In this paper un-binned statistical tools for analyzing the cosmic ray energy spectrum are developed and illustrated with a simulated data set. The methods are designed to extract accurate and precise model parameter estimators in the presence of statistical and systematic energy errors. Two robust methods are used to test for the presence of flux suppression at the highest energies: the Tail-P...

We describe an extension of the fixed-b approach introduced by Kiefer and Vogelsang (2005) to the empirical likelihood estimation framework. Under fixed-b asymptotics, the empirical likelihood ratio statistic evaluated at the true parameter converges to a nonstandard yet pivotal limiting distribution that can be approximated numerically. The impact of the bandwidth parameter and kernel choice i...

We show that the log likelihood ratio statistic factors in such a way that the expansion of its square root may be simply computed symbolically by machine. If the parameter of interest, , is scalar, the root of the log likelihood ratio statistic is simply expressed as the product of ^ ? times a scalar expression. The resulting product has a form that permits the symbolic computation of its mome...

Consider likelihood ratio test of a simple null hypothesis in a multiparameter exponential family. We study the asymptotic expansion of the null distribution of log likelihood ratio statistic to an arbitrary order. Bartlett correctability of the O(n ) term is well known. We show that higher order terms exhibit a similar simplicity. Moreover we give a combinatorially explicit expression for all ...

Data collected from response adaptive designs are dependent. Traditional statistical methods need to be justified for the use in response adaptive designs. This paper generalizes the Rao’s score test to response adaptive designs and introduces a generalized score statistic. Simulation is conducted to compare the statistical powers of the Wald, the score, the generalized score and the likelihood...

It is well known that likelihood ratio statistic is Bartlett correctable. We consider decomposition of a likelihood ratio statistic into 1 degree of freedom components based on sequence of nested hypotheses. We give a proof of the fact that the component likelihood ratio statistics are distributed mutually independently up to the order O(1=n) and each component is independently Bartlett correct...

The behavior of maximum likelihood estimates (MLE’s) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate n (slower than the usual √...

Existing methods for comparing the means of two independent skewed log-normal distributions do not perform well in a range of small-sample settings such as a small-sample bioavailability study. In this article, we propose two likelihood-based approaches-the signed log-likelihood ratio statistic and modified signed log-likelihood ratio statistic-for inference about the ratio of means of two inde...

In this paper, we consider the problem of testing the equality of multivariate normal populations when the data set has missing observations with a two-step monotone pattern. The likelihood ratio test (LRT) statistic for the simultaneous testing of the mean vectors and the covariance matrices is given under the condition of two-step monotone missing data. An approximate modified likelihood rati...

In this note we give a proof of the Wilks theorem for the empirical likelihood ratio for the right censored data, when the hypothesis are formulated in terms of p estimating equations or mean functions. In particular we show that the empirical likelihood ratio test statistic is equal to a quadratic form similar to a Hotelling’s T 2 statistic, plus a small error.