Rao's Statistic Research Articles

Investigating non-Gaussianity in Cosmic Microwave Background temperature maps using spherical harmonic phases

In the era of precision cosmology, accurate estimation of cosmological parameters is based upon the implicit assumption of the Gaussian nature of Cosmic Microwave Background (CMB) radiation. Therefore, an important scientific question to ask is whether the observed CMB map is consistent with Gaussian prediction. In this work, we extend previous studies based on CMB spherical harmonic phases (SHP) to examine the validity of the hypothesis that the temperature field of the CMB is consistent with a Gaussian random field (GRF). The null hypothesis is that the corresponding CMB SHP are independent and identically distributed in terms of a uniform distribution in the interval [0, 2π] [1,2]. We devise a new model-independent method where we use ordered and non-parametric Rao's statistic, based on sample arc-lengths to comprehensively test uniformity and independence of SHP for a given ℓ mode and independence of nearby ℓ mode SHP. We performed our analysis on the scales limited by spherical harmonic modes ≤ 128, to restrict ourselves to signal-dominated regions. To find the non-uniform or dependent sets of SHP, we calculate the statistic for the data and 10000 Monte Carlo simulated uniformly random sets of SHP and use 0.05 and 0.001 α levels to distinguish between statistically significant and highly significant detections. We first establish the performance of our method using simulated Gaussian, non-Gaussian CMB temperature maps, along with observed non-Gaussian 100 and 143 GHz Planck channel maps. We find that our method, performs efficiently and accurately in detecting phase correlations generated in all of the non-Gaussian simulations and observed foreground contaminated 100 and 143 GHz Planck channel temperature maps. We apply our method on Planck satellite mission's final released CMB temperature anisotropy maps- COMMANDER, SMICA, NILC, and SEVEM along with WMAP 9 year released ILC map. We report that SHP corresponding to some of the m-modes are non-uniform, some of the ℓ mode SHP and neighboring mode pair SHP are correlated in cleaned CMB maps. The detection of non-uniformity or correlation in the SHP indicates the presence of non-Gaussian signals in the foreground minimized CMB maps.

A Bayesian ILC Method for CMB B-mode Posterior Estimation and Reconstruction of Primordial Gravity Wave Signal

Abstract The cosmic microwave background (CMB) radiation B-mode polarization signal contains the unique signature of primordial metric perturbations produced during the inflation. The separation of the weak CMB B-mode signal from strong foreground contamination in observed maps is a complex task, and proposed new-generation low-noise satellite missions compete with the weak signal level of this gravitational background. In this paper, for the first time, we employ a foreground model-independent internal linear combination (ILC) method to reconstruct the CMB B-mode signal using simulated observations over large angular scales of the sky of six frequency bands of the future-generation CMB mission Probe of Inflation and Cosmic Origins (PICO). We estimate the joint CMB B-mode posterior density following the interleaving Gibbs steps of the B-mode angular power spectrum and cleaned map samples using the ILC method. We extend and improve the earlier reported Bayesian ILC method to analyze weak CMB B-mode reconstruction by introducing noise-bias corrections at two stages during the ILC weight estimation. By performing 200 Monte Carlo simulations of the Bayesian ILC method, we find that our method can reconstruct the CMB signals and the joint posterior density accurately over large angular scales of the sky. We estimate the Blackwell–Rao statistics of the marginal density of the CMB B-mode angular power spectrum and use them to estimate the joint density of the tensor-to-scalar ratio r and a lensing power spectrum amplitude A lens . Using 200 Monte Carlo simulations of the delensing approach, we find that our method can achieve an unbiased detection of the primordial gravitational-wave signal r with more than 8σ significance for levels of r ≥ 0.01 for input frequency maps with realistic lensing and foregrounds with constant spectral indices.

Annotations on the Relationship Among Discriminant Functions

Different forms of discriminant functions and the essence of their appearances were considered in this study. Various forms of classification problems were also considered, and in each of the cases mentioned, classification from simple functions of the observational vector rather than complicated regions in the higher-dimensional space of the original vector were made. Ever since the emergence of the Linear Discriminant Function (LDF) by Fisher, several other classification statistics have emerged and violation of condition of equal variance covariance matrix for Linear Discriminant Function (LDF) results to Quadratic Discriminant Function (QDF). While the Best Linear Discriminant Function (BLDF) is referred to Best Sample Discriminant Function (BSDF) when the parameters are estimated from a sample and also optimal in the same sense as Quadratic Discriminant Function (QDF), Rao statistic is best for discriminating between options that are close each other. The relationships among the classification statistics examined were established: Among the methods of classification statistics considered, Anderson’s (W) and Rao’s (R) statistics are equivalent when the two sample sizes n₁ and n₂ are equal, and when a constant is equal to 1, W, R and John-Kudo’s (Z) classification statistics are asymptotically comparable. A linear relationship is also established between W and Z classification.

Sequential Comparison of d Populations and Related Tables

ANOVA tests are used to compare several populations in terms of the mean parameter. In this study some new sequential tests are proposed as alternative procedures to the sequential F-test of Siegmund (1980). The tests are based on large sample approximations of the distribution of Rao's statistic. The empirical power of the tests is evaluated in a simulation study. In the Appendix extensive tables about the distribution of the maximum of the Bessel process are provided. They are useful for sequential tests that use the chi-square distribution in the fixed sample size setting.

Approximate confidence interval for the new extreme value distribution

PurposeExact confidence interval estimation for the new extreme value model is often impractical. This paper seeks to evaluate the accuracy of approximate confidence intervals for the two‐parameter new extreme value model.Design/methodology/approachThe confidence intervals of the parameters of the new model based on likelihood ratio, Wald and Rao statistics are evaluated and compared through the simulation study. The criteria used in evaluating the confidence intervals are the attainment of the nominal error probability and the symmetry of lower and upper error probabilities.FindingsThis study substantiates the merits of the likelihood ratio, the Wald and the Rao statistics. The results indicate that the likelihood ratio‐based intervals perform much better than the Wald and Rao intervals.Originality/valueExact interval estimates for the new model are difficult to obtain. Consequently, large sample intervals based on the asymptotic maximum likelihood estimators have gained widespread use. Intervals based on inverting likelihood ratio, Rao and Wald statistics are rarely used in commercial packages. This paper shows that the likelihood ratio intervals are superior to intervals based on the Wald and the Rao statistics.

Rao's statistic for homogeneity of multiple parameter

The paper proposes Rao's statistic for testing homogeneity of parameters of k populations and gives the asymptotic expansions of its distribution under null hypothesis and of its power function under contiguous alternatives.

A characterization of inequality measures based on bilateral inequality

This article deals with inequality measures on income distributions. These measures are assumed to fulfil common axioms like the Principle of Progressive Transfers, Normalization, and Symmetry. Strict Monotonicity will be defined and is included in the basic set of axioms. If the measures are especially based on bilateral comparisons and if two additional conditions are fulfilled the bilateral inequality measure must be proportional to the absolute distance of the two incomes involved. It is remarkable that all these conditions do not give an answer whether an inequality measure for two incomes should be a quasi-convex or a quasi-concave function. This question is controversely discussed in economic research. Our surprising answer, however, is given by use of RAO's statistic concept defining the diversity of a population.

Polynomial splines and their tensor products in extended linear modeling: 1994 Wald memorial lecture

Analysis of variance type models are considered for a regression function or for the logarithm of a probability function, conditional probability function, density function, conditional density function, hazard function, conditional hazard function or spectral density function. Polynomial splines are used to model the main effects, and their tensor products are used to model any interaction components that are included. In the special context of survival analysis, the baseline hazard function is modeled and nonproportionality is allowed. In general, the theory involves the $L_2$ rate of convergence for the fitted model and its components. The methodology involves least squares and maximum likelihood estimation, stepwise addition of basis functions using Rao statistics, stepwise deletion using Wald statistics and model selection using the Bayesian information criterion, cross-validation or an independent test set. Publicly available software, written in C and interfaced to S/S-PLUS, is used to apply this methodology to real data.

Detection of inhomogeneities in sets of NIR spectra

Methods for the detection of outliers and clusters in data sets for multivariate calibration are discussed in this study. Replicate as well as sample outliers have been investigated. The Cochran test applied to the sum of absorbances is shown to identify outlying replicates. Sample outliers were detected on latent variables (PCs) or by using Rao's statistic combined with Grubbs' and Dixon's tests. The conclusions were then verified by the modelling. The tendency to cluster was evaluated by Hopkins statistic.

Distribution-free tests for symmetry based on the linear signed rank statistics

Tests based on the linear signed rank statistics are proposed for testing the symmetry of a continuous distribution about a known median. Among the linear signed rank statistics, the Wilcoxon test has gotten much attention in the nonparametric literature. In this article we propose to use the scores such as the squares of the Wilcoxon scores and the normal scores. The proposed statistics are distribution-free under the null hypothesis. Critical values of the proposed test statistics at several selected levels for sample sizes n = 10(1)20, 25, 30 are obtained. Through Monte Carlo studies we compare the performance of our proposed tests with the Hill and Rao's statistic, McWilliams's runs statistic and Wilcoxon signed rank statistic. The results show that the proposed tests are more powerful than the competitive tests under a broad class of alternatives.

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion ( T L ( s ) ), Watson statistic ( T W ( s ) ) and Rao statistic ( T R ( s ) ) for testing H 0 s : μ ∈ V (a given subspace) against H 1 s : μ ∉ V , based on a sample of size n from a p -variate Langevin distribution M p ( μ , κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ −1 . For the case when κ is unknown, it is shown that T R (s) ≧ T L (s) ≧ T W (s) in their powers up to the order κ −1 .

On tests for the mean direction of the Langevin distribution

The asymptotic expansions of the distribution of a sum of independent random vectors with Langevin distribution are given. The power functions of the likelihood ratio criterion, Watson statistic, Rao statistic and the modified Wald statistic for testing the hypothesis of the mean direction are obtained asymptotically and a numerical comparison is made.

On the second order local comparison between perturbed maximum likelihood estimators and Rao's statistic as test statistics

A bias-adjusted maximum likelihood estimator (mle), which has been shown to possess certain optimality criteria as an estimate of θ (see Ghosh, J. K., Sinha, B. K., and Joshi, S. N. (1982). In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, Eds.), Vol. 1, pp. 403–456. Academic Press, Orlando, FL) is compared with Rao's statistic as a test statistic for the standard two-sided testing problem. It is shown that Rao's statistic is locally superior to any bias-adjusted mle in the sense of Chandra and Joshi (1983, Sankhyā Ser. A 45 226–246). A second interpretation of a conjecture of C. R. Rao is proposed and Rao's statistic is shown to be superior as a test statistic according to the new interpretation as well. The last fact provides an interesting supplement to the results of Chandra and Joshi (1983, Sankhyā Ser. A 45 226–246). Furthermore, a partial reason for the inferiority of the likelihood ratio and Wald's statistic to Rao's statistic is supplied and certain regularity assumptions of the last paper are eliminated. Finally, the local powers of certain modified versions of Rao's and Wald's statistics (see Skovgaard, I. M. (1985). Ann. Statist. 13 534–551) are studied.

On testing hypothesis of covariance matrices under elliptical population

The power functions of the likehood ratio criteria and Rao statistics for testing hypotheses of the structures of covariance matrices of normal population are derived asymptotically and are compared numerically under an elliptical population.

Asymptotic expansions of the distributions of some test statistics

A modified Wald statistic for testing simple hypothesis against fixed as well as local alternatives is proposed. The asymptotic expansions of the distributions of the proposed statistic as well as the Wald and Rao statistics under both the null and alternative hypotheses are obtained. The powers of these statistics are compared and its is shown that for special structures of parameters some statistics have same power in the sence of order\({1 \mathord{\left/ {\vphantom {1 {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }}\). The results obtained are applied for testing the hypothesis about the covariance matrix of the multivariate normal distribution and it is shown that none of the tests based on the above statistics is uniformly superior.

On the optimality of rao's statistic

This paper establishes the asymptotic optimality of Rao's test within a very wide class of tests that Includes the likeli hood ratio test and Wald's test. An expression for the defici¬ency of the tests in this class relative to Rao's test has also been obtained