M. L. Tiku

Robustness of MML estimators based on censored samples and robust test statistics

Abstract We investigate the efficiences of Tikus (1967) modified maximum likelihood estimators μc and σc (based on symmetrically censored normal samples) for estimating the location and scale parameters μ and σ of symmetric non-normal distributions. We show that μc and σc are jointly more efficient than x and s for long-tailed distributions (kurtosis β 2 ∗ = μ 4 μ 2 2 >4.2, β 2 ∗ = 4.2 for the Logistic), and always more efficient than the trimmed mean μT and the matching sample estimate σT of σ. We also show that μc and σc are jointly at least as efficient as some of the more prominent “robust” estimators (Gross, 1976). We show that the statistic t c = μ c m σ c , m = n −2r + 2rβ (r is the number of observations censored on each side of the sample and β is a constant), is robust and powerful for testing an assumed value of μ. We define a statistic Tc (based on μc andσc) for testing that two symmetric distributions are identical and show that Tc is robust and generally more poweerful than the well-known nonparametric statistics (Wilcoxon, normal-score, Kolmogorov-Smirnov), against the important location-shift alternatives. We generalize the statistic Tc to test that k symmetric distibutions are identical. The asymptotic distributions of tc and Tc are normal, under some very general regularity conditions. For small samples, the upper (lower) percentage points of tc and Tc are shown to be closely approximated by Students t-distributions. Besides, the statistics μc and σc (and hence tc and Tc) are explicit and simple functions of sample observations and are easy to compute.

A new method of estimation for location and scale parameters

Abstract A new method of estimating the location and scale parameters is presented. The resulting estimators are shown to be remarkably efficient. To compute these estimators, only the expected values of order statistics are needed.

Tables of the Power of the F-Test

Abstract The values of the power of the F-test corresponding to the degrees of freedom (f 1, f 2) and non-centrality parameter λ are tabulated for Type I error α =0.005, 0.01, 0.025, 0.05, for the following values of the parameters: and

Time Series Models in Non‐Normal Situations: Symmetric Innovations

We consider AR(q) models in time series with non‐normal innovations represented by a member of a wide family of symmetric distributions (Students t). Since the ML (maximum likelihood) estimators are intractable, we derive the MML (modified maximum likelihood) estimators of the parameters and show that they are remarkably efficient. We use these estimators for hypothesis testing, and show that the resulting tests are robust and powerful.

Power Function of the F-Test Under Non-Normal Situations

Abstract The values of the power of the F-test employed in analysis-of-variance are calculated under non-normal situations and compared with the normal-theory values of the power. Non-normality seems to have little effect on the power of the F-test.

Multiple Linear Regression Model Under Nonnormality

Abstract We consider multiple linear regression models under nonnormality. We derive modified maximum likelihood estimators (MMLEs) of the parameters and show that they are efficient and robust. We show that the least squares esimators are considerably less efficient. We compare the efficiencies of the MMLEs and the M estimators for symmetric distributions and show that, for plausible alternatives to an assumed distribution, the former are more efficient. We provide real-life examples.

Estimating parameters in autoregressive models in non-normal situations: symmetric innovations

The estimation of coe±cients in a simple regression model with autocorrelated errors is considered. The underlying distribution is assumed to be symmetric, one of Students t family for illustration. Closed form estimators are obtained and shown to be remarkably e±cient and robust. Skew distributions will be considered in a future paper.

NONNORMAL REGRESSION. II. SYMMETRIC DISTRIBUTIONS

Salient features of a family of short-tailed symmetric distributions, introduced recently by Tiku and Vaughan [1], are enunciated. Assuming the error distribution to be one of this family, the methodology of modified likelihood is used to derive MML estimators of parameters in a linear regression model. The estimators are shown to be efficient, and robust to inliers. This paper is essentially the first to achieve robustness to inliers. The methodology is extended to long-tailed symmetric distributions and the resulting estimators are shown to be efficient, and robust to outliers. This paper should be read in conjunction with Islam et al. [2]who develop modified likelihood methodology for skew distributions in the context of linear regression.

Time series models with asymmetric innovations

We consider AR(q) models in time series with asymmetric innovations represented by two families ofdistributions: (i) gamma with support IR : (0, ∞), and (ii) generalized logistic with support IR:(-∞,∞). Since the ML (maximum likelihood) estimators are intractable, we derive the MML (modified maximum likelihood) estimators of the parameters and show that they are remarkably efficient besides being easy to compute. We investigate the efficiency properties of the classical LS (least squares) estimators. Their efficiencies relative to the proposed MML estimators are very low.

Robust statistics for testing mean vectors of multivariate distributions

We develop a ‘robust’ statistic T2 R, based on Tikus (1967, 1980) MML (modified maximum likelihood) estimators of location and scale parameters, for testing an assumed meam vector of a symmetric multivariate distribution. We show that T2 R is one the whole considerably more powerful than the prominenet Hotelling T2 statistics. We also develop a robust statistic T2 D for testing that two multivariate distributions (skew or symmetric) are identical; T2 D seems to be usually more powerful than nonparametric statistics. The only assumption we make is that the marginal distributions are of the type (1/σk)f((x-μk)/σk) and the means and variances of these marginal distributions exist.