Linear Combination Of Means Research Articles

GP3: Gaussian Process Path Planning for Reliable Shortest Path in Transportation Networks

This paper investigates the reliable shortest path (RSP) problem in Gaussian process (GP) regulated transportation networks. Specifically, the RSP problem that we are targeting at is to minimize the (weighted) linear combination of mean and standard deviation of the path’s travel time. With the reasonable assumption that the travel times of the underlying transportation network follow a multi-variate Gaussian distribution, we propose a Gaussian process path planning (GP3) algorithm to calculate the a priori optimal path as the RSP solution. With a series of equivalent RSP problem transformations, we are able to reach a polynomial time complexity algorithm with guaranteed solution accuracy. Extensive experimental results over various sizes of realistic transportation networks demonstrate the superior performance of GP3 over the state-of-the-art algorithms.

Objective Bayesian testing for the linear combinations of normal means

This study considers objective Bayesian testing for the linear combinations of the means of several normal populations. We propose solutions based on a Bayesian model selection procedure to this problem in which no subjective input is considered. We first construct suitable priors to test the linear combinations of means based on measuring the divergence between competing models (so-called divergence-based priors). Next, we derive the intrinsic priors for which the Bayes factors and model selection probabilities are well defined. Finally, the behavior of the Bayes factors based on the DB priors, intrinsic priors, and classical test are compared in a simulation study and an example.

Asymptotic inequalities and comparison of classical means

Inequalities between linear combinations of means are the subject of interest for decades. In this paper we propose a new approach to this subject, using the concept of asymptotical expansion of means. This enables us to find necessary conditions and the optimal values for coefficients in order that the inequality between linear combination of three or four means will be valid. We restrict ourselfs to the study of the most common classical mean, leaving detailed study of parametric means to the future works.

Inferences on Linear Combinations of Normal Means with Unknown and Unequal Variances

This paper considers the problem of making inferences on a linear combination of means from independent normal distributions with unknown variances. While standard inference procedures require the assumption that the variances of the normal distributions are all equal or have known ratios, in this paper procedures are developed which allow the variances to take any unknown values. Bounds are developed for the confidence levels of confidence intervals, and for the sizes and p-values of hypothesis tests, which are valid for any values of the variances. These bounds are sharp in the sense that they are attained for certain limiting values of the variances. Applications to the one-way linear model are illustrated, and the Behrens-Fisher problem, which is a special case with just two means and for which the popular approximate Welch method can be employed, is also discussed. Some examples and simulations of the implementation of the procedures are provided, and comparisons are made with other procedures.

Identification of target clusters by using the restricted normal mixture model

This paper addresses the problem of identifying groups that satisfy the specific conditions for the means of feature variables. In this study, we refer to the identified groups as “target clusters” (TCs). To identify TCs, we propose a method based on the normal mixture model (NMM) restricted by a linear combination of means. We provide an expectation–maximization (EM) algorithm to fit the restricted NMM by using the maximum-likelihood method. The convergence property of the EM algorithm and a reasonable set of initial estimates are presented. We demonstrate the method's usefulness and validity through a simulation study and two well-known data sets. The proposed method provides several types of useful clusters, which would be difficult to achieve with conventional clustering or exploratory data analysis methods based on the ordinary NMM. A simple comparison with another target clustering approach shows that the proposed method is promising in the identification.

Generalized Cornish-Fisher expansions

Cornish-Fisher expansions about the normal distribution provide accurate approximations for distributions of estimates and also for the level in the nominal error of confidence intervals. However, there is an advantage is expanding about a skew distribution like the chi-square, since the first order approximations become second order if the skewness is matched. Higher order approximations are also simplified. We demonstrate the method by approximating the distribution of standardized and Studentized linear combinations of means.

Simultaneous Inference on All Linear Combinations of Means with Heteroscedastic Errors

We proposed a statistical method to construct simultaneous confidence intervals on all linear combinations of means without assuming equal variance where the classical Scheffé′s simultaneous confidence intervals no longer preserve the familywise error rate (FWER). The proposed method is useful when the number of comparisons on linear combinations of means is extremely large. The FWERs for proposed simultaneous confidence intervals under various configurations of mean variances are assessed through simulations and are found to preserve the predefined nominal level very well. An example of pairwise comparisons on heteroscedastic means is given to illustrate the proposed method.

Noninformative priors for linear combinations of the normal means

In this paper, we develop noninformative priors for linear combinations of the means under the normal populations. It turns out that among the reference priors the one-at-a-time reference prior satisfies a second order probability matching criterion. Moreover, the second order probability matching priors match alternative coverage probabilities up to the second order and are also HPD matching priors. Our simulation study indicates that the one-at-a-time reference prior performs better than the other reference priors in terms of matching the target coverage probabilities in a frequentist sense.

A more practical scheffe-type multiple comparison procedure for commonly encountered numbers of comparisons

Multiple comparison procedures are important tools used in the analysis and interpretation of linear combinations of means from several populations. These procedures are used for two different types of comparisons: 1) the comparisons of all possible pairs of means and 2) testing a set of “g” comparisons. The Scheffe procedure is one of several techniques available for multiple comparisons but is generally regarded as too conservative for most practical analyses. Some authors have suggested ad hoc adjustments to the significance level to overcome the conservative nature of the Scheffe method. A heuristic approach is proposed to achieve the same objective which is quite satisfactory for commonly encountered numbers of comparisons Simulations clearly indicate that the modification of the Scheffe test is always superior to the unmodi-fied Scheffe and has acceptable experimentwise error rates and more power than the Bonferroni test for the investigation of a moderate number of comparisons.

Statistics for the Life Sciences.

1. Introduction. Statistics and the Life Sciences. Examples and Overview. 2. Description of Populations and Samples. Introduction. Frequency Distributions: Techniques for Data. Frequency Distributions: Shapes and Examples. Descriptive Statistics: Measures of Center. Boxplots. Measures of Dispersion. Effect of Transformation of Variables (Optional). Samples and Populations: Statistical Inference. Perspective. 3. Random Sampling, Probability, and the Binomial Distribution. Probability and the Life Sciences. Random Sampling. Introduction to Probability. Probability Trees. Probability Rules (Optional). Density Curves. Random Variables. The Binomial Distribution. Fitting a Binomial Distribution to Data (Optional). 4. The Normal Distribution. Introduction. The Normal Curves. Areas Under a Normal Curve. Assessing Normality. The Continuity Correction (Optional). Perspective. 5. Sampling Distributions. Basic Ideas. Dichotomous Observations. Quantitative Observations. Illustration of the Central Limit Theorem (Optional). The Normal Approximation to the Binomial Distribution (Optional). Perspective. 6. Confidence Intervals. Statistical Estimation. Standard Error of the Mean. Confidence Interval. Planning a Study to Estimate. Conditions for Validity of Estimation Methods. Confidence Interval for a Population Proportion. Perspective and Summary. 7. Comparison of Two Independent Samples. Introduction. Standard Error of (y1 - y2). Confidence Interval. Hypothesis Testing: The t-test. Further Discussion of the t-test. One-Tailed Tests. More on Interpretation of Statistical Significance. Planning for Adequate Power (Optional). Student's t: Conditions and Summary. More on Principles of Testing Hypotheses. The Wilcoxon-Mann-Whitney Test. Perspective. 8. Statistical Principles of Design. Introduction. Observational Studies. Experiments. Restricted Randomization: Blocking and Stratification. Levels of Replication. Sampling Concerns (Optional). Perspective. 9. Comparison of Paired Samples. Introduction. The Paired-Sample t-Test and Confidence Interval. The Paired Design. The Sign Test. The Wilcoxon Signed-Rank Test. Further Considerations in Paired Experiments. Perspective. 10. Analysis of Categorical Data. Inference for Proportions: The Chi-Squared Goodness-of-Fit Test. The Chi-Squared Test for a 2 X 2 Contingency Table. Independence and Association in a 2 X 2 Contingency Table. Fisher's Exact Test (Optional). The r x k Contingency Table. Applicability of Methods. Confidence Interval for a Difference Between Proportions. Paired Data and 2 X 2 Tables (Optional). Relative Risk and the Odds Ratio (Optional). Summary of Chi-Squared Tests. 11. Comparing the Means of Many Independent Samples. Introduction. The Basic Analysis of Variance. The Analysis of Variance Model (Optional). The Global F-Test. Applicability of Methods. Two-Way ANOVA (Optional). Linear Combinations of Means (Optional). Multiple Comparisons (Optional). Perspective. 12. Linear Regression and Correlation. Introduction. The Fitted Regression Line. Parametric Interpretation of Regression: The Linear Model. Statistical Inference Concerning B1. The Correlation Coefficient. Guidelines for Interpreting Regression and Correlation. Perspective. Summary of Formulas. 13. A Summary of Inference Methods. Introduction. Data Analysis Samples. Appendices. Chapter Notes. Statistical Tables. Answers to Selected Exercises. Index. Index of Examples.

Equivariant two-stage estimation of a noral mean

We consider estimation of a normal mean based on a sample of independent identically distributed observations from a normal population with unknown variance. A two stage procedure involves first taking an initial sample, on the basis of that sample deciding on the size of a second sample, and then estimating the mean based on the total sample. The loss is a linear combination of mean squared error and number of observations taken. We restrict attention to procedures which are shift invariant. We show that a complete class of procedures consists of those which choose the second sample size monotonically in the initial sample variance, obtain a class of natural Generalized Bayes procedures and an asymptotically minimax procedure among procedures with given initial sample size, and show that this latter procedure is admissible among the full class of equivariant procedures.

Bayesian Estimation in Multivariate Analysis

Abstract : The Bayes approach to Multivariate Analysis taken previously by Geisser and Cornfield (JRSS Series B, 1963 No. 2, pp. 368-376) is extended and given a more comprehensive treatment. Posterior joint and marginal densities are derived for vector means, linear combinations of means; simple and partial variances; simple, partial and multiple correlation coefficients. Also discussed are the posterior distributions of the canonical correlations and of the principal components. For the general multivariate linear hypothesis, it is demonstrated that the joint Bayesian posterior region for the elements of the regression matrix is equivalent to the usual confidence region for these parameters. The joint predictive density of a set of future observations generated by the linear hypothesis is obtained thus enabling one to specify the probability that a set of future observations will be contained in a particular region. (Author)

Interval Estimation for Linear Combinations of Means

Abstract The simultaneous interval estimation for a given set of linear combinations of means whose estimates are jointly normally distributed is treated. In order to give the exact confidence intervals, t 2 M , the maximum of several quasi-independent t 2-statistics, is introduced. Using the inclusion-exclusion formula, a procedure and tables are given for obtaining a good approximation to the exact upper percentage point of t 2 M . As illustrative examples, the procedure is applied to the interval estimations for parameters in the model of a randomized block design and for coefficients in a normal regression equation.