On a measure of information gain for regression models in survival analysis

Papers dealing with measures of predictive power in survival analysis have seen their independence of censoring, or their estimates being unbiased under censoring, as the most important property. We argue that this property has been wrongly understood. Discussing the so-called measure of information gain, we point out that we cannot have unbiased estimates if all values, greater than a given time τ, are censored. This is due to the fact that censoring before τ has a different effect than censoring after τ. Such τ is often introduced by design of a study. Independence can only be achieved under the assumption of the model being valid after τ, which is impossible to verify. But if one is willing to make such an assumption, we suggest using multiple imputation to obtain a consistent estimate. We further show that censoring has different effects on the estimation of the measure for the Cox model than for parametric models, and we discuss them separately. We also give some warnings about the usage of the measure, especially when it comes to comparing essentially different models.     

A Note On Weighted Distributions

A representation of the "original" random variable (r.v.) in terms of the "weighted" r.v. is given and the Inverse Gaussian distribution is characterized through a distributional property the "weighted" r.v. observed under 1ength biased sampling.     

An approach to measure distance between compositional diet estimates containing essential zeros

For many applications involving compositional data, it is necessary to establish a valid measure of distance, yet when essential zeros are present traditional distance measures are problematic. In quantitative fatty acid signature analysis (QFASA), compositional diet estimates are produced that often contain many zeros. In order to test for a difference in diet between two populations of predators using the QFASA diet estimates, a legitimate measure of distance for use in the test statistic is necessary. Since ecologists using QFASA must first select the potential species of prey in the predator's diet, the chosen measure of distance should be such that the distance between samples does not decrease as the number of species considered increases, a property known in general as subcompositional coherence. In this paper we compare three measures of distance for compositional data capable of handling zeros, but not satisfying some of the well-accepted principles of compositional data analysis. For compositional diet estimates, the most relevant of these is the property of subcompositionally coherence and we show that this property may be approximately satisfied. Based on the results of a simulation study and an application to real-life QFASA diet estimates of grey seals, we recommend the chi-square measure of distance.     

Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law

When measuring units are expensive or time consuming, while ranking them is relatively easy and inexpensive, it is known that ranked set sampling (RSS) is preferable to simple random sampling (SRS). Many authors have suggested several extensions of RSS. As a variation, Al-Saleh and Al-Kadiri [Double ranked set sampling, Statist. Probab. Lett. 48 (2000), pp. 205–212] introduced double ranked set sampling (DRSS) and it was extended by Al-Saleh and Al-Omari [Multistage ranked set sampling, J. Statist. Plann. Inference 102 (2002), pp. 273–286] to multistage ranked set sampling (MSRSS). The entropy of a random variable (r.v.) is a measure of its uncertainty. It is a measure of the amount of information required on the average to determine the value of a (discrete) r.v.. In this work, we discuss entropy estimation in RSS design and aforementioned extensions and compare the results with those in SRS design in terms of bias and root mean square error (RMSE). Motivated by the above observed efficiency, we continue to investigate entropy-based goodness-of-fit test for the inverse Gaussian distribution using RSS. Critical values for some sample sizes determined by means of Monte Carlo simulations are presented for each design. A Monte Carlo power analysis is performed under various alternative hypotheses in order to compare the proposed testing procedure with the existing methods. The results indicate that tests based on RSS and its extensions are superior alternatives to the entropy test based on SRS.     

SB-robustness of directional mean for circular distributions

In this paper we study the robustness of the directional mean (a.k.a. circular mean) for different families of circular distributions. We show that the directional mean is robust in the sense of finite standardized gross error sensitivity (SB-robust) for the following families: (1) mixture of two circular normal distributions, (2) mixture of wrapped normal and circular normal distributions and (3) mixture of two wrapped normal distributions. We also show that the directional mean is not SB-robust for the family of all circular normal distributions with varying concentration parameter. We define the circular trimmed mean and prove that it is SB-robust for this family. In general the property of SB-robustness of an estimator at a family of probability distributions is dependent on the choice of the dispersion measure. We introduce the concept of equivalent dispersion measures and prove that if an estimator is SB-robust for one dispersion measure then it is SB-robust for all equivalent dispersion measures. Three different dispersion measures for circular distributions are considered and their equivalence studied.     

Tests for serial correlation in mean and variance of a sequence of time series objects

In this era of Big Data, large-scale data storage provides the motivation for statisticians to analyse new types of data. The proposed work concerns testing serial correlation in a sequence of sets of time series, here referred to as time series objects. An example is serial correlation of monthly stock returns when daily stock returns are observed. One could consider a representative or summarized value of each object to measure the serial correlation, but this approach would ignore information about the variation in the observed data. We develop Kolmogorov–Smirnov-type tests with the standard bootstrap and wild bootstrap Ljung–Box test statistics for serial correlation in mean and variance of time series objects, which take the variation within a time series object into account. We study the asymptotic property of the proposed tests and present their finite sample performance using simulated and real examples.     

Cumulative ratio information based on general cumulative entropy

In this paper, we suggest an extension of the cumulative residual entropy (CRE) and call it generalized cumulative entropy. The proposed entropy not only retains attributes of the existing uncertainty measures but also possesses the absolute homogeneous property with unbounded support, which the CRE does not have. We demonstrate its mathematical properties including the entropy of order statistics and the principle of maximum general cumulative entropy. We also introduce the cumulative ratio information as a measure of discrepancy between two distributions and examine its application to a goodness-of-fit test of the logistic distribution. Simulation study shows that the test statistics based on the cumulative ratio information have comparable statistical power with competing test statistics.     

The Effectiveness of Risk Scores: the Logit Rank Plot

A risk score s for event E is a function of covariates with the property that P ( E | s ) is an increasing function of s . Motivated by applications in medicine and in criminology, we suggest the logit rank plot as a good way of summarizing the effectiveness of such a score. Explicitly, plot logit{ P ( E | s )} against logit( r ), where r is the proportional rank of s in a sample or population. The slope of this plot gives an overall measure of effectiveness, and the logit rank transformation provides a common basis on which different risk scores can be compared. Some practical and theoretical aspects are discussed.     

A Multistage Chi-square Test for Measurement of the Degree of Association in Two-way Table

The classical chi-square test for independence cannot convey additional information and the degree of the association of two factors in two-way contingency table. Besides, measures of association by contingency coefficient need to be used with care, because the association measures depend on the number of rows r and the number of columns c. This article proposes a multistage chi-square test to measure the degree of the association between the two factors in two-way contingency table. We also give simulation and real examples to assess the performance of the proposed method. The results show that our proposed method can effectively investigate the degree of association of two factors in two-way contingency table.     

Forecasting Accuracy of Alternative Techniques: A Comparison of U.S. Macroeconomic Forecasts

This article introduces and discusses a new measure of the relative economic affluence (REA) between income distributions with different means. The REA measure D is applied to the U.S. white and black household income distributions of 1967 and 1979. The measure D shows that the REA of the white households with respect to the black households decreased from 1967 to 1979. This conclusion contrasts with those obtained by applications of distance or quasi-distance functions. It is shown in this study that REA measures and distance functions address different and relevant issues. An REA measure deals with the relation “more affluent than” and defines a partial strict ordering over the set of pairs of income distributions—that is, the relation is asymmetric and transitive—whereas a distance function accounts for the dissimilarity between distributions without imposing an ordering relation and hence fulfills the symmetry property.     

A new measure of association

This paper presents a new measure of association. It is applicable to polytomies of either categorical or numerical type. It has the desirable property of being 0 if and only if the polytomies are independent. Its properties are studied and compared to those of existing measures. An interpretation of it is given. One situation where it is particularly useful is in measuring the ability to predict one polytomy given knowledge of the other. An example is given where the proposed measure is more relevant in describing the degree of association between two polytomies than are any of the existing measures. The corresponding sample quantity is presented and its asymptotic properties are studied. A discussion of its use in inference is given. The test for independence based on this measure is contrasted with the chi-square test.     

On a new type of sufficiency in undominated models

If a family of measures is dominated by a σ-finite measure, then the classical Fisher formulation of sufficiency and the pairwise sufficiency coincide. However, in undominated models, these definitions are essentially different. In particular, the Fisher formulation lacks the basic property: if a σ-algebra is sufficient, then also any larger σ-algebra ? is sufficient. We introduce a new definition of sufficiency, based on the concept of randomization, which has the property described above. We show that in the undominated case, our definition implies pairwise sufficiency and that the converse does not hold. If we assume that the underlying measurable space is a standard Borel space, then Fisher sufficiency implies our new formulation, and the converse implication does not hold.     

The efficiency of price income situations,the real average income — a characterization

Since efficiency represents a measure of the average welfare level of the society it is often used as a synonym of “mean income” in (traditional) welfare measurement theory and neglects price information. In this note, we introduce a rather general concept of efficiency taking price information into account. Requiring some reasonable properties for efficiency judgements we look for real indicators, namely efficiency measures. By this axiomatic approach, we easily characterize a special class of (sequences of) functions that will turn out to be the real average income.     

On Two Forms of Fisher's Measure of Information

ABSTRACT

Fisher's information number is the second moment of the “score function” where the derivative is with respect to x rather than Θ. It is Fisher's information for a location parameter, and also called shift-invariant Fisher information. In recent years, Fisher's information number has been frequently used in several places regardless of parameters of the distribution or of their nature. Is this number a nominal, standard, and typical measure of information? The Fisher information number is examined in light of the properties of classical statistical information theory. It has some properties analogous to those of Fisher's measure, but, in general, it does not have good properties if used as a measure of information when Θ is not a location parameter. Even in the case of location parameter, the regularity conditions must be satisfied. It does not possess the two fundamental properties of the mother information, namely the monotonicity and invariance under sufficient transformations. Thus the Fisher information number should not be used as a measure of information (except when Θ a location parameter). On the other hand, Fisher's information number, as a characteristic of a distribution f(x), has other interesting properties. As a byproduct of its superadditivity property a new coefficient of association is introduced.     

Assessing departure from dose linearity under a repeated measures incomplete block design

Dose proportionality/linearity is a desirable property in pharmacokinetic studies. Various methods have been proposed for its assessment. When dose proportionality is not established, it is of interest to evaluate the degree of departure from dose linearity. In this paper, we propose a measure of departure from dose linearity and derive an asymptotic test under a repeated measures incomplete block design using a slope approach. Simulation studies show that the proposed method has a satisfactory small sample performance in terms of size and power.     

Uncertainty,information, and disagreement of economic forecasters

ABSTRACT

An information framework is proposed for studying uncertainty and disagreement of economic forecasters. This framework builds upon the mixture model of combining density forecasts through a systematic application of the information theory. The framework encompasses the measures used in the literature and leads to their generalizations. The focal measure is the Jensen–Shannon divergence of the mixture which admits Kullback–Leibler and mutual information representations. Illustrations include exploring the dynamics of the individual and aggregate uncertainty about the US inflation rate using the survey of professional forecasters (SPF). We show that the normalized entropy index corrects some of the distortions caused by changes of the design of the SPF over time. Bayesian hierarchical models are used to examine the association of the inflation uncertainty with the anticipated inflation and the dispersion of point forecasts. Implementation of the information framework based on the variance and Dirichlet model for capturing uncertainty about the probability distribution of the economic variable are briefly discussed.     

On the choice of optimality criteria in comparing statistical designs

We formulate in a reasonable sense a class of optimality functionals for comparing feasible statistical designs available in a given setup. It is desired that the optimality functionals reflect symmetric measures of the lack of information contained in the designs being compared. In view of this, Kiefer's (1975) universal optimality criterion is seen to rest on stringent conditions, some of which can be relaxed while preserving optimality (in an extended sense) of the so-called balanced designs.     

Discrete approximations to the Csiszár,Renyi, and Fisher measures of information

The problem of loss of information due to the discretization of data and its estimate is studied for various measures of information. The results of Ghurye and Johnson (1981) are generalized and supplemented for the Csiszár and Renyi measures of information as well as for Fisher's information matrix.     

On the geometric behaviour of multidimensional location measures

We consider several generic geometric properties of location measures in the multivariate setting. Our study addresses the representativity of location measures by considering possible generalizations of the so-called Cauchy mean value property in a general framework mainly based on Φ-means.Our study shows that some caution is needed when using intuitive arguments because the mean is, in some sense, the only “intuitively well-behaved” multivariate location measure.     

Poisson bounds for moments of shot noise processes

Starting from general moment formulas for shot noise processes (v(t)), for which a simple proof is given by means of the CAMPBELL theorem concerning marked point pro¬cesses, and using certain monotonicity property, bounds are obtained for E[v(t)]² resp. for the variance Var v(t) under the additional assumption that the intershot times are inde¬pendent random variables with distribution having the property 1SFBXJE (NWTJE)