Approximations of the Distributions of Test Statistics for Homogeneity of a Product Multinomial Model

Statistics R ^a based on power divergence can be used for testing the homogeneity of a product multinomial model. All R ^a have the same chi-square limiting distribution under the null hypothesis of homogeneity. R ⁰ is the log likelihood ratio statistic and R ¹ is Pearson's X ² statistic. In this article, we consider improvement of approximation of the distribution of R ^a under the homogeneity hypothesis. The expression of the asymptotic expansion of distribution of R ^a under the homogeneity hypothesis is investigated. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, a new approximation of the distribution of R ^a is proposed. A moment-corrected type of chi-square approximation is also derived. By numerical comparison, we show that both of the approximations perform much better than that of usual chi-square approximation for the statistics R ^a when a ≤ 0, which include the log likelihood ratio statistic.     

A New Test Procedure for Decreasing Mean Residual Life

A new test statistic for testing the strict DMRL property of life distribution is developed. The asymptotic normality is established and the comparison between the test proposed and some other related ones in literature is conducted through evaluating the Pitman's asymptotic relative efficiency. Edge-worth expansion is also employed to improve the accuracy of the convergence rate of the test statistic. Some numerical results are presented as well to demonstrate the performance and the asymptotic normality of the new testing procedure.     

Edgeworth Expansion and Bootstrap Approximation for Studentized Product-Limit Estimator with Truncated and Censored Data

The Yule-Walker estimators of the AR coefficients of a causal multidimensional AR model are obtained by replacing the autocovariances with their estimators in the Yule-Walker equations. It is shown that only unbiased-type estimators of the autocovariances yield consistency of the Yule-Walker estimators. Also, the asymptotic joint distribution of the Yule-Walker estimators is presented.     

Edgeworth Expansion for Linear Regression Processes with Long-Memory Errors

This article provides an Edgeworth expansion for the distribution of the log-likelihood derivative LLD of the parameter of a time series generated by a linear regression model with Gaussian, stationary, and long-memory errors. Under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, an Edgeworth expansion of the density as well as the distribution function of a vector of centered and normalized derivatives of the plug-in log-likelihood PLL function of arbitrarily large order is established. This is done by extending the results of Lieberman et al. (2003 Lieberman , O. , Rousseau , J. , Zucker , D. M. ( 2003 ). Valid edgeworth expansions for the maximum likelihood estimator of the parameter of a stationary. gaussian, strongly dependent processes. it Ann. Statist. 31:586–612 . [Google Scholar]), who provided an Edgeworth expansion for the Gaussian stationary long-memory case, to our present model, which is a linear regression process with stationary Gaussian long-memory errors.     

Asymptotic Expansion of the Non Null Distribution of the Two-Sample t-Statistic Under Non Normality with Application in Power Comparison

This article studies the non null distribution of the two-sample t-statistic, or Welch statistic, under non normality. The asymptotic expansion of the non null distribution is derived up to n ^?1, where n is the pooled sample size, under general conditions. It is used to compare the power with that obtained by normal theory method. A simple technique is recommended to achieve more power through a monotone transformation in practice.     

Approximate Marginal Posterior Distributions Using Asymptotic Modes

This article gives asymptotic expansions for marginal posterior distributions with asymptotic modes of order n ^?2, and shows their validity. In addition, by using the asymptotic expansion, an approximate central posterior credible interval is derived.     

A Short-Cut Derivation for the Solution of Autoregressive Models from Sharp Algebraic Arguments

This article uses algebraic arguments to cast light on the solution of vector autoregressive models in the presence of unit roots. First, the linear case and then the multi-lag specification are investigated. Clear-cut representations of the model solutions are obtained, closed-form expressions of the coefficient matrices are provided, and integration features and cointegration mechanisms for stationarity recovery are elucidated.     

Second-Order Powers of a Class of Tests in the Presence of a Nuisance Parameter

The second-order local powers of a broad class of asymptotic chi-squared tests are considered in a composite case where both the parameter of interest and the nuisance parameter are possibly multidimensional for which no assumption has been made regarding global parametric orthogonality or curved exponentiality. The main result is that the second-order (point-by-point) local power identity holds if approximate third cumulants of a square-root version of the (modified) test statistic in the class vanish up to the second-order, which is an extension of Kakizawa (2010a Kakizawa , Y. ( 2010a ). Second-order power comparison of tests . Commun. Statist. Theor. Meth. 39 : 1424 – 1436 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the absence of the nuisance parameter. It is also shown that in the presence of the nuisance parameter, such a third cumulant condition does not always imply the second-order local unbiasedness of the resulting test. Then, the adjusted likelihood ratio test by Mukerjee (1993b Mukerjee , R. ( 1993b ). An extension of the conditional likelihood ratio test to the general multiparameter case . Ann. Inst. Statist. Math. 45 : 759 – 771 .[Crossref], [Web of Science ®] , [Google Scholar]) can be interpreted as the second-order local unbiased modification after applying the third cumulant condition.     

A Study of Expansions of Posterior Distributions

Johnson (1970 Johnson , R. ( 1970 ). Asymptotic expansions associated with posterior distributions . Ann. Math. Statist. 41 : 851 – 864 .[Crossref] , [Google Scholar]) obtained expansions for marginal posterior distributions through Taylor expansions. Here, the posterior expansion is expressed in terms of the likelihood and the prior together with their derivatives. Recently, Weng (2010 Weng , R. C. ( 2010 ). A Bayesian Edgeworth expansion by Stein's Identity . Bayesian Anal. 5 ( 4 ): 741 – 764 .[Crossref], [Web of Science ®] , [Google Scholar]) used a version of Stein's identity to derive a Bayesian Edgeworth expansion, expressed by posterior moments. Since the pivots used in these two articles are the same, it is of interest to compare these two expansions.

We found that our O(t ^?1/2) term agrees with Johnson's arithmetically, but the O(t ^?1) term does not. The simulations confirmed this finding and revealed that our O(t ^?1) term gives better performance than Johnson's.     

On the distribution of lrt for testing the equality of exponential distributions

In this paper, an asymptotic expansion of the distribution' of the likelihood ratio criterion for testing the equality of p one-parameter exponential distributions is obtained for unequal sample sizes. The expansion is obtained up to the order of n^-3 with the second term of the order of n^-2 so that the first term of this expansion alone should provide an excellent approximation to the distribution for moderately large values of n, where n is the combined sample size.