Nirian Martín - Academia.edu (original) (raw)

Papers by Nirian Martín

We consider nested sequences of hierarchical loglinear models when expected frequencies are subje... more We consider nested sequences of hierarchical loglinear models when expected frequencies are subject to linear constraints and we study the problem of finding the model in the the nested sequence that is able to explain more clearly the given data. It will be necessary to give a method to estimate the parameters of the loglinear models and also a procedure to choose the best model among the models considered in the nested sequence under study. These two problems will be solved using the φ -divergence measures. We estimate the unknown parameters using the minimum φ -divergence estimator (Martín and Pardo [8]) which can be considered as a generalization of the maximum likelihood estimator (Haber and Brown [5]) and we consider a φ -divergence test statistic (Martín [7]) that generalize the likelihood ratio test as well as the chi-square test statistic, for testing two nested loglinear models.

Kybernetika -Praha-

ABSTRACT

Mathematics and Computers in Simulation, 2015

In this paper new test statistics are introduced and studied for the important problem of testing... more In this paper new test statistics are introduced and studied for the important problem of testing hypothesis that involves inequality constraint on proportions when the sample comes from independent binomial random variables: Wald type and phi-divergence based teststatistics. As a particular case of phi-divergence based test-statistics, the classical likelihood ratio test is considered. An illustrative example is given and the performance of all of them for small and moderate sample sizes is analyzed in an extensive simulation study.

We propose two families of maximally selected phi-divergence tests for studying change point loca... more We propose two families of maximally selected phi-divergence tests for studying change point locations when the unknown probability vectors of a sequence of multinomial random variables, with possibly different sizes, are piecewise constant. In addition, these test-statistics are valid to estimate the location of the change-point. Two variants of the first family are considered by following two versions of the Darling- Erdös' formula. Under the no changes null hypothesis, we derive their limit distributions, extreme value and Gaussian-type respectively. We pay special attention to the checking the accuracy of these limit distributions in case of finite sample sizes. In such a framework, a Monte Carlo analysis shows the possibility of improving the behaviour of the test-statistics based on the likelihood ratio and chi-square tests introduced in Horváth and Serbinowska (1995). The data of the classical Lindisfarne Scribes problem are used in order to apply the proposed test-statis...

Statistics, 2015

In testing of hypothesis the robustness of the tests is an important concern. Generally, the maxi... more In testing of hypothesis the robustness of the tests is an important concern. Generally, the maximum likelihood based tests are most efficient under standard regularity conditions, but they are highly non-robust even under small deviations from the assumed conditions. In this paper we have proposed generalized Wald-type tests based on minimum density power divergence estimators for parametric hypotheses. This method avoids the use of nonparametric density estimation and the bandwidth selection. The trade-off between efficiency and robustness is controlled by a tuning parameter β. The asymptotic distributions of the test statistics are chi-square with appropriate degrees of freedom. The performance of the proposed tests are explored through simulations and real data analysis.

Metrika, 2014

Statistical techniques are used in all branches of science to determine the feasibility of quanti... more Statistical techniques are used in all branches of science to determine the feasibility of quantitative hypotheses. One of the most basic applications of statistical techniques in comparative analysis is the test of equality of two population means, generally performed under the assumption of normality. In medical studies, for example, we often need to compare the effects of two different drugs, treatments or preconditions on the resulting outcome. The most commonly used test in this connection is the two sample t-test for the equality of means, performed under the assumption of equality of variances. It is a very useful tool, which is widely used by practitioners of all disciplines and has many optimality properties under the model. However, the test has one major drawback; it is highly sensitive to deviations from the ideal conditions, and may perform miserably under model misspecification and the presence of outliers. In this paper we present a robust test for the two sample hypothesis based on the density power divergence measure , and show that it can be a great alternative to the ordinary two sample t-test. The asymptotic properties of the proposed tests are rigorously established in the paper, and their performances are explored through simulations and real data analysis.

The estimation of the radiological inventory of Nuclear Facilities to be dismantled is a process ... more The estimation of the radiological inventory of Nuclear Facilities to be dismantled is a process that included information related with the physical inventory of all the plant and radiological survey. Estimation of the radiological inventory for all the components and civil structure of the plant could be obtained with mathematical models with statistical approach. A computer application has been developed

When an underlying logit based order dose-response model is considered with small or moderate sam... more When an underlying logit based order dose-response model is considered with small or moderate sample sizes, the Cochran-Armitage (CA) test represents the most efficient test in the framework of the test-statistics applied with asymptotic distributions for testing monotone proportions. The Wald and likelihood ratio (LR) test have much worse behaviour in type error I in comparison with the CA test. It suffers, however, from the weakness of not maintaining the nominal size. In this paper a family of test-statistics based on φ-divergence measures is proposed and their asymptotic distribution under the null hypothesis is obtained either for one-sided or two-sided hypothesis testing. A numerical example based on real data illustrates that the proposed test-statistics are simple for computation and moreover, the necessary goodness-of-fit test-statistic are easily calculated from them. The simulation study shows that the test based on the Cressie and Read (Journal of the Royal Statistical Society, Series B, 46, 440-464, 1989) divergence measure usually provides a better nominal size than the CA test for small and moderate sample sizes.

In regression models involving economic variables such as income, log transformation is typically... more In regression models involving economic variables such as income, log transformation is typically taken to achieve approximate normality and stabilize the variance. However, often the interest is predicting individual values or means of the variable in the original scale. Back transformation of predicted values introduces a non-negligible bias. Moreover, assessing the uncertainty of the actual predictor is not straightforward. In this paper, a nested error model for the log transformation of the target variable is considered. Nested error models are widely used for estimation of means in subpopulations with small sample sizes (small areas), by linking all the areas through common parameters. These common parameters are estimated using the overall set of sample data, which leads to much more efficient small area estimators. Analytical expressions for the best predictors of individual values of the original variable and of small area means are obtained under the nested error model with log transformation of the target variable. Empirical best predictors are defined by estimating the unknown model parameters in the best predictors. Exact mean squared errors of the best predictors and second order approximations to the mean squared errors of the empirical best predictors are derived. Mean squared error estimators that are second order correct are also obtained. An example with Spanish data on living conditions illustrates the procedures.

The main purpose of this paper is to present new families of test statistics for studying the pro... more The main purpose of this paper is to present new families of test statistics for studying the problem of goodness-of-fit of some data to a latent class model for binary data. The families of test statistics introduced are based on phi-divergence measures, a natural extension of maximum likelihood. We also treat the problem of testing a nested sequence of latent class models for binary data. For these statistics, we obtain their asymptotic distribution. Finally, a simulation study is carried out in order to compare the efficiency, in the sense of the level and the power, of the new statistics considered in this paper for sample sizes that are not big enough to apply the asymptotical results. MSC: Primary 62F03; 62F05; Secondary 62H15

We consider a robust version of the classical Wald test statistics for testing simple and composi... more We consider a robust version of the classical Wald test statistics for testing simple and composite null hypotheses for general parametric models. These test statistics are based on the minimum density power divergence estimators instead of the maximum likelihood estimators. An extensive study of their robustness properties is given though the influence functions as well as the chi-square inflation factors. It is theoretically established that the level and power of these robust tests are stable against outliers, whereas the classical Wald test breaks down. Some numerical examples confirm the validity of the theoretical results.

In any parametric inference problem, the robustness of the procedure is a real concern. A procedu... more In any parametric inference problem, the robustness of the procedure is a real concern. A procedure which retains a high degree of efficiency under the model and simultaneously provides stable inference under data contamination is preferable in any practical situation over another procedure which achieves its efficiency at the cost of robustness or vice versa. The density power divergence family of provides a flexible class of divergences where the adjustment between efficiency and robustness is controlled by a single parameter β. In this paper we consider general tests of parametric hypothesis based on the density power divergence. We establish the asymptotic null distribution of the test statistic and explore its asymptotic power function. Numerical results illustrate the performance of the theory developed.

Journal of Statistical Computation and Simulation, 2014

In this paper new families of test statistics are introduced and studied for the problem of compa... more In this paper new families of test statistics are introduced and studied for the problem of comparing two treatments in terms of the likelihood ratio order. The considered families are based on phi-divergence measures and arise as natural extensions of the classical likelihood ratio test and Pearson test statistics. It is proven that their asymptotic distribution is a common chi-bar random variable. An illustrative example is presented and the performance of these statistics is analysed through a simulation study. Through a simulation study it is shown that, for most of the proposed scenarios adjusted to be small or moderate, some members of this new family of test-statistic display clearly better performance with respect to the power in comparison to the classical likelihood ratio and the Pearson's chi-square test while the exact size remains closed to the nominal size.

Sankhya B, The Indian Journal of Statististics, 2012

In this paper we consider Poisson loglinear models with linear constraints (LMLC) on the expected... more In this paper we consider Poisson loglinear models with linear constraints (LMLC) on the expected table counts. Multinomial and product multinomial loglinear models can be obtained by considering that some marginal totals (linear constraints on the expected table counts) have been prefixed in a Poisson loglinear model. Therefore with the theory developed in this paper, multinomial and product multinomial loglinear models can be considered as a particular case. To carry out inferences on the parameters in the LMLC an information-theoretic approach is followed from which the classical maximum likelihood estimators and Pearson chi-square statistics for goodness-of fit are obtained. In addition, nested hypotheses are proposed as a general procedure for hypothesis testing. Through a simulation study the appropriateness of proposed inference tools is illustrated.

Journal of Multivariate Analysis, 2014

When some treatments are ordered according to the categories of an ordinal categorical variable (... more When some treatments are ordered according to the categories of an ordinal categorical variable (e.g., extent of side effects) in a monotone order, one might be interested in knowing wether the treatments are equally effective or not. One way to do that is to test if the likelihood ratio order is strictly verified. A method based on log-linear models is derived to make statistical inference and phi-divergence test-statistics are proposed for the test of interest. Focussed on loglinear modeling, the theory associated with the asymptotic distribution of the phi-divergence test-statistics is developed. An illustrative example motivates the procedure and a simulation study for small and moderate sample sizes shows that it is possible to find phi-divergence test-statistic with an exact size closer to nominal size and higher power in comparison with the classical likelihood ratio.

Statistics: A Journal of Theoretical and Applied Statistics (in Press), 2014

In Econometrics, imposing restrictions without assuming underlying distributions to modelize comp... more In Econometrics, imposing restrictions without assuming underlying distributions to modelize complex realities is a valuable methodological tool. However, if a subset of restrictions were not correctly specified, the usual test-statistics for correctly specified models tend to reject erronously a simple null hypothesis. In this setting, we may say that the model suffers from misspecification. We study the behavior of empirical phidivergence test-statistics, introduced in Balakrishnan et al. (2015), by using the exponential tilted empirical likelihood estimators of Schennach (2007), as a good compromise between efficiency of the significance level for small sample sizes and robustness under misspecification. JEL classification: C12; C14

Advances in Mathematical and Statistical Modeling, 2008