。
She insists that her analysis is correct.
Literal
She [topic-は] her own analysis [subject-が] correct [quotative-と] insists.
~と言い張る ('to insist stubbornly that X, to keep claiming X') combines the quotative と with 言い張る (literally 'speak-stretch' = to maintain a position despite pushback). The plain present 言い張る gives a habitual or characterizing reading — she's the kind of person who insists, or she's still insisting. 自分の分析 frames the analysis as her own, lending the insistence its 'this is mine and I'm not backing down' edge.