[Paper Review] Rare Event Statistics Applied to Fast Radio Bursts

Scott Vander Wiel, Sarah Burke-Spolaor+3 more|arXiv (Cornell University)|Dec 2, 2016

Pulsars and Gravitational Waves Research2 references9 citations

TL;DR

This paper applies rare event statistics, particularly the negative binomial model for Poisson counts with uncertain rates, to analyze fast radio burst (FRB) detection surveys. It provides Bayesian and frequentist methods to estimate FRB rates, assess detection probabilities, and compare rates across surveys, finding strong evidence for higher FRB rates at high Galactic latitudes and quantifying required observing times for detection with high confidence.

ABSTRACT

Statistical interpretation of sparsely sampled event rates has become vital for new transient surveys, particularly those aimed at detecting fast radio bursts (FRBs). We provide an accessible reference for a number of simple, but critical, statistical questions relevant for current transient and FRB research and utilizing the negative binomial model for counts in which the count rate parameter is uncertain or randomly biased from one study to the next. We apply these methods to re-assess and update results from previous FRB surveys, finding as follows. 1) Thirteen FRBs detected across five high-Galactic-latitude (> 30$^\circ$) surveys are highly significant $(p = 5 imes 10^{-5})$ evidence of a higher rate relative to the single FRB detected across four low-latitude (< 5$^\circ$) surveys, even after accounting for effects that dampen Galactic plane sensitivity. High- vs. mid-latitude (5 to 15$^\circ$) is marginally significant $(p = 0.03)$. 2) A meta analysis of twelve heterogeneous surveys gives an FRB rate of 2866 sky$^{-1}$ day$^{-1}$ above 1 Jy at high Galactic latitude (95% confidence 1121 to 7328) and 285 sky$^{-1}$ day$^{-1}$ at low/mid latitudes (95% from 48 to 1701). 3) Using the Parkes HTRU high-latitude setup requires 193 observing hours to achieve 50% probability of detecting an FRB and 937 hours to achieve 95% probability, based on the ten detections of (Champion et al. 2016) and appropriately accounting for uncertainty in the unknown Poisson rate. 4) Two quick detections at Parkes from a small number of high-latitude fields (Ravi et al. 2015; Petroff et al. 2015) tentatively favor a look long survey style relative to the scan wide HTRU survey, but only at $p = 0.07$ significance.

Motivation & Objective

• Address statistical challenges in estimating rare event rates from sparse FRB detection data across multiple surveys with small counts or zero detections.
• Provide accessible, practical statistical tools for researchers to estimate FRB rates, predict detection times, and compare rates across different survey configurations.
• Quantify uncertainty in FRB rate estimates due to unknown or variable underlying Poisson rates, especially when survey sensitivity and sky coverage differ.
• Assess whether FRB rates differ significantly between high- and low/mid-Galactic-latitude surveys, accounting for Galactic plane sensitivity effects.
• Enable meta-analysis of heterogeneous FRB surveys to derive a consolidated, confidence-interval-anchored estimate of the FRB rate function.

Proposed method

• Use the negative binomial distribution to model count data when the Poisson rate parameter is uncertain or varies across surveys, allowing for overdispersion.
• Apply Bayesian inference with a neutral gamma prior (α=1/3, β=0) to estimate posterior distributions for the FRB rate, enabling direct probability statements about detection times.
• Implement frequentist prediction bounds using the F-distribution quantile to compute conservative 95% prediction intervals for future counts, particularly for detecting at least one pulse.
• Use likelihood ratio tests to evaluate whether multiple surveys share a common underlying rate, testing for consistency across high- and low-latitude samples.
• Apply negative binomial regression to model the FRB rate as a function of flux sensitivity, Galactic latitude, and extra-Poisson variation, estimating a source count index.
• Compare Bayesian and frequentist approaches for detection time estimation, favoring the Bayesian method for interpretability and better calibration.

Experimental results

Research questions

• RQ1 How many observing hours are required with the Parkes multibeam receiver to achieve a 50% or 95% probability of detecting an FRB, given existing detection counts and uncertainty in the rate?
• RQ2 Is the FRB detection rate significantly higher at high Galactic latitudes (>30°) compared to low-latitude surveys (<5°), even after accounting for reduced sensitivity in the Galactic plane?
• RQ3 Do surveys with longer individual pointings (‘look long’) detect FRBs at a higher rate than wide-area surveys (‘scan wide’), based on available detection data?
• RQ4 Can a common underlying FRB rate be assumed across a collection of heterogeneous surveys, or is there significant variation requiring a more flexible model?
• RQ5 What is the estimated FRB rate as a function of flux sensitivity, and how does it vary with Galactic latitude, based on combined survey data?

Key findings

• Thirteen FRBs detected in five high-latitude surveys (|b| > 30°) provide strong evidence (p = 5 × 10⁻⁵) of a higher FRB rate compared to one FRB in four low-latitude surveys (|b| < 5°), even after correcting for Galactic plane sensitivity suppression.
• A meta-analysis of twelve heterogeneous surveys yields an estimated FRB rate of 2,866 sky⁻¹ day⁻¹ (95% CI: 1,121 to 7,328) at high Galactic latitudes and 285 sky⁻¹ day⁻¹ (95% CI: 48 to 1,701) at low/mid latitudes.
• For the Parkes HTRU survey, 193 hours of observation are required to achieve a 50% probability of detecting an FRB, and 937 hours for 95% probability, based on ten detections and rate uncertainty modeling.
• Two quick detections from small high-latitude fields (Ravi et al., 2015; Petroff et al., 2015) suggest a 'look long' strategy may outperform 'scan wide' HTRU, but this is only marginally significant (p = 0.07).
• A likelihood ratio test shows that a common rate is plausible for high-latitude and combined low/mid-latitude surveys, but not for the full set under Euclidean scaling assumptions.
• Negative binomial regression estimates a rate function consistent with other published results, with a source count index that reflects flux sensitivity and Galactic latitude dependence, while accounting for extra-Poisson variation.