DeepAI AI Chat
Log In Sign Up

Density functions for QuickQuant and QuickVal

by   James Allen Fill, et al.

We prove that, for every 0 ≤ t ≤ 1, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the tth quantile in a randomly ordered list has a Lipschitz continuous density function f_t that is bounded above by 10. Furthermore, this density f_t(x) is positive for every x > min{t, 1 - t} and, uniformly in t, enjoys superexponential decay in the right tail. We also prove that the survival function 1 - F_t(x) = ∫_x^∞f_t(y) dy and the density function f_t(x) both have the right tail asymptotics exp [-x ln x - x lnln x + O(x)]. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.


page 1

page 2

page 3

page 4


On the tails of the limiting QuickSort density

We give upper and lower asymptotic bounds for the left tail and for the ...

QuickSort: Improved right-tail asymptotics for the limiting distribution, and large deviations

We substantially refine asymptotic logarithmic upper bounds produced by ...

A new life of Pearson's skewness

In this work we show how coupling and stochastic dominance methods can b...

Strong Asymptotic Properties of Kernel Smooth Density and Hazard Function Estimation for Right Censoring NA Data

The paper considers kernel estimation of the density function together w...

Generalization Error Bounds via mth Central Moments of the Information Density

We present a general approach to deriving bounds on the generalization e...

Exponential confidence region based on the projection density estimate. Recursivity of these estimations

We investigate the famous Tchentzov's projection density statistical est...

Right-truncated Archimedean and related copulas

The copulas of random vectors with standard uniform univariate margins t...