# Bounds on the mills ratio

How do I show the following bounds on the mills ratio :

$$\frac{1}{x}- \frac{1}{x^3} < \frac{1-\Phi(x)}{\phi(x)} < \frac{1}{x}- \frac{1}{x^3} +\frac{3}{x^5} \ \ \ \ \ \ \$$ for $$\ \ \ x>0$$ where $$\Phi()$$ is the CDF of the Normal distribution , and $$\phi()$$ is the density function of the Normal distribution ?

Also , is there a similar bound when $$x < 0$$ ?

I am aware of the proof of the fact that the mills ratio is bounded below by $$\frac{x}{1+x^2}$$ and above by $$\frac{1}{x}$$ , but I am unable to prove this inequality .

Here's a sketch and a link for how I prove it. Let $$f(x) = - \left( \frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5}\right) \phi(x) .$$

Now show (lemma): $$\frac{df}{dx} = \left(1 + \frac{15}{x^6}\right)\phi(x)$$.

(To prove this, use that $$\frac{d\phi}{dx} = - x \phi(x)$$, the product rule, and some cancellation.)

Now suppose $$x > 0$$:

\begin{align*} 1 - \Phi(x) &= \int_{t=x}^{\infty} \phi(x) dx \\ &\leq \int_{t=x}^{\infty} \left(1 + \frac{15}{x^6}\right) \phi(x) dx \\ &= \lim_{t\to\infty} f(t) - f(x) \\ &= - f(x) \\ &= \left(\frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5} \right) \phi(x) . \end{align*}

Using the next term in the series gives $$f(x) = -\left(\frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5} - \frac{15}{x^7}\right)\phi(x)$$ and $$\frac{df}{dx} = \left(1 - \frac{105}{x^8}\right)\phi(x)$$. Notice because of the alternating positive/negative terms, $$\frac{df}{dx}$$ is now $$\phi$$ times something less than one, so plugging it into the same proof gives a lower bound on $$1 - \Phi(x)$$. I have a blog post on the general form of this (sorry to not point you to a more formal reference).

This is a special case of Proposition 1.3.

The case $$x<0$$ obtains from this by symmetry: $$\Phi(-z)=1-\Phi(z)$$ and $$\phi(-z)=1-\phi(z)$$ for all $$z$$.

• +1 Could you please include a proof of Proposition 1.3 for the sake of completeness . – John Apr 26 at 16:31
• @John : The proof of the identity from which your inequalities immediately follow is the following sentence on page 3 of the referenced paper: "Identity (1.4) is immediate from the recurrence relation [...], which in turn is obtained by integration by parts". – Iosif Pinelis Apr 26 at 16:46