## Asymptotic distribution of the Pearson chi-square statistic

Imagen tomada de ResearchGate.

I recently learned of a fairly succinct proof for the asymptotic distribution of the Pearson chi-square statistic (from Chapter 9 of Reference 1), which I share below.

First, the set-up: Assume that we have \$latex n\$ independent trials, and each trial ends in one of \$latex J\$ possible outcomes, which we label (without loss of generality) as \$latex 1, 2, dots, J\$. Assume that for each trial, the probability of the outcome being \$latex j\$ is \$latex p_j > 0\$. Let \$latex n_j\$ denote that number of trials that result in outcome \$latex j\$, so that \$latex sum_{j=1}^J n_j = n\$. Pearson’s \$latex chi^2\$-statistic is defined as

\$latex begin{aligned} chi^2 = sum_{text{cells}} dfrac{(text{obs} – text{exp})^2}{text{exp}} = sum_{j=1}^J dfrac{(n_j – np_j)^2}{np_j}. end{aligned}\$

Theorem. As \$latex n rightarrow infty\$, \$latex chi^2 stackrel{d}{rightarrow} chi_{J-1}^2\$, where \$latex stackrel{d}{rightarrow}\$ denotes convergence in distribution.

Before proving the theorem, we prove a lemma that we will…

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## General chi-square tests

Imagen tomada de Lifeder.

In this previous post, I wrote about the asymptotic distribution of the Pearson \$latex chi^2\$ statistic. Did you know that the Pearson \$latex chi^2\$ statistic (and the related hypothesis test) is actually a special case of a general class of \$latex chi^2\$ tests? In this post we describe the general \$latex chi^2\$ test. The presentation follows that in Chapters 23 and 24 of Ferguson (1996) (Reference 1). I’m leaving out the proofs, which can be found in the reference.

(Warning: This post is going to be pretty abstract! Nevertheless, I think it’s worth a post since I don’t think the idea is well-known.)

Let’s define some quantities. Let \$latex Z_1, Z_2, dots in mathbb{R}^d\$ be a sequence of random vectors whose distribution depends on a \$latex k\$-dimensional parameter \$latex theta\$ which lies in a parameter space \$latex Theta\$. \$latex Theta\$ is assumed to be a non-empty open subset…

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## GENERALIDADES SOBRE LA TEORÍA ESTADÍSTICA DE ENCUESTAS POR MUESTREO. PARTE II

La imagen del encabezado ha sido tomada de QuestionPro.

## The wealth of nations

Marx’s first sentence in Capital Volume One is: “The wealth of those societies in which the capitalist mode of production prevails, presents itself as an “immense accumulation of commodities”, its unit being a single commodity.” (Moore and Aveling translation).  So, from the beginning, Marx makes a distinction between wealth in societies and how it appears […]

The wealth of nations

## Kolmogorov’s strong law of large numbers

The strong law of large numbers (SLLN) is usually stated in the following way: Theorem: For such that the ‘s are independent and identically distributed (i.i.d.) with finite mean , as , What if the ‘s are independent but not identically distributed? Can we say anything in that setting? We can if we add a […]

Kolmogorov’s strong law of large numbers