The exact confidence interval for the scale parameter and the mvue of the laplace distribution

The exact distribution of the sample median, and of the maximum likelihood estimator of the scale parameter of the Laplace distribution is derived. Tables of Teans, variances and the distribution functions of the corresponding dislributions are evaluacted. Exact ,solutions to the problem of confidence interval and hypothesrs testing for the scale paramrter are provided. The minimum variance unbiased estimator (MVUE) of the p.d.f. of the Laplace distribution when the location parameter is known is also given.     

The likelihood ratio test foe the homogeneity of the variances in a covariance matrix with block compound symmetry

For the problem of testing the homogeneity of the variances in a covariance matrix with a block compound symmetric structure, the likelihood ratio test is derived in this paper, A modification of the test that allows its distribution to be better approximated by the chi-square distribution is also considered, Formulae for calculating approximate sample size and power are derived, Small sample performances of these tests in the case of two dependent bivariate or trivariate normals are compared to each other and to the competing tests by simulating levels of significance and powers, and recommendation is made of the ones that have good performance, The recommended tests are then demonstrated in an illustrative example.     

Estimation of the multi-criteria worths of the alternatives in a hierarchical structure of comparisons

The aggregated worths of the alternatives, when compared with respect to several criteria, are estimated in a hierarchical comparisons model introduced by Saaty (1980). A multiplicative model is used for the paired comparisons data which are collected in a ratio scale in this set-up in any level of this hierarchy. An iterative scheme is found for the maximum likelihood estimation of the worth parameters in this multiplicative model. The iterative values are shown to be convergent monotonically to the estimates. We also obtain the asymptotic dispersion matrix of the maximum likelihood estimates of the relative worths of the alternatives according to a single criterion as well as those according to the over-all suitability when compared under several criteria. A numerical example is presented to illustrate the method developed in this paper. Simulation techniques are employed to find the average number of iterations required for the convergence of the above iterative scheme.     

Detecting a change in variances

The likelihood ratio test for testing for a change in a sequence of variances of normal populations is derived. The alternative hypothesis considered is of a one-sided nature. For the test, the conservativeness of the Sidak bound is shown and the asymptotic version of the Sidak bound is also constructed. These bound are compared with the Bonferroni bound and the Worsley bound, using the Monte Carlo method. Finally Hsu's data of stock market returns is reanalysed, using the test.     

The left-truncated Weibull distribution: theory and computation

This paper studies the two-parameter, left-truncated Weibull distribution (LTWD) with known, fixed, positive truncation pointT. Important hitherto unknown statistical properties of the LTWD are derived. The asymptotic theory of the maximum likelihood estimates (MLEs) is invoked to develop parameter confidence intervals and regions. Numerical methods are described for computing the MLEs and for evaluating the exact, asymptotic variances and covariances of the MLEs. An illustrative example is given.     

Computational algorithms for the factor model

Algorithms for computing the maximum likelihood estimators and the estimated covariance matrix of the estimators of the factor model are derived. The algorithms are particularly suitable for large matrices and for samples that give zero estimates of some error variances. A method of constructing estimators for reduced models is presented. The algorithms can also be used for the multivariate errors-in-variables model with known error covariance matrix.     

Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood

Likelihood ratio tests for fixed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation. Bartlett-type adjustments, using an approximate decomposition of the data, are developed for the test statistics. A simulation study is used to compare properties of the test statistics proposed, with or without adjustment, with a Wald test. A proposed test statistic constructed by dropping fixed terms from the full fixed model is shown to give a better approximation to the asymptotic χ²-distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the χ²-approximation for the proposed tests substantially.     

Testing equality of population variances the robust way

The purpose of this paper is to investigate the robustness (stability of Type I error to deviations from normality) and power properties of various tests for testing equality of population variances. It is shown that the tests based on Tiku’ s (1967, 1980, 1982) MML estimators have good robustness properties and are the most powerful overall.     

Behrens-fisher problem under the assumption of homogeneous coefficients of variation

The powers of the likelihood ratio (LR) test and an “asymptotically (in some sense) optimum” invariant test are examined and compared by simulation techniques with those of several other relevant tests for the problem of testing the equality of two univariate normal population means under the assumption of heterogeneous variances but homogeneous coefficients of variation. It is seen that the LR test is highly satisfactory for all values of the coefficient of variation and the “asymptotically optimum” invariant test, which is computationally much simpler than the LR test, is a reasonably good competitor for cases where the value of the coefficient of variation is greater than or equal to 3. Also, a     

Comparing Two Population Means and Variances: A Parametric Robust Way

Abstract

This article introduces a parametric robust way of comparing two population means and two population variances. With large samples the comparison of two means, under model misspecification, is lesser a problem, for, the validity of inference is protected by the central limit theorem. However, the assumption of normality is generally required, so that the inference for the ratio of two variances can be carried out by the familiar F statistic. A parametric robust approach that is insensitive to the distributional assumption will be proposed here. More specifically, it will be demonstrated that the normal likelihood function can be adjusted for asymptotically valid inferences for all underlying distributions with finite fourth moments. The normal likelihood function, on the other hand, is itself robust for the comparison of two means so that no adjustment is needed.     

On the Convergence of the Monte Carlo Maximum Likelihood Method for Latent Variable Models

While much used in practice, latent variable models raise challenging estimation problems due to the intractability of their likelihood. Monte Carlo maximum likelihood (MCML), as proposed by Geyer & Thompson (1992 ), is a simulation-based approach to maximum likelihood approximation applicable to general latent variable models. MCML can be described as an importance sampling method in which the likelihood ratio is approximated by Monte Carlo averages of importance ratios simulated from the complete data model corresponding to an arbitrary value of the unknown parameter. This paper studies the asymptotic (in the number of observations) performance of the MCML method in the case of latent variable models with independent observations. This is in contrast with previous works on the same topic which only considered conditional convergence to the maximum likelihood estimator, for a fixed set of observations. A first important result is that when is fixed, the MCML method can only be consistent if the number of simulations grows exponentially fast with the number of observations. If on the other hand, is obtained from a consistent sequence of estimates of the unknown parameter, then the requirements on the number of simulations are shown to be much weaker.     

A comparison of multivariate signal discrimination techniques

This paper investigates several techniques to discriminate two multivariate stationary signals. The methods considered include Gaussian likelihood ratio tests for variance equality, a chi-squared time-domain test, and a spectral-based test. The latter two tests assess equality of the multivariate autocovariance function of the two signals over many different lags. The Gaussian likelihood ratio test is perhaps best viewed as principal component analyses (PCA) without dimension reduction aspects; it can be modified to consider covariance features other than variances via dimension augmentation tactics. A simulation study is constructed that shows how one can make inappropriate conclusions with PCA tests, even when dimension augmentation techniques are used to incorporate non-zero lag autocovariances into the analysis. The various discrimination methods are first discussed. A simulation study then illuminates the various properties of the methods. In this pursuit, calculations are needed to identify several multivariate time series models with specific autocovariance properties. To demonstrate the applicability of the methods, nine US and Canadian weather stations from three distinct regions are clustered. Here, the spectral clustering perfectly identified distinct regions, the chi-squared test performed marginally, and the PCA/likelihood ratio method did not perform well.     

Estimates for mean and dispersion effects in unreplicated factorial designs

One important goal of experimentation in quality improvement is to minimize the variability of a product or process around a target mean value. Factors which affect variances as well as factors that affect the mean can be identified using the analysis of mean and dispersion. Box and Meyer (1986b) proposed a method of model identification and maximum likelihood estimation for mean and dispersion effects from unreplicated designs. In this article, we address two problems associated with MLE’s. First, asymptotic variance of MLE's for dispersion effects which can be used to judge the significance of factors can be misleading. A possible explanation is provided; simulation results also indicate that the asymptotic, variance underestimates.     

Two-Step Residual-Based Estimation of Error Variances for Generalized Least Squares in Split-Plot Experiments

In split-plot experiments, estimation of unknown parameters by generalized least squares (GLS), as opposed to ordinary least squares (OLS), is required, owing to the existence of whole- and subplot errors. However, estimating the error variances is often necessary for GLS. Restricted maximum likelihood (REML) is an established method for estimating the error variances, and its benefits have been highlighted in many previous studies. This article proposes a new two-step residual-based approach for estimating error variances. Results of numerical simulations indicate that the proposed method performs sufficiently well to be considered as a suitable alternative to REML.     

MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR MODELS WITH EQUI-CORRELATED RANDOM ERRORS

Necessary and sufficient conditions for the existence of maximum likelihood estimators of unknown parameters in linear models with equi‐correlated random errors are presented. The basic technique we use is that these models are, first, orthogonally transformed into linear models with two variances, and then the maximum likelihood estimation problem is solved in the environment of transformed models. Our results generalize a result of Arnold, S. F. (1981) [The theory of linear models and multivariate analysis. Wiley, New York]. In addition, we give necessary and sufficient conditions for the existence of restricted maximum likelihood estimators of the parameters. The results of Birkes, D. & Wulff, S. (2003) [Existence of maximum likelihood estimates in normal variance‐components models. J Statist Plann. Inference. 113 , 35–47] are compared with our results and differences are pointed out.     

Approximations to the powers of the likelihood ratio tests: the loop ordering and slippage alternatives

Approximations to the power functions of the likelihood ratio tests of homogeneity of normal means against the simple loop ordering at slippage alternatives are considered. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a simple loop ordering is appropriate. The accuracy of the several moment approximations are studied for the case of known variances and it is found that for powers in the range typically of interest, the two-moment approximation seems quite adequate. Approximations based on mixtures of noncentral F variables are developed for the case of unknown variances. The critical values of the test statistics are also tabulated for selected levels of significance.     

Maximum likelihood estimation for the negative multinomial log-linear model

A log-linear model is defined for multiway contingency tables with negative multinomial frequency counts. The maximum likelihood estimator of the model parameters and the estimator covariance matrix is given. The likelihood ratio test for the general log-linear hypothesis also is presented.     

Semiparametric Inference for the Two‐way Layout Under Order Restrictions

There are many situations in which a researcher would like to analyse data from a two‐way layout. Often, the assumptions of linearity and normality may not hold. To address such situations, we introduce a semiparametric model. The model extends the well‐known density ratio model from the one‐way to the two‐way layout and provides a useful framework for semiparametric analysis of variance type problems under order restrictions. In particular, the likelihood ratio order is emphasized. The model enables highly efficient inference without resorting to fully parametric assumptions or the use of transformations. Estimation and testing procedures under order restrictions are developed and investigated in detail. It is shown that the model is robust to misspecification, and several simulations suggest that it performs well in practice. The methodology is illustrated using two data examples; in the first, the response variable is discrete, whereas in the second, it is continuous.     

Testing homoosenetty of t normal means against ordered alternatives in 2 groups having uncommon and unknown variance

Given two series of normal populations with variances unknown and common within the series but possibly differ between the series, the likelihood ratio test of simultaneous homogeneity of means against sinpie order alternative is considered. Sole approxirations Par the null distribution of thetest statistic are also proposed.     

Odds ratio inference from stratified samples

One of the common problems encountered in applied statistics is that of comparing two proportions from stratified samples. One approach to this problem is via inference on the corresponding odds ratio. In this paper, the various point and interval estimators of and hypothesis testing procedures for a common odds ratio from multiple 2 ×2 tables are reviewed. Based On research to date, the conditional maximum likelihood and Mantel-Haenszel estimators are recommended as the point estimators of choice. Neither confidence intervals nor hypothesis testing metthods have been studied as well as the point estimators, but there is a confidence interval method associated with the Mantel-Haenszel estimator that is a good choice.