when I try to install it can't install Longman Dictionary.
How are you trying to install it, and what happens when you try?
Maybe Apple can remove an app from AppStore but, it must be possible to download it from 'cloud'. LDOCE has been removed not only from AppStore but also from iCloud.
No. If it's removed from the App Store, it's gone. Only if you have a backup somewhere can you get it back. iCloud is not separate from the App Store when it comes to redownloading purchased content.
LDOCE app had released by Enfour Inc. not by Pearson.
The LDOCE and other Longman dictionaries are owned by Pearson, as you'll see if you look at the other apps in the App Store. Enfour may have done the development for Pearson and partnered with Enfour, or the title may have been released under the Enfor name in some countries.
Regardless, whoever owns the app has pulled it from the App Store. If you don't have a working backup copy, then your only recourse will be to contact Pearson/Enfour and ask if the app will ever be returning.
Regards.