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Some Recent Results in the Asymptotic Theory of Statistical Estimation

One of the simplest results in asymptotic theory of estimation is the Hajek-Le Cam asymptotic minimax theorem. Besides being simple, it has many applications. We review the theorem and give brief indications on some applications. The theorem is called Hajek-Le Cam because it was proved by Hâjek (1972) for the asymptotically normal (more precisely LAN) case. There was a previous theorem by Le Cam (1953). Hajek's result was substantially extended in Le Cam (1979). Section 2 below gives a summary of definitions and notation. Section 3 reviews the asymptotic minimax theorem. Section 4 indicates how the theorem can be applied to problems recently studied by Donoho and Liu (1990), by M. Low (1989) and by Golubev and Nussbaum (1990). For further applications of the asymptotic minimax theorem, see Millar (1983).