Bias correction and improved residuals for non-exponential family nonlinear models

In this paper we derive general formulae for the biases to order n ^?1 of the parameter estimates in a general class of nonlinear regression models, where n is the sample size. The formulae are related to those of Cordeiro and McCullagh (1991) and Paula (1992) and may be viewed as extensions of their results, Correction factors are derived for the score and deviance component residuals in these models. The practical use of such corrections is illustrated for the log-gamma model.     

Improved maximum-likelihood estimation of the shape parameter in the Nakagami distribution

We develop and evaluate analytic and bootstrap bias-corrected maximum-likelihood estimators for the shape parameter in the Nakagami distribution. This distribution is widely used in a variety of disciplines, and the corresponding estimator of its scale parameter is trivially unbiased. We find that both ‘corrective’ and ‘preventive’ analytic approaches to eliminating the bias, to O(n ^?2), are equally, and extremely, effective and simple to implement. As a bonus, the sizeable reduction in bias comes with a small reduction in the mean-squared error. Overall, we prefer analytic bias corrections in the case of this estimator. This preference is based on the relative computational costs and the magnitudes of the bias reductions that can be achieved in each case. Our results are illustrated with two real-data applications, including the one which provides the first application of the Nakagami distribution to data for ocean wave heights.     

Improved Score Tests in Symmetric Linear Regression Models

The class of symmetric linear regression models has the normal linear regression model as a special case and includes several models that assume that the errors follow a symmetric distribution with longer-than-normal tails. An important member of this class is the t linear regression model, which is commonly used as an alternative to the usual normal regression model when the data contain extreme or outlying observations. In this article, we develop second-order asymptotic theory for score tests in this class of models. We obtain Bartlett-corrected score statistics for testing hypotheses on the regression and the dispersion parameters. The corrected statistics have chi-squared distributions with errors of order O(n ^?3/2), n being the sample size. The corrections represent an improvement over the corresponding original Rao's score statistics, which are chi-squared distributed up to errors of order O(n ^?1). Simulation results show that the corrected score tests perform much better than their uncorrected counterparts in samples of small or moderate size.     

Generalized bartlett correction

This paper provides a general method of modifying a statistic of interest in such a way that the distribution of the modified statistic can be approximated by an arbitrary reference distribution to an order of accuracy of O(n ^-1/2) or even O(n ^-1). The reference distribution is usually the asymptotic distribution of the original statistic. We prove that the multiplication of the statistic by a suitable stochastic correction improves the asymptotic approximation to its distribution. This paper extends the results of the closely related paper by Cordeiro and Ferrari (1991) to cope with several other statistical tests. The resulting expression for the adjustment factor requires knowledge of the Edgeworth-type expansion to order O(n^-1) for the distribution of the unmodified statistic. In practice its functional form involves some derivatives of the reference distribution. Certain difference between the cumulants of appropriate order in n of the unmodified statistic and those of its first-order approximation, and the unmodified statistic itself. Some applications are discussed.     

BARTLETT-TYPE CORRECTIONS FOR TWO-PARAMETER EXPONENTIAL FAMILY MODELS

ABSTRACT

Cordeiro and Ferrari[1] Cordeiro, G.M. and Ferrari, S.L.P. 1991. A Modified Score Test Statistic Having Chi-Squared Distribution to Order n^?1. Biometrika, 78: 573–582. [Web of Science ®] , [Google Scholar] obtained a Bartlett-type correction to the score statistic that is given by a polynomial of second degree in the statistic itself with coefficients that depend on cumulants of log–likelihood derivatives. Although the corrected statistic has good size properties, it is not always a monotone transformation of the original statistic. Recently, some monotone transformations of the score statistic have been proposed as alternatives to the polynomial transformation. In this paper we derive simple formulae for various modified score statistics for testing a scalar parameter in two-parameter exponential models which do not require knowledge of cumulants. The formulae are readily applicable to cover many important and commonly used distributions and involve only trivial operations on certain functions and their derivatives.     

Modified p-Value of Two-Sided Test for Normal Distribution with Restricted Parameter Space

This article proposes a modified p-value for the two-sided test of the location of the normal distribution when the parameter space is restricted. A commonly used test for the two-sided test of the normal distribution is the uniformly most powerful unbiased (UMPU) test, which is also the likelihood ratio test. The p-value of the test is used as evidence against the null hypothesis. Note that the usual p-value does not depend on the parameter space but only on the observation and the assumption of the null hypothesis. When the parameter space is known to be restricted, the usual p-value cannot sufficiently utilize this information to make a more accurate decision. In this paper, a modified p-value (also called the rp-value) dependent on the parameter space is proposed, and the test derived from the modified p-value is also shown to be the UMPU test.     

C472. The expected value of a conditional variance: An upper bound

The article derives Bartlett corrections for improving the chi-square approximation to the likelihood ratio statistics in a class of symmetric nonlinear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. In this paper we present, in matrix notation, Bartlett corrections to likelihood ratio statistics in nonlinear regression models with errors that follow a symmetric distribution. We generalize the results obtained by Ferrari, S. L. P. and Arellano-Valle, R. B. (1996). Modified likelihood ratio and score tests in linear regression models using the t distribution. Braz. J. Prob. Statist., 10, 15–33, who considered a t distribution for the errors, and by Ferrari, S. L. P. and Uribe-Opazo, M. A. (2001). Corrected likelihood ratio tests in a class of symmetric linear regression models. Braz. J. Prob. Statist., 15, 49–67, who considered a symmetric linear regression model. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions.     

Corrected asymptotic distribution of statistics based on the multinomial law

Investigators and epidemiologists often use statistics based on the parameters of a multinomial distribution. Two main approaches have been developed to assess the inferences of these statistics. The first one uses asymptotic formulae which are valid for large sample sizes. The second one computes the exact distribution, which performs quite well for small samples. They present some limitations for sample sizes N neither large enough to satisfy the assumption of asymptotic normality nor small enough to allow us to generate the exact distribution. We analytically computed the 1/N corrections of the asymptotic distribution for any statistics based on a multinomial law. We applied these results to the kappa statistic in 2×2 and 3×3 tables. We also compared the coverage probability obtained with the asymptotic and the corrected distributions under various hypothetical configurations of sample size and theoretical proportions. With this method, the estimate of the mean and the variance were highly improved as well as the 2.5 and the 97.5 percentiles of the distribution, allowing us to go down to sample sizes around 20, for data sets not too asymmetrical. The order of the difference between the exact and the corrected values was 1/N² for the mean and 1/N³ for the variance.     

A note on bias reduction of maximum likelihood estimates for the scalar skew t distribution

The skew t distribution is a flexible parametric family to fit data, because it includes parameters that let us regulate skewness and kurtosis. A problem with this distribution is that, for moderate sample sizes, the maximum likelihood estimator of the shape parameter is infinite with positive probability. In order to try to solve this problem, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. In this note we prove that the resulting modified maximum likelihood estimator is always finite, considering the degrees of freedom as known and greater than or equal to 2.     

The Pearson Score Statistic for Multinomial-Poisson Models

The score statistic S² is commonly used for general likelihood-based inference. Pearson’s Chi-squared statistic X² = ∑(O ? E)²/E is ubiquitous in contingency table inference. Because tests and confidence intervals based on S² have been shown to work well in practice and theory and because X² has such a simple and intuitively appealing form, it is of interest to know when S² is identical to X² and when X² has an approximate Chi-squared distribution. Toward these ends, this paper gives a simple proof that S² = X² for the broad class of multinomial-Poisson distributions when the alternative hypothesis is unrestricted in a certain sense. This paper also gives a sufficient condition under which the null distribution of the Pearson score statistic is approximately Chi-squared. Several examples illustrate the utility of the results and counter-examples highlight the importance of the sufficient conditions of the results.     

A note on LeCam’s bound for the distance between the Poisson binomial and the Poisson distribution

n possibly different success probabilities p ₁, p ₂, ..., p _n is frequently approximated by a Poisson distribution with parameter λ = p ₁ + p ₂ + ... + p _n . LeCam's bound p ² ₁ + p ² ₂ + ... + p _n ² for the total variation distance between both distributions is particularly useful provided the success probabilities are small. The paper presents an improved version of LeCam's bound if a generalized d-dimensional Poisson binomial distribution is to be approximated by a compound Poisson distribution. Received: May 10, 2000; revised version: January 15, 2001     

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order O_p(T^-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed.     

Three Bartlett-type corrections for score statistics in symmetric nonlinear regression models

We present simple matrix formulae for corrected score statistics in symmetric nonlinear regression models. The corrected score statistics follow more closely a χ ² distribution than the classical score statistic. Our simulation results indicate that the corrected score tests display smaller size distortions than the original score test. We also compare the sizes and the powers of the corrected score tests with bootstrap-based score tests.     

A Bayesian analysis for the Wilcoxon signed-rank statistic

A Bayesian analysis is provided for the Wilcoxon signed-rank statistic (T⁺). The Bayesian analysis is based on a sign-bias parameter φ on the (0, 1) interval. For the case of a uniform prior probability distribution for φ and for small sample sizes (i.e., 6 ? n ? 25), values for the statistic T⁺ are computed that enable probabilistic statements about φ. For larger sample sizes, approximations are provided for the asymptotic likelihood function P(T⁺|φ) as well as for the posterior distribution P(φ|T⁺). Power analyses are examined both for properly specified Gaussian sampling and for misspecified non Gaussian models. The new Bayesian metric has high power efficiency in the range of 0.9–1 relative to a standard t test when there is Gaussian sampling. But if the sampling is from an unknown and misspecified distribution, then the new statistic still has high power; in some cases, the power can be higher than the t test (especially for probability mixtures and heavy-tailed distributions). The new Bayesian analysis is thus a useful and robust method for applications where the usual parametric assumptions are questionable. These properties further enable a way to do a generic Bayesian analysis for many non Gaussian distributions that currently lack a formal Bayesian model.     

Some Second-Order Asymptotics for Extreme Value Linear Regression Models

ABSTRACT

In this article we derive finite-sample corrections in matrix notation for likelihood ratio and score statistics in extreme-value linear regression models. We consider three corrected score tests that perform better than the usual score test. We also derive general formulae for second-order biases of maximum likelihood estimates of the linear parameters. Some simulations are performed to compare the likelihood ratio and score statistics with their modified versions and to illustrate the bias correction.     

Marginal information for expectation parameters

The authors consider a special case of inference in the presence of nuisance parameters. They show that when the orthogonalized score function is a function of a statistic S, no Fisher information for the interest parameter is lost by using the marginal distribution of S rather than the full distribution of the observations. Therefore, no information for the interest parameter is recovered by conditioning on an ancillary statistic, and information will be lost by conditioning on an approximate ancillary statistic. This is the case for regular multivariate exponential families when the interest parameter is a subvector of the expectation parameter and the statistic is the maximum likelihood estimate of the interest parameter. Several examples are considered, including the 2 × 2 table.     

Statistical inference on the drift parameter in fractional Brownian motion with a deterministic drift

In statistical inference on the drift parameter a in the fractional Brownian motion W^H_t with the Hurst parameter H ∈ (0, 1) with a constant drift Y^H_t = at + W^H_t, there is a large number of options how to do it. We may, for example, base this inference on the properties of the standard normal distribution applied to the differences between the observed values of the process at discrete times. Although such methods are very simple, it turns out that more appropriate is to use inverse methods. Such methods can be generalized to non constant drift. For the hypotheses testing about the drift parameter a, it is more proper to standardize the observed process, and to use inverse methods based on the first exit time of the observed process of a pre-specified interval until some given time. These procedures are illustrated, and their times of decision are compared against the direct approach. Other generalizations are possible when the random part is a symmetric stochastic integral of a known, deterministic function with respect to fractional Brownian motion.