Why did Mrs. Peckletide wish to kill a tiger?

As with most of Saki's short stories, the rationale behind the motivation and actions of many of his characters is often as shallow, empty, and senseless as the characters are themselves. 
Mrs. Peckletide is no exception. The only reason why she would want to commit the atrocious act of killing a tiger, which is particularly more shocking from the eye of the modern reader who is more aware of the rights of animals, is to keep up and compete with another woman, Loona Bimberton. 
It happens that Mrs. Bimberton went on a trip in an "aeroplane" with an Algerian aviator and would not be quiet about it. Hence, 

only a personally procured tiger-skin and a heavy harvest of Press photographs could successfully counter that sort of thing. 

Moreover, she had plans to brag about all of this in front of Bimberton. She would put the tiger as a rug in her house, offer a tea in Bimberton's honor, and even give Bimberton a tiger claw brooch on her birthday. In all, it was all for the sake of pride and to "one-up" Bimberton and her air excursion.

In a world that is supposed to be chiefly swayed by hunger and by love Mrs. Packletide was an exception; her movements and motives were largely governed by dislike of Loona Bimberton.
http://www.eastoftheweb.com/short-stories/UBooks/MrsPac.shtml