GENERALIZED PIVOTS GENERALIZED PIVOTS FOR VARIANCE COMPONENTS FOR - PowerPoint PPT Presentation

GENERALIZED PIVOTS GENERALIZED PIVOTS FOR VARIANCE COMPONENTS FOR VARIANCE COMPONENTS IN MIXED LINEAR MODELS IN MIXED LINEAR MODELS Barbora Arendacká

THE MODEL: Y = n-dimensional vector of observations X, A 1 = known matrices 2 I  , ~ N n  0,  2 I  u ~ N q  0,  1 cov  u, = 0 2 ,  2  = unknown parameters  ,  1 2 ≥ 0,  2  0  1 T W 1 = A 1 A 1 R  A 1 ⊈ R  X  Further we suppose 2 THE TASK: construction of confidence intervals on  1

Invariance with respect to the group of translations in mean: T = M X = I n − X  X T X  − X T B X B X maximal invariant T B X = I n − rank  X  B X T W 1 B X V 1 = B X Minimal sufficient statistics: Olsen, Seely, Birkes (1976) spectral decomposition of V 1 r V 1 = ∑ i = 1  i E i  1  2  ...  r ≥ 0  1 , .... ,  r

Balanced one-way random effects model Y ij = i  ij ,i = 1,... , I , j = 1,... ,J 2  ,  ij ~ N  0,  2  , mutually independent  i ~ N  0,  1 2 ~ J  1 2  2  I − 1 2 U 1 = S B between-groups sum of squares 2 ~ 2  I  J − 1  2 U 2 = S W within-group sum of squares  1 = J ,  2 = 0 U 1 , ... ,U r General case 2   r 2 U r ~ supposing is not too restrictive

2 Construction of confidence intervals on  1 U 1 U r − 1 , U r 2 , ...... , 2 2 ~  1 ~  r − 1 2 ~  r 2  2  2  2   1  1  r − 1  1  U = U 1 , ... ,U r  Pivot Pivot 2  R  U ,  1 - distribution independent of all parameters find q 1 , q 2 such that (1- α )100% confidence set

Difficulties caused by nuisance parameters generalized inference Tsui, Weerahandi (1989), Weerahandi (1993) Extension of “classical” methods for testing and constructing confidence intervals in such a way that the quantities these methods are based on (pivots and test statistics) are let to be FUNCTIONS not only of the RESPECTIVE RANDOM VECTOR and the parameter of interest but of the OBSERVED DATA and ALL THE PARAMETERS as well The only requirement: - for each fixed data the generalized quantities behave like their “classical” counterparts

GENERALIZED PIVOTS U = U 1 , ... ,U r  u = u 1 , ... ,u r  2 ,  2  with properties: Generalized pivot: a function R  U ,u ,  1 For each observed u can find q 1 ,q 2 (1- α )100% confidence set

A drawback The confidence set is constructed for fixed u , 2 ,  2  R  U ,u ,  1 the distribution of usually depends on u , and so the set preserves the frequency properties of an (1 −α ) 100% confidence set only conditionally, for given u, and its ACTUAL CONFIDENCE LEVEL must be CHECKED BY SIMULATIONS On the other hand Inclusion of observed data makes construction of expressions with distribution independent of nuisance parameters much easier

An example of a generalized pivot u i U i observed 2 = Q i ~  i 2  2  2  2  value  i  1  i  1 u r U r observed 2 2 = V ~  r value 2   2 u i   u i − Q i u r r − 1 ∑ i = 1 r − 1 R = ∑ i = 1  u i − u r   2  2 u r  i  1 V 2 =  1  i u i r − 1 ∑ i = 1 r − 1 ∑ i = 1  i Q i 2  2  i  1 2 r obs = 1

 u i − Q i u r r − 1 R = ∑ i = 1 Computing the bounds of the generalized interval  V r − 1 ∑ i = 1  i Q i For fixed u P  q / 2 ≤ R ≤ q 1 −/ 2 = 1 − C  u = {  1 2 ≤ q 1 −/ 2 } = {  1 2 ≤ q 1 −/ 2 } 2 ;q / 2 ≤ R  u ,u ,  1 2 ; q / 2 ≤ 1 Monte Carlo methods Simulate the distribution of R by generating a large number of Q 1 ,...., Q r-1 ,V – independent, chi squared distributed random variables Numerical integration + numerical solving of the equation Q i  i q 1 −/ 2  u r r − 1 r − 1 1 −/ 2 = P  R ≤ q 1 −/ 2 = P  ∑ i = 1 V ≥ ∑ i = 1 u i  Negative bounds are put equal to zero

Unless r =2 , R is far from being unique Zhou, Mathew (1994) c i  u i − Q i u r r − 1 ∑ i = 1  V R = nonnegative constants c i r − 1 ∑ i = 1  i c i Q i My suggestions for the choice of c i inspired by test statistics 2  1 for testing the nullity of : c i = i c i = 1 / i c i = 1

Another and not the only other generalized pivot Park, Burdick (2003) u i u i U i r − 1 r − 1 r − 1 r − 1 ∑ i = 1 = ∑ i = 1 ∑ i = 1 2 = ∑ i = 1 Q i 2  u r 2 2  u r  2   i  1  i  1  i  1 V U r the observed values are equal 2  1 R defined as a solution for of the non-linear equation 2 - observed value is  1

Computation of bounds of the interval: Monte Carlo methods or numerical approach Again, constants can be added c i ≠ 1 However, with the computational attractiveness is lost r − 1 ∑ i = 1 2 Q i ~  r − 1 = ∑ i = 1  i

THANK YOU FOR YOUR ATTENTION THANK YOU FOR YOUR ATTENTION