Article

Simple relaxed conditional likelihood

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

When the data are sparse but not exceedingly so, we face a trade-off between bias and precision that makes the usual choice between conducting either a fully unconditional inference or a fully conditional inference unduly restrictive. We propose a method to relax the conditional inference that relies upon commonly available computer outputs. In the rectangular array asymptotic setting, the relaxed conditional maximum likelihood estimator has smaller bias than the unconditional estimator and smaller mean square error than the conditional estimator.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... Yet Bartolucci and Pigini (2019) show that the finite sample performance of the resulting APE estimator is superior to that of the panel jackknife with short-T , while the two estimators are comparable with moderately long panels. 8 The result is a special case of Theorem 1 by Hanfelt and Wang (2014), who extend the results in Hahn and Newey (2004) to derive asymptotic properties for a general class of estimators based on what they call a "relaxed" conditional likelihood. The CML estimator emerges as a special case when, as in our case, responses are distributed according to the regular exponential family. ...
Article
Full-text available
We propose a multiple-step procedure to compute average partial effects (APEs) for fixed-effects static and dynamic logit models estimated by (pseudo) conditional maximum likelihood. As individual effects are eliminated by conditioning on suitable sufficient statistics, we propose evaluating the APEs at the maximum likelihood estimates for the unobserved heterogeneity, along with the fixed- T consistent estimator of the slope parameters, and then reducing the induced bias in the APEs by an analytical correction. The proposed estimator has bias of order O(T2)O(T^{-2}) O ( T - 2 ) , it performs well in finite samples and, when the dynamic logit model is considered, better than alternative plug-in strategies based on bias-corrected estimates for the slopes, especially in panels with short T . We provide a real data application based on labour supply of married women.
Article
We consider inference for a scalar parameter Ψ in the presence of one or more nuisance parameters. The nuisance parameters are required to be orthogonal to the parameter of interest, and the construction and interpretation of orthogonalized parameters is discussed in some detail. For purposes of inference we propose a likelihood ratio statistic constructed from the conditional distribution of the observations, given maximum likelihood estimates for the nuisance parameters. We consider to what extent this is preferable to the profile likelihood ratio statistic in which the likelihood function is maximized over the nuisance parameters. There are close connections to the modified profile likelihood of Barndorff‐Nielsen (1983). The normal transformation model of Box and Cox (1964) is discussed as an illustration.
Article
This paper extends the projected score methods of C. G. Small and D. L. McLeish [ibid. 76, No. 4, 693-703 (1989; Zbl 0681.62008)]. It is shown that the conditional score function may be approximated, with arbitrarily small stochastic error, in terms of a natural basis for the space of centred likelihood ratios. The utility of using this basis is established by identifying a U-statistic representation theorem and a class of expectation identities for the basis elements, making higher order asymptotics more tractable. The results are applied to a canonical exponential family model, where it is shown that the projected scores with estimated nuisance parameters can provide an accurate approximation to the conditional score function.
Article
This paper examines statistical methods based upon estimating functions, i.e. functions of both the parameter and data that are designed to permit inference about an unknown parameter in a statistical model. We explore reductions of such estimating functions by projection. This reduction, analogous to the process of Rao–Blackwellization, may be used either to increase the power of a test, the efficiency of a point estimator, or alternatively to render an inference function insensitive to the value of a nuisance parameter. In the case where a complete sufficient statistic exists for a parameter of interest the methods reduce to increasing sensitivity through Rao–Blackwellization. When this same parameter is regarded as a nuisance parameter, the techniques lead us to condition on the complete sufficient statistic for this parameter. However the techniques are seen to be more widely applicable than for models permitting reduction through complete sufficiency. Examples involving mixture models will be developed.
Article
This paper concerns the efficiency of the conditional likelihood method for inference in models which include nuisance parameters. A new concept of ancillarity, asymptotic weak ancillarity, is introduced. It is shown that the conditional maximum likelihood estimator and the conditional score test of θ, the parameter of interest, are asymptotically equivalent to their unconditional counterparts, and hence are asymptotically efficient, provided that the conditioning statistic is asymptotically weakly ancillary. The key assumption that the conditioning statistic is asymptotically weakly ancillary is verified when the underlying distribution is from exponential families. Some illustrative examples are given.
Article
The conditional score function has previously been shown to generate the optimal estimating equation for a parameter of interest when the conditioning statistic is complete and sufficient for the nuisance parameters (Godambe, 1976). The present paper generalizes these results to partial likelihood factorizations and then examines the nature of the problem when the appropriate conditioning statistics depend on the parameter of interest. In this case, globally optimal estimating functions are impossible. A weaker criterion of optimal weighting leads to a class of estimated conditional score functions which satisfy an information equality.
Article
SUMMARY Godambe (1976) put forward two concepts of ancillarity in the presenmce of nuisance paremeters. In this paper they areunified and extended concept of Fisher information.
Article
Cox & Reid (1987) proposed the technique of orthogonalizing parameters, to deal with the general problem of nuisance parameters, within fully parametric models. They obtained a large-sample approximation to the conditional likelihood. Along the same lines Davison (1988) studied generalized linear models. In the present paper we deal with the problem of nuisance parameters, within a semiparametric setup which includes the class of distributions associated with generalized linear models. The technique used is that of optimum orthogonal estimating functions (Godambe & Thompson, 1989). The results are related to those of Cox & Reid (1987).
Article
In settings where the full probability model is not specified, consider a general estimating function g(&thgr;, &lgr;; y) that involves not only the parameters of interest, &thgr;, but also some nuisance parameters, &lgr;. We consider methods for reducing the effects on g of fitting nuisance parameters. We propose Cox--Reid-type adjustment to the profile estimating function, g(&thgr;, &lgr;ˆ-sub-&thgr;; y), that reduces its bias by two orders. Typically, only the first two moments of the response variable are needed to form the adjustment. Important applications of this method include the estimation of the pairwise association and main effects in stratified, clustered data and estimation of the main effects in a matched pair study. A brief simulation study shows that the proposed method considerably reduces the impact of the nuisance parameters. Copyright Biometrika Trust 2003, Oxford University Press.
Article
It is well known, at least through many examples, that when there are many nuisance parameters modified profile likelihoods often perform much better than the profile likelihood. Ordinary asymptotics almost totally fail to deal with this issue. For this reason, we study asymptotic properties of the profile and modified profile likelihoods in models for stratified data in a two-index asymptotics setting. This means that both the sample size of the strata, m, and the dimension of the nuisance parameter, q, may increase to infinity. It is shown that in this asymptotic setting modified profile likelihoods give improvements, with respect to the profile likelihood, in terms of consistency of estimators and of asymptotic distributional properties. In particular, the modified profile likelihood based statistics have the usual asymptotic distribution, provided that 1/m = o(q-super- - 1/3), while the analogous condition for the profile likelihood is 1/m = o(q-super- - 1). Copyright Biometrika Trust 2003, Oxford University Press.
Article
A conditional method is presented that renders an estimating function insensitive to nuisance parameters. The approach is a generalisation of the conditional score method to a general estimating function context and does not require complete specification of the probability model. We exploit the informal relationship between general estimating functions and score functions to derive simple generalisations of sufficient and partially ancillary statistics, referred to as G-sufficient and G-ancillary statistics, respectively. These two types of statistic are defined in a manner that does not require complete knowledge of the probability model and thus are more suitable for use with estimating functions. If we condition on a G-sufficient statistic for the nuisance parameters, the resulting conditional estimating function is insensitive to nuisance parameters and in particular achieves the plug-in unbiasedness property. Furthermore, if the conditioning argument is also G-ancillary for the parameters of interest, then the conditional estimating function possesses an attractive optimality property. Copyright Biometrika Trust 2003, Oxford University Press.
Article
In a parametric model the maximum likelihood estimator of a parameter of interest &psgr; may be viewed as the solution to the equation l′-sub-p(&psgr;) &equals; 0, where l-sub-p denotes the profile <?Pub Caret>loglikelihood function. It is well known that the estimating function l′-sub-p(&psgr;) is not unbiased and that this bias can, in some cases, lead to poor estimates of &psgr;. An alternative approach is to use the modified profile likelihood function, or an approximation to the modified profile likelihood function, which yields an estimating function that is approximately unbiased. In many cases, the maximum likelihood estimating functions are unbiased under more general assumptions than those used to construct the likelihood function, for example under first- or second-moment conditions. Although the likelihood function itself may provide valid estimates under moment conditions alone, the modified profile likelihood requires a full parametric model. In this paper, modifications to l′-sub-p(&psgr;) are presented that yield an approximately unbiased estimating function under more general conditions. Copyright Biometrika Trust 2002, Oxford University Press.
Article
The paper considers a rectangular array asymptotic embedding for multistratum data sets, in which both the number of strata and the number of within-stratum replications increase, and at the same rate. It is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non-zero mean in its asymptotic normal distribution. By using a projection operator on the score function, an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the Cramér-Rao lower bound. The adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator. Copyright 2003 Royal Statistical Society.
Article
Fixed effects estimators of panel models can be severely biased because of the well-known incidental parameters problem. We show that this bias can be reduced by using a panel jackknife or an analytical bias correction motivated by large T. We give bias corrections for averages over the fixed effects, as well as model parameters. We find large bias reductions from using these approaches in examples. We consider asymptotics where T grows with n, as an approximation to the properties of the estimators in econometric applications. We show that if T grows at the same rate as n, the fixed effects estimator is asymptotically biased, so that asymptotic confidence intervals are incorrect, but that they are correct for the panel jackknife. We show T growing faster than n-super-1/3 suffices for correctness of the analytic correction, a property we also conjecture for the jackknife. Copyright The Econometric Society 2004.
  • BARTLETT
  • LIANG
  • LINDSAY