Maximum likelihood estimation for the endpoint and exponent of a distribution

Consider a random sample from a regularly varying distribution
function with a finite right endpoint parameter and an exponent parameter of
regular variation. The primary interest of the paper is to estimate both
the endpoint and the exponent. Since the distribution is semiparametric
and the endpoint and the exponent reveal asymptotic properties of the
right tail for the distribution, estimates for both parameters involve
only a few largest observations from the sample. The conventional maximum
likelihood method can be used to estimate both parameters, see
e.g., Hall (1982) for regular cases when the exponent is greater than 2, and Smith (1987),
Dress, Ferreira and de Haan (2004) and Peng and Qi (2009) for nonregular
cases when the exponent is between (1,2). The global maximum of the likelihood
function doesn't exist if one allows the exponent on (0,1], and a local
maximum exists with probability tending to one only if it is greater than 1.
Therefore, the maximum likelihood method fails when it is on (0,1]. In
this paper we propose a new likelihood method to estimate both parameters.
The use of the new method requires no prior information on the exponent,
the likelihood function is always bounded, and the estimates from this new
likelihood exist in all cases. We present the asymptotic distributions for
the estimates from the new method. Our simulation study shows that the
proposed method has better finite sample properties than the conventional
maximum likelihood method in regular cases.