## Understanding Kullback – Leibler Divergence

It is easy to measure distance between two points. But what about measuring distance between two distributions? Good question. Long answer. Welcome the Kullback – Leibler Divergence measure.

The motivation for thinking about the Kullback – Leibler Divergence measure is that you can pick up questions such as: “how different was the behavior of the stock market this year compared with the average behavior?”. This is a rather different question than the trivial “how was the return this year compared to the average return?”.

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## LASSO, LASSO, LASSO

LASSO stands for Least Absolute Shrinkage and Selection Operator. It was first introduced 21 years ago by Robert Tibshirani (Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B). In 2004 the four statistical masters: Efron, Hastie, Johnstone and Tibshirani joined together to write the paper Least angle regression published in the Annals of statistics. It is that paper that sent the LASSO to the podium. The reason? they removed a computational barrier. Armed with a new ingenious geometric interpretation, they presented an algorithm for solving the LASSO problem. The algorithm is as simple as solving an OLS problem, and with computer code to accompany their paper, the LASSO was set for its liftoff*.

The LASSO overall reduces model complexity. It does this by completely excluding some variables, using only a subset of the original potential explanatory variables. Since this can add to the story of the model, the reduction in complexity is a desired property. Clarity of authors’ exposition and well rehashed computer code are further reasons for the fully justified, full fledged LASSO flareup.

This is not a LASSO tutorial. Google-search results, undoubtedly refined over years of increased popularity, are clear enough by now. Also, if you are still reading this I imagine you already know what is the LASSO and how it works. To continue from this point, what follows is a selective list of milestones from the academic literature- some theoretical and practical extensions.

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## Density Estimation Using Regression

Density estimation using regression? Yes we can!

I like regression. It is one of those simple yet powerful statistical methods. You always know exactly what you are doing. This post is about density estimation, and how to get an estimate of the density using (Poisson) regression.

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## Statistical Shrinkage

Machine estimated reading time:

Shrinkage in statistics has increased in popularity over the decades. Now statistical shrinkage is commonplace, explicitly or implicitly.

But when is it that we need to make use of shrinkage? At least partly it depends on signal-to-noise ratio.

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## Top countries in poker (Test equality of proportions using bootstrap)

Every once in a while I play poker online. The poker site allows you to ask for tournament history. You get an email which contains hundreds summaries (I open several tables at once so have quite some history), a typical summary looks as follows:

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## Understanding False Discovery Rate

False Discovery Rate is an unintuitive name for a very intuitive statistical concept. The math involved is as elegant as possible. Still, it is not an easy concept to actually understand. Hence i thought it would be a good idea to write this short tutorial.

We reviewed this important topic in the past, here as one of three Present-day great statistical discoveries, here in the context of backtesting trading strategies, and here in the context of scientific publishing. This post target the casual reader, explaining the concept of False Discovery Rate in plain words.

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## A few words about outliers

In statistics, outliers are as thorny topic as it gets. Is it legitimate to treat the observations seen during global financial crisis as outliers? or are those simply a feature of the system, and as such are integral part of a very fat tail distribution?

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## Trim your mean

The mean is arguably the most commonly used measure for central tendency, no no, don’t fall asleep! important point ahead.

We routinely compute the average as an estimate for the mean. All else constant, how much return should we expect the S&P 500 to deliver over some period? the average of past returns is a good answer. The average is the Maximum Likelihood (ML) estimate under Gaussianity. The average is a private case of least square minimization (a regression with no explanatory variables). It is a good answer. BUT:

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## Optimism of the Training Error Rate

We all use models. We all continuously working to improve and validate our models. Constant effort is made trying to estimate: how good our model actually is?

A general term for this estimate is error rate. Low error rate is better than high error rate, it means our model is more accurate.

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## On Central Moments

Sometimes I read academic literature, and often times those papers contain some proofs. I usually gloss over some innocent-looking assumptions on moments’ existence, invariably popping before derivations of theorems or lemmas. Here is one among countless examples, actually taken from Making and Evaluating Point Forecasts:

## Why statistical bootstrap

I often write about bootstrap (here an example and here a critique). I refer to it here as one of the most consequential advances in modern statistics. When I wrote that last post I was searching the web for a simple explanation to quickly show how useful bootstrap is, without boring the reader with the underlying math. Since I was not content with anything I could find, I decided to write it up, so here we go.

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## Laws of large numbers

The laws of large numbers are the cornerstones of asymptotic theory. ‘Large numbers’ in this context does not refer to the value of the numbers we are dealing with, rather, it refers to a large number of repetitions (or trials, or experiments, or iterations). This post takes a stab at explaining the difference between the strong law of large numbers (SLLN) and the weak law of large numbers (WLLN). I think it is important, not amply clear to most, and I will need it as a reference in future posts.

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## Forecast averaging example

Especially in economics/econometrics, modellers do not believe their models reflect reality as it is. No, the yield curve does NOT follow a three factor Nelson-Siegel model, the relation between a stock and its underlying factors is NOT linear, and volatility does NOT follow a Garch(1,1) process, nor Garch(?,?) for that matter. We simply look at the world, and try to find an apt description of what we see.

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## What is measurement error bias?

Errors-in-variables, or measurement error situation happens when your right hand side variable(s); your $x$ in a $y_t = \alpha + \beta x_t + \varepsilon_t$ model is measured with error. If $x$ represents the price of a liquid stock, then it is accurately measured because the trading is so frequent. But if $x$ is a volatility, well, it is not accurately measured. We simply don’t yet have the power to tame this variable variable.

Unlike the price itself, volatility estimates change with our choice of measurement method. Since no model is a perfect depiction of reality, we have a measurement error problem on our hands.

Ignoring measurement errors leads to biased estimates and, good God, inconsistent estimates.

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## ASA statement on p-values

There are many problems with p-values, and I too have chipped in at times. I recently sat in a presentation of an excellent paper, to be submitted to the highest ranked journal in the field. The authors did not conceal their ruthless search for those mesmerizing asterisks indicating significance. I was curious to see many in the crowd are not aware of current history in the making regarding those asterisks.

The web is now swarming with thought-provoking discussions about the recent American Statistical Association (ASA) statement on p-values. Despite their sincere efforts, there are still a lot of back-and-forth over what they actually mean. Here is how I read it.

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