Skip to main content
Cross Validated

Return to Answer

added 98 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89

The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:
\begin{align*} 1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1} \end{align*} where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.

$(1)$ is a corollary of the inequality (for fixed $x > 0$) \begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \Phi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}

To prove $(2)$, notice the obvious inequality: \begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \end{align*}\begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \tag{3} \end{align*} Integrating three expressions above from $x$ to $\infty$ yields $(2$) -- note that each term in $(2)$ is the primitive function of each term in $(3)$ times $-1$.

The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:
\begin{align*} 1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1} \end{align*} where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.

$(1)$ is a corollary of the inequality (for fixed $x > 0$) \begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \Phi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}

To prove $(2)$, notice the obvious inequality: \begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \end{align*} Integrating three expressions above from $x$ to $\infty$ yields $(2$).

The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:
\begin{align*} 1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1} \end{align*} where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.

$(1)$ is a corollary of the inequality (for fixed $x > 0$) \begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \Phi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}

To prove $(2)$, notice the obvious inequality: \begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \tag{3} \end{align*} Integrating three expressions above from $x$ to $\infty$ yields $(2$) -- note that each term in $(2)$ is the primitive function of each term in $(3)$ times $-1$.

deleted 3 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89

The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:
\begin{align*} 1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1} \end{align*} where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.

$(1)$ is a corollary of the inequality (for fixed $x > 0$) \begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \varphi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}\begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \Phi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}

To prove $(2)$, notice the obvious inequality: \begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \end{align*} Integrating three expressions above from $x$ to $\infty$ yields $(2$).

The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:
\begin{align*} 1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1} \end{align*} where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.

$(1)$ is a corollary of the inequality (for fixed $x > 0$) \begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \varphi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}

To prove $(2)$, notice the obvious inequality: \begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \end{align*} Integrating three expressions above from $x$ to $\infty$ yields $(2$).

The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:
\begin{align*} 1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1} \end{align*} where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.

$(1)$ is a corollary of the inequality (for fixed $x > 0$) \begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \Phi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}

To prove $(2)$, notice the obvious inequality: \begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \end{align*} Integrating three expressions above from $x$ to $\infty$ yields $(2$).

Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89

The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:
\begin{align*} 1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1} \end{align*} where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.

$(1)$ is a corollary of the inequality (for fixed $x > 0$) \begin{align*} (x^{-1} - x^{-3})\varphi(x) < 1 - \varphi(x) < x^{-1}\varphi(x). \tag{2} \end{align*}

To prove $(2)$, notice the obvious inequality: \begin{align*} (1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. \end{align*} Integrating three expressions above from $x$ to $\infty$ yields $(2$).