臺灣博碩士論文加值系統

常態分佈的假設經常被廣泛的運用於統計分析模型。然而，在資料具偏斜特性之分析，常態分配的假設並非合適。本研究試著從Azzalini(1985)所提出的偏斜常態分佈，常態分佈為其一特例，的架構上，提出一些統計推論方法及其應用。值得一提，儘管在文獻上已提出許多有關偏斜常態模型在機率上的特性，不過在統計推論裡，卻存在某些的困難度。事實上，在探究參數之最大概似估計量中，該模型下複雜的概似函數會導致出在數值運算上的問題。

在本文中，我們首先針對Monti(2003)所提出的兩種方法中所牽涉之常數k如何選擇的問題進行探討。Monti宣稱利用此兩種方法於偏斜常態模型下尋求參數估計量，可以減低在數值運算上的困難度，然而，Monti文中只呈現採用k=n樣本數的方式來估計參數的比較結果。因此，在此研究中，我們首先考慮從已知參數最大概似估計量的模型：指數和常態模型著手討論“k=n”此一選擇的適當性；緊接著，理所當然地，我們也會在偏斜常態模型下論述此議題。從我們的模擬結果指出，對於給定一些k≠n的設定下，相對於k=n，利用此兩種方法可以得到表現比較好的估計值。因此，我們的研究結果強烈質疑 Monti(2003)採取k=n的選擇並非是最適當。

其次，我們探討在偏斜常態模型下的一些統計推論問題。更明確地說，我們提供了在標準化的偏斜常態誤差分佈的建構下，母體平均值之單邊或雙邊漸近信賴區間。此推論問題看似基本，卻是統計推論問題首重的核心議題。綜合理論推導及模擬的結果，我們所提出的雙邊漸近信賴區間，在適中到大樣本數之下，表現是令人滿意的。我們也發現所提的漸近信賴區間，特別是單邊，的覆蓋機率，會受到偏斜參數是正或負和值的大小而有明顯的變化。

最後，我們以偏斜常態模型為基礎，利用從MEPS裡健保費用的資料來說明，如何應用母體平均值之雙邊漸近信賴區間。事實上，Yu(2005)已利用此筆資料，從常態、伽瑪、對數常態分佈之下的信賴區間進行分析與比較。透過分析和模擬的結果，我們成功地運用偏斜常態模型於實際資料分析。更值得一提的是，在合理的樣本數下，我們所提出的雙邊漸近信賴區間的覆蓋機率比
Yu(2005)所考慮之三種信賴區間的覆蓋機率高。

The assumption of normality is widely applied in statistical models, however it is sometimes hard to be justified when analyzing real data. The skew normal distribution is therefore proposed as an extension of the normal distribution that includes a skewness parameter. Despite its many probabilistic properities being derived, there are still certain difficulties in statistical inferences under skew normal models. In particular, the complicated form of the likelihood function under a skew normal model causes numerical issues in finding maximum likelihood estimators (MLE) of parameters.

In this study, we first study the choice of a crucial constant k to the two methods provided in Monti (2003) to ease above mentioned numerical difficulty in finding maximum likelihood estimators for skew normal parameters; k=n the sample size is used by Monti. The problem is scrutinized, first, under exponential, normal models
where MLE's are known, then, of course, under the skew normal model. Our simulation results indicate that these two methods give better estimators in terms of mean squared error, with choices of k≠n for many of our studied cases, where some are under the skew normal model. Thus, taking into account our findings, the selection k=n taken in
Monti (2003) is seriously questioned.

Next, in this work, we take on the statistical inference issue, under a skew normal model, from a basic but important framework. More precisely, we provide one-sided or two-sided asymptotic confidence intervals for the mean of a population with standardized skew normal error distribution. Based on theoretical derivations and
simulation results, we find that the performance of
two-sided confidence interval is satisfactory for moderate to large sample sizes. Also, the coverage probabilities of confidence intervals, especially one-sided, vary drastically with the skewness parameters.

Finally, we illustrate how to apply the asymptotic
two-sided confidence interval for the population mean under a skew normal model to analyze the health care expenditure data from MEPS, which is analyzed, from confidence interval aspect, in Yu (2005) under normal, gamma, and lognormal models. The results we have here indicate that, successfully, we implement the skew normal model for real data analyses, and better off, our asymptotic two-sided confidence interval has higher coverage probabilities than those considered in Yu (2005) for reasonable sample sizes.

1 Introduction
2 Some notes on the methods proposed in Monti (2003) for
estimating parameters under a SN model
3 Asymptotic confidence intervals for the mean of a
population with standardized SN-error
4 An application of using a SN model to analyze the health
care expenditure data from MEPS
5 Conclusions
Appendix

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