Statistical Hypothesis Testing Problem Research Articles

TESTING STATISTICAL HYPOTHESES UNDER FUZZY DATA AND BASED ON A NEW SIGNED DISTANCE

This paper deals with the problem of testing statisticalhypotheses when the available data are fuzzy. In this approach, wefirst obtain a fuzzy test statistic based on fuzzy data, and then,based on a new signed distance between fuzzy numbers, we introducea new decision rule to accept/reject the hypothesis of interest.The proposed approach is investigated for two cases: the casewithout nuisance parameters and the case with nuisance parameters.This method is employed to test the hypotheses for the mean of anormal distribution with known/unknown variance, the variance of anormal distribution, the difference of means of two normaldistributions with known/unknown variances, and the ratio ofvariances of two normal distributions.

Detecting Localized Categorical Attributes on Graphs

Do users from Carnegie Mellon University form social communities on Facebook? Do signal processing researchers from tightly collaborate with each other? Do Chinese restaurants in Manhattan cluster together? These seemingly different problems share a common structure: an attribute that may be localized on a graph. In other words, nodes activated by an attribute form a subgraph that can be easily separated from other nodes. In this paper, we thus focus on the task of detecting localized attributes on a graph. We are particularly interested in categorical attributes such as attributes in online social networks, ratings in recommender systems and viruses in cyber-physical systems because they are widely used in numerous data mining applications. To solve the task, we formulate a statistical hypothesis testing problem to decide whether a given attribute is localized or not. We propose two statistics: graph wavelet statistic and graph scan statistic, both of which are provably effective in detecting localized attributes. We validate the robustness of the proposed statistics on both simulated data and two real-world applications: high air-pollution detection and keyword ranking in a co-authorship network collected from IEEE Xplore. Experimental results show that the proposed graph wavelet statistic and graph scan statistic are effective and efficient.

A Note on the Connection between Likelihood Inference, Bayes Factors, and P-Values

The p-value is widely used for quantifying evidence in a statistical hypothesis testing problem. A major criticism, however, is that the p-value does not satisfy the likelihood principle. In this paper, we show that a p-value assessment of evidence can indeed be defined within the likelihood inference framework. Included within this framework is a link between a p-value and the likelihood ratio statistic. Thus, a link between a p-value and the Bayes factor can also be highlighted. The connection between p-values and likelihood based measures of evidence broaden the use of the p-value and deepen our understanding of statistical hypothesis testing.

On methods for correcting for the look-elsewhere effect in searches for new physics

The search for new significant peaks over a energy spectrum often involves a statistical multiple hypothesis testing problem. Separate tests of hypothesis are conducted at different locations over a fine grid producing an ensemble of local p-values, the smallest of which is reported as evidence for the new resonance. Unfortunately, controlling the false detection rate (type I error rate) of such procedures may lead to excessively stringent acceptance criteria. In the recent physics literature, two promising statistical tools have been proposed to overcome these limitations. In 2005, a method to ``find needles in haystacks'' was introduced by Pilla et al. [1], and a second method was later proposed by Gross and Vitells [2] in the context of the ``look-elsewhere effect'' and trial factors. We show that, although the two methods exhibit similar performance for large sample sizes, for relatively small sample sizes, the method of Pilla et al. leads to an artificial inflation of statistical power that stems from an increase in the false detection rate. This method, on the other hand, becomes particularly useful in multidimensional searches, where the Monte Carlo simulations required by Gross and Vitells are often unfeasible. We apply the methods to realistic simulations of the Fermi Large Area Telescope data, in particular the search for dark matter annihilation lines. Further, we discuss the counter-intuitive scenario where the look-elsewhere corrections are more conservative than much more computationally efficient corrections for multiple hypothesis testing. Finally, we provide general guidelines for navigating the tradeoffs between statistical and computational efficiency when selecting a statistical procedure for signal detection.

Interval Estimation and Hypothesis Testing

Mathematical statistics is a branch of mathematics has extensive application of interval estimation and hypothesis testing, which are two important problems of statistical inference. As two important statistical inference methods, interval estimation and hypothesis testing problem is more and more widely used in the field of economic management, finance and insurance, scientific research, engineering technology, the science of decision functions are recognized by more and more people. Can go further to establish mutual influence and communication between the interval estimation and hypothesis testing, can use the theory to explain the problem of interval estimation of parameter hypothesis test, this is an important problem to improve the statistical inference theory. Therefore, the basis on the internal relations between the interval estimation and hypothesis test for deep research, explain the problem of hypothesis testing and interval estimation from the point of view, discusses the difference and connection between the two.

Assessing compatibility of two laboratories: formulations as a statistical hypothesis testing problem

A decision problem frequently occurring in metrology is that of compatibility of data obtained by two (or several) different laboratories, methods or instruments. One laboratory can be a prestigious national metrology institute whose data are considered to be a gold standard or a certified reference material interval. When each laboratory presents its results in the form of a coverage interval for the measurand, several statistical approaches to this conformity assessment problem are reviewed including the classical ‘equality of means’ hypotheses tests. A new compatibility hypothesis is formulated in terms of consistency of laboratory results and compliance with a maximum permitted uncertainty. The power functions of these tests are compared numerically. The Kullback–Leibler information number is suggested as a directional (asymmetric) interchangeability index.

Testing fuzzy hypotheses based on vague observations: a p-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.

Improving the characterization of the alternative hypothesis via minimum verification error training with applications to speaker verification

Speaker verification is usually formulated as a statistical hypothesis testing problem and solved by a log-likelihood ratio (LLR) test. A speaker verification system's performance is highly dependent on modeling the target speaker's voice (the null hypothesis) and characterizing non-target speakers’ voices (the alternative hypothesis). However, since the alternative hypothesis involves unknown impostors, it is usually difficult to characterize a priori. In this paper, we propose a framework to better characterize the alternative hypothesis with the goal of optimally distinguishing the target speaker from impostors. The proposed framework is built on a weighted arithmetic combination (WAC) or a weighted geometric combination (WGC) of useful information extracted from a set of pre-trained background models. The parameters associated with WAC or WGC are then optimized using two discriminative training methods, namely, the minimum verification error (MVE) training method and the proposed evolutionary MVE (EMVE) training method, such that both the false acceptance probability and the false rejection probability are minimized. Our experiment results show that the proposed framework outperforms conventional LLR-based approaches.

Incorporating gene networks into statistical tests for genomic data via a spatially correlated mixture model

It is a common task in genomic studies to identify a subset of the genes satisfying certain conditions, such as differentially expressed genes or regulatory target genes of a transcription factor (TF). This can be formulated as a statistical hypothesis testing problem. Most existing approaches treat the genes as having an identical and independent distribution a priori, testing each gene independently or testing some subsets of the genes one by one. On the other hand, it is known that the genes work coordinately as dictated by gene networks. Treating genes equally and independently ignores the important information contained in gene networks, leading to inefficient analysis and reduced power. We propose incorporating gene network information into statistical analysis of genomic data. Specifically, rather than treating the genes equally and independently a priori in a standard mixture model, we assume that gene-specific prior probabilities are correlated as induced by a gene network: while the genes are allowed to have different prior probabilities, those neighboring ones in the network have similar prior probabilities, reflecting their shared biological functions. We applied the two approaches to a real ChIP-chip dataset (and simulated data) to identify the transcriptional target genes of TF GCN4. The new method was found to be more powerful in discovering the target genes.

ROBUST TEST BASED ON NONLINEAR REGRESSION QUANTILE ESTIMATORS

In this paper we consider the problem of testing statistical hypotheses for unknown parameters in nonlinear regression models and propose three asymptotically equivalent tests based on regression quantiles estimators, which are Wald test, Lagrange Multiplier test and Likelihood Ratio test. We also derive the asymptotic distributions of the three test statistics both under the null hypotheses and under a sequence of local alternatives and verify that the asymptotic relative efficiency of the proposed test statistics with classical test based on least squares depends on the error distributions of the regression models. We give some examples to illustrate that the test based on the regression quantiles estimators performs better than the test based on the least squares estimators of the least absolute deviation estimators when the disturbance has asymmetric and heavy-tailed distribution.

Compressed video processing for cut detection

One of the challenging problems in video databases is the organisation of video information. Segmenting a video into a number of clips and characterising each clip has been suggested as one mechanism for organising video information. This approach requires a suitable method to automatically locate cut points (boundaries between consecutive camera shots in a video). Several existing techniques solve this problem using uncompressed video. Since video is increasingly being captured, moved, and stored in compressed form, there is a need for detecting shot boundaries directly in compressed video. The authors address this issue and show certain feature extraction steps in MPEG compressed video that allow the implementation of most of the existing cut detection methods developed for uncompressed video for MPEG video stream. They also examine the performance of three tests for cut detection by viewing the problem of cut detection as a statistical hypothesis testing problem. As the experimental results indicate, the statistical hypothesis testing approach permits fast and accurate detection of video cuts.

Blunder Detection and Data Snooping in LS and Robust Adjustments

There are many procedures for adjusting data and detecting the presence of blunders in a set of observations. Most such procedures involve examining the adjustment results for residuals whose magnitude is in some sense “large.” In data snooping, each residual is divided by its own standard deviation, resulting in a statistic whose distribution is known. Thus blunder detection becomes a statistical hypothesis testing problem. In iterated data snooping, only the observation with the largest normalized residual is deleted at each iteration. The residuals may be either from a conventional least‐squares LS adjustment or from an “L1 norm” adjustment that seeks to minimize the sum of absolute values of residuals. Iterated data snooping with the LS residuals is at least as effective as any other method of detecting blunders. We show by example that it can produce much better results than the L1 norm adjustment. Both forms of data snooping are superior to the many empirical or approximate methods that are still widely used.

The asymptotically pointwise optimal testing rule and its competitors – an empirical bayes approach

In this paper,the performance of asymptotically pointwise optimal procedures is studied for fixed c, the cost per unit sample. Our treatment is focused on a statistical hypothesis testing problem, namely,the linear loss two-action problem. Using the empirical Bayes approach, we construct a sequential testing procedure which dominates the A.P.O. testing procedure when the number of auxiliary data available is large. It is shown that this domination occurs for most natural cases of the A.P.O. stopping rule. The comparison of the two procedures is made by comparing their respective Bayes risks.

Automated inspection of textile fabrics using textural models

The authors discuss the problem of textile fabric inspection using the visual textural properties of the fabric. The problem is to detect and locate the various kinds of defects that might be present in a given fabric sample based on an image of the fabric. Stochastic models are used to model the visual fabric texture. The authors use the Gaussian Markov random field to model the texture image of nondefective fabric. The inspection problem is cast as a statistical hypothesis testing problem on statistics derived from the model. The image of the fabric patch to be inspected is partitioned into nonoverlapping windows of size N*N where each window is classified as defective or nondefective based on a likelihood ratio test of size alpha . The test is recast in terms of the sufficient statistics associated with the model parameters. The sufficient statistics are easily computable for any sample. The authors generalize the test when the model parameters of the fabric are assumed to be unknown. >

The fuzzy decision problem: An approach to the problem of testing statistical hypotheses with fuzzy information

This paper is devoted to the problems of testing statistical hypotheses about an experiment, when the available information from its sampling is `vague'. When the information supplied by the experimental sampling is exact, the problems of testing statistical hypotheses about the experiment can be regarded as a particular statistical decision problem. In addition, decision procedures may be used in problems of testing hypotheses. In a similar manner, the problem of testing statistical hypotheses about an experiment when the available sample information is vague, is approached in this paper as a particular fuzzy decision problem (as defined by Tanaka, Okuda and Asai). This approach assumes that the previous information about the experiment can be expressed by means of certain conditional probabilistic information, whereas the present information about it can be expressed by means of fuzzy information. The preceding framework allows us to extend the notion of risk function and some nonfuzzy decision procedures to the fuzzy case, and particularize them to the problem of testing. Finally, several illustrative examples are presented.

Likelihood Ratio Test, Wald Test, and Kuhn-Tucker Test in Linear Models with Inequality Constraints on the Regression Parameters

This paper considers the problem of testing statistical hypotheses in linear regression models with inequality constraints on the regression coefficients. The Kuhn-Tucker multiplier test statistic is defined and its relationships with the likelihood ratio test and the Wald test are examined. It is shown, in particular, that these relationships are the same as in the equality constrained case. It is emphasized, however, that their common asymptotic distribution is a mixture of chi-square distributions under the null hypothesis.

Chernoff bounds for discriminating between two markov processes

We study a statistical hypothesis testing problem, where a sample function of a Markov process with one of two sets of known parameters is observed over a finite time interval. When a log likelihood ratio test is used to discriminate between the two sets of parameters, we give bounds on the probability of choosing an incorrect hypothesis, and on the total probability of error, for both discrete and continuous time parameter, and discrete and continuous state space. The asymptotic behavior of the bounds is examined as the observation interval becomes infinite.

Modern statistical approaches to reception in communication theory

When reception in the theory of communication is recognized as a problem in statistical inference, system design and system analysis appear as the counterparts of designing and evaluating statistical tests. This paper discusses the optimum properties of designs based on statistical decision theory from the risk point of view, and from that of information theory. Connections between risk and information loss are established, which result in a unified theory of system design. This includes Minimax methods capable in principle of handling all degrees of a priori knowledge of signal and noise statistics, new methods for comparing actual and ideal systems for the same purpose, and new interpretations of previously used formulations as special cases of the more general theory. Both detection and extraction of signals in noise are considered, the former as a problem of testing statistical hypotheses and the latter as one of estimating parameters. Formulation of the general reception problem as a decision operation is followed by a summary of statistical decision theory from the risk point of view, with some examples of Bayes and Minimax tests and optimum classes of decision rules. Applications to detection show the optimum nature of likelihood ratio receivers as a class, and indicate methods for defining the minimum detectable signal and for comparing system performance. As an illustration, curves of Bayes and Minimax risk are given for detection of a pulsed carrier in noise. Applications to extraction show the nature of optimum extraction and the roles of the mean square error and maximum likelihood criteria from the more general point of view of risk theory. Conditions under which information loss is an extremum in detection and extraction are established, and information loss itself as a criterion of performance is compared with that of the risk measure.

IX. On the problem of the most efficient tests of statistical hypotheses

The problem of testing statistical hypotheses is an old one. Its origin is usually connected with the name of Thomas Bayes, who gave the well-known theorem on the probabilities a posteriori of the possible “causes" of a given event. Since then it has been discussed by many writers of whom we shall here mention two only, Bertrand and Borel, whose differing views serve well to illustrate the point from which we shall approach the subject. Bertrand put into statistical form a variety of hypotheses, as for example the hypothesis that a given group of stars with relatively small angular distances between them as seen from the earth, form a “system” or group in space. His method of attack, which is that in common use, consisted essentially in calculating the probability, P, that a certain character, x , of the observed facts would arise if the hypothesis tested were true. If P were very small, this would generally be considered as an indication that the hypothesis, H, was probably false, and vice versa . Bertrand expressed the pessimistic view that no test of this kind could give reliable results. Borel, however, in a later discussion, considered that the method described could be applied with success provided that the character, x , of the observed facts were properly chosen—were, in fact, a character which he terms “en quelque sorte remarquable.”