Applications of hyperellipsoidal prediction regions

Olive (Internat J Stat Probab 2:90–100, 2013) developed a large sample $$100(1-\delta )\%$$ 100 ( 1 - δ ) % nonparametric prediction region for a future $$m \times 1$$ m × 1 test vector $$\varvec{y}_f$$ y f given past training data $$\varvec{y}_1,\ldots , \varvec{y}_n$$ y 1 , … , y n . Consider predicting an $$m \times 1$$ m × 1 future test response vector $$\varvec{y}_f$$ y f , given $$\varvec{x}_f$$ x f and past training data $$(\varvec{x}_1,\varvec{y}_1),\ldots , (\varvec{x}_n,\varvec{y}_n)$$ ( x 1 , y 1 ) , … , ( x n , y n ) . For the multivariate linear regression model $$\varvec{y}_i = \varvec{B}^T \varvec{x}_i + \varvec{\epsilon }_i$$ y i = B T x i + ϵ i , let the pseudodata $$\varvec{w}_i = \hat{\varvec{y}}_f + \hat{\varvec{\epsilon }}_i$$ w i = y ^ f + ϵ ^ i for $$i = 1,\ldots , n$$ i = 1 , … , n where the $$\hat{\varvec{\epsilon }}_i$$ ϵ ^ i are the residual vectors. Under mild regularity conditions, applying the (Olive in Internat J Stat Probab 2:90–100, 2013) prediction region to the pseudodata gives a large sample $$100(1-\delta )\%$$ 100 ( 1 - δ ) % nonparametric prediction region for $$\varvec{y}_f$$ y f . Suppose there is an $$m \times 1$$ m × 1 statistic $$T_n$$ T n such that $$\sqrt{n} (T_n - \varvec{\mu }) \mathop {\rightarrow }\limits ^{D} N_m(\varvec{0}, \varvec{\Sigma }_T)$$ n ( T n - μ ) → D N m ( 0 , Σ T ) . Under regularity conditions, applying the (Olive in Internat J Stat Probab 2:90–100, 2013) prediction region to the bootstrap sample $$T^*_1,\ldots , T^*_B$$ T 1 ∗ , … , T B ∗ gives a large sample $$100(1-\delta )\%$$ 100 ( 1 - δ ) % confidence region for the parameter vector $$\varvec{\mu }$$ μ .