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Reduced Bias Factor Distribution to Reduce the Likelihood Estimate Bias of Small Sample Sizes

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A new method is developed by the author to reduce the bias of the maximum likelihood estimates (MLE) with small sample sizes. The new method is based on a special case of the Fréchet distribution. The special case of the Fréchet distribution is now referred as the “New Distribution” in this article and/or the “Reduce Bias Factor” Distribution. The cumulative distribution function (CDF) of the “New Distribution” is the factor that decreases the bias in distribution parameter estimates to improve data analysis and reliability/lifetime prediction accuracy when using maximum likelihood estimates (MLE) particularly for small sample sizes. The new distribution is very flexible and versatile with two parameters, i.e. Scale and Shape to fit for relevant scenarios. This function support any life distributions such as Weibull, Normal, Log Normal and or others distributions as needed. The new distribution is capable of describing most of the correction factor formulas among them the well known correction factors such as the C4 for Normal and Log Normal Distributions Sigma by William Gossett (Refs. 1, 2, and 3), who was the Chief Brewmaster for the Guinness Breweries.
William Gossett also invented the Monte Carlo simulation method for statistical distribution analysis. Dr. Robert B. Abernethy used Monte Carlo later to develop corrections for the Weibull MLE Median and Mean Beta approximately C4^3.5 and C4^6 respectively, see (Refs. 1, 2, and 3). Extensive Monte Carlo simulations and transposed linear regressions by the author provide the basis for the conclusions in this paper. The results herein apply to complete samples starting from, but not limited to, Beta values of 0.5 up to Beta value of 10 with 1200 sets of samples of 2 up to 300. The author is using the new method to reduce the MLE bias with censored data, never the less the author does encourage additional research into censored data.

Keywords: RAMS 2010 Proceedings - Statistics - Weibull

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