After rejecting the null hypothesis in the analysis of variance, the next step is to make the pairwise comparisons to find out differences in means. The purpose of this paper is threefold. The foremost aim is to suggest expression for calculating decision limit that enables us to collect the test and pairwise comparisons in one step. This expression is proposed as the ratio of between square for each treatment and within sum of squares for all treatments. The second aim is to obtain the exact sampling distribution of the proposed ratio under the null hypothesis. The exact sampling distribution is derived as the beta distribution of the second type. The third aim is to use beta distribution and adjusted p values to create decision limit. Therefore, reject the null hypothesis of equal means if any adjusted point falls outsides the decision limit. Simulation study is conducted to compute Type one error. The results show that the proposed method controls the type one error near nominal values using Benjamini-Hochberg adjusted p-values. Two applications are given to show the benefits of the proposed method.
Statistical dependence between hypotheses poses a significant challenge to the stability of large scale multiple hypotheses testing. Ignoring it often results in an unacceptably large spread in the false positive proportion even though the average value is acceptable [21, 39, 40, 49]. However, the statistical dependence structure of data is often unknown. Using a generic signalprocessing model, Bayesian multiple testing, and simulations, we demonstrate that the variance of the false positive proportion can be substantially reduced even under unknown short range dependence. We do this by modeli…
We propose simultaneous mean-variance regression for the linear estimation and approximation of conditional mean functions. In the presence of heteroskedasticity of unknown form, our method accounts for varying dispersion in the regression outcome across the support of conditioning variables by using weights that are jointly determined with the mean regression parameters. Simultaneity generates outcome predictions that are guaranteed to improve over ordinary least-squares prediction error, with corresponding parameter standard errors that are automatically valid. Under shape misspecification o…
The two-level normal hierarchical model (NHM) has played a critical role in the theory of small area estimation (SAE), one of the growing areas in statistics with numerous applications in different disciplines. In this paper, we address major well-known shortcomings associated with the empirical best linear unbiased prediction (EBLUP) of a small area mean and its mean squared error (MSE) estimation by considering an appropriate model variance estimator that satisfies multiple properties. The proposed model variance estimator simultaneously (i) improves on the estimation of the related shrinkag…
Standard ANOVA is among the most widely used tests in the life sciences and beyond. Several alternatives are proposed to provide simultaneous confidence intervals, ensure tight control of FWER, be robust to variance heterogeneity, avoid pre-testing for global effect (for one-way designs) or irrelevant interaction (for multi-way designs) prior to multiple comparison procedures. On the basis of selected examples, the R programs are provided for this purpose.
This paper deals with a general class of transformation models that contains many important semiparametric regression models as special cases. It develops a self-induced smoothing for the maximum rank correlation estimator, resulting in simultaneous point and variance estimation. The self-induced smoothing does not require bandwidth selection, yet provides the right amount of smoothness so that the estimator is asymptotically normal with mean zero (unbiased) and variance-covariance matrix consistently estimated by the usual sandwich-type estimator. An iterative algorithm is given for the varia…
The widespread availability of high-dimensional biological data has made the simultaneous screening of many biological characteristics a central problem in computational biology and allied sciences. While the dimensionality of such datasets continues to grow, so too does the complexity of biomarker identification from exposure patterns in health studies measuring baseline confounders; moreover, doing so while avoiding model misspecification remains an issue only partially addressed. Efficient estimators capable of incorporating flexible, data adaptive regression techniques in estimating releva…
We consider the equivalent problems of estimating the residual variance, the proportion of explained variance $η$ and the signal strength in a high-dimensional linear regression model with Gaussian random design. Our aim is to understand the impact of not knowing the sparsity of the regression parameter and not knowing the distribution of the design on minimax estimation rates of $η$. Depending on the sparsity $k$ of the regression parameter, optimal estimators of $η$ either rely on estimating the regression parameter or are based on U-type statistics, and have minimax rates depending on $k$. …
Most comparisons of treatments or doses against a control are performed by the original Dunnett single step procedure \cite{Dunnett1955} providing both adjusted p-values and simultaneous confidence intervals for differences to the control. Motivated by power arguments, unbalanced designs with higher sample size in the control are recommended. When higher variance occur in the treatment of interest or in the control, the related per-pairs power is reduced, as expected. However, if the variance is increased in a non-affected treatment group, e.g. in the highest dose (which is highly significant)…
This work develops non-asymptotic theory for estimation of the long-run variance matrix and its inverse, the so-called precision matrix, for high-dimensional time series under general assumptions on the dependence structure including long-range dependence. The estimation involves shrinkage techniques which are thresholding and penalizing versions of the classical multivariate local Whittle estimator. The results ensure consistent estimation in a double asymptotic regime where the number of component time series is allowed to grow with the sample size as long as the true model parameters are sp…
A multiple filter test (MFT) for the analysis and detection of rate change points in point processes on the line has been proposed recently. The underlying statistical test investigates the null hypothesis of constant rate. For that purpose, multiple filtered derivative processes are observed simultaneously. Under the null hypothesis, each process $G$ asymptotically takes the form \begin{align*} G \sim L, \end{align*} while $L$ is a zero-mean Gaussian process with unit variance. This result is used to derive a rejection threshold for statistical hypothesis testing. The purpose of this paper is…
Concentration inequalities have become increasingly popular in machine learning, probability, and statistical research. Using concentration inequalities, one can construct confidence intervals (CIs) for many quantities of interest. Unfortunately, many of these CIs require the knowledge of population variances, which are generally unknown, making these CIs impractical for numerical application. However, recent results regarding the simultaneous bounding of the probabilities of quantities of interest and their variances have permitted the construction of empirical CIs, where variances are replac…
Standard Monte Carlo computation is widely known to exhibit a canonical square-root convergence speed in terms of sample size. Two recent techniques, one based on control variate and one on importance sampling, both derived from an integration of reproducing kernels and Stein's identity, have been proposed to reduce the error in Monte Carlo computation to supercanonical convergence. This paper presents a more general framework to encompass both techniques that is especially beneficial when the sample generator is biased and noise-corrupted. We show our general estimator, which we call the doub…
Based on $X \sim N_d(θ, σ^2_X I_d)$, we study the efficiency of predictive densities under $α-$divergence loss $L_α$ for estimating the density of $Y \sim N_d(θ, σ^2_Y I_d)$. We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension $d$, the variances $σ^2_X$ and $σ^2_Y$, the choice of loss $L_α$; $α\in (-1,1)$. The findings also apply to a large number of plug-in densities, as well as for restricted par…
Joint spectral embeddings facilitate analysis of multiple network data by simultaneously mapping vertices in each network to points in Euclidean space where statistical inference is then performed. In this work, we consider one such joint embedding technique, the omnibus embedding of arXiv:1705.09355 , which has been successfully used for community detection, anomaly detection, and hypothesis testing tasks. To date the theoretical properties of this method have only been established under the strong assumption that the networks are conditionally i.i.d. random dot product graphs. Herein, we tak…
Inference and prediction under the sparsity assumption have been a hot research topic in recent years. However, in practice, the sparsity assumption is difficult to test, and more importantly can usually be violated. In this paper, to study hypothesis test of any group of parameters under non-sparse high-dimensional linear models, we transform the null hypothesis to a testable moment condition and then use the self-normalization structure to construct moment test statistics under one-sample and two-sample cases, respectively. Compared to the one-sample case, the two-sample additionally require…
The issue of variance components testing arises naturally when building mixed-effects models, to decide which effects should be modeled as fixed or random. While tests for fixed effects are available in R for models fitted with lme4, tools are missing when it comes to random effects. The varTestnlme package for R aims at filling this gap. It allows to test whether any subset of the variances and covariances corresponding to any subset of the random effects, are equal to zero using asymptotic property of the likelihood ratio test statistic. It also offers the possibility to test simultaneously …
The naive importance sampling (IS) estimator generally does not work well in examples involving simultaneous inference on several targets, as the importance weights can take arbitrarily large values, making the estimator highly unstable. In such situations, alternative multiple IS estimators involving samples from multiple proposal distributions are preferred. Just like the naive IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. The selection of these proposal distributions is the focus of this article. We propose three methods: (i) …
We show that the variance of the Monte Carlo estimator that is importance sampled from an exponential family is a convex function of the natural parameter of the distribution. With this insight, we propose an adaptive importance sampling algorithm that simultaneously improves the choice of sampling distribution while accumulating a Monte Carlo estimate. Exploiting convexity, we prove that the method's unbiased estimator has variance that is asymptotically optimal over the exponential family.
