Noir is so often defined solely in the male, hard-boiled, detective vein and male writers that it is difficult to describe female noir. I'll take a stab though.

1. What is female? I'd define female noir as written by women.

2. What is noir? This is harder, as "noir" is not so much a genre as a sensibility, and one defined by critics, and for movies at that:
"In 1946 a Paris retrospective of American films embargoed during the war clearly revealed this trend toward visibly darker, more cynical crime melodramas. It was noted by several Gallic critics who christened this new type of Hollywood product “film noir,” or black film, in literal translation.

Few, if any of the artists in Hollywood who made these films called them “noir” at the time. But the vivid co-mingling of lost innocence, doomed romanticism, hard-edged cynicism, desperate desire, and shadowy sexuality that was unleashed in those immediate post-war years proved hugely influential... To this day the debate goes on as to whether “noir” is a film genre, circumscribed by its content, or a style of storytelling, identified by its visual attributes.(http://www.filmnoirfoundation.org/fil...)

In the classic films of the "noir" style, a prevalent trope of interest is the femme fatale:
The equation of female sexuality and power with evil was not only a cinematic sign, it clearly drew on wider currents. The world of film noir is a world in which women are frequently beyond the home and family, the woman being absorbed not simply in a job or career, but, that ultimate transgression, her 'self'. Viewed from this angle, these films indirectly express male fears of the disturbing female mobility that had been released by the domestic upheaval of war, economic change, new patterns of consumption and the subsequent questioning of pre-existing roles. (https://books.google.com/books?id=SP0...)
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3. What then is "female noir"?

It is fiction written by women that engages with noir themes/tropes from a female perspective. Noir is found in many genres; female noir is found in many genres --

- crime fiction (featuring antiheroines), ex. Natsuo Kirino's Out
*"The Bitter Women of Japanese noir" http://thenewinquiry.com/essays/the-b...
A housewife assists in dismembering the body of an unfaithful husband, a female office worker plots a traceless murder, and a group of schoolgirls help a high school outcast escape arrest for matricide. They derive no benefit from their stereotypes; they are too hard-bitten, too unattractive, too young to be taken seriously, or too old to be socially desirable. Their stories are usually ignored, and their voices yield contempt. In the U.S., we like stories of horror or violence that speak to concerns about sexuality and emasculation, but we do not have as much interest in female antiheroes. There exists a fine yet definite line between the serial killer protagonist who gets to head a premium cable show and the twisted fantasies of a woman steeped in a lifetime of frustrated ambitions... This is a side of Japan that isn’t often shown—the lives of women who aren’t cute or child-like, the ones who can’t afford to depend on anyone and aren’t really liked enough anyway, who work on the fringes of society and look forward to the bleakest of futures

- (non-crime) antiheroine literature, see the "Teenage girls are SCARY" subgenre: ex. Megan Abbott's The Fever, The End of Everything
Much of the potency and continued metaphorical usefulness of noir, of course, comes from its bright palette of heightened emotional states: the desperate patsy, the conflicted killer, the frantic housewife who accidentally hit and killed a drifter and then buried him in a shallow grave in her backyard. Teenagers, of course, exist in a perpetual (and self-perpetuated) frenzy of heightened emotions. Each new crush is a world-ending love affair; every subtle shift in allegiance is a poisonous betrayal, punishable by rough expulsion from the herd.(http://www.slate.com/articles/arts/bo...)

- suspense, including domestic suspense: ex. Margaret Millar's The Fiend, Dorothy Sanxay Holding's The Blank Wall
*From a review of Sarah Weinman's Troubled Daughters, Twisted Wives: Stories from the Trailblazers of Domestic Suspense:
[Domestic suspense] penetrates the veneer of familial serenity to the dark side of homelife, exposing the creepy conundrums inherent in what are perceived as “women’s issues” and the female domain. During the genre’s heydey, from the post-war years until the early seventies, the concept of the nuclear family and the idea that “a woman’s place is in the home” had a stranglehold on the American psyche. The doyennes of Domestic Suspense strove to subvert the idealized family in their fiction and disturb their readers out of complacency. (http://www.bookslut.com/the_bombshell...)
*Relevant: https://lareviewofbooks.org/essay/the...
http://news.nationalpost.com/arts/boo...

- thrillers, including psychological thrillers: ex. Dorothy G. Hughes' The Expendable Man
*"Hughes chose for herself a different challenge: a white woman, she would tell stories about and from the points of view of others—psychotic men, black men, Spanish men, Native Americans; jazz musicians, fashionable women, soldiers, doctors. The creation of difference itself was her subject. Her books were widely praised for their atmospheres of fear and suspense, and criticized when they reached, as the New York Times said of “The Fallen Sparrow,” “toward conflict and situations that are rather beyond the usual whodunit scheme.” But this is Hughes’s point. It is not whodunit, but who-ness itself, that she’s after. By this I do not mean that she asks why—specific motives are as mulish and unanswerable as sin. Crime was never Hughes’s interest, evil was, and to be evil, for her, is to be intolerant of others, of the very fact of the existence of something outside the self. (http://www.newyorker.com/books/page-t...

- mysteries, ex. Vera Caspary's Laura
*Wilkie Collins, Vera Caspary and the Evolution of the Casebook Novel is too long to quote extensively, but the gist is about how Vera Caspary drew Collin's tradition of sensation novels and the gothic novel, as well as contemporary noir films and tropes in order to create her own version of "New-Woman-noir":
It is the merger of this new-woman character, imperiled by her assertiveness in a man's world, that reconstructs the female Gothic and the sensational new woman into a modern setting...
https://books.google.com/books?id=_-P...

- war fiction/post-war fiction, especially spy thrillers (think Notorious, The Third Man), ex. Elizabeth Bowen's The Heat of the Day
*http://digitool.library.mcgill.ca/R/?...

- hard-boiled fiction, ex. Christa Faust's Money Shot
This is what people usually think of as "noir" fiction (see: Raymond Chandler and his PI/detective fiction). However not all hard-boiled fiction is, in fact noir.
*Interview with three female writers hardboiled/neo-noir fiction (including Abbott and Faust), wherein, interestingly enough, Faust herself thinks of her work as "more hardboiled than noir." http://www.bookslut.com/features/2008...

- pulp fiction, ex. Dare Me by Megan Abbott, again (same link: http://www.slate.com/articles/arts/bo...)
*What distinguishes pulp written by women, Ms. Tenzer said in an interview, "are images that counter the standard, conventional myths about American womanhood." (http://www.nytimes.com/2004/01/01/boo...) - from article on Feminist Press's "Femme Fatale" series:(http://www.feministpress.org/books/fp...

- melodrama
I can't think an example right now because when men write melodrama, it is still "noir". When women write melodrama, it's "soap opera" or "women's film." Anyway, think of James M. Cain's Mildred Pierce. Again thinking in the film realm, some of the popular "Gainsborough melodrama" pictures may qualify, specifically psychological-gothic-melodrama hybrid The Madonna of the Seven Moons, possibly antiheroine-melodrama The Wicked Lady.
*Talks about Madonna of the Seven Moons as film noir: https://books.google.com/books?id=U3C...
*More on the hybridity of British film noir: https://books.google.com/books?id=CTQ...

- horror and gothic
See note above. Rebecca is a primary example. Think The Night of the Hunter. Possibly V.C. Andrew's Flowers in the Attic?
*Gothic influences on film noir: https://books.google.com/books?id=ixV...
*More on the imposed gendered separation between gothic/noir films: As critics begun claiming some films as representatives of the invented genre Film Noir, the ones left out had to be fitted in another category. Some of these rejected films usually brought women as protagonists and thus a new classification started... Here is an example of how difficult genre separation can become, this group of films, which share similar style and context of production, are assessed as two distinct genres. The basic criteria formerly employed to justify this division lies in the gender of the protagonist. By definition we have that Gothic Noir/Female Gothic films present heroines in the central role while Film Noir movies bring men as protagonists. (http://studiesinfiction.blogspot.com/...)
*"Southern noir/grit lit": http://www.crimefictionlover.com/2013...

- Also retro-noir, which deals directly with images/tropes from noir films, such as the "femme fatale", ex. A Hell of a Woman: An Anthology of Female Noir e. Megan Abbott

4. I used mostly older examples because it's easier to find scholarship and that's how I roll. While I was googling I found this new Library of America series which includes many of the novels I already mentioned: Women Crime Writers: Four Suspense Novels of the 1940s & Women Crime Writers: Four Suspense Novels of the 1950s. It even has a site with more reading: http://womencrime.loa.org/

Gone Girl is basically the baseline for recent female noir. No one can agree on what to call it yet, but basically it seems to be something of a cross between the domestic thrillers and antiheroine genres already mentioned above:

-"Marriage thriller"
http://www.theguardian.com/books/book...

- "chick noir"
http://www.telegraph.co.uk/culture/bo...
http://www.independent.co.uk/arts-ent...

- "suburban noir"
http://www.knowledgelost.org/literatu...