Non-normal bivariate distributions: estimation and hypothesis testing

Qunsiyeh, Sahar Botros
When using data for estimating the parameters in a bivariate distribution, the tradition is to assume that data comes from a bivariate normal distribution. If the distribution is not bivariate normal, which often is the case, the maximum likelihood (ML) estimators are intractable and the least square (LS) estimators are inefficient. Here, we consider two independent sets of bivariate data which come from non-normal populations. We consider two distinctive distributions: the marginal and the conditional distributions are both Generalized Logistic, and the marginal and conditional distributions both belong to the Student’s t family. We use the method of modified maximum likelihood (MML) to find estimators of various parameters in each distribution. We perform a simulation study to show that our estimators are more efficient and robust than the LS estimators even for small sample sizes. We develop hypothesis testing procedures using the LS and the MML estimators. We show that the latter are more powerful and robust. Moreover, we give a comparison of our tests with another well known robust test due to Tiku and Singh (1982) and show that our test is more powerful. The latter is based on censored normal samples and is quite prominent (Lehmann, 1986). We also use our MML estimators to find a more efficient estimator of Mahalanobis distance. We give real life examples.


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İslam, Muhammed Qamarul; Yildirim, F (2001-01-01)
In a linear regression model of the type y = thetaX + e, it is often assumed that the random error e is normally distributed. In numerous situations, e.g., when y measures life times or reaction times, e typically has a skew distribution. We consider two important families of skew distributions, (a) Weibull with support IR: (0, infinity) on the real line, and (b) generalised logistic with support IR: (-infinity, infinity). Since the maximum likelihood estimators are intractable in these situations, we deriv...
Hypothesis testing in one-way classification AR(1) model with Student’s t innovations: An application to a real life data
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In this study, we estimate the model parameters in one-way classification AR (1) model when the distribution of the error terms is independently and identically distributed (iid) Student’s t. Maximum likelihood (ML) methodology is used in the estimation procedure. We also introduce the F statistic based on the ML estimators of the parameters for testing the equality of the treatment means. See also Yıldırım (2017) (M.S. Thesis, METU, Ankara, Continue) and Şenoğlu and Bayrak (2016) (Linear Contrasts in one-w...
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Can Mutan, Oya; Tiku, Moti Lal; Department of Statistics (2010)
Let X and Y be two random variables such that Y depends on X=x. This is a very common situation in many real life applications. The problem is to estimate the location and scale parameters in the marginal distributions of X and Y and the conditional distribution of Y given X=x. We are also interested in estimating the regression coefficient and the correlation coefficient. We have a cost constraint for observing X=x, the larger x is the more expensive it becomes. The allowable sample size n is governed by a...
Inference of Autoregressive Model with Stochastic Exogenous Variable Under Short-Tailed Symmetric Distributions
Bayrak, Ozlem Tuker; Akkaya, Ayşen (2018-12-01)
In classical autoregressive models, it is assumed that the disturbances are normally distributed and the exogenous variable is non-stochastic. However, in practice, short-tailed symmetric disturbances occur frequently and exogenous variable is actually stochastic. In this paper, estimation of the parameters in autoregressive models with stochastic exogenous variable and non-normal disturbances both having short-tailed symmetric distribution is considered. This is the first study in this area as known to the...
Applications of estimation techniques on genetic and other types of data
Aslan, Murat; Akkaya, Ayşen; Department of Statistics (2003)
The parameters of genetic and other types of data, particularly with small samples, are estimated by using method of moments, least squares, minimum chi- square, maximum likelihood and modified maximum likelihood estimation methods. These methods are also compared in terms of their efficiencies and robustness property.
Citation Formats
S. B. Qunsiyeh, “Non-normal bivariate distributions: estimation and hypothesis testing,” Ph.D. - Doctoral Program, Middle East Technical University, 2007.