偏態 - 模擬與交易.9

在對稱分布的情況下，均數(Mean)、眾數(Mode)、中數(Middle)，三個敘述統計量彼此相等。也就是說，在不對稱有所偏態的情況下，除去眾數不討論，右偏之下，極端值(大)會加大均數的數值，而有Mean>Middle；反之，左偏則有Middle>Mean，因此本文利用Mean的平方與Middle的平方取差值，作為統計檢定量。透過標準常態分配亂數來模擬，制定一個查表值，用以確認偏態的訊號，以8筆資料為例，檢定統計量 = Mean的平方-Middle的平方，95%的顯著門檻值為0.71

Excel VBA參考程式碼
Public RankSort(1 To 8) As Variant
Const FixConst As Variant = 0.398942284 '1/sqr(2*3.1415926)

Public Sub MeanModdle()
'模擬常態分配隨機變數的順序統計量期望值
Dim seed, seedCalc, nextSeed, NorValue As Variant
Dim ArrNorRandom(1 To 8), x(1 To 100000) As Variant
Dim iith, testNum, xTemp, yTemp As Long

For testNum = 1 To 100000
iith = 1
Do
seed = Int(Rnd() * 1000000)
seedCalc = ((seed - 500000) * 0.000001) * 8
'模擬4個標準差範圍的 N(0,1) 常態分配變數
NorValue = Format(FixConst * Exp(-0.5 * seedCalc * seedCalc), "0.000000")
'常態分配機率函數的數值
nextSeed = 0
nextSeed = Int(0.001 * (seed * seed))
nextSeed = Format(0.000001 * nextSeed - Int(0.000001 * nextSeed), "0.000000")
nextSeed = Int(1000000 * nextSeed)
nextSeed = Int(((23 + iith * 0.01) * nextSeed + 1011 * iith)) Mod 999999
nextSeed = Format(nextSeed * 0.00001 * FixConst, "0.000000")

If (nextSeed) < NorValue Then
ArrNorRandom(iith) = seedCalc
iith = iith + 1
End If
Loop While iith <= 8

'泡沫排序法
Dim swapTemp, ii, jj As Variant
For jj = 0 To 6
For ii = 1 To 7 - jj
If ArrNorRandom(ii) > ArrNorRandom(ii + 1) Then
swapTemp = ""
swapTemp = ArrNorRandom(ii + 1)
ArrNorRandom(ii + 1) = ArrNorRandom(ii)
ArrNorRandom(ii) = swapTemp
End If
Next ii
Next jj

xTemp = (ArrNorRandom(4) + ArrNorRandom(5)) * 0.5 'Middle
yTemp = (ArrNorRandom(1) + ArrNorRandom(2) + ArrNorRandom(3) + ArrNorRandom(4) + ArrNorRandom(5) + ArrNorRandom(6) + ArrNorRandom(7) + ArrNorRandom(8)) / 8
x(testNum) = yTemp * yTemp - xTemp * xTemp
Next 'For testNum = 1 To 100000

For jj = 0 To 4999 '資料數據共計10萬筆，求解第95001序位
For ii = 1 To 100000 - 1 - jj
If x(ii) > x(ii + 1) Then
xTemp = 0
xTemp = x(ii + 1)
x(ii + 1) = x(ii)
x(ii) = xTemp
End If
Next ii
Next jj

Debug.Print x(95001)
' 0.709336686576
' 0.706396410096
' 0.708935433984
End Sub