Noncentral F-distribution Research Articles

Thurstonian model for the four-interval oddity task

The four-interval oddity task is an extension of the Triangle method, which is the three-interval oddity task and is one of the most broadly used conventional sensory discrimination methods. It is a special case of the m-interval oddity task or the m-alternative odd-man-out task where m = 4. It is also a special situation of the unspecified ‘M + N’ with M = 3 and N = 1. Some new forms of the analytical psychometric function for the four-interval oddity task based on a normal distribution function, a bivariate normal distribution function, and a non-central F-distribution function are provided originally in this paper. The performance of the four-interval oddity task in both difference testing and similarity/equivalence testing is explored and compared with some unspecified forced-choice sensory discrimination methods including the Triangle. Tables and R codes are presented and provided for estimations of the probability of correct response, Pc, Thurstonian discriminal distance δ or d´, and the B value for estimating the variance of d´ for the four-interval oddity task.

Proposed specific heat capacity model for a concrete wall containing phase change material (PCM) under field experiment conditions

This study compares (1) the specific heat capacity (CP) model obtained from the differential scanning calorimetry (DSC) measurement, (2) the conventional CP model using the step function and normal distribution function, and (3) the newly proposed CP model using the non-central F-distribution function. The CP models were implemented into the numerical conductive heat transfer model for the PCM-containing Portland cement (PC) concrete wall. Except for the heat capacity of PCM, other experimental data such as density, thermal conductivity, heat flux, or inner temperature profiles were obtained from the previous studies to build the numerical model. Through the validation process, the CP model using the non-central F-distribution function was selected as the best model, including the thermal properties of PCM. The proposed model was applied to the parametric study to investigate the best thermal efficiency, such as thermal comfort and diurnal temperature difference when applying PCM to PC concrete walls.

Inferences on location parameters based on independent multivariate skew normal distributions

PurposeThe purpose of this study is to extend the classical noncentral F-distribution under normal settings to noncentral closed skew F-distribution for dealing with independent samples from multivariate skew normal (SN) distributions.Design/methodology/approachBased on generalized Hotelling's T2 statistics, confidence regions are constructed for the difference between location parameters in two independent multivariate SN distributions. Simulation studies show that the confidence regions based on the closed SN model outperform the classical multivariate normal model if the vectors of skewness parameters are not zero. A real data analysis is given for illustrating the effectiveness of our proposed methods.FindingsThis study’s approach is the first one in literature for the inferences in difference of location parameters under multivariate SN settings. Real data analysis shows the preference of this new approach than the classical method.Research limitations/implicationsFor the real data applications, the authors need to remove outliers first before applying this approach.Practical implicationsThis study’s approach may apply many multivariate skewed data using SN fittings instead of classical normal fittings.Originality/valueThis paper is the research paper and the authors’ new approach has many applications for analyzing the multivariate skewed data.

A Method for Detecting Incipient Faults in Satellites Based on Dynamic Linear Discriminant Analysis.

Timely detection and treatment of possible incipient faults in satellites will effectively reduce the damage and harm they could cause. Although much work has been done concerning fault detection problems, the related questions about satellite incipient faults are little addressed. In this paper, a new satellite incipient fault detection method was proposed by combining the ideas of deviation in unsupervised fault detection methods and classification in supervised fault detection methods. First, the proposed method uses dynamic linear discriminant analysis (LDA) to find an optimal projection vector that separates the in-orbit data from the normal historical data as much as possible. Second, under the assumption that the parameters obey a multidimensional Gaussian distribution, it applies the normal historical data and the optimal projection vector to build a normal model. Finally, it employs the noncentral F-distribution to test whether a fault has occurred. The proposed method was validated using a numerical simulation case and a real satellite fault case. The results show that the method proposed in this paper is more effective at detecting incipient faults than traditional methods.

AN IMPROVED TEST OF THE SQUARED SHARPE RATIO

AbstractThe sample squared Sharpe ratio (SSR) is a critical statistic of the risk-return tradeoff. We show that sensitive upper-tail probabilities arise when the sample SSR is employed to test the mean-variance efficiency under different test statistics. Assuming the error's normality with a nonzero mean, we integrate the sample SSR and the arbitrage regression into a noncentral chi-square (χ2) test. We find that the distribution of the sample SSR based on the regression error is to the left of the F-distribution when assuming the returns' normality. Compared to two benchmarks that use the noncentral F-distribution and the central F-statistic, the χ2-statistic is more effective, competitive, significant, and locally robust when used to reject the upper-tailed mean-variance efficiency test using the usual parameters (sample size, portfolio size, and SSR).

On the computation and inversion of the cumulative noncentral beta distribution function

The computation and inversion of the noncentral beta distribution Bp,q(x, y) (or the noncentral F-distribution, a particular case of Bp,q(x, y)) play an important role in different applications. In this paper we study the stability of recursions satisfied by Bp,q(x, y) and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large. We also consider the inversion problem of finding x or y when a value of Bp,q(x, y) is given. We provide approximations to x and y which can be used as starting values of methods for solving nonlinear equations (such as Newton) if higher accuracy is needed.

A statistical test and sample size recommendations for comparing community composition following PCA.

Many investigations of anthropogenic and natural impacts in ecological systems attempt to detect differences in ecological variables or community composition. Frequently, ordination procedures such as principal components analysis (PCA) or canonical correspondence analysis (CCA) are used to simplify such complex data sets into a set of primary factors that express the variation across the original variables. Scatterplots of the first and second principal components are then used to visually inspect for differences in community composition between treatment groups. We present a multidimensional extension of analysis of variance based on an analysis of distance (ANODIS) that can be used to formally test for differences in community composition using 1, 2, or more dimensions of a PCA or CCA of the original sample observations. The statistical tests of significance are based on F-statistics adapted for the analysis of this multidimensional data. Because the analysis is parametric, power and sample size calculations useful in the design of field studies can be readily computed. The use of ANODIS is illustrated using bivariate PCA scatterplots from three published studies. Statistical power calculations using the noncentral F-distribution are illustrated.

The Association between SNPs and a Quantitative Trait: Power Calculation.

A power calculation for a study with a quantitative outcome requires information on the outcome distribution under the alternative hypothesis. Researchers face challenges when they concisely specify alternative distributions in genetic studies because power depends on genotype frequencies and the average effect of each genotype. In GWAS, investigators evaluate hundreds of thousands of associations; therefore it is unrealistic to specify gene frequencies and gene effects for each test and some simplification is needed. Software packages are available to calculate power, but many of them have limited flexibility and / or may have a steep learning curve. In this review, we describe to researchers and graduate students the essentials of a power calculation for testing for an association between a quantitative trait and genotypes. In addition, we provide them with the codes of the different available software packages—free and commercial—to calculate this power. The calculations can be carried out using virtually any computer language that computes the cumulative distribution function of a non-central F-distribution.

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

The computations involving the noncentral-F distribution are notoriously difficult to implement properly in floating-point arithmetic: Catastrophic loss of precision, floating-point underflow and overflow, drastically increasing computation time and program hang-ups, and instability due to numerical cancellation have all been reported. It is therefore recommended that existing statistical packages are cross-checked, and the present paper proposes a numerical algorithm precisely for this purpose. To the best of our knowledge, the proposed method is the first method that can compute the noncentrality parameter of the noncentral-F distribution with guaranteed accuracy over a wide parameter range that spans the range relevant for practical applications. Although the proposed method is limited to cases where the the degree of freedom of the denominator of the F test statistic is even, it does not affect its usefulness significantly: All of those algorithmic failures and inaccuracies that we can still reproduce today could have been prevented by simply cross-checking against the proposed method. Two numerical examples are presented where the intermediate computations went wrong silently, but the final result of the computations seemed nevertheless plausible, and eventually erroneous results were published. Cross-checking against the proposed method would have caught the numerical errors in both cases. The source code of the algorithm is available on GitHub, together with self-contained command-line executables. These executables can read the data to be cross-checked from plain text files, making it easy to cross-check any statistical software in an automated fashion.

Noncentralities Induced in Regression Diagnostics

Anomalies persist in the use of deletion diagnostics in regression. Tests for outliers under subset deletions utilize the R–Fisher statistics, each having a noncentral F-distribution with noncentrality parameter as a function of shifts only at deleted rows in the index set I. Numerous studies examine empirical outcomes of these diagnostics in random experiments. In contrast, studies here are probabilistic, examining distributions behind those empirical outcomes and tracking the effects of shifts at nondeleted rows. By allowing shifts at nondeleted rows in a set J, in addition to traditional shifts at deleted rows in I, is shown to have a doubly noncentral F-distribution. By removing the unnecessary restriction that shifts occur only at deleted rows, these findings support constructs akin to power curves in tracking probabilities of masking or swamping as shifts evolve. In addition, “regression effects” among outliers may have unforeseen consequences. A dichotomy of shifts is discovered as projections into...

GNSS carrier phase ambiguity resolution based on integrity restriction in ambiguity domain

Carrier phase ambiguity resolution of Global Navigation Satellite System (GNSS) is a key technology for high-precision navigation and positioning, and it is a challenge for applications which require both high accuracy and high integrity. This paper proposes efficient ambiguity resolution methods based on integrity restriction using Fixed Failure rate Ratio Test (FF-RT) and Doubly Non-central F-distribution Ratio Test (DNF-RT), and derives the related processing models and numerical algorithms compared with the traditional Ratio Test (RT) method. Firstly, the integer ambiguity resolution and validation procedures, especially the Least squares AMBiguity Decorrelation Adjustment (LAMBDA) estimation and RT validation are analyzed. Then the quality evaluation using success rate, the FF-RT method using Integer Aperture (IA) estimation and the NDF-RT method are proposed. Lastly, the simulation and analysis for LAMBDA using RT, FF-RT and DNF-RT methods are performed. Simulation results show that in case of unbiased scenario FF-RT and DNF-RT have similar performances, which are significantly better than RT. In case of biased scenario it is difficult for FF-RT to predict the biased success rate thus it should not be used for bias detection, while DNF-RT can detect biases in most cases except for the biases are approximate or equal to integer, which has the important benefit for early detection of potential threat to the position solution.

Power of Global Test in Deformation Analysis

AbstractThere are two kinds of global test procedures in deformation analysis; χ2-test (CT) and F-test (FT). This study discusses their power functions. The CT is more powerful than the other one in an analytical point of view. However, it requires an accurate knowledge on the a priori variance of unit weight. Therefore, in practice, the FT is mostly chosen. Despite its common usage, a χ2-power function is considered in the sensitivity design of deformation networks. It is claimed in this study that the F-distribution’s power function should be taken into account, if, in reality, the FT will be applied. Thereby, some boundary values deduced from the noncentral F-distribution to be used in sensitivity analysis are computed and tabulated. Furthermore, a simulation for a monitoring network is designed, and it is shown that the mean success rates of the two testing procedures are identical with their own powers known beforehand. This numerical experiment depicts that one should consider the related power func...

Evaluating the improvement in diagnostic utility from adding new predictors

Multiple diagnostic tests and risk factors are commonly available for many diseases. This information can be either redundant or complimentary. Combining them may improve the diagnostic/predictive accuracy, but also unnecessarily increase complexity, risks, and/or costs. The improved accuracy gained by including additional variables can be evaluated by the increment of the area under (AUC) the receiver-operating characteristic curves with and without the new variable(s). In this study, we derive a new test statistic to accurately and efficiently determine the statistical significance of this incremental AUC under a multivariate normality assumption. Our test links AUC difference to a quadratic form of a standardized mean shift in a unit of the inverse covariance matrix through a properly linear transformation of all diagnostic variables. The distribution of the quadratic estimator is related to the multivariate Behrens-Fisher problem. We provide explicit mathematical solutions of the estimator and its approximate non-central F-distribution, type I error rate, and sample size formula. We use simulation studies to prove that our new test maintains prespecified type I error rates as well as reasonable statistical power under practical sample sizes. We use data from the Study of Osteoporotic Fractures as an application example to illustrate our method.

Noncentral F-Distribution for an M-ary Phase Shift Keying Wedge-Shaped Region

This letter presents an alternative analytical expression for computing the probability of an M-ary phase shift keying (MPSK) wedge-shaped region in an additive white Gaussian noise channel. The expression is represented by the cumulative distribution function of known noncentral F- distribution. Computer simulation results demonstrate the validity of our analytical expression for the exact computation of the symbol error probability of an MPSK system with phase error.

Maximum likelihood estimation of reverberation chamber direct-to-scattered ratio

The exact maximum likelihood estimator of the direct-to-scattered ratio of the Rician channel in a reverberation chamber is derived and its performance analysed. It is shown that the estimator obeys a non-central F-distribution, the mean value and variance of which are derived. For well stirred chambers, it is shown that the systematic error in the estimated ratio may be significant.

Small-scale variations of cross-polar discrimination in Ricean fading channels

Cross-polarised antenna systems are an attractive way to reduce equipment size while maintaining low inter-antenna correlation. The cross-polar discrimination (XPD) is an important parameter of cross-polarised systems. Starting from the definition of Ricean fading channels, it is shown that the small-scale variations of the XPD have a doubly non-central F-distribution. The simulated XPD distribution obtained as a function of the Ricean parameters is successfully compared with measurement results for validation.

A simple method for calculating the statistical power for detecting a QTL located in a marker interval

We developed a simple method for calculating the statistical power for detecting a QTL located in an interval flanked by two markers. The statistical method for QTL detection is assumed to be the Haley and Knott's simple regression method of interval mapping. This method allows us to answer one of the fundamental questions in designing a QTL mapping experiment: What is the minimum marker density required to detect a QTL explaining a certain heritable proportion of the phenotypic variance (denoted by h(2)) with a power gamma under a Type I error alpha in an F(2) or other mating designs with a sample size n? Computing the statistical power only requires the ability to evaluate a non-central F-distribution function and the inverse function of this distribution.

On the computation of the noncentral F and noncentral beta distribution

Unfortunately many of the numerous algorithms for computing the comulative distribution function (cdf) and noncentrality parameter of the noncentral F and beta distributions can produce completely incorrect results as demonstrated in the paper by examples. Existing algorithms are scrutinized and those parts that involve numerical difficulties are identified. As a result, a pseudo code is presented in which all the known numerical problems are resolved. This pseudo code can be easily implemented in programming language C or FORTRAN without understanding the complicated mathematical background. Symbolic evaluation of a finite and closed formula is proposed to compute exact cdf values. This approach makes it possible to check quickly and reliably the values returned by professional statistical packages over an extraordinarily wide parameter range without any programming knowledge. This research was motivated by the fact that a very useful table for calculating the size of detectable effects for ANOVA tables contains suspect values in the region of large noncentrality parameter values compared to the values obtained by Patnaik's 2-moment central-F approximation. The cause is identified and the corrected form of the table for ANOVA purposes is given. The accuracy of the approximations to the noncentral-F distribution is also discussed.

Signal detection for communications in the underwater acoustic environment

Signal detection is a critical stage in the implementation of any effective communications system. The underwater acoustic environment, particularly in the presence of underwater vehicles, presents significant challenges to reliable detection without excessive false alarms. While there is often sufficient signal-to-noise ratio with respect to stationary broad-band background noise to permit reliable operation, the presence of strong event-like interference signals such as narrow-band signals and impulsive broad-band signals complicates the detection problem significantly. Frequency-hopped signals interleaved with quiescent bands are proposed as the basis of a robust detection system. These signals also make robust detection possible in a multi-access communications system. Two new detection algorithms that exploit the particular structure of these frequency-hopped signals are developed and their performance is analyzed. This analysis uses a modification of the doubly noncentral F-distribution that has not been used previously for such analysis. This distribution makes possible the direct calculation of probabilities of detection and false-alarm under interference and signal scenarios that cannot be analyzed using the better known noncentral F-distribution. With this analysis, the two developed detectors are shown to offer superior performance to that of either the CFAR detector or the binary data sequence detector. Experimental data confirms the theoretically derived results.

A Comparison of Approximations to Percentiles of the Noncentral F-Distribution. By: Hardeo Sahai.

A Comparison of Approximations to Percentiles of the Noncentral F-Distribution. By: Hardeo Sahai.