• Journal of the Korean Statistical Society
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• v.7 no.1
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• pp.3-10
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• 1978

Let $X_i,...,X_n$ be a random sample from a distribution with cumulants $K_1, K_2,...$. The statistic $t = \frac{\sqrt{x}(\bar{X}-K_1)}{S}$ has the well-known 'student' distribution with $\nu = n-1$ degrees of freedom if the $X_i$ are normally distributed (i.e., $K_i = 0$ for $i \geq 3$). An Edgeworth series expansion for the distribution of t when the $X_i$ are not normally distributed is obtained. The form of this expansion is Prob $(t

• Journal of the Korean Statistical Society
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• v.22 no.2
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• pp.235-247
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• 1993

The predictive density function of a potential future observation and its first four moments are obtained in this paper to study the effects of a non-normal prior of the unknown mean of a normal population. The derived predictive density function is modified to study changes in utility curves, used to choose the optimum treatment from a given set of treatments, at a given level of stimulus due to slight deviations from normality of the prior distribution. Numerical illustrations are provided to exhibit some effectsl.

• Journal of the Korean Society for Marine Environment & Energy
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• v.13 no.2
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• pp.112-120
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• 2010

In this paper time series wave data measured continuously for 24 hours during a storm in Yura Sea Area are used to investigate statistical characteristics of nonlinear waves. The exceedance probability of wave height is compared using the Rayleigh distribution and the Edgeworth-Rayleigh (ER) distribution. Wave data which show stationary state for 10 hours contain 4600 waves approximately. The Gram-Chalier distribution fits the probability of wave elevation better than the Gaussian distribution. The Rayleigh ($H_{rms}$) distribution follows the exceedance probability of wave height in general and predicts the probability of freak waves well. The ER distribution overpredicts the exceedance probability of wave heights and the occurrence of freak waves. If wave data measured for 30 minute period which contains 250 waves are used, the ER distribution can predict the occurrence probability of freak waves well. But it overpredicts the probability of overall wave height If no freak wave occurs, the Rayleigh ($H_{rms}$) distribution agrees well with wave height distribution for the most of wave height ranges. The wave height distribution of freak waves of which height are less than 10 m shows similar tendency compared with freak waves greater than 10 m. The value of $H_{max}/H_{1/3}$ is related to the kurtosis of wave elevation. It seems that there exists threshold value of the kurtosis for the occurrence of freak waves.

• International Journal of Automotive Technology
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• v.7 no.7
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• pp.859-866
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• 2006

This study employed the perturbation method, the Edgeworth series, the reliability optimization, the reliability sensitivity technique and the robust design to present a practical and effective approach for the robust reliability design of vehicle components with arbitrary distribution parameters on the condition of known first four moments of original random variables. The theoretical formulae of the robust reliability design for vehicle components with arbitrary distribution parameters are obtained. The reliability sensitivity is added to the reliability optimization design model and the robust reliability design is described as a multi-objection optimization. On the condition of known first four moments of original random variables, the respective program can be used to obtain the robust reliability design parameters of vehicle components with arbitrary distribution parameters accurately and quickly.