Measure of departure from symmetry for the analysis of collapsed square contingency tables with ordered categories

For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper considers the symmetry model for collapsed square contingency tables and proposes a measure to represent the degree of departure from symmetry. The proposed measure is defined as the arithmetic mean of submeasures each of which represents the degree of departure from symmetry for each collapsed 3×3 table. Each submeasure also represents the mean of power-divergence or diversity index for each collapsed table. Examples are given.     

Measure of departure from conditional marginal homogeneity for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Tomizawa et al. (S. Tomizawa, N. Miyamoto, and N. Ashihara, Measure of departure from marginal homogeneity for square contingency tables having ordered categories, Behaviormetrika 30 (2003), pp. 173–193.) and Tahata et al. (K. Tahata, T. Iwashita, and S. Tomizawa, Measure of departure from symmetry of cumulative marginal probabilities for square contingency tables with ordered categories, SUT J. Math., 42 (2006), pp. 7–29.) considered the measures which represent the degree of departure from the marginal homogeneity (MH) model. The present paper proposes a measure that represents the degree of departure from the conditional MH, given that an observation will fall in one of the off-diagonal cells of the table. The measure proposed is expressed by using the Cressie–Read power-divergence or the Patil–Taillie diversity index, which is applied for the conditional cumulative marginal probabilities given that an observation will fall in one of the off-diagonal cells of the table. When the MH model does not hold, the measure is useful for seeing how far the conditional cumulative marginal probabilities are from those with an MH structure and for comparing the degree of departure from MH in several tables. Examples are given.     

Power-divergence-type measure of departure from diagonals-parameter symmetry for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Goodman considered the diagonals-parameter symmetry (DPS) model. This paper proposes a measure to represent the degree of departure from the DPS model. The proposed measure is expressed by applying Read and Cressie’s power-divergence or Patil and Taillie’s diversity index. The measure would be useful for comparing the degree of departure from the DPS model in several tables. Examples are given.     

Marginal asymmetry model for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, this paper proposes a model which indicates the structure of marginal asymmetry. The model states that the absolute values of logarithm of ratio of the cumulative probability that an observation will fall in row category i or below and column category i+1 or above to the corresponding cumulative probability that the observation falls in column category i or below and row category i+1 or above are constant for every i. We deal with the estimation problem for the model parameter and goodness-of-fit tests. Also we discuss the relationships between the model and a measure which represents the degree of departure from marginal homogeneity. Examples are given.     

Marginal inhomogeneity models for square contingency tables with nominal categories

For the analysis of square contingency tables with nominal categories, this paper proposes two kinds of models that indicate the structure of marginal inhomogeneity. One model states that the absolute values of log odds of the row marginal probability to the corresponding column marginal probability for each category i are constant for every i. The other model states that, on the condition that an observation falls in one of the off-diagonal cells in the square table, the absolute values of log odds of the conditional row marginal probability to the corresponding conditional column marginal probability for each category i are constant for every i. These models are used when the marginal homogeneity model does not hold, and the values of parameters in the models are useful for seeing the degree of departure from marginal homogeneity for the data on a nominal scale. Examples are given.     

Tests of odds ratio homogeneity with improved power in sparse fourfold tables

Some new tests of odds ratio homogeneity for fourfold tables are compared with the mixture model score test in the sparse-data case (many tables, small margins per table). Based on general empirical Bayes inequalities, the new tests have competitive power for 1:R matched designs, and superior power for more balanced designs.     

On entropy of a Pareto distribution in the presence of outliers

ABSTRACT

The aim of this paper is obtaining the amount of information there exists in the Pareto distribution in the presence of outliers. For the sake of this purpose, Shannon entropy, ?-entropy, Fisher information, and Kullback–Leibler distance are computed. Furthermore, a section has been devoted to compare these quantities in these two cases of the Pareto distribution (with outliers and the homogenous case). At the end of this paper, two actual examples, which are related to insurance companies, are brought. A brief summary of which is done in this work is also reported.     

Testing for Homogeneity in Mixture Using Weighted Relative Entropy

In this article, we propose a test for homogeneity based on Kullback–Leibler information (also known as relative entropy). Though widely used in hypothesis testing problems, Kullback–Leibler information is not desirable to many researchers in the context of mixture because of its complicated form. In this article, a weighted relative entropy test (WE test), which has closed form expression in terms of the parameter estimators, is proposed. Theoretical results show that this test is consistent. Some simulation results demonstrate that the WE test is better than some leading tests when the mixture components come from normal distribution, and is competitive with them in the Poisson case. The usage of the test is illustrated in an example with data about acidity index of lakes.     

Covariate selection for accelerated failure time data

Selection of appropriate predictors for right censored time to event data is very often encountered by the practitioners. We consider the ?₁ penalized regression or “least absolute shrinkage and selection operator” as a tool for predictor selection in association with accelerated failure time model. The choice of the penalizing parameter λ is crucial to identify the correct set of covariates. In this paper, we propose an information theory-based method to choose λ under log-normal distribution. Furthermore, an efficient algorithm is discussed in the same context. The performance of the proposed λ and the algorithm is illustrated through simulation studies and a real data analysis. The convergence of the algorithm is also discussed.