A method for estimating lorenz curves

This paper proposes a new method for estimating the parameters of Lorenz Curves (LC’s) and fitting LC’s to observed data. The method is very general. It is applicable to any family of LC’s as long as it is given in closed form which is often the case in practice. The method can also be applied to either the LC or to its associated distribution. The estimators are easy to compute as they are obtained one at a time by solving only one equation in one unknown and in many cases the solutions are given in closed-forms. An additional advantage, that is not shared with the currently used method of estimation, is that the method is invariant as to the specification of which variable is written as a function of the other in the LC form. The method is applied to the most commonly suggested LC’s families. An example of real-life data is used to illustrate the methodology. A simulation study is performed to study the properties of the proposed estimators and to compare them with existing ones. The results seem to indicate that the proposed estimators have good properties and they often perform much better than the existing ones.     

Estimation for two-parameter weibull distribution and extreme-value distribution under multiply type-II censoring

Compared to Type-II censoring, multiply Type-II censoring is a more general, yet mathematically and numerically much more complicated censoring scheme. For multiply Type II censored data from a two-parameter Weibull distribution, we propose several estimators, including MLE, approximate MLE, and estimators corresponding to the BLUE and BLIE from estimating parameters in extreme-value distribution. An approximately unbiased estimator for the shape parameter is also proposed which has the smallest MSE. Numerical examples show that this estimator is the best in terms of bias and MSE. Numerical examples also show that the approximate MLE which admits a closed form is better for estimating the scale parameter.     

Approximating the distribution of a renewal process using generalized poisson distributions

The exact distribution of a renewal counting process is not easy to compute and is rarely of closed form. In this article, we approximate the distribution of a renewal process using families of generalized Poisson distributions. We first compute approximations to the first several moments of the renewal process. In some cases, a closed form approximation is obtained. It is found that each family considered has its own strengths and weaknesses. Some new families of generalized Poisson distributions are recommended. Theorems are obtained determining when these variance to mean ratios are less than (or exceed) one without having to find the mean and variance. Some numerical comparisons are also made.     

Analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable. We develop the essential methodology for estimating the model parameters via maximum likelihood method. The general form of the maximum likelihood estimator is obtained in color closed form. Adjusted treatment effects and adjusted covariate effects are given, too. We also provide the asymptotic distribution of the proposed estimators. A simulation study and a real world application are also performed to illustrate and evaluate the proposed methodology.     

An alternative version of zero-inflated logarithmic series distribution and some of its applications

An important drawback of the standard logarithmic series distribution (LSD) in several practical applications is that it excludes the zero observation from its support. The LSD with non-negative support is not much studied in the literature. Recently Kumar and Riyaz [On the zero-inflated LSD and its modification. Statistica (accepted for publication). 2013] considered a distribution in this respect namely ‘zero-inflated logarithmic series distribution (ZILSD)’. Through this paper we propose an alternative form of the ZILSD and study some of its properties. We obtain expressions for its probability-generating function, mean and variance, and develop certain recurrence relations for its probabilities, raw moments and factorial moments. The parameters of the model are estimated by the method of moments and the method of maximum likelihood, and certain test procedures are considered for testing the significance of the additional parameter of the distribution. The distribution has been fitted to certain real-life data sets for illustrating its usefulness compared with certain existing models available in the literature. Further, a simulation study is conducted for assessing the performance of the estimators.     

Nonparametric estimation of bivariate distribution under right truncation with application to panic disorder

In several studies, investigators are interested in estimating the bivariate distribution of the onset ages of a generic disorder in successive generations. The empirical distribution is inappropriate for this purpose due to truncation: only parent–child pairs with onset ages prior to the ages at interview were included in the sample. In this paper, we propose a simple nonparametric estimator for the underlying bivariate distribution of the onset ages. Compared with the existing estimators, the proposed estimator has a closed form and smaller biases when estimating marginal distributions. A real example is used to illustrate this estimator.     

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.     

Estimation of parameters of the gped

Janardan (1973) introduced the generalized Polya-Eggenberger distribution as a limiting form of the generalized Markov-Polya distribution (GMPD), Ja¬nardan (1998) derived GPED formally by means of Lagrange's expansion and discussed its various properties systematically. Here, a new urn model is pro¬vided for the GPED. Moment estimators of the parameters are given in closed form. Maximum hkelihood estimators are also given. Some apphcations are provided.     

Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations

It is well-known that maximum likelihood (ML) estimators of the two parameters in a gamma distribution do not have closed forms. This poses difficulties in some applications such as real-time signal processing using low-grade processors. The gamma distribution is a special case of a generalized gamma distribution. Surprisingly, two out of the three likelihood equations of the generalized gamma distribution can be used as estimating equations for the gamma distribution, based on which simple closed-form estimators for the two gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estimators. The study consolidates this conjecture by establishing the asymptotic behaviors of the new estimators. In addition, the closed-forms enable bias-corrections to these estimators. The bias-correction significantly improves the small-sample performance.     

Multivariate tail conditional expectation for scale mixtures of skew-normal distribution

In practice, a financial or actuarial data set may be a skewed or heavy-tailed and this motivates us to study a class of distribution functions in risk management theory that provide more information about these characteristics resulting in a more accurate risk analysis. In this paper, we consider a multivariate tail conditional expectation (MTCE) for multivariate scale mixtures of skew-normal (SMSN) distributions. This class of distributions contains skewed distributions and some members of this class can be used to analyse heavy-tailed data sets. We also provide a closed form for TCE in a univariate skew-normal distribution framework. Numerical examples are also provided for illustration.     

Parameter estimates and tests of fit for infinite mixther distrirutions

Al though mixtures form a rich class of probability models, they often present difficulties for statistical inference. Likelihood functions are sometimes unbounded at certain values of the parameters, and densities often have no closed form. These features complicate hoth maximum-likelihood estimation and tests of fit based on the empirical distribution function. New inferential methods using sample characteristic functions (Cfs) and moment generating functions (MGFs) seem well-suited to mixtures. since these transforms often take simple form/ This paper reports a simulation study of the properties of estimators and tests of fit based on CFs, MGFs, and sample moments when applied to three specific families of thick tailed mixture distributios.     

Normal quantile estimation based on minimizing the error in predicted distribution functions

Estimators of the form [Xbar] + kS for estimating the p quantile of a normal distribution are studied when k is chosen to either minimize the mean square error in the predicted distribution function or to make the predicted distribution function unbiased for p. Here, [Xbar] and S are the usual sample mean and standard deviation, respectively, and the predicted distribution function is the true (normal) distribution function evaluated at the estimated quantile.

These k values are presented for various sample sizes and values of p, and application to warranty determination is discussed.     

Bayes estimates of mixing proportions in finite mixture distributions

With reference to the problem of estimating the mixing proportions in a finite mixture distribution with known components, employing Dirichlet prior, closed form expressions for the posterior means and variances are obtained. To avoid the difficulties in computing the estimates, an approximation procedure is introduced. Numerical studies carried out for normal mixtures indicate the closeness of the approximations and their superiority over the maximum likelihood estimates at least in the case of small samples.     

Generalized Inverse Gamma Distribution and its Application in Reliability

In this article, we introduce a new reliability model of inverse gamma distribution referred to as the generalized inverse gamma distribution (GIG). A generalization of inverse gamma distribution is defined based on the exact form of generalized gamma function of Kobayashi (1991). This function is useful in many problems of diffraction theory and corrosion problems in new machines. The new distribution has a number of lifetime special sub-models. For this model, some of its statistical properties are studied. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. We also demonstrate the usefulness of this distribution on a real data set.     

Time series models based on the unrestricted skew-normal process

The standard location and scale unrestricted (or unified) skew-normal (SUN) family studied by Arellano-Valle and Genton [On fundamental skew distributions. J Multivar Anal. 2005;96:93–116] and Arellano-Valle and Azzalini [On the unification of families of skew-normal distributions. Scand J Stat. 2006;33:561–574], allows the modelling of data which is symmetrically or asymmetrically distributed. The family has a number of advantages suitable for the analysis of stochastic processes such as Auto-Regressive Moving-Average (ARMA) models, including being closed under linear combinations, being able to satisfy the consistency condition of Kolmogorov’s theorem and providing the guarantee of the existence of such a SUN stochastic process. The family is able to be represented in a hierarchical form which can be used for the ease of simulation. In addition, it facilitates an EM-type algorithm to estimate the model parameters. The performances and suitability of the proposed model are demonstrated on simulations and using two real data sets in applications.     

Abel series distributions with applications to fluctuations of sample functions of stochastic processes

A two-parameter class of discrete distributions, Abel series distributions, generated by expanding a suitable pa,rametric function into a series of Abel polynomials is discussed. An Abel series distribution occurs in fluctuations of sample functions of stochastic processes and has applications in insurance risk, queueing, dam and storage processes. The probability generating function and the factorial moments of the Abel series distributions are obtained in closed forms. It is pointed out that the name of the generalized Poisson distribution of Consul and Jain is justified by the form of its generating function. Finally it is shown that this generalized Poisson distribution is the only member of the Abel series distributions which is closed under convolution.     

A generalization of the log-series distribution

A new generalized logarithmic series distribution (GLSD) with two parameters is proposed.The proposed model is flexible enough to describe short-tailed as well as long-tailed data.Some recurence relations for its probabilities and the factorial moments are presente.These recurrence relations are utilized to obtain the minimum chi-square estimators for the parmaters.Maximum likelihood estimators and some other estimators based on first few moments and probabilities are also suggested.Asymptotic relative efficiency of some of these estimators is also obtained and compared.Two test statistics based on the minimum chi-square estimators fo testing some hypotheses regarding the GLSD are proposed.The fit of the model and the application of the test statistics are exemplified by some data sets.Finally, a graphical method is suggested for differentiating between the ordinary logarithmic series distribution and the GLSD.     

Maximum Likelihood Estimation of Parameters in a Mixture Model

The estimation of parameters of the log normal distribution based on complete and censored samples are considered in the literature. In this article, the problem of estimating the parameters of log normal mixture model is considered. The Expectation Maximization algorithm is used to obtain maximum likelihood estimators for the parameters, as the likelihood equation does not yield closed form expression. The standard errors of the estimates are obtained. The methodology developed here is then illustrated through simulation studies. The confidence interval based on large-sample theory is obtained.     

A non-iterative least squares estimation of missing values in experimental designs

In this paper an attempt has been made to obtain a systematic method of estimating the missing values in experimental designs. When the observations are missing in a particular pattern (in RBD and LSD) explicit expressions are given for the estimators of the missing values. This procedure is compared with Yate's iterative procedure by numerical examples.     

Log-gamma-generated families of distributions

We introduce two new general families of continuous distributions, generated by a distribution F and two positive real parameters α and β which control the skewness and tail weight of the distribution. The construction is motivated by the distribution of k-record statistics and can be derived by applying the inverse probability integral transformation to the log-gamma distribution. The introduced families are suitable for modelling the data with a significantly skewed and heavy-tailed distribution. Various properties of the introduced families are studied and a number of estimations and data fitness on real data are given to illustrate the results.