對稱型分位數管制圖

Abstract

假定某一產品之特徵以變數X表示，並假設X之分配函數為q F 及分位數為 - 1 (a ),0 < a < 1. q F 分位數管制圖 (Grimshaw及Alt (1997))之概念為令 ( ) ( ( ), ( ),..., 1 ( )) 2 1 1 1 m Q a F a F a F a q q q = - - - 之分位數向量為控制參數。當我們有一估計 量(a ) n Q 且滿足n(Q ( ) - Q( )) ® N(0, å ) n a a 時， n(Q (a ) Q(a ))' 1 (Q (a ) Q(a )) n n - å - - 將會趨近於 2m c ，因此我們可藉由此一分配 建立管制界限。在式中通常Q(a )及å 為未知，我們可以用training sample來估計。 令μ為一location參數(如中位數，平均數等)，下面為一摺疊式分配函數 G (a) = P( X - £ a),a ³ 0. s m 在r下之對稱分位數為{ ( ), ( )} ( ( ), ( )). F 1 r F 1 r G 1 r G 1 r s s s s - - - + = m - - m + - 其中( ) inf{ : ( ) }. G 1 r a G a r s s - = ³ 可以驗證當q F 為對稱分配時， ) 2 F 1 (r) F 1 (1 r s - - = - - q 及) 2 F 1 (r) F 1 (1 r s - + = - + q ，因此兩個分位數相同且它們對應的 估計量將估計相同參數。 令X ,..., X n 1 為一隨機樣本，取一μ的估計量mˆ ，我們可以定義樣本摺疊式 分配函數為å = = - £ n sn i i I X a n G a 1 ( ) 1 ( mˆ )及G 1 (r) inf{a :G (a) r}. sn sn - = ³ ，因此r 對稱型分位數為{ ( ), ( )} ( ˆ ( ), ˆ ( )). F 1 r F 1 r G 1 r G 1 r sn sn sn sn - - - + = m - - m + - 當給定 k r ,..., r 1 ，就 會有m=2k個對稱分位數，我們令(a ) sn Q 為此一對稱分位數的向量，令Q(α)為 其母體分位數向量。 我們將完成下面工作： (1) 導出n(Q (a ) Q(a )) sn - 的Bahadur展開式並列出近似分配。 (2) 建造對稱型分位數的管制圖。 (3) 設定幾個常用分配計算對稱型及Grimshaw及Alt的經驗型分位數的近似變 異數。 (4) 再比較兩種管制圖的Average run length (ARL)。 (5) 將做一實際資料分析。

Let's assume that the characteristic of a product is denoted by X which has a distribution function Fq with quantile - 1 (a ),0 < a < 1. q F The concept of quantile control chart of Grimshaw and Alt (1997) is setting a quantile vector ( ) ( ( ), ( ),..., 1 ( )) 2 1 1 1 m Q a F a F a F a q q q = - - - as a target to monitor. When we have an estimator (a ) n Q that satisfies n(Q ( ) - Q( )) ® N(0, å ) n a a , we know that n(Q (a ) Q(a ))' 1 (Q (a ) Q(a )) n n - å - - converges to 2m c that allows us to construct a control limit. In case that Q(a ) and å are unknown, we may estimate them from a training sample. Let μbe a location parameter such as median or mean. We define the folded cumulative distribution function G (a) = P( X - £ a),a ³ 0. s m The r symmetric quantile pair is {F 1 (r),F 1 (r)} ( G 1 (r), G 1 (r)) s s s s - - - + = m - - m + - where G 1 (r) inf{a :G (a) r} s s - = ³ . We may show that when q F is symmetric ) 2 F 1 (r) F 1 (1 r s - - = - - q and ) 2 F 1 (r) F 1 (1 r s - + = - + q . Hence symmetric quantile and the classical quantile are identical. Let n X ,..., X 1 be a random sample. Let mˆ be an estimate of μ. We may define the sample folded cumulative distribution function as å = = - £ n sn i i I X a n G a 1 ( ) 1 ( mˆ ) and G 1 (r) inf{a :G (a) r} sn sn - = ³ . Hence, a sample r symmetric quantile pair is {F 1 (r),F 1 (r)} ( ˆ G 1 (r), ˆ G 1 (r)) sn sn sn sn - - - + = m - - m + - . Given k r ,..., r 1 , we can compute m=2k symmetric quantiles. Let (a ) sn Q be the vector of symmetric quantiles, and let Q(α) be the corresponding vector of population quantiles. We are going to accomplish the following tasks： (1) Derive a Bahadur representation for n(Q (a ) Q(a )) sn - and provide its asymptotic distribution. (2) Construct the symmetric quantile control chart. (3) Considering several distributions, we will compare the asymptotic variances of the empirical quantile and symmetric quantile. (4) Derive and compare the average run length (ARL) for two considered quantile control charts. (5) Conduct a data analysis.