# Variance of Normal Order Statistics

Suppose we have $$X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$$ with $$n > 50$$, and let $$X_{(1)}, \cdots, X_{(n)}$$ be the associated order statistics.

Are there any references pointing to formulas that specify (or estimate) the variance of the $$k^{\textrm{th}}$$ order statistic, i.e. $$\textrm{Var}(X_{(k)})$$, for $$1 \leq k \leq n$$?

I am looking for formulas can be explicitly evaluated without resorting back to numerical integration.

I have seen this question: Approximate order statistics for normal random variables, which attracted a number of answers on approximate formula for the mean of the $$k^{\textrm{th}}$$ order statistic, e.g. Blom (1958) (and modifications based on work by Harter (1961) and Elfving (1947)).

Searching on Google also yields a number of work on the variance / standard deviation for the $$k^\textrm{th}$$ order statistics for small $$n$$, all of them opting for a computation/tabulation route. They are: Godwin(1949), Teichroew (1956), and Parrish (1992).

I have also attempted to derive something myself following the method sketched by @probabilityislogic in this question, along the lines of:

$$\textrm{Var}(X_{(k)}) = \int_{-\infty}^{\infty} \left(x-\mathbb{E}(X)\right)^2 \frac{n!}{(k-1)!(n-k)!} f_X(x)[1-F_X(x)]^{n-k}[F_X(x)]^{k-1}\;\textrm{d}x$$, with lower case $$f$$ denoting the PDF of a r.v. $$X$$, and upper case $$F$$ denoting the CDF.

Noting $$x = F^{-1}(F_X(x))$$ and using the substitution $$u = F_X(x)$$ (with corresponding r.v. in upper case $$U$$), we have: $$\textrm{Var}(X_{(k)}) = \int_{0}^{1} \left(F_X^{-1}(u) - \mathbb{E}\left(F_X^{-1}(U)\right)\right)^2\,\mathcal{B}(u | k, n-k+1) \; \textrm{d}u$$, where $$\mathcal{B}$$ denotes the PDF of a beta distribution (not the beta function). This can be rewritten as the variance based on the beta distribution:

$$\textrm{Var}(X_{(k)}) = \textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right)$$

Approximating the RHS using (the variance bit of the) delta method, we get: $$\textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right) \approx \textrm{Var}_{\mathcal{B}(u | k, n-k+1)}(U) \cdot \left[(F_X^{-1})'\left(\mathbb{E}_{\mathcal{B}(u | k, n-k+1)}(U)\right)\right]^2$$, where the prime denotes the derivative.

The first term of the product is standard result. For the second part, given $$(F^{-1})'(\cdot)=\frac{1}{f(F^{-1})(\cdot)}$$ (see e.g. this), we then have: $$\textrm{Var}_{\mathcal{B}(u | k, n-k+1)}\left(F_X^{-1}(u)\right) \approx \frac{k(n-k+1)}{(n+1)^2(n+2)} \frac{1}{\left(f\left(F^{-1}\left(\frac{k}{n+1}\right)\right)\right)^2}$$

Substituting in the standard normal distribution ($$\phi$$ for PDF, $$\Phi$$ for CDF) and scaling it to the desired variance $$\sigma^2$$ we arrive at: $$\textrm{Var}(X_{(k)}) \approx \frac{k(n-k+1)}{(n+1)^2(n+2)} \frac{1}{\left(\phi\left(\Phi^{-1}\left(\frac{k}{n+1}\right)\right)\right)^2}$$, which we can throw in a $$\sigma^2$$ term in front if it is not a standard normal.

But surely someone must have derived that somewhere (again I am looking for some references), and there is a high chance that I have gone off track somewhere above.

The author stated the variance of the $$k^{\textrm{th}}$$ order statistics can be estimated as:
$$\textrm{Var}(X_{(k)}) \approx \frac{p(1-p)}{(n+2)(f(\theta))^2},$$
where $$f(\cdot)$$ is the PDF of $$X$$, $$p = \frac{k}{n+1}$$, and $$\theta$$ is the $$p^\textrm{th}$$ quantile of the distribution.
Applying to the normal case, we have $$\theta = \Phi^{-1} (\frac{k}{n+1})$$, and one can easily verify the referenced variance estimate equates to the variance derived in the original question with some rearrangement of terms.