# Optimal upper bounds on expected kth record values from IGFR distributions

## TL;DR

This paper derives optimal upper bounds for the expected values of the kth record in sequences of i.i.d. random variables with distributions in the IGFR family, including ID and IFR, using the projection method.

## Contribution

It introduces new bounds for kth record expectations within the IGFR distribution class, expanding understanding of record values under these distribution constraints.

## Key findings

- Derived bounds expressed in standard deviation units.
- Bounds applicable to distributions with increasing density or failure rate.
- Utilized the projection method for optimal bounds.

## Abstract

The paper concerns the optimal upper bounds on the expectations of the kth record values (k >= 1) centered about the sample mean. We consider the case, when the records are based on the infinite sequence of the independent identically distributed random variables, which distribution function is restricted to the family of distributions with the increasing generalized failure rate (IGFR). Such a class can be defined in terms of the convex orders of some distribution functions. Particularly important examples of IGFR class are the distributions with the increasing density (ID) and increasing failure rate (IFR). Presented bounds were obtained with use of the projection method, and are expressed in the scale units based on the standard deviation of the underlying distribution function.