Jeb Bush trounced the Republican presidential field in fundraising in the first major report of the 2016 campaign, but Ben Carson, Ted Cruz and Marco Rubio posted solid numbers, suggesting they will have financial backing needed to compete.

Others lagged. Rand Paul collected an average of about $62,000 a day in contributions — less than one-tenth what Mr. Bush averaged since he announced his campaign — and Mike Huckabee and Rick Santorum, the past two winners of the Iowa caucuses, trailed even further.

The numbers, which were due at the Federal Election Commission before Thursday, cover the fundraising period from April 1 through June 30, when most of the major Republican candidates announced their bids for the party nomination.

Mr. Bush, a former Florida governor who announced in mid-June, collected $11.4 million in 16 days, averaging more than $700,000 a day.

The next closest was Mr. Carson, a retired neurosurgeon, who announced in early May and raised about $8.5 million, averaging $146,000 a day. Mr. Cruz and Mr. Rubio, U.S. senators from Texas and Florida, respectively, each averaged about $110,000 a day.

Doug Watts, a spokesman for Mr. Carson, said his boss showed he has the fundraising chops to stay in the race.

“We understood that people questioned our ability to make the grade because of his non-politician status,” Mr. Watts said. “I think we have proven to a lot of people who were looking for some demonstration with 230,000 donors that I think we don’t have to prove much about the viability of our candidacy.”

Republican observers, though, said Mr. Carson’s spending rate is higher than some of his rivals, which could hurt him in the long run.

Anthony Corrado, a political science professor at Colby University in Maine, said the reports proved that Mr. Bush can raise lots of money quickly through a well-established fundraising network with many donors willing to shell out the $2,700 maximum contribution.

The Campaign Finance Institute, a nonpartisan think tank that tracks money in politics, said 88 percent of the $11.4 million Mr. Bush raised over two weeks came from donors who contributed the maximum of $2,700 to his primary bid.

“This represents the highest percentage of max out dollars we have seen at this point in the cycle, including the 75 percent George W. Bush posted in 2003 and the 74 percent of Mitt Romney in 2011,” the institute said in a press release.

Mr. Carson and Mr. Cruz, meanwhile, are pulling in more small donations from the grass roots — 67 percent of Mr. Carson’s donors gave less than $200 to his campaign, according to the Campaign Finance Institute.

The presidential campaign of Sen. Lindsey Graham of South Carolina raised $3.7 million, $1.5 million of which he transferred from his Senate campaign account.

Mr. Corrado said the reports provide the most insight into candidates who have been in the race for a most of the quarter. That is probably bad news for Mr. Paul, a senator from Kentucky who entered the race in early April and had most of the period to try to match his father’s fundraising prowess. Instead, he was in the middle of the pack and trailed the three other sitting senators in the presidential race.

“He is attracting small donors, but not as much as some of the other candidates,” Mr. Corrado said. “So, I think the attention he received earlier this year [through his Senate filibuster] and the fact that he was one of the first to announce and had a relative empty field in front of him, you would have to say it is not the fundraising total they were looking for.”

The Paul campaign did not respond to an email seeking comment.

Mr. Watts said he was surprised by Mr. Paul’s first report.

“I was thrown a little bit because Rand Paul raised substantially less than we did,” he said. “I know he has a very loyal following, but given that he has effectively been running for a couple of years, and that he is also leveraging the long-held, loyal supporters of his father, it did surprise me that he didn’t have a stronger showing.”

Also lagging was former Texas Gov. Rick Perry, who raised a little more than $1 million with nearly a full month’s worth of campaigning.

Govs. Scott Walker of Wisconsin, and Chris Christie of New Jersey announced their campaigns after the quarter closed and did not have to file reports.

Ohio Gov. John Kasich is expected to enter the race next week, and former Virginia Gov. James S. Gilmore recently announced that he would join the fray next month, rounding out the list of expected entrants.

The Campaign Finance Institute said the presidential candidates have raised more than $130 million, with $65 million going to Republicans and $65.2 million to Democrats.

The group said that marks an increase from the 2012 campaign season, when the candidates raised $84 million, but trailed the $298 million that had been raised at this point in the 2008 race — the last contest without an incumbent.

“However, when all Super PAC money is counted at the end of the month, the combined candidate plus Super PAC fundraising will likely exceed what was raised through June 30, 2007,” the group said.

Some PACs have teased their numbers already. The pro-Bush Right to Rise PAC said it had raised $103 million to aid his bid. Pro-Cruz groups said they had raised nearly $37 million, and pro-Perry PACs said they had pulled in $16.8 million.

The Cruz campaign said the combination of the campaign fundraising reports and super PAC success show that the freshman senator is the “most well-funded conservative in the race.”

“We know that people are ready for change and ready to support the campaign that represents change,” said Jeff Roe, manager of the Cruz campaign. “Cruz has said that since the beginning of the campaign, and we now have the data to prove it.”

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