Effects of Ignoring Correlations When Computing Sample Chi-Square. John W. Fowler February 26, 2012
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1 Effects of Ignorng Correlatons When Computng Sample Ch-Square John W. Fowler February 6, 0 It can happen that ch-square must be computed for a sample whose elements are correlated to an unknown extent. We denote the sample elements as x, to. In general, when there are correlatons, the form of ch-square s χ j w x x x x ( where the w are the elements of the nverse of the covarance matrx Ω, Ω σ σσ K σσ σσ σ L σσ M M O M σ σ σ σ L σ ( If the covarance matrx s not known numercally, the only formula avalable for computng chsquare s that for the case of zero off-dagonal correlaton. Snce ths uncorrelated form s erroneous when non-zero off-dagonal correlatons are beng gnored, we attach a subscrpt e: var χ χ e ( x x σ As shown below, both forms (Equatons and have the expectaton value, but the erroneous form has a larger varance. The mean and varance of the correct form are well known to be and, respectvely, whereas the varance for the erroneous form s e + + Ths s found by ntegratng the moments of the erroneous ch-square over the jont densty functon for the correlated Gaussan random varables nvolved. In general, for degrees of freedom, the jont densty s p ( x e χ / / ( π Ω ( ( (5
2 where χ and Ω are used formally as defned n Equatons and, and Ω s the determnant of the covarance matrx. The varance of the erroneous ch-square n Equaton can be wrtten more smply by usng the fact that the dagonal correlatons are always unty and are pcked up once n the summaton below, whereas the others are pcked up twce: χ e χ var χ e ote that ths s just twce the square of the Frobenus norm of the correlaton matrx. If there are no off-dagonal correlatons, Equaton shows (and less obvously, Equaton 6 that the varance wll reduce to, and the ch-square wll not be erroneous. Otherwse, any non-zero off-dagonal correlaton can only ncrease the varance above. The fact that the expectaton value of the erroneous ch-square s, the same as for the correct chsquare, can be seen as obvous from the fact that the former s smply the sum of terms, each of whch has the expectaton value, snce each term has the expectaton value of the numerator n the denomnator. The varance of the erroneous ch-square s not as obvous, snce the expectaton value of the square depends on the underlyng densty functon. For notatonal brevty, defne χ e z where the z are generally correlated zero-mean unt-varance Gaussan random varables. Then obvously The varance s where e z z z σ e e e z zj z zj z z j j For any combnaton of and j, the form of the expectaton value <z z j > depends only on the propertes of z and z j, namely ther correlaton and the fact that both are zero-mean unt-varance Gaussan random varables, so we can compute the expectaton value of ths product usng a bvarate jont Gaussan densty functon,.e., Equaton 5 wth all means zero and all varances unty and all the random varables ntegrated out except the th and j th : (6 (7 ( (9 (0 z z z z p( z, z dz dz j j j j + (
3 ote that for j, ths reduces to <z >, the well known value for the th moment of a zero-mean unt-varance Gaussan random varable. Usng Equaton n Equaton 0, ( e ( + χ + ( and Equaton 9 becomes χ χ e e ( whch s just Equaton 6, hence equvalent to Equaton. Smlar computatons can be used to show that the thrd raw moment s ( χe + ( + jk + k + jk k k ( jk k jk k k jk k j k k j k 6 + jk k k jk k ( Denotng the n th raw moment m n and the n th central moment µ n, the thrd central moment s μ m m m + m (5 So the thrd central moment of the erroneous ch-square s μ k + + k jk k jk k (6 The skewness s Equaton (6 dvded by the rght-hand sde of ether Equaton or 6 (or rased to the / power:
4 jk k k skew e If all the off-dagonal correlatons are set to zero, then / (7 m + μ m μ skew e ( skew / ( Contnung to the fourth raw moment, we ntegrate z z j z k z m over the jont densty functon to obtan <z z j z k z m >, whch s then summed over all four ndexes to gve us m e k m ( k m jk jm km ( km k jm m jk ( k jk m jm k m km jk jm km ( k jm km m jk km k m jk jm k m + + k jk k k m km k jm km (9 The fourth central moment s μ + m m m 6m m m (0 The fourth central moment of the erroneous ch-square s therefore
5 μ + + Expandng and cancelng, k jk k ( km k jm km μ + k m k jk k k m k m km k jm km ( ( So the kurtoss s kurt e k m ( km + k jm km ( If all the off-dagonal correlatons are set to zero, then μ + kurt e + + ( kurt (
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