# Second Order Expansions for Sample Median with Random Sample Size

## TL;DR

This paper develops second order asymptotic expansions for the sample median when the sample size is random, extending classical results to more realistic scenarios where sample size varies unpredictably.

## Contribution

It introduces novel second order Chebyshev–Edgeworth and Cornish–Fisher expansions for the median with a specific type of random sample size, advancing asymptotic theory.

## Key findings

- Derived second order expansions for median with random sample size
- Applied expansions to Student's t- and Laplace distributions
- Enhanced understanding of median's asymptotic behavior under randomness

## Abstract

In practice, we often encounter situations where a sample size is not defined in advance and can be a random value. The randomness of the sample size crucially changes the asymptotic properties of the underlying statistic. In the present paper second order Chebyshev--Edgeworth and Cornish--Fisher expansions based of Student's $t$- and Laplace distributions and their quantiles are derived for sample median with random sample size of a special kind.