Small subsets of data with disproportionate influence on model outcomes can have dramatic impacts on conclusions, with a few data points sometimes overturning key findings. While recent work has developed methods to identify these most influential sets, no formal theory exists to determine when their influence reflects genuine problems rather than natural sampling variation. We address this gap by developing a principled framework for assessing the statistical significance of most influential sets. Our theoretical results characterize the extreme value distributions of maximal influence and enable rigorous hypothesis tests for excessive influence, replacing current ad-hoc sensitivity checks. We demonstrate the practical value of our approach through applications across economics, biology, and machine learning benchmarks.

Entitled son, 21, of top lawyer mows down police with his Mercedes G-Wagen...as he smiles in his mugshot Trump'humiliates' speaker Mike Johnson in private conversation as government shutdown rumbles on Tupac's humiliating intimate disfigurement revealed... and how his lies to cover it up led to his murder'I'm Madeline - and this is what I have to say to Lily Allen': Read world exclusive reveal of mother who had affair with star's husband David Harbour, how it started and how she feels about THOSE texts being exposed Loved up Katy Perry holds hands with Justin Trudeau as they officially confirm romance while celebrating the singer's birthday in Paris Furrow-browed boyfriend'strangled girlfriend and set her house on fire while newborn baby was inside' I've uncovered my husband's filthy Viagra habit: But, warns DEAR JANE, one thing YOU are doing is making it so much worse I've started having heart palpitations. Jackie Kennedy's revenge romance with American political icon: Revealed for first time in titillating love letters, the man who helped her cope with JFK's cheating The night that haunted a Wisconsin town forever... and the little girl whose trick-or-treat next door ended in horror Why going gray may save you from CANCER... as scientists make bombshell breakthrough Brazen demands for flying private REVEALED by the woman paid to fulfill them: 'Answer is always yes' They sneered at Trump's'eagle graveyards' - but now Biden's hated windmills crippling an American legend are haunting the US military Kim Kardashian's just been caught in a despicable lie. She can cry all she wants... there's no hiding the truth now: CAROLINE BULLOCK Tua Tagovailoa's swollen eye sparks concern after Dolphins QB woke up with mystery illness on day of Falcons game JD Vance's wife is given secret role in Trump's deal-making inner circle: 'I'll have Usha look at it' The Biden blunder that allowed an alleged October 7 'monster' to become a restaurant worker in Louisiana How I reversed my hair loss and lost 8 stone aged 45 - without weight-loss jabs. Urgent warning over cannabis as UK's top psychiatrist warns it isn't safe for young brains still developing It may seem like a relatively harmless right of passage. But cannabis isn't safe for young brains still developing, the UK's top psychiatrist has warned.

Entitled son, 21, of top lawyer mows down police with his Mercedes G-Wagen...as he smiles in his mugshot Tupac's humiliating intimate disfigurement revealed... and how his lies to cover it up led to his murder Trump'humiliates' speaker Mike Johnson in private conversation as government shutdown rumbles on'I'm Madeline - and this is what I have to say to Lily Allen': Read world exclusive reveal of mother who had affair with star's husband David Harbour, how it started and how she feels about THOSE texts being exposed Loved up Katy Perry holds hands with Justin Trudeau as they officially confirm romance while celebrating the singer's birthday in Paris Furrow-browed boyfriend'strangled girlfriend and set her house on fire while newborn baby was inside' I've uncovered my husband's filthy Viagra habit: But, warns DEAR JANE, one thing YOU are doing is making it so much worse I've started having heart palpitations. Jackie Kennedy's revenge romance with American political icon: Revealed for first time in titillating love letters, the man who helped her cope with JFK's cheating The night that haunted a Wisconsin town forever... and the little girl whose trick-or-treat next door ended in horror Why going gray may save you from CANCER... as scientists make bombshell breakthrough Brazen demands for flying private REVEALED by the woman paid to fulfill them: 'Answer is always yes' They sneered at Trump's'eagle graveyards' - but now Biden's hated windmills crippling an American legend are haunting the US military Kim Kardashian's just been caught in a despicable lie. She can cry all she wants... there's no hiding the truth now: CAROLINE BULLOCK Tua Tagovailoa's swollen eye sparks concern after Dolphins QB woke up with mystery illness on day of Falcons game JD Vance's wife is given secret role in Trump's deal-making inner circle: 'I'll have Usha look at it' The Biden blunder that allowed an alleged October 7 'monster' to become a restaurant worker in Louisiana How I reversed my hair loss and lost 8 stone aged 45 - without weight-loss jabs. A growing network of at least 5,000 data centers across the US is becoming a hidden public health threat, scientists have warned. That is because the energy-hungry backbone of artificial intelligence pumps out dangerous pollutants that can cause asthma, cancer and even death.

Three Chinese nationals have been arrested in Georgia on suspicion of attempting to illegally purchase 2kg of uranium. Lasha Maghradze, deputy head of the nation's State Security Service (SSG), told a news briefing the group planned to pay $400,000 (£300,570) for the nuclear material in the capital, Tblisi, before transporting it to China via Russia. The alleged plot was unearthed by intelligence agents while one member of the group was attempting to buy the radioactive substance on the black market, he said. The three pleaded not guilty at a court in Tblisi and have been placed in custody to prevent them fleeing the country, according to public broadcaster Georgia Today. They face up to five years in prison under a provision of Georgia's criminal code banning the purchasing of nuclear material.

Two arrested over Louvre'heist of the century' after raid of France's £76million crown jewels Tupac's humiliating intimate disfigurement revealed... and how his lies to cover it up led to his murder Virginia Giuffre's ex boyfriend says she was terrified'shaking with fear' after sex with Prince Andrew fearing'something would happen to her' I've started having heart palpitations. Psychotherapist explains why No Kings rallies consisted of mostly'educated white women' The'marry me' sex move that'll make even the most commitment-phobic of men beg to see you again... and it worked for THREE of my friends Inside Prince Harry and Meghan's final night of freedom at a Halloween party with Princess Eugenie just hours before news of their relationship broke Kristen Bell's friends turn on her with savage disclosures: Insiders reveal poisonous whispers behind her back... as she goes into full diva mode Influencer, 23, speaks out after being arrested for'running interstate drug smuggling network' Kim Kardashian's just been caught in a despicable lie. She can cry all she wants... there's no hiding the truth now: CAROLINE BULLOCK Meghan Markle's fashion faux pas that shocked onlookers during her first overseas tour just months after marrying Prince Harry Inside Andrew's family summit: How Fergie wailed and'melted down' at title loss, Beatrice and Eugenie were'blindsided' and now daughters' assets face'ethics check' to avoid more scandal: BARBARA DAVIES Flamboyant art dealer whose'fake Warhols fooled' Florida's elite is out on bond and cashing in on a flashy new fad Californians being urged to take up arms to deal with'aggressive' invasive species attacking children A disturbing weather pattern could wreak havoc across the US at the end of this year's hurricane season, experts have warned. November tropical storms may be affected by La Nin a, according to Matthew Rosencrans, the lead hurricane seasonal forecaster with the National Oceanic and Atmospheric Administration (NOAA). La Nin a is part of a natural climate cycle known as El Nin o-Southern Oscillation Neutral (ENSO), which alternates between warmer and cooler seawater along the equator in the Pacific Ocean.

Things to Do in L.A. Tap to enable a layout that focuses on the article. California will be the first state to ban most law enforcement, including federal immigration agents, from covering their faces while conducting official business under a bill signed by Gov. Gavin Newsom on Saturday. This is read by an automated voice. Please report any issues or inconsistencies here . SACRAMENTO -- Though hailed by some for signing new laws to combat antisemitism in California schools, Gov. Gavin Newsom expressed enough reservations about the bills to urge state lawmakers to make some changes.

Game Theory What's behind bans on away fans? Maccabi Tel Aviv fans have been banned from Villa Park for their Europa League game against Aston Villa. But who decides when away fans are banned and what constitutes a high-risk game? Samantha Johnson looks at the reasons and the politics behind banning orders in European football. Afghan Women's Team: The Fight to Play U-20 World Cup Who would you play for?

Tupac's humiliating intimate disfigurement revealed... and how his lies to cover it up led to his murder I've started having heart palpitations. 'Black Ivy League' university looks to expand into crime-riddled Oakland Kristen Bell's friends turn on her with savage disclosures: Insiders reveal poisonous whispers behind her back... as she goes into full diva mode Shooting leaves two dead and 11 injured at large house party with'underage people' in North Carolina Kim Kardashian's just been caught in a despicable lie. She can cry all she wants... there's no hiding the truth now: CAROLINE BULLOCK The'marry me' sex move that'll make even the most commitment-phobic of men beg to see you again... and it worked for THREE of my friends Prosecutor who declined to charge Letitia James with bank fraud fired after'mishandling evidence' Californians being urged to take up arms to deal with'aggressive' invasive species attacking children Inside Andrew's family summit: How Fergie wailed and'melted down' at title loss, Beatrice and Eugenie were'blindsided' and now daughters' assets face'ethics check' to avoid more scandal: BARBARA DAVIES LIZ JONES: I was devastated when my husband cheated. But here's the reason part of me was secretly glad that every woman over-50 will understand Psychotherapist explains why No Kings rallies consisted of mostly'educated white women' Tree optical illusion messes with your mind - you can see the squirrel but can you spot the cat in 30 seconds? Turn off the lights, burn a candle, look into the mirror and say the magic words: 'Bloody Mary, Bloody Mary, Bloody Mary'.

Is Trump losing patience with Putin? Will sanctions against Russian oil giants hurt Putin? How much of Europe's oil still comes from Russia? Russian overnight attack on Ukraine's Kyiv kills at least 3, wounds dozens At least three people have been killed and dozens wounded in an overnight Russian air attack on Kyiv, according to the mayor of the Ukrainian capital, as Russia's war on Ukraine approaches its four-year mark. Mayor Vitali Klitschko said on Sunday that "several" Russian drones were operating over the city, and warned people to "remain in shelters".

Is Trump losing patience with Putin? Will sanctions against Russian oil giants hurt Putin? How much of Europe's oil still comes from Russia? Russia launched a drone attack on the Ukrainian capital, Kyiv, in the early hours of Sunday, striking residential buildings and wounding at least five people, including two children, according to officials. Russian missile and drone attacks on Ukraine overnight on Saturday killed at least two people and wounded 12 others, Ukrainian Prime Minister Yulia Svyrydenko said on X.