scholarly journals Privacy-Aware Distributed Hypothesis Testing

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 665 ◽  
Author(s):  
Sreejith Sreekumar ◽  
Asaf Cohen ◽  
Deniz Gündüz
Keyword(s):  
Error Probability ◽  
Type I Error ◽  
Single Letter ◽  
Type I ◽  
Type Ii ◽  
Type Ii Error ◽  
Error Exponent ◽  
Trade Offs ◽  
Type I Error Probability ◽  
Binary Hypothesis

A distributed binary hypothesis testing (HT) problem involving two parties, a remote observer and a detector, is studied. The remote observer has access to a discrete memoryless source, and communicates its observations to the detector via a rate-limited noiseless channel. The detector observes another discrete memoryless source, and performs a binary hypothesis test on the joint distribution of its own observations with those of the observer. While the goal of the observer is to maximize the type II error exponent of the test for a given type I error probability constraint, it also wants to keep a private part of its observations as oblivious to the detector as possible. Considering both equivocation and average distortion under a causal disclosure assumption as possible measures of privacy, the trade-off between the communication rate from the observer to the detector, the type II error exponent, and privacy is studied. For the general HT problem, we establish single-letter inner bounds on both the rate-error exponent-equivocation and rate-error exponent-distortion trade-offs. Subsequently, single-letter characterizations for both trade-offs are obtained (i) for testing against conditional independence of the observer’s observations from those of the detector, given some additional side information at the detector; and (ii) when the communication rate constraint over the channel is zero. Finally, we show by providing a counter-example where the strong converse which holds for distributed HT without a privacy constraint does not hold when a privacy constraint is imposed. This implies that in general, the rate-error exponent-equivocation and rate-error exponent-distortion trade-offs are not independent of the type I error probability constraint.

Hypothesis Testing in Business Administration

2020 ◽  
Author(s):  
Rand R. Wilcox
Keyword(s):  
Hypothesis Testing ◽  
Empirical Evidence ◽  
Confidence Intervals ◽  
Error Probability ◽  
Type I Error ◽  
Type I ◽  
Type Ii ◽  
Practical Concern ◽  
Type I Error Probability ◽  
The Difference

Hypothesis testing is an approach to statistical inference that is routinely taught and used. It is based on a simple idea: develop some relevant speculation about the population of individuals or things under study and determine whether data provide reasonably strong empirical evidence that the hypothesis is wrong. Consider, for example, two approaches to advertising a product. A study might be conducted to determine whether it is reasonable to assume that both approaches are equally effective. A Type I error is rejecting this speculation when in fact it is true. A Type II error is failing to reject when the speculation is false. A common practice is to test hypotheses with the type I error probability set to 0.05 and to declare that there is a statistically significant result if the hypothesis is rejected. There are various concerns about, limitations to, and criticisms of this approach. One criticism is the use of the term significant. Consider the goal of comparing the means of two populations of individuals. Saying that a result is significant suggests that the difference between the means is large and important. But in the context of hypothesis testing it merely means that there is empirical evidence that the means are not equal. Situations can and do arise where a result is declared significant, but the difference between the means is trivial and unimportant. Indeed, the goal of testing the hypothesis that two means are equal has been criticized based on the argument that surely the means differ at some decimal place. A simple way of dealing with this issue is to reformulate the goal. Rather than testing for equality, determine whether it is reasonable to make a decision about which group has the larger mean. The components of hypothesis-testing techniques can be used to address this issue with the understanding that the goal of testing some hypothesis has been replaced by the goal of determining whether a decision can be made about which group has the larger mean. Another aspect of hypothesis testing that has seen considerable criticism is the notion of a p-value. Suppose some hypothesis is rejected with the Type I error probability set to 0.05. This leaves open the issue of whether the hypothesis would be rejected with Type I error probability set to 0.025 or 0.01. A p-value is the smallest Type I error probability for which the hypothesis is rejected. When comparing means, a p-value reflects the strength of the empirical evidence that a decision can be made about which has the larger mean. A concern about p-values is that they are often misinterpreted. For example, a small p-value does not necessarily mean that a large or important difference exists. Another common mistake is to conclude that if the p-value is close to zero, there is a high probability of rejecting the hypothesis again if the study is replicated. The probability of rejecting again is a function of the extent that the hypothesis is not true, among other things. Because a p-value does not directly reflect the extent the hypothesis is false, it does not provide a good indication of whether a second study will provide evidence to reject it. Confidence intervals are closely related to hypothesis-testing methods. Basically, they are intervals that contain unknown quantities with some specified probability. For example, a goal might be to compute an interval that contains the difference between two population means with probability 0.95. Confidence intervals can be used to determine whether some hypothesis should be rejected. Clearly, confidence intervals provide useful information not provided by testing hypotheses and computing a p-value. But an argument for a p-value is that it provides a perspective on the strength of the empirical evidence that a decision can be made about the relative magnitude of the parameters of interest. For example, to what extent is it reasonable to decide whether the first of two groups has the larger mean? Even if a compelling argument can be made that p-values should be completely abandoned in favor of confidence intervals, there are situations where p-values provide a convenient way of developing reasonably accurate confidence intervals. Another argument against p-values is that because they are misinterpreted by some, they should not be used. But if this argument is accepted, it follows that confidence intervals should be abandoned because they are often misinterpreted as well. Classic hypothesis-testing methods for comparing means and studying associations assume sampling is from a normal distribution. A fundamental issue is whether nonnormality can be a source of practical concern. Based on hundreds of papers published during the last 50 years, the answer is an unequivocal Yes. Granted, there are situations where nonnormality is not a practical concern, but nonnormality can have a substantial negative impact on both Type I and Type II errors. Fortunately, there is a vast literature describing how to deal with known concerns. Results based solely on some hypothesis-testing approach have clear implications about methods aimed at computing confidence intervals. Nonnormal distributions that tend to generate outliers are one source for concern. There are effective methods for dealing with outliers, but technically sound techniques are not obvious based on standard training. Skewed distributions are another concern. The combination of what are called bootstrap methods and robust estimators provides techniques that are particularly effective for dealing with nonnormality and outliers. Classic methods for comparing means and studying associations also assume homoscedasticity. When comparing means, this means that groups are assumed to have the same amount of variance even when the means of the groups differ. Violating this assumption can have serious negative consequences in terms of both Type I and Type II errors, particularly when the normality assumption is violated as well. There is vast literature describing how to deal with this issue in a technically sound manner.

scholarly journals BANKRUPTCY PREDICTION MODEL WITH ZETAc OPTIMAL CUT-OFF SCORE TO CORRECT TYPE I ERRORS

2005 ◽  
Vol 7 (1) ◽  
pp. 41 ◽  
Author(s):  
Mohamad Iwan
Keyword(s):  
Type I Error ◽  
Bankruptcy Prediction ◽  
Type I ◽  
Type Ii ◽  
Type Ii Error ◽  
Prior Probabilities ◽  
Prediction Result ◽  
Optimum Cutting ◽  
Classification Errors ◽  
One Year

This research examines financial ratios that distinguish between bankrupt and non-bankrupt companies and make use of those distinguishing ratios to build a one-year prior to bankruptcy prediction model. This research also calculates how many times the type I error is more costly compared to the type II error. The costs of type I and type II errors (cost of misclassification errors) in conjunction to the calculation of prior probabilities of bankruptcy and non-bankruptcy are used in the calculation of the ZETAc optimal cut-off score. The bankruptcy prediction result using ZETAc optimal cut-off score is compared to the bankruptcy prediction result using a cut-off score which does not consider neither cost of classification errors nor prior probabilities as stated by Hair et al. (1998), and for later purposes will be referred to Hair et al. optimum cutting score. Comparison between the prediction results of both cut-off scores is purported to determine the better cut-off score between the two, so that the prediction result is more conservative and minimizes expected costs, which may occur from classification errors.  This is the first research in Indonesia that incorporates type I and II errors and prior probabilities of bankruptcy and non-bankruptcy in the computation of the cut-off score used in performing bankruptcy prediction. Earlier researches gave the same weight between type I and II errors and prior probabilities of bankruptcy and non-bankruptcy, while this research gives a greater weigh on type I error than that on type II error and prior probability of non-bankruptcy than that on prior probability of bankruptcy.This research has successfully attained the following results: (1) type I error is in fact 59,83 times more costly compared to type II error, (2) 22 ratios distinguish between bankrupt and non-bankrupt groups, (3) 2 financial ratios proved to be effective in predicting bankruptcy, (4) prediction using ZETAc optimal cut-off score predicts more companies filing for bankruptcy within one year compared to prediction using Hair et al. optimum cutting score, (5) Although prediction using Hair et al. optimum cutting score is more accurate, prediction using ZETAc optimal cut-off score proved to be able to minimize cost incurred from classification errors.

Is It Really Robust?

Methodology ◽  
2010 ◽  
Vol 6 (4) ◽  
pp. 147-151 ◽  
Author(s):  
Emanuel Schmider ◽  
Matthias Ziegler ◽  
Erik Danay ◽  
Luzi Beyer ◽  
Markus Bühner
Keyword(s):  
Goodness Of Fit ◽  
Type I Error ◽  
Effect Sizes ◽  
Random Numbers ◽  
Type I ◽  
Type Ii ◽  
Type Ii Error ◽  
Normality Assumption ◽  
Different Types ◽  
Factor Type

Empirical evidence to the robustness of the analysis of variance (ANOVA) concerning violation of the normality assumption is presented by means of Monte Carlo methods. High-quality samples underlying normally, rectangularly, and exponentially distributed basic populations are created by drawing samples which consist of random numbers from respective generators, checking their goodness of fit, and allowing only the best 10% to take part in the investigation. A one-way fixed-effect design with three groups of 25 values each is chosen. Effect-sizes are implemented in the samples and varied over a broad range. Comparing the outcomes of the ANOVA calculations for the different types of distributions, gives reason to regard the ANOVA as robust. Both, the empirical type I error α and the empirical type II error β remain constant under violation. Moreover, regression analysis identifies the factor “type of distribution” as not significant in explanation of the ANOVA results.

scholarly journals On the Performances of Trend and Change-Point Detection Methods for Remote Sensing Data

2020 ◽  
Vol 12 (6) ◽  
pp. 1008 ◽  
Author(s):  
Ana Militino ◽  
Mehdi Moradi ◽  
M. Ugarte
Keyword(s):  
Error Probability ◽  
Type I Error ◽  
Remote Sensing Data ◽  
Change Point Detection ◽  
Change Points ◽  
Detection Methods ◽  
Type I ◽  
Power Of The Test ◽  
Type I Error Probability ◽  
Point Detection

Detecting change-points and trends are common tasks in the analysis of remote sensing data. Over the years, many different methods have been proposed for those purposes, including (modified) Mann–Kendall and Cox–Stuart tests for detecting trends; and Pettitt, Buishand range, Buishand U, standard normal homogeneity (Snh), Meanvar, structure change (Strucchange), breaks for additive season and trend (BFAST), and hierarchical divisive (E.divisive) for detecting change-points. In this paper, we describe a simulation study based on including different artificial, abrupt changes at different time-periods of image time series to assess the performances of such methods. The power of the test, type I error probability, and mean absolute error (MAE) were used as performance criteria, although MAE was only calculated for change-point detection methods. The study reveals that if the magnitude of change (or trend slope) is high, and/or the change does not occur in the first or last time-periods, the methods generally have a high power and a low MAE. However, in the presence of temporal autocorrelation, MAE raises, and the probability of introducing false positives increases noticeably. The modified versions of the Mann–Kendall method for autocorrelated data reduce/moderate its type I error probability, but this reduction comes with an important power diminution. In conclusion, taking a trade-off between the power of the test and type I error probability, we conclude that the original Mann–Kendall test is generally the preferable choice. Although Mann–Kendall is not able to identify the time-period of abrupt changes, it is more reliable than other methods when detecting the existence of such changes. Finally, we look for trend/change-points in land surface temperature (LST), day and night, via monthly MODIS images in Navarre, Spain, from January 2001 to December 2018.

When Studies are in Error: Basic Statistical Vocabulary Needed to Understand Clinical Studies

1996 ◽  
Vol 1 (1) ◽  
pp. 25-28 ◽  
Author(s):  
Martin A. Weinstock
Keyword(s):  
Null Hypothesis ◽  
Statistical Power ◽  
Critical Appraisal ◽  
Type I Error ◽  
Statistical Significance ◽  
P Value ◽  
Type I ◽  
Type Ii ◽  
Type Ii Error ◽  
Error Type

Background: Accurate understanding of certain basic statistical terms and principles is key to critical appraisal of published literature. Objective: This review describes type I error, type II error, null hypothesis, p value, statistical significance, a, two-tailed and one-tailed tests, effect size, alternate hypothesis, statistical power, β, publication bias, confidence interval, standard error, and standard deviation, while including examples from reports of dermatologic studies. Conclusion: The application of the results of published studies to individual patients should be informed by an understanding of certain basic statistical concepts.

Automatic Real-Time Identification of Fingerprint Images Using Wavelets and Gradient of Gaussian

1997 ◽  
Vol 07 (05) ◽  
pp. 433-440 ◽  
Author(s):  
Woo Kyu Lee ◽  
Jae Ho Chung
Keyword(s):  
Wavelet Transform ◽  
Real Time ◽  
Type I Error ◽  
Recognition Algorithm ◽  
Fingerprint Recognition ◽  
Type I ◽  
Type Ii ◽  
Type Ii Error ◽  
Local Orientation ◽  
Real Time Identification

In this paper, a fingerprint recognition algorithm is suggested. The algorithm is developed based on the wavelet transform, and the dominant local orientation which is derived from the coherence and the gradient of Gaussian. By using the wavelet transform, the algorithm does not require conventional preprocessing procedures such as smoothing, binarization, thining and restoration. Computer simulation results show that when the rate of Type II error — Incorrect recognition of two different fingerprints as identical fingerprints — is held at 0.0%, the rate of Type I error — Incorrect recognition of two identical fingerprints as different ones — turns out as 2.5% in real time.

A New and Simpler Approximation for ANOVA Under Variance Heterogeneity

1994 ◽  
Vol 19 (2) ◽  
pp. 91-101 ◽  
Author(s):  
Ralph A. Alexander ◽  
Diane M. Govern
Keyword(s):  
Type I Error ◽  
Order Approximation ◽  
Error Rates ◽  
Type I ◽  
Type Ii ◽  
Type Ii Error ◽  
Tail Probabilities ◽  
Type I Error Rates ◽  
Heterogeneity Of Variance ◽  
The Face

A new approximation is proposed for testing the equality of k independent means in the face of heterogeneity of variance. Monte Carlo simulations show that the new procedure has Type I error rates that are very nearly nominal and Type II error rates that are quite close to those produced by James’s (1951) second-order approximation. In addition, it is computationally the simplest approximation yet to appear, and it is easily applied to Scheffé (1959) -type multiple contrasts and to the calculation of approximate tail probabilities.

Sign in / Sign up

Export Citation Format

Share Document