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Let $\{\hat{X}_n\}$ be a sequence of estimators that converges in probability to the constant $\bar{x}$, which I take to mean that, for any $\epsilon > 0$, $\lim \limits_{n \to \infty} \Pr(|\hat{X}_n - \bar{x}| > \epsilon) = 0$.

I have also seen people write $(\hat{X}_n - \bar{x}) = o_p(1)$ and have seen some people refer to it as equivalent to the statement that $\hat{X}_n$ is consistent for $\bar{x}$.

What is the difference between these two statements?

Is it correct that $(\hat{X}_n - \bar{x}) = o_p(1)$ is equivalent to $\lim \limits_{n \to \infty} \Pr(|\hat{X}_n - \bar{x}| \geq \epsilon) = 0$? And could one also write that $(\hat{X}_n - \bar{x}) = o_p(1)$ is equivalent to $\hat{X}_n = \bar{x} + o_p(1)$?

Thank you for any and all help!

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2 Answers 2

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Yes, $X_n-\bar x=o_p(1)$ is just notation for saying "For every $\epsilon>0$, $$\lim_{N\to\infty}P\left(|X_n-\bar x|>\epsilon \right)=0"$$

More generally, $X_n-Z_n=o_p(Y_n)$ is notation for saying "For every $\epsilon>0$, $$\lim_{N\to\infty}P\left(|X_n-Z_n|>\epsilon |Y_n| \right)=0"$$

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And could one also write that $(\hat{X}_n - \bar{x}) = o_p(1)$ is equivalent to $\hat{X}_n = \bar{x} + o_p(1)$?

I do not know whether this is conventional, but you can do it.


Interpretation in terms of quantile functions

We can argue about these notations with $o_p$ and $O_p$ in the same way as for $o$ and $O$ when we make an interpretation that connects the two.

With a particular interpretation you can turn the probabilistic expressions ${o}_p(f(n))$ or ${O}_p(f(n))$ into an expression with the more well known ${o}(f_n)$ or ${O}(f_n)$.

We can interpret

$$X_n = {o}_p(f(n)) \quad \text{for }n \to \infty$$

in terms of the quantile functions for $X_n$

$$\forall p: \lbrace Q_{X_n}(p) = {o}(f(n)) \quad \text{for }n \to \infty \rbrace$$

which means

$$\forall p: \lim_{n \to \infty} \frac{Q_{X_n}(p)}{f(n)} = 0$$

For example: in the figure below we show some quantiles functions of the mean $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$, where $X_i \sim N(\mu,\sigma^2)$ with $\mu = 0.5$ and $\sigma =1$

convergence of quantile functions

The convergence of the quantile functions means that for every for any $0 < p \leq 1$, there is an $n$ such that $Pr(\vert X_n \vert > \epsilon) \leq p $


Interpretation of equal sign

With this view, in terms of quantiles, the notations $o_p$ and $O_p$ follow the same rules as the notations $o$ and $O$ in 'regular' mathematics.

For these notations, there are some debates on how the equality sign '$=$' must/can be interpreted. See for instance

The equality sign '$=$' is not the typical equality and should be interpreted as a short-handed notation for 'is element of', or in terms of symbols '$\in$', and should not be interpreted as an 'identity'.

The acceptance and use of notations with an algebraic appearance like $\hat{X}_n = \mu_{x} + o_p(1)$ is a matter of taste, but it is certainly not wrong. You may however need to keep in mind that you do not confuse the reader.

To contemplate the meaning of $\hat{X}_n = \mu_{x} + o_p(1)$, the reader might need to imagine eventually the 'difference' $\hat{X}_n - \mu_{x} = o_p(1)$. So you could write it out like that from the start.

Which notation you use might depend on the intuition that you want to stress.

  • The function is some other function plus/minus some bounds: $$f(x) = g(x) + O(h(x))$$
  • The difference between the function and some other function is within some bounds: $$f(x) - g(x) = O(h(x))$$

To me it is a nuanced difference and the notations like $f(x) = g(x) + O(h(x))$ are not wrong. They are actually being used a lot (this sounds like an argumentum ad populum, but it is not a fallacy when we are arguing about conventions and use of interpretations that might be confusing when they are uncommon).

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