scholarly journals Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic

Stats ◽  
2020 ◽  
Vol 3 (3) ◽  
pp. 330-342
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Degrees Of Freedom ◽  
Exact Distribution ◽  
Statistical Testing ◽  
Estimation Methods ◽  
Weighted Sum ◽  
Confidence Estimation ◽  
Chi Square ◽  
Mixing Coefficient ◽  
Sample Variances ◽  
Bridge Distribution

We prove that the Behrens–Fisher statistic follows a Student bridge distribution, the mixing coefficient of which depends on the two sample variances only through their ratio. To this end, it is first shown that a weighted sum of two independent normalized chi-square distributed random variables is chi-square bridge distributed, and secondly that the Behrens–Fisher statistic is based on such a variable and a standard normally distributed one that is independent of the former. In case of a known variance ratio, exact standard statistical testing and confidence estimation methods apply without the need for any additional approximations. In addition, a three pillar bridges explanation is given for the choice of degrees of freedom in Welch’s approximation to the exact distribution of the Behrens–Fisher statistic.

Statistical Justification of Pearson's Criterion for Testing a Complex Hypothesis on the Uniform Distribution

2018 ◽  
pp. 45-53
Author(s):  
T. V. Oblakova
Keyword(s):  
Uniform Distribution ◽  
Degrees Of Freedom ◽  
Likelihood Function ◽  
Random Number Generator ◽  
Maximum Likelihood Estimates ◽  
Chi Square ◽  
Main Hypothesis ◽  
Pearson Criterion ◽  
Complex Hypothesis ◽  
Uniform Law

The paper is studying the justification of the Pearson criterion for checking the hypothesis on the uniform distribution of the general totality. If the distribution parameters are unknown, then estimates of the theoretical frequencies are used [1, 2, 3]. In this case the quantile of the chi-square distribution with the number of degrees of freedom, reduced by the number of parameters evaluated, is used to determine the upper threshold of the main hypothesis acceptance [7]. However, in the case of a uniform law, the application of Pearson's criterion does not extend to complex hypotheses, since the likelihood function does not allow differentiation with respect to parameters, which is used in the proof of the theorem mentioned [7, 10, 11].A statistical experiment is proposed in order to study the distribution of Pearson statistics for samples from a uniform law. The essence of the experiment is that at first a statistically significant number of one-type samples from a given uniform distribution is modeled, then for each sample Pearson statistics are calculated, and then the law of distribution of the totality of these statistics is studied. Modeling and processing of samples were performed in the Mathcad 15 package using the built-in random number generator and array processing facilities.In all the experiments carried out, the hypothesis that the Pearson statistics conform to the chi-square law was unambiguously accepted (confidence level 0.95). It is also statistically proved that the number of degrees of freedom in the case of a complex hypothesis need not be corrected. That is, the maximum likelihood estimates of the uniform law parameters implicitly used in calculating Pearson statistics do not affect the number of degrees of freedom, which is thus determined by the number of grouping intervals only.

scholarly journals Crossover Interference in Arabidopsis

Genetics ◽  
2002 ◽  
Vol 160 (4) ◽  
pp. 1631-1639 ◽  
Author(s):  
G P Copenhaver ◽  
E A Housworth ◽  
F W Stahl
Keyword(s):  
Arabidopsis Thaliana ◽  
Degrees Of Freedom ◽  
Double Strand Breaks ◽  
Crossing Over ◽  
Chi Square ◽  
Strand Breaks ◽  
Crossover Interference

AbstractThe crossover distribution in meiotic tetrads of Arabidopsis thaliana differs from those previously described for Drosophila and Neurospora. Whereas a chi-square distribution with an even number of degrees of freedom provides a good fit for the latter organisms, the fit for Arabidopsis was substantially improved by assuming an additional set of crossovers sprinkled, at random, among those distributed as per chi square. This result is compatible with the view that Arabidopsis has two pathways for meiotic crossing over, only one of which is subject to interference. The results further suggest that Arabidopsis meiosis has >10 times as many double-strand breaks as crossovers.

scholarly journals Estimation of Wave Period from Pitch and Roll of a Lidar Buoy

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1310
Author(s):  
Andreu Salcedo-Bosch ◽  
Francesc Rocadenbosch ◽  
Miguel A. Gutiérrez-Antuñano ◽  
Jordi Tiana-Alsina
Keyword(s):  
Degrees Of Freedom ◽  
Translational Motion ◽  
Cutoff Frequency ◽  
Peak Level ◽  
Wave Period ◽  
Estimation Methods ◽  
Zero Crossing ◽  
Power Spectral ◽  
Period Estimation ◽  
Wind Lidar

This work proposes a new wave-period estimation (L-dB) method based on the power-spectral-density (PSD) estimation of pitch and roll motional time series of a Doppler wind lidar buoy under the assumption of small angles (±22 deg) and slow yaw drifts (1 min), and the neglection of translational motion. We revisit the buoy’s simplified two-degrees-of-freedom (2-DoF) motional model and formulate the PSD associated with the eigenaxis tilt of the lidar buoy, which was modelled as a complex-number random process. From this, we present the L-dB method, which estimates the wave period as the average wavelength associated to the cutoff frequency span at which the spectral components drop off L decibels from the peak level. In the framework of the IJmuiden campaign (North Sea, 29 March–17 June 2015), the L-dB method is compared in reference to most common oceanographic wave-period estimation methods by using a TriaxysTM buoy. Parametric analysis showed good agreement (correlation coefficient, ρ = 0.86, root-mean-square error (RMSE) = 0.46 s, and mean difference, MD = 0.02 s) between the proposed L-dB method and the oceanographic zero-crossing method when the threshold L was set at 8 dB.

scholarly journals Confronting Theories of Firm Growth in Light of Degrees-of-Freedom Analysis

2015 ◽  
Vol 22 (74) ◽  
pp. 385-404
Author(s):  
Sérgio Fernando Loureiro Rezende ◽  
Ricardo Salera ◽  
José Márcio de Castro
Keyword(s):  
Degrees Of Freedom ◽  
Firm Growth ◽  
Dynamic Capabilities ◽  
Statistical Tests ◽  
Explanatory Power ◽  
Stage Theory ◽  
Chi Square ◽  
Growth Optimum ◽  
Chi Square Test ◽  
Do So

This article aims to confront four theories of firm growth – Optimum Firm Size, Stage Theory of Growth, The Theory of the Growth of the Firm and Dynamic Capabilities – with empirical data derived from a backward-looking longitudinal qualitative case of the growth trajectory of a Brazilian capital goods firm. To do so, we employed Degree of Freedom-Analysis for data analysis. This technique aims to test the empirical strengths of competing theories using statistical tests, in particular Chi-square test. Our results suggest that none of the four theories fully explained the growth of the firm we chose as empirical case. Nevertheless, Dynamic Capabilities was regarded as providing a more satisfactory explanatory power.

scholarly journals Location Selection for Smog Towers Using Zadeh’s Z-Numbers Integrated with WASPAS

2021 ◽  
Author(s):  
Janani Bharatraj
Keyword(s):  
Air Pollution ◽  
Fuzzy Sets ◽  
Degrees Of Freedom ◽  
Assessment Method ◽  
Weighted Sum ◽  
Membership Degree ◽  
Residential Areas ◽  
Location Selection ◽  
Degree Of Membership ◽  
Product Assessment

Fuzzy sets have been extensively researched and results have been developed based on the extensions of fuzzy sets. In this chapter, fuzzy sets and its extensions are discussed. Z-numbers along with weighted sum product assessment method is used to obtain a feasible solution to the location selection problem for installation of smog towers in a densely populated locality. The degrees of freedom namely degree of membership, degree of non-membership and the degree of hesitancy have been expressed as Zadeh’s Z-number with probability quotient for the degrees. Further, ranking of the alternatives based on Z-numbers and WASPAS to allocate smog towers to residential areas stricken by air pollution.

The Application of the Chi-Square Test of Goodness-of-Fit to Mortality Data Graduated by Summation Formulae.

1971 ◽  
Vol 97 (2-3) ◽  
pp. 325-330 ◽  
Author(s):  
J. H. Pollard
Keyword(s):  
Degrees Of Freedom ◽  
Goodness Of Fit ◽  
Summation Formula ◽  
Mortality Data ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Summation Formulae ◽  
Chi Square Test

In his paper of 1941, Seal included details of some experiments he performed in an attempt to estimate the appropriate number of degrees of freedom for the chi-square goodness-of-fit test of a summation formula graduation. These results are referred to by Tetley and by Benjamin and Haycocks in their textbooks when they mention the difficulty of determining the number of degrees of freedom or mean chi-square value.

Query 23: Degrees of Freedom for Chi-Square

Technometrics ◽  
1967 ◽  
Vol 9 (3) ◽  
pp. 489 ◽  
Author(s):  
Herman Chernoff
Keyword(s):  
Degrees Of Freedom ◽  
Chi Square
Sign in / Sign up

Export Citation Format

Share Document