show that Benford's law has matured and is now taken much more seriously than it was before.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of research papers dealing with Benford's law. There are some introductory survey papers, some papers on the the theory, but most discuss applications in diverse domains.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/steven-j-miller" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Steven J. Miller</a></li><li class="vocabulary-links field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691147611 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 75.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">464</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10527.html" title="Link to web page">http://press.princeton.edu/titles/10527.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62-02</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62e10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62E10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/60f15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">60F15</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/60g57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">60G57</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/62p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62P05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/62p20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62P20</a></li></ul></span>Tue, 21 Jul 2015 15:23:42 +0000Adhemar Bultheel46320 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/benfords-law-theory-and-applications#commentsAn Introduction to Benford's Law
https://euro-math-soc.eu/review/introduction-benfords-law
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The law is named after Benford but it was actually Newcomb who discovered in 1881 that logarithm tables were more worn out in the pages starting with smaller digits. Benford rediscovered this fact only in 1938. The law says that in many collections of numerical data, the probability that the first significant digit is $d$ is given by $\log_{10}(1+1/d)$. It has been generalized in the sense that the probability that the first significant digit is $d_1$, the second $d_2$ and so on until the $m$th is $d_m$ is given by $\log_{10}(1+(\sum_{j=1}^m 10^{m-j}d_j)^{-1})$. So the probability that the first digit is 1 is about 30.1%, and it decreases to 17.6% for a 2, etc., till only 4.6% for 9. Software code to test the law on a set of 10000 Fibonacci numbers can be found at the <a href="http://rosettacode.org/wiki/Benford%27s_law" target="_blank">rosettacode.org</a> site.</p>
<p>
However not all sets of data satisfy the law. For one thing, the law is asymptotic, which means that it holds for an infinite set of data. But even then, not all data sets need to satisfy the law. An obvious example are telephone book numbers, but also square roots do not follow the law. So it is important to know to what extent the law is satisfied or not. This can be measured by a number Δ that is for example the maximal deviation from the theoretical probability by one of the digits considered.</p>
<p>
Because the law is simple and counter intuitive, it has been popular among mathematical hobbyists. But a proper study of the phenomenon requires rigorous mathematics defining random variables, probability spaces, stochastic processes, etc. All this is introduced in this book and the reader should be willing to assimilate all of it to get a better grip on the phenomenon.</p>
<p>
The book has numerous tables and examples illustrating the law, each time with a Δ-value. The approach taken here illustrates that the law is catching on and gets a broader and more serious attention. The authors have compiled a <a href="http://www.benfordonline.net" target="_blank">database</a> of literature on the subject.</p>
<p>
The law has been applied in fraud detection, justice, research data, game theory, etc. However, before the conclusion should have any guarantee of correctness, the rules should be clearly understood. There is a chapter on applications in this book but it is not the main objective. More can be found in another book with research papers that is appearing simultaneously. See the accompanying <a href="/review/benfords-law-theory-and-applications" target="_blank">review</a> of <em>Benford's Law: Theory and Applications </em> (Princeton University Press, 2015) edited by S.J. Miller. Another recent book by A.E. Kossovsky is <a href="/review/benfords-law" target="_blank">Benford's Law</a> (World Scientific, 2014) but this is much less mathematical and more speculative.</p>
<p>
If you want to understand all the mathematics behind the law and are prepared to accept all the necessary theory, this is a marvelous and excellent introduction you might be looking for. If you are less patient and/or are already well trained in the mathematics, you might want to read a chapter in the book by Miller mentioned above where the authors give a 40 page summary of the theory in this introduction.</p>
<p>
One idiosyncratic particularity took some time for me to get used to. Floating point numbers are only given with 4 significant digits and written with equality signs, even though they are not strict equalities. For example <em>π</em>=3.141 meaning that <em>π</em> has a value between 3.141 and 3.142. For Δ, only 2 digits are used. So Δ = 0.00 means that Δ is between 0.00 and 0.01.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a sound mathematical introduction to the statistics of Benford's law.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/arno-berger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Arno Berger</a></li><li class="vocabulary-links field-item odd"><a href="/author/theodore-p-hil" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Theodore P. Hil</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691163062 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 75.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">256</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10526.html" title="Link to web page">http://press.princeton.edu/titles/10526.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62-02</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62e10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62E10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/60f15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">60F15</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/60g57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">60G57</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/62p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62P05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/62p20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62P20</a></li></ul></span>Tue, 21 Jul 2015 15:17:42 +0000Adhemar Bultheel46319 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/introduction-benfords-law#commentsDynamic Mixed Models for Familial Longitudinal Data
https://euro-math-soc.eu/review/dynamic-mixed-models-familial-longitudinal-data
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The book focuses on the analysis of familial and longitudinal data by means of dynamic mixed models. These data are correlated as (a) the responses of the members of a particular family share a common random family effect and (b) the repeated responses of the same individual are not independent. These familial and longitudinal correlation structures play a crucial role in the analysis of this type of data and greatly condition the model estimates and the rest of the statistical inferences. The book consists of eleven chapters. The first exhaustively lists the most relevant antecedents on familiar and longitudinal models in the last three decades. Chapters 2 and 3 give an overview of the analysis of longitudinal data by linear models (fixed and mixed, respectively). The rest of the chapters are divided in four pairs and each pair focusses on the study of count data and binary data, respectively, using distinct kind of models. In particular, the following: familial models (chapters 4 and 5), longitudinal models (chapters 6 and 7), longitudinal mixed models (in the following pair of chapters), and finally familial longitudinal models (in the last pair). The different models are discussed in depth with a special care for the technicalities behind the countless amounts of different correlation models. This means that the statistical background of the potential readers of the book needs to be rather advanced and the reading becomes occasionally arduous. This drawback is alleviated by the fact that the book also broaches the analysis of a considerable number of real life data sets, mainly within the context of biostatistics and econometrics. Researchers of these two areas along with applied statistics undoubtedly constitute the main target of the book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Teófilo Valdés Sánchez</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense de Madrid</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The book focuses on the analysis of familial and longitudinal data by means of dynamic mixed models.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/brajendra-c-sutradhar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">brajendra c. sutradhar</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2011</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even"> 1441983414, 9781441983411</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62j12" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62j12</a></li></ul></span>Tue, 02 Apr 2013 13:57:51 +0000Anonymous45499 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/dynamic-mixed-models-familial-longitudinal-data#commentsStatistics for Non-Statisticians
https://euro-math-soc.eu/review/statistics-non-statisticians
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>As the author claims in the preface of the book, everything is nowadays supported by numbers. Hence, if numbers are such an important source of information, the art of dealing and assessing with statistical data material is becoming more and more essential in our society.<br />
The spirit of this book is a first course for practitioners. The word “first” means that the basic concepts of classical Statistics are all motivated from their fundamentals whereas “practitioner” (or non-statistician, as indicated in the own title) means that the presentation does not cover the mathematical theory behind those concepts but practical information and useful details thought, v.g., for planning and interpreting surveys.<br />
The book is organized in nine chapters. They successively introduce the basics in descriptive Statistics (that is, presentation, collection and description of data) and the main notions in classical analytical Statistics (confidence interval, linear regression, hypothesis testing, samplings,...). The emphasis is put on practical situations where all these ideas can be applied. This emphasis is intensified by a frequent (excessive?) use of exclamation marks found in those sentences containing the essential information. Additionally, every chapter shows an implementation of its topics within the framework of the statistical freeware Calc. The final chapter contains a summary of Probability and Statistics, a glossary of terms and, like most books in this field, tables of useful distributions.<br />
The presentation of the book, both visually and mathematically, is good and well organized and may provide a pleasant (and useful) reading for either statisticians and non-statisticians.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Marco CASTRILLON LOPEZ</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense de Madrid, Spain</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>A quick and applied introduction to Statistics</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/bierger-madsen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">bierger madsen</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2011</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-642-17655-5</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62-01</a></li></ul></span>Wed, 22 Feb 2012 10:39:38 +0000Anonymous45444 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/statistics-non-statisticians#commentsModels for Intensive Longitudinal Data
https://euro-math-soc.eu/review/models-intensive-longitudinal-data
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Classical longitudinal analysis is mainly focused on examples to some ten occasions. New technologies lead to longitudinal databases with a considerably higher intensity and a substantially larger volume of data, for which the new term ‘intensive longitudinal data’ (ILD) is used. The number of occasions in ILD may be hundreds or thousands. However, the main difference between ILD and other models pertain to the scientific motivations for collecting ILD, the nature of hypotheses about them and the complex features of the data. The main themes in ILD modeling are: (i) the complexity and variety of individual trajectories, (ii) the role of time as a covariate, (iii) effects found in the covariance structure, (iv) relationships that change over time, (v) interest in autodependence and regulatory mechanisms. </p>
<p>The book is a collection of eleven papers (arranged as chapters) written by different authors. The introductory chapters focus on multilevel models and on marginal modelling through generalized estimating equations. Later chapters describe methodological tools from item response theory, functional data analysis, time series, state-space modeling, stochastic differential equations, engineering control systems, and models of point processes. Theory is illustrated on real data drawn from psychology, studies of smoking and alcohol use, brain imaging and traffic engineering. Some authors have supplied programs and source code examples. They are available at a website accompanying the book. By the way, the formula on page 118, line 6, should read sm=c22m-1+c22m. The remark on page 130 that the order p of an autoregressive process is often determined heuristically should be complemented by another remark that the order p is also often determined using AIC, BIC and similar criteria. This collection contains many interesting models and practical examples. The volume can be attractive reading for statisticians working in biostatistics and behavioural and social sciences.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">ja</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/t-walls" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">t. a. walls</a></li><li class="vocabulary-links field-item odd"><a href="/author/jl-schafer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j.l. schafer</a></li><li class="vocabulary-links field-item even"><a href="/author/eds" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">eds.</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press-oxford" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press, oxford</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-19-517344-4</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 38,99</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span>Sun, 23 Oct 2011 12:26:06 +0000Anonymous40054 at https://euro-math-soc.euThe Oxford Dictionary of Statistical Terms
https://euro-math-soc.eu/review/oxford-dictionary-statistical-terms
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The first edition of this dictionary appeared in 1957 and contained 1,700 terms. It became a well-respected reference book. The revision for the current sixth edition started in 1998. A website was constructed where all the terms of the fifth edition were listed. The statistical community suggested 320 terms for elimination and 1,067 terms for addition. It was decided to eliminate 265 entries and to add 640 new entries. The sixth edition now contains 3,540 terms. For some entries a reference to the literature is given. The list of references is quite long (pp. 439-498) and it is one of the positive features of the dictionary. The readers and users of the dictionary are invited to send their comments and suggestions to a website of the International Statistical Institute. The address of the website is introduced in the preface.<br />
I would just like to make the comment that I am puzzled by the explanation of the term Rayleigh distribution presented on page 339: “A χ2 distribution with two degrees of freedom, so called because it was considered by Rayleigh in some physical situations”. In statistical textbooks and papers we read something different. For example, in Lindgren B. W. (1993): Statistical Theory, 4th edition, Chapman and Hall, p. 186, we find the following information: “if X and Y are iid N(0,σ2) random variables, then R=√(X2+Y2) has the Rayleigh distribution. Since X2+Y2 has the χ2distribution with two degrees of freedom, the distribution of R cannot be χ¬¬¬¬2 ”. Despite this small point, I must say that the book covers a broad area of mathematical statistics and it is a useful reference for fast orientation on statistical terms.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">ja</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/y-dodge" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">y. dodge</a></li><li class="vocabulary-links field-item odd"><a href="/author/ed" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ed.</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press-oxford" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press, oxford</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-19-920613-1 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 12,99</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span>Sat, 22 Oct 2011 17:58:01 +0000Anonymous39988 at https://euro-math-soc.euIntroduction to Rare Event Simulation
https://euro-math-soc.eu/review/introduction-rare-event-simulation
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a very interesting monograph that attempts to present a unified theory of rare events simulation. Two basic tools used are importance sampling and the theory of large deviations. This framework allows an assortment of simulation problems to be viewed from a single unified perspective and gives a great deal of insight into the fundamental nature of rare events simulation. After a short summary of random number generation and simulation of selected stochastic models (such as Markov chains, processes and fields) the author presents basic results of large deviation theory and importance sampling methodology. A key part of the book is chapter 5, which deals with the large deviation theory of importance sampling. It shows the way to cover efficiently rare events simulations and includes, in the form of examples, many important models from different fields of statistics. The rest of the book is devoted to special applications of the methodology, including conditional importance sampling, Chernoff's bound for rare events simulation, level crossing and queuing models. One small point I would like to make is that, unlike the author, I am a bit skeptical about the blind simulation as described in chapter 12. I recommend the book to everybody who is interested in rare events and/or Monte Carlo simulation.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jant</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-bucklew" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j. a. bucklew</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-new-york-springer-series-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer, new york: springer series in statistics</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2004</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-387-20078-9</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 74,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span>Sat, 22 Oct 2011 17:50:21 +0000Anonymous39984 at https://euro-math-soc.euMathematical Statistics: Exercises and Solutions
https://euro-math-soc.eu/review/mathematical-statistics-exercises-and-solutions
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is a companion to the author’s textbook ‘Mathematical Statistics’ (2nd ed., Springer, 2003), which contains over 900 exercises. This collection consists of 400 exercises and their solutions. Most of the exercises (over 95%) are introduced in the cited textbook. The reader should have a good knowledge of advanced calculus, real analysis and measure theory. The book is divided into seven chapters: 1. Probability theory (measure and integral, distribution functions, random variables), 2. Fundamentals of statistics (sufficiency, risk functions, admissibility, consistency, Bayes rule), 3. Unbiased estimation (uniformly minimum variance unbiased estimators, Fisher information, U-statistics, linear models), 4. Estimation in parametric models (conjugate priors, posterior distributions, minimum risk invariant estimators, least squares estimators, maximum likelihood estimators, asymptotic relative efficiency), 5. Estimation in nonparametric models (Mallows’ distance, influence function, L-functionals, Hodges-Lehmann estimator), 6. Hypothesis test (uniformly most powerful tests, likelihood ratio tests), 7. Confidence sets (Fieller’s confidence sets, pivotal quantity, uniformly most accurate confidence sets).<br />
The collection is a stand-alone book. It is written very rigorously and solutions are presented in detail. It can be recommended as a source of solved problems for teachers and students of advanced mathematical statistics.<br />
I will finish with two remarks. First, I think that it may be of some interest to reproduce a very simple exercise (Ex. 9, p. 7): Let F be a cumulative distribution function on the real line and a a real number. Show that . Second, the reviewer would like to point out a misprint to prove that he studied the book: the expression mk.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">ja</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-shao" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j. shao</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-berlin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer, berlin</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-387-24970-2</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 39,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span>Fri, 21 Oct 2011 12:54:50 +0000Anonymous39931 at https://euro-math-soc.euStochastics. Introduction to Probability and Statistics
https://euro-math-soc.eu/review/stochastics-introduction-probability-and-statistics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a translation of the third edition of the German textbook “Stochastik”. It presents fundamental ideas and results of both probability theory and statistics. The areas covered from probability are: principles of modelling chance, stochastic standard models, conditional probabilities and independence, expectation and variance, the law of large numbers and the central limit theorem and Markov chains. The book covers the following areas of mathematical statistics: estimation, confidence regions, the normal distribution, hypothesis testing, asymptotic tests and rank tests, regression models and the analysis of variance. The basic notions and theorems are accompanied by interesting illustrative examples. At the end of each chapter there is a collection of problems offering applications, additions and supplements to the text. The book is primarily aimed at students of mathematics but also to scientists with an interest in the mathematical side of stochastics. Knowledge of abstract mathematics including elements of measure theory is assumed. The author gives a recommendation of how to successfully study the book. The book is well-written and mathematically oriented students and researchers will certainly find that it provides a high level introduction to probability theory and mathematical statistics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">mahu</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/h-o-georgii" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">h.-o. georgii</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/walter-de-gruyter-berlin-de-gruyter-textbook" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">walter de gruyter, berlin: de gruyter textbook</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2008</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-11-019145-5 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 39.95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span>Sat, 01 Oct 2011 12:42:55 +0000Anonymous39862 at https://euro-math-soc.eu