Estimating hazard rates from $δ$-records in discrete distributions

TL;DR

This paper develops a nonparametric framework for estimating the discrete hazard rate using $\delta$-records from a single sequence of i.i.d. observations on $\mathbb{Z}_+$. It derives the explicit MLE $\hat{h}_{j,k} = \dfrac{V^k_j}{1+\sum_{\ell=0}^k V^{k-\ell}_{j+\ell}}$ and provides its exact distribution, exact moments, and asymptotic properties, enabling confidence intervals and goodness-of-fit testing via likelihood ratios. The methodology is extended to monotone hazard-rate estimation and demonstrated on real data sets (defective-item times and earthquake occurrences), illustrating practical nonparametric inference without requiring multiple samples. The paper also discusses potential extensions to continuous distributions, Bayesian approaches, and incorporating additional record-derived statistics, positioning $\delta$-records as a powerful tool for reliability and anomaly-detection studies.

Abstract

This paper focuses on nonparametric statistical inference of the hazard rate function of discrete distributions based on $δ$-record data. We derive the explicit expression of the maximum likelihood estimator and determine its exact distribution, as well as some important characteristics such as its bias and mean squared error. We then discuss the construction of confidence intervals and goodness-of-fit tests. The performance of our proposals is evaluated using simulation methods. Applications to real data are given, as well. The estimation of the hazard rate function based on usual records has been studied in the literature, although many procedures require several samples of records. In contrast, our approach relies on a single sequence of $δ$-records, simplifying the experimental design and increasing the applicability of the methods.

Figures (6)

• Figure 1: Expectation of $\hat{h}_{j,k}$ as a function of $p=h_j$ for $k=1,2,3,4$ and a Geometric distribution $F$ together with the identity function (black).
• Figure 2: Estimates and $95\%$ confidence intervals for $h_6$ for $100$ independent sequences following a negative binomial distribution $\mathrm{NegBin}(5;0.8)$ and $k=3,4$. The intervals that contain the value $h_6\approx 0.6914$ are plotted in blue and those that do not are plotted in red.
• Figure 3: Estimator $\hat{h}_{j,k}^{\text{inc.}}$ for $X_1\sim\mathrm{Poi}(6)$ with $n = 5$, $k = 3$ and $r_{n} = 18$.
• Figure 4: Values of the estimators $\hat{h}_{j,k}$ in Table \ref{['table_estimators_xiegoh']} along with the MLE of the probability of success $p$ for the entire sample.
• Figure 5: Comparison between observed relative frequency of earthquakes with magnitudes of 7.5 and above from 1950 to 2023 (Table \ref{['data_earthquakes']}) and the adjusted Poisson model.

Theorems & Definitions (31)

• Proposition 3.1
• proof
• Remark 3.2
• Theorem 3.3
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Proposition 3.6
• proof