A Generalized Version of Neyman-Pearson Lemma for Testing Fuzzy Hypotheses Based on r-Level Sets

In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts.

In this article, we first redefine some concepts in testing of fuzzy hypotheses and then introduce a generalized version of Neyman-Pearson lemma for testing fuzzy hypotheses using r-levels. Finally, two numerical examples are presented to demonstrate the proposed approach.     

Fuzzy p-value in testing fuzzy hypotheses with crisp data

In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp. We first introduce the notion of fuzzy p-value, by applying the extension principle and then present an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level, based on a comparison of two fuzzy sets. Numerical examples are also provided to illustrate the approach.     

A new generation of process capability indices

In quality control, we may confront imprecise concepts. One case is a situation in which upper and lower specification limits (SLs) are imprecise. If we introduce vagueness into SLs, we face quite new, reasonable and interesting processes, and the ordinary capability indices are not appropriate for measuring the capability of these processes. In this paper, similar to the traditional process capability indices (PCIs), we develop a fuzzy analogue by a distance defined on a fuzzy limit space and introduce PCIs, where instead of precise SLs we have two membership functions for upper and lower SLs. These indices are necessary when SLs are fuzzy, and they are helpful for comparing manufacturing process with fuzzy SLs. Some interesting relations among these introduced indices are proved. Numerical examples are given to clarify the method.     

Contingency tables with fuzzy information

ABSTRACT

This paper extends the classical methods of analysis of a two-way contingency table to the fuzzy environment for two cases: (1) when the available sample of observations is reported as imprecise data, and (2) the case in which we prefer to categorize the variables based on linguistic terms rather than as crisp quantities. For this purpose, the α-cuts approach is used to extend the usual concepts of the test statistic and p-value to the fuzzy test statistic and fuzzy p-value. In addition, some measures of association are extended to the fuzzy version in order to evaluate the dependence in such contingency tables. Some practical examples are provided to explain the applicability of the proposed methods in real-world problems.     

Testing fuzzy hypotheses based on vague observations: a <Emphasis Type="Italic">p</Emphasis>-value approach

This paper deals with the problem of testing statistical hypotheses when both the hypotheses and data are fuzzy. To this end, we first introduce the concept of fuzzy p-value and then develop an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level. Numerical examples are provided to illustrate the approach for different cases.     

Bayesian (mean) most powerful tests

A fundamental theorem in hypothesis testing is the Neyman‐Pearson (N‐P) lemma, which creates the most powerful test of simple hypotheses. In this article, we establish Bayesian framework of hypothesis testing, and extend the Neyman‐Pearson lemma to create the Bayesian most powerful test of general hypotheses, thus providing optimality theory to determine thresholds of Bayes factors. Unlike conventional Bayes tests, the proposed Bayesian test is able to control the type I error.     

Testing for Structural Instability in Moment Restriction Models: An Info-Metric Approach

In this paper, we develop an info-metric framework for testing hypotheses about structural instability in nonlinear, dynamic models estimated from the information in population moment conditions. Our methods are designed to distinguish between three states of the world: (i) the model is structurally stable in the sense that the population moment condition holds at the same parameter value throughout the sample; (ii) the model parameters change at some point in the sample but otherwise the model is correctly specified; and (iii) the model exhibits more general forms of instability than a single shift in the parameters. An advantage of the info-metric approach is that the null hypotheses concerned are formulated in terms of distances between various choices of probability measures constrained to satisfy (i) and (ii), and the empirical measure of the sample. Under the alternative hypotheses considered, the model is assumed to exhibit structural instability at a single point in the sample, referred to as the break point; our analysis allows for the break point to be either fixed a priori or treated as occuring at some unknown point within a certain fraction of the sample. We propose various test statistics that can be thought of as sample analogs of the distances described above, and derive their limiting distributions under the appropriate null hypothesis. The limiting distributions of our statistics are nonstandard but coincide with various distributions that arise in the literature on structural instability testing within the Generalized Method of Moments framework. A small simulation study illustrates the finite sample performance of our test statistics.     

Robust inference by influence functions

We first review briefly some basic approaches to robust inference and discuss the role and the place of some key concepts (influence function, breakdown point, robustness versus efficiency, etc.). We then discuss in some detail recent results on robust testing in general multivariate parametric models. Recent applications include inference in logistic regression and testing for non-nested hypotheses.     

Evaluating lifetime performance for the Pareto model with censored and imprecise information

A lifetime capability index L _tp has been proposed to measure the business lifetime performance, wherein output lifetime measurements are assumed to be precise from the Pareto model with censored information. In the present study, we study a more realistic situation where the lifetime output data are imprecise. The approach developed by Buckley [Fuzzy system, Soft Comput. 9 (2005), pp. 757–760; Fuzzy statistics: Regression and prediction, Soft Comput. 9 (2005), pp. 769–775] incorporated with some extensions (a set of confidence intervals, one on top of the other), is used to construct the triangular-shaped fuzzy number for the fuzzy estimate of the L _tp. With the sampling distribution of the unbiased estimator of the L _tp, two useful fuzzy inference criteria, its critical value and fuzzy p-value are obtained to assess the lifetime performance. The presented methodology can handle the lifetime performance assessment on the condition that sample lifetime data are involved with imprecise information, classifying the lifetime performance with the three-decision rule. With different preset requirements and a certain degree of imprecise data, we also develop a four quadrants decision-making plot where managers can easily simultaneously visualize several important features of lifetime performance for making a decision. An example of business lifetime data is given to illustrate the applicability of the proposed method.     

On the Hypothesis of No Interaction on a Linear Scale in Contingency Tables

The purpose of this paper is to review briefly the three main formulations of no Interaction hypotheses in contingency tables and to consider the formulation on a linear scale in some detail.More specifically we (i) present a situation in 2×2 tables where such a formulation may be more appropriate than others, (ii) study the geometry for this problem, (iii) give contrast-type or parametric ANOVA type formulations in the general n-dimensional tables, (iv) discuss estimation and testing procedures and (v) consider collapsibility of contingency tables in relation to the hypotheses of no interaction on a linear scale.     

Percentage points of the statistics for testing hypotheses on mean vectors of multivariate normal distributions with missing observations

A problem of testing of hypotheses on the mean vector of a multivariate normal distribution with unknown and positive definite covariance matrix is considered when a sample with a special, though not unusual, pattern of missing observations from that population is available. The approximate percentage points of the test statistic are obtained and their accuracy has been checked by comparing them with some exact percentage points which are calculated for complete samples and some special incomplete samples. The approximate percentage points are in good agreement with exact percentage points. The above work is extended to the problem of testing the hypothesis of equality of two mean vectors of two multivariate normal distributions with the same, unknown covariance matrix     

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.     

The direct use of likelihood for significance testing

An approach to significance testing by the direct interpretation of likelihood is defined, developed and distinguished from the traditional forms of tail-area testing and Bayesian testing. The emphasis is on conceptual issues. Some theoretical aspects of the new approach are sketched in the two cases of simple vs. simple hypotheses and simple vs. composite hypotheses.     

An Estimation Method of the Pair Potential Function for Gibbs Point Processes on Spheres

An estimation method for pairwise interaction potential of a stationary Gibbs point process is introduced by considering the case of observations located on a sphere. It is based both on Fourier decomposition of the potential and on minimum contrast estimation. It is defined when many independent realizations of the process are available. Consistency and asymptotic normality are proved for the resulting estimators. The method enables derivation of the choice of the potential function by embedded hypotheses testing. The method is applied to independent observations of root locations on internodes around stem of maize roots. The internodes are described as circles and we focus on the interaction function associated with the potential. Since a model with too many components seems to fail, we choose a sequential procedure based on embedded hypotheses testing to build a simpler model.     

Generalized Likelihood Ratio Tests and Uniformly Most Powerful Tests

For testing a one-sided hypothesis in a one-parameter family of distributions, it is shown that the generalized likelihood ratio (GLR) test coincides with the uniformly most powerful (UMP) test, assuming certain monotonicity properties for the likelihood function. In particular, the equivalence of GLR tests and UMP tests holds for one-parameter exponential families. In addition, the relationship between GLR and UMPU (UMP unbiased) tests is considered when testing two-sided hypotheses.     

Statistical test based on intuitionistic fuzzy hypotheses

In this paper, the classical statistical test based on intuitionistic fuzzy hypotheses in relation to the underlying population parametric is extended. In this approach, the type-I, type-II, power of test, and p-value are extended for intuitionistic fuzzy hypotheses. Throughout the paper, some applied examples are provided for both parametric and non parametric cases to clarify the discussions.     

Nonparametric Bayes factors based on empirical likelihood ratios

Bayes methodology provides posterior distribution functions based on parametric likelihoods adjusted for prior distributions. A distribution-free alternative to the parametric likelihood is use of empirical likelihood (EL) techniques, well known in the context of nonparametric testing of statistical hypotheses. Empirical likelihoods have been shown to exhibit many of the properties of conventional parametric likelihoods. In this paper, we propose and examine Bayes factors (BF) methods that are derived via the EL ratio approach. Following Kass and Wasserman (1995), we consider Bayes factors type decision rules in the context of standard statistical testing techniques. We show that the asymptotic properties of the proposed procedure are similar to the classical BF's asymptotic operating characteristics. Although we focus on hypothesis testing, the proposed approach also yields confidence interval estimators of unknown parameters. Monte Carlo simulations were conducted to evaluate the theoretical results as well as to demonstrate the power of the proposed test.     

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model''s parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.     

Inferences on the ratio of two generalized variances: independent and correlated cases

Statistical inferences about the dispersion of multivariate population are determined by generalized variance. In this article, we consider constructing a confidence interval and testing the hypotheses about the ratio of two independent generalized variances, and the ratio of two dependent generalized variances in two multivariate normal populations. In the case of independence, we first propose a computational approach and then obtain an approximate approach. In the case of dependence, we give an approach using the concepts of generalized confidence interval and generalized p value. In each case, simulation studies are performed for comparing the methods and we find satisfactory results. Practical examples are given for each approach.     

Mixture Models in View of Evidential Analysis

In some practical inferential situations, it is needed to mix some finite sort of distributions to fit an adequate model for multi-modal observations. In this article, using evidential analysis, we determine the sample size for supporting hypotheses about the mixture proportion and homogeneity. An Expectation-Maximization algorithm is used to evaluate the probability of strong misleading evidence based on modified likelihood ratio as a measure of support.