U-tests for variance components in linear mixed models

Abstract

We propose a U-statistics-based test for null variance components in linear mixed models and obtain its asymptotic distribution (for increasing number of units) under mild regularity conditions that include only the existence of the second moment for the random effects and of the fourth moment for the conditional errors. We employ contiguity arguments to derive the distribution of the test under local alternatives assuming additionally the existence of the fourth moment of the random effects. Our proposal is easy to implement and may be applied to a wide class of linear mixed models. We also consider a simulation study to evaluate the behaviour of the U-test in small and moderate samples and compare its performance with that of exact F-tests and of generalized likelihood ratio tests obtained under the assumption of normality. A practical example in which the normality assumption is not reasonable is included as illustration.

Appendix

1.1 Proof of the contiguity result and of (12)

Denoting the densities of Y _ij under ${\mathcal {H}}_{1n}$ and ${\mathcal {H}}_{0}$, respectively, by q _ij and p _ij we will prove the contiguity of the sequence q _ij to p _ij , using the LeCam’s Second Lemma (Hájek et al. 1999). For details on contiguity and Lecam’s lemmas, see Hájek et al. (1999, Chap. 7) or Sen and Pedroso-de Lima (2011), for example.

Without loss of generality, consider that e _ij and b _i are absolutely continuous with densities, respectively, given by

where f _e and f _b do not depend on unknown parameters. Under model (7), the likelihood of Y _ij is

(30)

Expanding $f_{e_{ij}}(y-\mu_{i}-b_{i})$ around b _i =0, we have

(31)

where $f_{e_{ij}}^{(1)}$ and $f_{e_{ij}}^{(2)}$ denote the first and second derivatives of $f_{e_{ij}}$, respectively. Substituting (31) in (30) and recalling that $\mathbb{E} [b_{i}]=0$ and $\operatorname{Var}[b_{i}]=\sigma_{b}^{2}$, we have

Therefore, the likelihood ratio associated to y _ij is

implying that the global log-likelihood is

After some (arduous) algebra is possible to show that

(32)

where $W_{i}=f_{e_{ij}}^{(2)}(Y_{i}-\mu_{i})/f_{e_{ij}}(Y_{i}-\mu_{i})=f_{e}^{(2)}(Y_{i}-\mu_{i})/f_{e}(Y_{i}-\mu_{i})$. Note that under ${\mathcal {H}}_{0}$, W _i denote independent random variables with null means and variances σ ², so that by the Central Limit Theorem, we have

(33)

with γ ²=δ ⁴ σ ²/4. On the other hand, using Khintchine´s Weak Law of Large Numbers, we conclude that

(34)

Therefore, substituting (34) and (33) in (32) and using Slutsky’s theorem, it follows that

and hence, by LeCam’s Second Lemma, contiguity holds for the considered local alternatives.

Since contiguity holds, we may appeal to LeCam’s third lemma (on the joint distribution of $l({\bf Y})$ and J _n ) under the null hypothesis and obtain the distribution of the latter under such contiguous alternatives. Incidentally, that does not need the computation of the variance of J _n under contiguous alternatives. Thus, the proof of (12) follows.

1.2 Proof of (18)

First note that the ordinary least-squares estimator $\widehat {\boldsymbol{\beta} } = ({\bf X}^{\top}{\bf X})^{-1} {\bf X}{\bf Y}$ may be re-expressed as

with

(35)

Furthermore, is possible shown that

(36)

where

(37)

with e _r representing th rth element of ${\bf e}$. Is possible show that γ is a p-dimensional vector with $\mathbb {E}[\boldsymbol{\gamma}]={\bf0}$ and $\operatorname{Var}[\boldsymbol{\gamma}]=O(M_{n})$, since that the elements of the elements of ${{\bf x}}_{s}{{\bf x}}_{t}^{\top}$, 1≤s,t≤n are uniformly bounded by (16), implying that $\boldsymbol{\gamma }=O_{p}(\sqrt{M_{n}})$. To proof (18), consider the following theorem on the uniform asymptotic linearity in probability of $T_{n\widehat{\beta}}$.

Theorem 1

Under the assumptions of model (13) and conditions (16)–(17), we have

(38)

as k(n)→∞.

Proof

By (35) and (36), we have

Recalling that $\boldsymbol{\gamma}=O_{p}(\sqrt{M_{n}})$, then

 □

Therefore, using the above result and Slutsky’s theorem it follows that (18) holds.

1.3 Proof of (19)

Under ${\mathcal {H}}_{1n}:\sigma_{b}^{2}=\delta^{2}/\sqrt{n}$, it follows that $T_{n\widehat{\beta}}=\sum_{1 \leq r < s \leq n}\eta_{nrs} (Y_{r}-{{\bf x}}_{r}^{\top}\widehat{\boldsymbol{\beta}})(Y_{s}-{{\bf x}}_{s}^{\top }\widehat{\boldsymbol{\beta}})$ can be rewritten as

(39)

where

Now note that under ${\mathcal {H}}_{0}: \sigma_{b}^{2}=0$, $T_{n\widehat{\beta }}= T_{n\widehat{\beta}}^{0}$ almost surely. Let u(s) denote a function that associates the subject’s index to the sth observation lexicographically ordered. The statistic A _n1 can be decomposed as

implying that $\mathbb{E}[A_{n1}]=0$. Similarly, we may be show that $\mathbb{E}[A_{n2}]=0$. Since , it follows that $\mathbb{E}[A_{n3}]=\frac{\delta^{2}}{2\sqrt{n}}(n^{2}-\sum_{i=1}^{k}n_{i}^{2})$. Recalling that $J_{n\widehat{\beta}}=T_{n\widehat{\beta}}/(S_{n}\sqrt {M_{n}})$, then by (39) we have

where $J_{n\widehat{\beta}}^{0}=T_{n\widehat{\beta}}^{0}/(S_{n}\sqrt{M_{n}})$ and $A_{ni}^{*}=A_{ni}/(S_{n}\sqrt{M_{n}})$ (i=1,2,3). We see that $J_{n\widehat{\beta}}^{0}$ has mean and variance asymptotic, given, respectively by 0 and 1.

Now, let

and note that ${{\bf T}}_{n1}^{*}=\sum_{1 \leq r < s \leq n}\eta_{nrs}b_{u(r)}{{\bf x}}_{s}$ is a p-dimensional vector with mean vector and covariance matrix, given, respectively, by

Considering (16), we may show that ${{\bf T}}_{n1}^{*}/\sqrt {M_{n}}=o_{p}(1)$, since that $\varSigma_{{{\bf T}}_{n1}^{*}}=o(M_{n})$. Similarly, is possible show also that

$$\frac{1}{\sqrt{M_n}}\sum_{1 \leq r < s \leq n}\eta_{nrs}b_{u(r)}e_s=o_p(1), $$

implying that

has mean and variance asymptotic equal to zero. Along the same lines, we may show that $A_{n2}^{*}$ also has mean and variance asymptotic equal to zero. Denoting lim _n→∞ M _n /n ³=λ, we have

Since b ₁,…,b _k are iid random variables and the n _i are bounded, using $\mathbb{E}[b_{1}^{4}]<\infty$, we have $\operatorname{Var}[A^{*}_{n3}] \rightarrow0$. Then, by the contiguity of the sequence of local alternative hypotheses ${\mathcal {H}}_{1n}:\sigma_{b}^{2}=\delta^{2}/\sqrt{n}$, and recalling that $S_{n}{\stackrel{{\mathbb{P}}}{\longrightarrow}}\sigma_{e}^{2}$, we have, under ${\mathcal {H}}_{1n}$,

implying that under ${\mathcal {H}}_{1n}:\sigma_{b}^{2}=\delta^{2}/\sqrt{n}$ the center of the normal distribution is shifted to the right by $\delta^{2} / (2\sigma_{e}^{2}\sqrt{\lambda})$.

The proof of (27) follows similarly and for this reason its is omitted.