On the complexity and approximation of non-unique probe selection using d-disjunct matrix

In this paper, we studied the MINimum-d-Disjunct Submatrix (MIN-d-DS), which can be used to select the minimum number of non-unique probes for viruses identification. We prove that MIN-d-DS is NP-hard for any fixed d. Using d-disjunct matrix, we present an O(log k)-approximation algorithm where k is an upper bound on the maximum number of targets hybridized to a probe. We also present a (1+(d+1)log n)-approximation algorithm to identify at most d targets in the presence of experimental errors. Our approximation algorithms also yield a linear time complexity for the decoding algorithms. The research of T. Znati was supported in part by National Science Foundation under grant CCF-0548895.     

Estimation of proportions by group testing with retesting of positive groups

Abstract

When estimating a proportion p by group testing, and it is desired to increase precision, it is sometimes impractical to obtain additional individuals but it is possible to retest groups formed from the individuals within the groups that test positive at the first stage. Hepworth and Watson assessed four methods of retesting, and recommended a random regrouping of individuals from the first stage. They developed an estimator of p for their proposed method, and, because of the analytic complexity, used simulation to examine its variance properties. We now provide an analytical solution to the variance of the estimator, and compare its performance with the earlier simulated results. We show that our solution gives an acceptable approximation in a reasonable range of circumstances.     

Two‐Sample Test Against One‐Sided Alternatives

Abstract. This paper proposes, implements and investigates a new non‐parametric two‐sample test for detecting stochastic dominance. We pose the question of detecting the stochastic dominance in a non‐standard way. This is motivated by existing evidence showing that standard formulations and pertaining procedures may lead to serious errors in inference. The procedure that we introduce matches testing and model selection. More precisely, we reparametrize the testing problem in terms of Fourier coefficients of well‐known comparison densities. Next, the estimated Fourier coefficients are used to form a kind of signed smooth rank statistic. In such a setting, the number of Fourier coefficients incorporated into the statistic is a smoothing parameter. We determine this parameter via some flexible selection rule. We establish the asymptotic properties of the new test under null and alternative hypotheses. The finite sample performance of the new solution is demonstrated through Monte Carlo studies and an application to a set of survival times.     

A relative error-based estimation with an increasing number of parameters

The least product relative error (LPRE) estimator and test statistic to test linear hypotheses of regression parameters in the multiplicative regression model are studied when the number of covariate variables increases with the sample size. Some properties of the LPRE estimator and test statistic are obtained such as consistency, Bahadur presentation, and asymptotic distributions. Furthermore, we extend the LPRE to a more general relative error criterion and provide their statistical properties. Numerical studies including simulations and two real examples show that the proposed estimation performs well.     

Hypothesis testing via a penalized-likelihood approach

It is illustrated in this paper that hypothesis testing procedures can be derived based on the penalized likelihood approach. Based on this point of view, many traditional hypothesis tests, including the two-sample mean test, score test, and Hotelling’s $T^2$ test are revisited under the penalized likelihood framework. Similar framework is also applicable to the empirical likelihood.     

“What If” analysis: Benefits of utilizing a “What If” analysis in excel

The “What If” analysis is applicablein research and heuristic situations that utilize statistical significance testing. One utility for the “What If” is in a pedagogical perspective; the “What If” analysis provides professors an interactive tool that visually represents examples of what statistical significance testing entails and the variables that affect the commonly misinterpreted p_CALCULATED value. In order to develop a strong understanding of what affects the p_CALCULATED value, the students tangibly manipulate data within the Excel sheet to create a visualize representation that explicitly demonstrates how variables affect the p_CALCULATED value. The second utility is primarily applicable to researchers. “What If” analysis contributes to research in two ways: (1) a “What If” analysis can be run a priori to estimate the sample size a researcher may wish to use for his study; and (2) a “What If” analysis can be run a posteriori to aid in the interpretation of results. If used, the “What If” analysis provides researchers with another utility that enables them to conduct high-quality research and disseminate their results in an accurate manner.     

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.     

Accurately sized test statistics with misspecified conditional homoskedasticity

We study the finite-sample performance of test statistics in linear regression models where the error dependence is of unknown form. With an unknown dependence structure, there is traditionally a trade-off between the maximum lag over which the correlation is estimated (the bandwidth) and the amount of heterogeneity in the process. When allowing for heterogeneity, through conditional heteroskedasticity, the correlation at far lags is generally omitted and the resultant inflation of the empirical size of test statistics has long been recognized. To allow for correlation at far lags, we study the test statistics constructed under the possibly misspecified assumption of conditional homoskedasticity. To improve the accuracy of the test statistics, we employ the second-order asymptotic refinement in Rothenberg [Approximate power functions for some robust tests of regression coefficients, Econometrica 56 (1988), pp. 997–1019] to determine the critical values. The simulation results of this paper suggest that when sample sizes are small, modelling the heterogeneity of a process is secondary to accounting for dependence. We find that a conditionally homoskedastic covariance matrix estimator (when used in conjunction with Rothenberg's second-order critical value adjustment) improves test size with only a minimal loss in test power, even when the data manifest significant amounts of heteroskedasticity. In some specifications, the size inflation was cut by nearly 40% over the traditional heteroskedasticity and autocorrelation consistent (HAC) test. Finally, we note that the proposed test statistics do not require that the researcher specify the bandwidth or the kernel.     

Testing exponentiality based on characterizations of the exponential distribution

In this paper, we first present two characterizations of the exponential distribution and next introduce three exact goodness-of-fit test for exponentiality. By simulation, the powers of the proposed tests under various alternatives are compared with the existing tests.     

Model fusion and multiple testing in the likelihood paradigm: shrinkage and evidence supporting a point null hypothesis

ABSTRACT

According to the general law of likelihood, the strength of statistical evidence for a hypothesis as opposed to its alternative is the ratio of their likelihoods, each maximized over the parameter of interest. Consider the problem of assessing the weight of evidence for each of several hypotheses. Under a realistic model with a free parameter for each alternative hypothesis, this leads to weighing evidence without any shrinkage toward a presumption of the truth of each null hypothesis. That lack of shrinkage can lead to many false positives in settings with large numbers of hypotheses. A related problem is that point hypotheses cannot have more support than their alternatives. Both problems may be solved by fusing the realistic model with a model of a more restricted parameter space for use with the general law of likelihood. Applying the proposed framework of model fusion to data sets from genomics and education yields intuitively reasonable weights of evidence.