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Student t-statistic distribution for non-Gaussian populations
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| The exact distribution of t(n-1) = √n X n[-μ/Sn is easily derived when the parent population is Gau (μ, σ), since the sample mean Xn and sample standard deviationSn are independent. However this is an exceptional situation, since, the. independence of Xn and S2 n is a characterization of the Gaussian populations. When Y isn't Gaussian, the exact distribution of Tn-1 = √n Y n-μ/Sn is difficult to compute, due to the dependence structure, tying the. sample, mean and variance. Our aim has been to investigate, for general parent Y with known skewness and kurtosis, whether there, exists one type in the Pearson system of distributions which better approximates Tn-1 = √n Y-μ/Sn in the specific sense, that it provides better approximations to the high quantiles of T n-1 than the corresponding quantiles of t(n-1). We show that the Tn-1 distribution for general parent can be approximated by a Pearson's type IV distribution, an unexpected result since. Student's t distributions is not, of Pearson's type IV. We also show that this new approximation is better because, skewness is taken into account. In fact, the covariance between Xn and S2n suggests a strong relation between the population skewness and the. attraction or repulsion behaviour between Xn and S2n. To support this statement some, simulation work is done. | 316.17 KB | Adobe PDF |
Authors
Advisor(s)
Abstract(s)
The exact distribution of t(n-1) = √n X n[-μ/Sn is easily derived when the parent population is Gau (μ, σ), since the sample mean Xn and sample standard deviationSn are independent. However this is an exceptional situation, since, the. independence of Xn and S2 n is a characterization of the Gaussian populations. When Y isn't Gaussian, the exact distribution of Tn-1 = √n Y n-μ/Sn is difficult to compute, due to the dependence structure, tying the. sample, mean and variance. Our aim has been to investigate, for general parent Y with known skewness and kurtosis, whether there, exists one type in the Pearson system of distributions which better approximates Tn-1 = √n Y-μ/Sn in the specific sense, that it provides better approximations to the high quantiles of T n-1 than the corresponding quantiles of t(n-1). We show that the Tn-1 distribution for general parent can be approximated by a Pearson's type IV distribution, an unexpected result since. Student's t distributions is not, of Pearson's type IV. We also show that this new approximation is better because, skewness is taken into account. In fact, the covariance between Xn and S2n suggests a strong relation between the population skewness and the. attraction or repulsion behaviour between Xn and S2n. To support this statement some, simulation work is done.
Description
EISBN - 978-1-4244-5733-5
Conference name - 32nd International Conference on Information Technology Interfaces, ITI 2010; Conference date - 21 June 2010 - 24 June 2010; Conference code - 81697
Conference name - 32nd International Conference on Information Technology Interfaces, ITI 2010; Conference date - 21 June 2010 - 24 June 2010; Conference code - 81697
Keywords
Attraction Delta method Pearson's type IV distributions Repulsion Skewness and kurtosis
Pedagogical Context
Citation
J. P. Martins, "Student t-statistic distribution for non-Gaussian populations," Proceedings of the ITI 2010, 32nd International Conference on Information Technology Interfaces, Cavtat, Croatia, 2010, pp. 563-568.
Publisher
IEEE Canada
CC License
Without CC licence