Archive for standard deviation

the joy of stats [book review]

Posted in Books, pictures, University life with tags , , , , , , , , , , , , on April 8, 2019 by xi'an

David Spiegelhalter‘s latest book, The Art of Statistics: How to Learn from Data, has made it to Nature Book Review main entry this week. Under the title “the joy of stats”,  written by Evelyn Lamb, a freelance math and science writer from Salt Lake City, Utah. (I noticed that the book made it to Amazon #1 bestseller, albeit in the Craps category!, which I am unsure is completely adequate!, especially since the book is not yet for sale on the US branch of Amazon!, and further Amazon #1 in the Probability and Statistics category in the UK.) I have not read the book yet and here are a few excerpts from the review, quoted verbatim:

“The book is part of a trend in statistics education towards emphasizing conceptual understanding rather than computational fluency. Statistics software can now perform a battery of tests and crunch any measure from large data sets in the blink of an eye. Thus, being able to compute the standard deviation of a sample the long way is seen as less essential than understanding how to design and interpret scientific studies with a rigorous eye.”

“…a main takeaway from the book is a sense of circumspection about our confidence in what is known. As Spiegelhalter writes, the point of statistical science is to ease us through the stages of extrapolation from a controlled study to an understanding of the real world, `and finally, with due humility, be able to say what we can and cannot learn from data’. That humility can be lacking when statistics are used in debates about contentious issues such as the costs and benefits of cancer screening.

an unbiased estimator of the Hellinger distance?

Posted in Statistics with tags , , , on October 22, 2012 by xi'an

Here is a question I posted on Stack Exchange a while ago:

In a setting where one observes X1,…,Xn distributed from a distribution with (unknown) density f, I wonder if there is an unbiased estimator (based on the Xi‘s) of the Hellinger distance to another distribution with known density f0, namely

\mathfrak{H}(f,f_0)=\left\{1-\int\sqrt{f_0(x)/(x)}\text{d}x\right\}^{1/2}
Now, Paulo has posted an answer that is rather interesting, if formally “off the point”. There exists a natural unbiased estimator of if not of H, based on the original sample and using the alternative representation
\mathfrak{H}^2(f,f_0)=1-\mathbb{E}_f[\sqrt{f_0(X)/f(X)}]

for the Hellinger distance. In addition, this estimator is guaranteed to enjoy a finite variance since

\mathbb{E}_f[\sqrt{f_0(X)/f(X)}^2]=1\,.

Considering this question again, I am now fairly convinced there cannot be an unbiased estimator of H, as it behaves like a standard deviation for which there usually is no unbiased estimator!