A general approximation to quantiles

Abstract

For many continuous distributions, a closed-form expression for their quantiles does not exist. Numerical approximations for their quantiles are developed on a distribution-by-distribution basis. This work develops a general approximation for quantiles using the Taylor expansion. Our method only requires that the distribution has a continuous probability density function and its derivatives can be derived to a certain order (usually 3 or 4). We demonstrate our unified approach by approximating the quantiles of the normal, exponential, and chi-square distributions. The approximation works well for these distributions.

A Note on the Accuracy of Normal Approximation of Random Quantities

“…In the above, K is the Berry-Esseen generic constant which was shown by van Beek [3] to be 0.7975. Yu and Zeltermam [4] developed numerical approximation to evaluate the quantiles for many of the continuous distributions using Taylor expansion. Hyndman and Fan [5] investigate the motivation of sample quantiles and their properties.…”

Approximate Quantiles for Stochastic Optimal Control of LTI Systems with Arbitrary Disturbances

“…For a random variable 𝒚 𝑖 , and a known quantile evaluation point 𝒚 * 𝑖 = Φ −1 𝒚 𝑖 ( 𝑝 0 ) for 𝑝 0 ∈ (0, 1), [35] proposes an iterative process that evaluates a Taylor series expansion of 𝑛 𝑑 terms at points that are an interval ℎ ∈ R + apart. A quantile approximation at 𝑝 𝑐+1 = 𝑝 𝑐 + ℎ is described by…”

Approximate Quantiles for Stochastic Optimal Control of LTI Systems with Arbitrary Disturbances

“…We use an approach that relies on a Taylor series expansion of the quantile [20]. For a random variable X, and an initial evaluation point p 0 ∈ (0, 1), [20] proposes an iterative process that evaluates a finite Taylor's series expansion at points that are an interval h ∈ R apart. With n d Taylor's series terms, a quantile approximation at p c+1 = p c + h is described by…”

“…with successive derivatives eliciting higher derivatives of φ X (•). Analytical expressions for the first four derivatives are provided in [20].…”

“…A similar reformulation can be posed for (14d). In the limit, as (19) becomes an affine overapproximation of Φ −1 (•), ( 20) is a tightening of ( 14) and Assumption 1 ensures the convexity of (20).…”