A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

The Siable-Law Model of Stock Returns

This study investigates the tail shapes of empirical distributions of returns on an extensive group of common stocks. The tails of the return distributions are found to be thinner than those of infinite variance stable distributions. Therefore, although homogeneity is evident in general, economic and statistical inferences drawn from stable-law parameters estimated from samples of stock returns may be misleading. This is in spite of the apparent overall similarity (in shape) between empirical and stable distributions.     

On a mixture autoregressive model

We generalize the Gaussian mixture transition distribution (GMTD) model introduced by Le and co-workers to the mixture autoregressive (MAR) model for the modelling of non-linear time series. The models consist of a mixture of K stationary or non-stationary AR components. The advantages of the MAR model over the GMTD model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series. The stationarity conditions and autocorrelation function are derived. The estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real data sets and compared with other competing models. The MAR models appear to capture features of the data better than other competing models do.     

Inference in affine shape theory under elliptical models

This paper studies the elliptical statistical affine shape theory under certain particular conditions on the evenness or oddness of the number of landmarks. In such a case, the related distributions are polynomials, and the inference is easily performed; as an example, a landmark data is studied, and the performance of the polynomial density versus the usual series density is compared.     

Wishart and Pseudo-Wishart distributions under elliptical laws and related distributions in the shape theory context

In this paper, we determine the density of a singular elliptically contoured matrix. From this, the study of Wishart and Pseudo-Wishart distributions, whether central or noncentral, whether singular or nonsingular, is extended to the case of elliptical models. Some elated distributions are studied in the context of shape theory. Particular attention is paid to singular size-and-shape and size-and-shape cone densities.     

The unit extended Weibull families of distributions and its applications

In this paper, two new general families of distributions supported on the unit interval are introduced. The proposed families include several known models as special cases and define at least twenty (each one) new special models. Since the list of well-being indicators may include several double bounded random variables, the applicability for modeling those is the major practical motivation for introducing the distributions on those families. We propose a parametrization of the new families in terms of the median and develop a shiny application to provide interactive density shape illustrations for some special cases. Various properties of the introduced families are studied. Some special models in the new families are discussed. In particular, the complementary unit Weibull distribution is studied in some detail. The method of maximum likelihood for estimating the model parameters is discussed. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Applications to the literacy rate in Brazilian and Colombian municipalities illustrate the usefulness of the two new families for modeling well-being indicators.     

Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

Survival distributions arising from two families and generated by transformations

Two families of distributions are introduced and studied within the framework of parametric survival analysis. The families are derived from a general linear form by specifying a function of the survival function with certain restrictions. Distributions within each family are generated by transformations of the survival time variable subject to certain restrictions. Two specific transformations were selected and, thus, four distributions are identified for further study. The distributions have one scale and two shape parameters and include as special cases the exponential, Weibull, log-logistic and Gompertz distributions. One of the new distributions, the modified Weibull, is studied in some detail.

The distributions are developed with an emphasis on those features that data analysts find especially useful for survivorship studies, A wide variety of hazard shapes are available. The survival, density and hazard functions may be written in simple algebraic forms. Parameter estimation is demonstrated using the least squares and maximum likelihood methods. Graphical techniques to assess goodness of fit are demonstrated. The models may be extended to include concmitant information.     

Confidence regions for Pareto parameters from a single and independent samples

[Abstract] Based on a single and on two independent samples, joint confidence regions for parameters of Pareto distributions are proposed with minimum volume properties and without assigning the confidence level to dimensions. In the one-sample case, comparisons are made to former simultaneous confidence sets for Pareto parameters by means of simulation and a real data set. The two-sample case is studied in various set-ups and comprises simultaneous confidence regions for the shape parameters, the scale parameters, and higher-dimensional vectors of these parameters, where common shape and common scale models are also considered.     

Tests for homogeneity of extreme value scale parameters

It is often assumed in situations in which life data from Weibull or extreme-value distributions are involved that data in different samples come from extreme-value distributions with the same scale parameter (equivalently, Weibull distributions with the same shape parameter). This paper proposes a number of tests for homogeneity for extreme-value scale parameters, based on a number of commonly used estimators for these scale parameters. Previous theoretical work and some simulation results provided here indicate that the null distributions of the test statistics proposed are well approximated by the x2 distribution under a wide range of conditions     

Small Sample Tests for Shape Parameters of Gamma Distributions

The introduction of shape parameters into statistical distributions provided flexible models that produced better fit to experimental data. The Weibull and gamma families are prime examples wherein shape parameters produce more reliable statistical models than standard exponential models in lifetime studies. In the presence of many independent gamma populations, one may test equality (or homogeneity) of shape parameters. In this article, we develop two tests for testing shape parameters of gamma distributions using chi-square distributions, stochastic majorization, and Schur convexity. The first one tests hypotheses on the shape parameter of a single gamma distribution. We numerically examine the performance of this test and find that it controls Type I error rate for small samples. To compare shape parameters of a set of independent gamma populations, we develop a test that is unbiased in the sense of Schur convexity. These tests are motivated by the need to have simple, easy to use tests and accurate procedures in case of small samples. We illustrate the new tests using three real datasets taken from engineering and environmental science. In addition, we investigate the Bayes’ factor in this context and conclude that for small samples, the frequentist approach performs better than the Bayesian approach.     

Longitudinal shape analysis by using the spherical coordinates

One of the important topics in morphometry that received high attention recently is the longitudinal analysis of shape variation. According to Kendall's definition of shape, the shape of object appertains on non-Euclidean space, making the longitudinal study of configuration somehow difficult. However, to simplify this task, triangulation of the objects and then constructing a non-parametric regression-type model on the unit sphere is pursued in this paper. The prediction of the configurations in some time instances is done using both properties of triangulation and the size of great baselines. Moreover, minimizing a Euclidean risk function is proposed to select feasible weights in constructing smoother functions in a non-parametric smoothing manner. These will provide some proper shape growth models to analysis objects varying in time. The proposed models are applied to analysis of two real-life data sets.     

A Starship Estimation Method for the Generalized λ Distributions

A development of the 'starship' method (Owen, 1988), a computer intensive estimation method, is presented for two forms of generalized λ distributions (gλd). The method can be used for the full parameter space and is flexible, allowing choice of both the form of the generalized λ distribution and of the nature of fit required. Some examples of its use in fitting data and approximating distributions are given. Some simulation studies explore the sampling distribution of the parameter estimates produced by this method for selected values of the parameters and consider comparisons with two other methods, for one of the gλd distributional forms, not previously so investigated. In the forms and parameter regions available to the other methods, it is demonstrated that the starship compares favourably. Although the differences between the methods, where available, tend to disappear with largersamples, the parameter coverage, flexibility and adaptability of the starship method make it attractive. However, the paper also demonstrates that care is needed when fitting and using such quantile-defined distributional families that are rich in shape, but have complex properties.     

Modelling lifetimes by quantile functions using Parzen's score function

The present paper introduces methods of constructing quantile functions as models of lifetimes with monotone and nonmonotone hazard functions. This is accomplished on the basis of the relationships the hazard quantile function has with the score function introduced by Parzen in connection with the tail heaviness of probability distributions. Three models illustrated here contain several existing models as particular cases. The appropriateness of the models in real situations is also demonstrated.     

NEW LARGE SAMPLE AND BOOTSTRAP METHODS ON SHAPE SPACES IN HIGH LEVEL ANALYSIS OF NATURAL IMAGES

This paper, dedicated to the 80th birthday of Professor C. R. Rao, deals with asymptotic distributions of Fréchet sample means and Fréchet total sample variance that are used in particular for data on projective shape spaces or on 3D shape spaces. One considers the intrinsic means associated with Riemannian metrics that are locally flat in a geodesically convex neighborhood around the support of a probability measure on a shape space or on a projective shape space. Such methods are needed to derive tests concerning variability of planar projective shapes in natural images or large sample and bootstrap confidence intervals for 3D mean shape coordinates of an ordered set of landmarks from laser images.     

Dependent models for observations which include angular ones

This paper discusses some stochastic models for dependence of observations which include angular ones. First, we provide a theorem which constructs four-dimensional distributions with specified bivariate marginals on certain manifolds such as two tori, cylinders or discs. Some properties of the submodel of the proposed models are investigated. The theorem is also applicable to the construction of a related Markov process, models for incomplete observations, and distributions with specified marginals on the disc. Second, two maximum entropy distributions on the cylinder are discussed. The circular marginal of each model is distributed as the generalized von Mises distribution which represents a symmetric or asymmetric, unimodal or bimodal shape. The proposed cylindrical model is applied to two data sets.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Modified proportional hazard rates and proportional reversed hazard rates models via Marshall–Olkin distribution and some stochastic comparisons

Adding parameters to a known distribution is a useful way of constructing flexible families of distributions. Marshall and Olkin (1997) introduced a general method of adding a shape parameter to a family of distributions. In this paper, based on the Marshall–Olkin extension of a specified distribution, we introduce two new models, referred to as modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models, which include as special cases the well-known proportional hazard rates and proportional reversed hazard rates models, respectively. Next, when two sets of random variables follow either the MPHR or the MPRHR model, we establish some stochastic comparisons between the corresponding order statistics based on majorization theory. The results established here extend some well-known results in the literature.     

Models for Dependent Extremes Using Stable Mixtures

Abstract.  This paper unifies and extends results on a class of multivariate extreme value (EV) models studied by Hougaard, Crowder and Tawn. In these models, both unconditional and conditional distributions are themselves EV distributions, and all lower-dimensional marginals and maxima belong to the class. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential-stable location mixtures (for Gumbel) or as power-stable scale mixtures (for non-Gumbel EV distributions). A third interpretation is through a peaks over thresholds model with a positive stable intensity. The mixing variables are used as a modelling tool and for better understanding and model checking. We study EV analogues of components of variance models, and new time series, spatial and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation.     

On the difference in inference and prediction between the joint and independent f-error models for seemingly unrelated regressions

We consider likelihood and Bayesian inferences for seemingly unrelated (linear) regressions for the joint niultivariate terror (e.g. Zellner, 1976) and the independent t-error (e.g. Maronna, 1976) models. For likelihood inference, the scale matrix and the shape parameter for the joint terror model cannot be consistently estimated because of the lack of adequate information to identify the latter. The joint terror model also yields the same MLEs for the regression coefficients and the scale matrix as for the independent normal error model. which are not robust against outliers. Further, linear hypotheses with respect

to the regression coefficients also give rise to the same mill distributions AS for the independent normal error model, though the MLE has a non-normal limiting distribution. In contrast to the striking similarities between the joint t-error and the independent normal error models, the independent f-error model yields AiLEs that are lubust against uuthers. Since the MLE of the shape parameter reflects the tails of the data distributions, this model extends the independent normal error model for modeling data distributions with relatively t hicker tails. These differences are also discussed with respect to the posterior and predictive distributions for Bayesian inference.