Posts Tagged ‘ mathematics ’

Christian Robert Shows that the Sample Median Cannot Be a Sufficient Statistic

Christian Robert Shows that the Sample Median Cannot Be a Sufficient Statistic

I am grateful to Christian Robert (Xi’an) for commenting on my recent Mathematical Statistics Lessons of the Day on sufficient statistics and minimally sufficient statistics. In one of my earlier posts, he wisely commented that the sample median cannot be a sufficient statistic.  He has supplemented this by writing on his own blog to show that […]

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Mathematical Statistics Lesson of the Day – Minimally Sufficient Statistics

Mathematical Statistics Lesson of the Day – Minimally Sufficient Statistics

In using a statistic to estimate a parameter in a probability distribution, it is important to remember that there can be multiple sufficient statistics for the same parameter.  Indeed, the entire data set, , can be a sufficient statistic – it certainly contains all of the information that is needed to estimate the parameter.  However, […]

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Can we try to make an adjustment?

November 14, 2014
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Can we try to make an adjustment?

In most of our data science teaching (including our book Practical Data Science with R) we emphasize the deliberately easy problem of “exchangeable prediction.” We define exchangeable prediction as: given a series of observations with two distinguished classes of variables/observations denoted “x”s (denoting control variables, independent variables, experimental variables, or predictor variables) and “y” (denoting […] Related posts: Don’t use correlation to track prediction performance Reading the Gauss-Markov theorem Bad…

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Multiple Linear Regression Revisited

November 10, 2014
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Multiple Linear Regression Revisited

Last night, I had a discussion about the integrative data analysis (closely related with the discussion of AOAS 2014 paper from Dr Xihong Lin’s group and JASA 2014 paper from Dr. Hongzhe Li’s group) with my friend. If some biologist gave you the genetic variants (e.g. SNP) data and the phenotype (e.g. some trait) data, […]

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Mathematical Statistics Lesson of the Day – Sufficient Statistics

Mathematical Statistics Lesson of the Day – Sufficient Statistics

*Update on 2014-11-06: Thanks to Christian Robert’s comment, I have removed the sample median as an example of a sufficient statistic. Suppose that you collected data in order to estimate a parameter .  Let be the probability density function (PDF)* for . Let be a statistic based on .  Let be the PDF for . If the […]

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Mathematics and Mathematical Statistics Lesson of the Day – Convex Functions and Jensen’s Inequality

Mathematics and Mathematical Statistics Lesson of the Day – Convex Functions and Jensen’s Inequality

Consider a real-valued function that is continuous on the interval , where and are any 2 points in the domain of .  Let be the midpoint of and .  Then, if then is defined to be midpoint convex. More generally, let’s consider any point within the interval .  We can denote this arbitrary point as where . […]

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Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

In 2 earlier tutorials that focused on exploratory data analysis in statistics, I introduced the conceptual background behind empirical cumulative distribution functions (empirical CDFs) how to plot  empirical cumulative distribution functions in 2 different ways in R There is actually an elegant theorem that provides a rigorous basis for using empirical CDFs to estimate the true CDF – and […]

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Lebesgue Measure and Outer Measure Problems

September 8, 2014
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More proving, still on Real Analysis. This is my solution and if you find any errors, do let me know.ProblemsLebesgue Measure: Let $\mu$ be set function defined for all set in $\sigma$-algebra $\mathscr{F}$ with values in $[0,\infty]$. Assume $\mu$ is ...

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Translation Invariant of Lebesgue Outer Measure

September 7, 2014
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Another proving problem, this time on Real Analysis.ProblemProve that the Lebesgue outer measure is translation invariant. (Use the property that, the length of an interval $l$ is translation invariant.) SolutionProof. The outer measure is translation ...

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Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

In 2 recent Statistics Lessons of the Day, I introduced Markov’s inequality. explained the motivation and intuition behind Markov’s inequality. Chebyshev’s inequality is just a special version of Markov’s inequality; thus, their motivations and intuitions are similar. Markov’s inequality roughly says that a random variable is most frequently observed near its expected value, .  Remarkably, it quantifies just […]

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