APPROXIMATE TEST FOR TESTING A NULL VARIANCE RATIO IN THE UNBALANCED ONE-WAY RANDOM MODEL

The approximate test for testing the significance of the random effect is presented in the unbalanced one-way random model in which both random effects and errors are from nonnormal universes. The test is based on the asymptotic distribution of the F -ratio. Under the condition that the number of groups tends to infinity while the average of powers of the group sizes is bounded, the asymptotic distribution of the F statistic is obtained. Robustness of the proposed test is given.


Introduction
We derive the approximate test for testing the signi…cance of the random e¤ect in the unbalanced one-way random e¤ects model where both random e¤ects and errors are from nonnormal universes. To derive the approximate test, we …rst obtain the asymptotic distribution of the F -ratio.
In literature there are two di¤erent methods to obtain the asymptotic distribution of the F -ratio. Akritas and Arnold (2000) and Akritas and Papadatos (2004) obtained asymptotic normality of the F -ratio from the di¤erence M S M SE and from the fact that M SE converges in probability to constant. Here, M S and M SE are the mean square for the random e¤ects and errors respectively. Westfall (1988) …rst derived the joint asymptotic distribution of M S and M SE and then used the delta method to obtain asymptotic normality of the F -ratio.
To get the asymptotic distribution of the F -ratio, we use the method of Westfall and establish the following asymptotic condition. The number of groups is large while the average of powers of the group sizes is bounded. This asymptotic condition may be viewed as modi…cation of the asymptotic condition established by Wesfall (1987,1988). He assumed that the number of groups is large while the group sizes are from a …nite set of positive integers.

SEVGI DEM IRCIO ¼ GLU AND BILGEHAN G ÜVEN
Also it is implicitly shown that the presented approximate test is robust for the size of the test in the balanced model does not depend on the fourth moment of the error term for the balanced case. The size of the test in the non normal case is same as it in the normal case. This paper di¤ers from the previous studies in three ways. A new asymptotic condition is established by modifying Westfall's asymptotic condition. Robustness of the asymptotic distribution of the F -ratio is analytically shown. Di¤erent distributions having positive, null and negative kurtosis are used in simulations.
This paper is organized as follows: Sec. 2 demonstrates the asymptotic condition and its consequences. Sec. 3 gives the asymptotic distribution of the F -ratio under the established asymptotic condition. Sec. 4 proposes the approximate test for testing signi…cance of the random e¤ects. Sec. 5 shows that the approximate test is robust. In Sec. 6 some numerical and simulated results are given to examine the accuracy of the approximate test.
Throughout the paper we shall use the following notations. If d N is a sequence of N and r is a real number then has a nonzero …nite limit as N ! 1.

The Model and Asymptotic
The unbalanced one-way random e¤ects model is: : : : t j = 1; 2; : : : where is an overall mean, i and e ij are random variables with zero means and variances 2 and 2 respectively. The model is appropriate for analyzing data involving t random treatments. The number of observation is N where N = P t i=1 n i . We shall address the problem of testing H 0 : = 0 vs. H 1 : > 0 where the ratio of variances is de…ned as = 2 = 2 . The statistic for testing H 0 is based on F N = M S =M SE (2) where M S = (t 1) 1 SS and M SE = (N t) 1 SSE. SS and SSE are the sum of squares for treatment and for error respectively and they are de…ned as Under the normality of the random e¤ects and the error terms, the test rejects H 0 when F N > F t 1;N t; where F 1; 2 ; denotes the 1 quantile of the F distribution with degrees of freedom 1 and 2 .
When the random e¤ects and error terms are from nonnormal universes, the approximate distribution of F N is used for testing problem presented above. With UNBALANCED ONE-WAY RANDO M M ODEL 25 the moment conditions that Ej i j 4+ < 1 and Eje ij j 4+ < 1 for some positive we establish the following asymptotic condition.
Asymptotic Condition. Consider a sequence of the model (1). The number of groups t tends to in…nity in such a way that the average of n p 1 , n p 2 , : : :, n p t is bounded where p 1. So there exists a real number M > 0 such that for all t. It is ensured by …nite group sizes.
We are free to put in order the levels of the random e¤ect among the (t + 1) levels. The group sizes can be ordered in the ascending order,i.e., n i n i+1 . Then The sequences t 1 P t i=1 n p i of t are bounded and monotone and than they have a …nite limit as t ! 1. The positive monotone sequence t 1 P t i=1 (1=n p i ) are bounded from both left by 0 and right by t 1 P t i=1 n p i So it has a …nite limit as t ! 1.
We have shown that (1=t) P t i=1 n i has a …nite limit as t ! 1 where (1=t) P t i=1 n i = N=t. Then t=N = O(1) implying that t and N are of the same order. So t can be replaced by N .
Thus we are ready to de…ne the following limits appearing in calculation of the asymptotic covariance matrix. They are: n p i for p = 1; 2: where a 2 (0; 1) since 0 < t < N .

Asymptotic Distribution of F N
In this section we derive the asymptotic distribution of F N in Eq. (2) where a variance ratio is considered to be positive. The derivation of the asymptotic distribution of F N is based on obtaining the joint asymptotic distribution of p N (M S ; M SE) and then applying the delta method.  Proof. We …rst derive the asymptotic covariance matrix of N 1=2 (SS ; SSE) 0 . Let We follow Searle's notation (see Searle 1987, p 212-213). SS and SSE in Eq. (3) can be expressed in a matrix notation as where symmetric idempotent matrices Q 1 and Q 2 are: Here I m and J m are matrices of identity and ones of the order m m respectively.
The model (1) is in a matrix notation as Y = 1 N + U + e where 1 m denotes a vector of ones of the order m 1, = ( 1 ; 2 ; : : : t ) 0 and e is de…ned similarly to Y . The matrix U of the order N t is de…ned as It follows that SS and SSE are rewritten as (6) and (7), the matrix U and the matrix U 0 Q 1 of the order t N is equal to fB ij g i=t;j=t i=1;j=1 where the matrix B ij of the order 1 n j is of the form Using Lemma 1 of Westfall (1987) that simpli…es calculation of covariance between quadratic forms in a vector of mean zero random variables, we get (12) and Cov(SS ; SSE) = 4 k e tr(Q 1 diag(Q 2 )): (13) Using Eqs. (6) and (9), we get the following traces From the asymptotic condition given in Sec.2. and Eqs. (14)-(17), we get and lim where V ar(SS ), V ar(SSE) and Cov(SS ; SSE) are given in Eqs. (11), (12) and (13) respectively and the limits a and p for lp 1; 2 are de…ned by Eq. (4). From these, the covariance matrix of N 1=2 (SS ; SSE) 0 is: converges in distribution to the bivariate normal distribution with zero-mean vector and the covariance matrix ACOV given in Eq. (5). (6) and (8), the matrix P can be written as Then with the aid of Eqs. (9) and (10), the (n i +1) (n j +1) symmetric submatrix P ij of P is written as ; e i2 ; : : : ; e ini ). Using the projection method for quadratic forms (see Akritas and Papadatos (2004), van der Vaart (1998) ch. 11), Q N is decomposed as It should be noted that U N is the sum of independent but not identical random variables and U N and V N are uncorrelated.
Observe that where the inequality is acquired by using Cauchy-Schwartz inequality. The moment conditions Ej i j 4+ < 1 and Eje ij j 4+ < 1 for some positive ensure that there exists a …nite and positive M such that (Ej i On the other hand, tr(P ii P ii ) = (1 (n i =N )) 2 (1 + n i ) 2 + n i 1 6n 4 i . It follows from these that By using Lemma 1 of Westfall (1987), c 2 N is calculated and it is equal to Then, using the asymptotic condition in Sec. 2, the following limit is obtained  converges in distribution to normal distribution with 0-mean and variance 2 F as N ! 1 where F N is as in Eq. (2), 2 F is: and the limits a and p for p = 1; 2 are in Eq. (4).
Proof. Let 5F N denote the vector of the partial derivatives of F N with respect to M S and M SE at their expectations. Then 5F N = (1= 2 ; [1 + a 1 ]= 2 ) 0 .
From the delta method, p N (F N [1 + a 1 ]) converges in distribution to normal distribution with zero mean and the variance 2 F = 5 0 F N ACOV 5F N where ACOV is in (5). The explicit form of 2 F is given in Eq. (24).

The Proposed Test
The sized approximate test rejects H 0 : = 0 when F N > u where F N is in Eq. (2) and u is the upper 1 quantile of the asymptotic null distribution of F N . Then, we have P (F N > u j = 0) = The asymptotic null distribution of p N (F N 1) determined from Theorem 3.3 is the normal distribution with zero mean and variance 2 0 where it is written as after some algebraic operation on Eq. (24). One …nds u and it is given by where z is the upper 1 quantile of the standard normal distribution. Finally the approximate power of the proposed test for a …nite sample size is: where denotes the cumulative standard normal distribution, 2 F and u are in Eqs. (24) and (26) respectively.

Robustness of the Test
The robustness of the asymptotic distribution of the F N statistic is valid only for the balanced models and it is de…ned as follows. The asymptotic null distribution of F N does not depend on the fourth moment of error. Proof. To show this, it is enough to show that the asymptotic null variance 2 0 in Eq. (25) is free of the kurtosis k e of error. When n i = n for all i, where n is …xed, we have N = tn and then 1=n i t 2 =N 2 g = 1=n 2 1=n 2 = 0: As indicated by (Akritas and Arnold 2000, p.221), (Sche¤e 1959, p.344), and Güven (2014) the asymptotic null distribution of F N is asymptotically robust with respect to departure from normality of error. So, for the balanced case, the size of the test is asymptotically robust to nonnormal error.

Numerical and Simulation Study
The power values of the approximate test are compared with the simulated power values for some selected distributions to i and e ij in order to check accuracy of the power of the approximate test.
Simulation is based on 1000 runs. In each run, F N is calculated from generating data. The number of F N exceeding u is divided by 1000 to get a power value of the approximate test. The simulated level of signi…cance of the test is obtained in getting simulated power value of the approximate test when = 0. Simply we skip the step 2 in generation of F N It is equivalently to simulate the level of signi…cance of the test for testing hypothesis of no …xed treatment e¤ects in the one-way ANOVA model.

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In Table 1. through 6, sizes and power values of the approximate test are very closer to simulated sizes and power values of the test for small values of . However,the di¤erences between them values slightly increase as the value of increases. It is also observed that both approximated and simulated power values of the test are higher for a large n i design than for a small n i design. So according to the simulation results, the test is more appropriate for a small variance ratio and large group sizes. Table 1 and 4 are for the null kurtosis case while the rest of the tables are for either the positive or negative kurtosis case. It is not detected any signi…cant rise or decline of power values of the approximate test in departing from the null kurtosis case. In comparison Tables 2 and 5 with Table 3 and 6, the power values of the test are higher for the negative kurtosis case than for the positive kurtosis case.

Conclusion
In the present paper we establish the approximate test for the hypothesis of zero variance ratio in the unbalanced one way random e¤ects model from non normal universes. As shown in Sec. 4. calculation of both the upper percentile point and a power value of the test can easily be accomplished. The test is robust for the balanced one way random e¤ects model. In the balanced case the null distribution of the test statistics F N ratio does not depend on the fourth moment of the error term.
The di¤erences between the calculated and generated sizes and power values are closer to a small design and lower variance ratios than a large design and higher variance ratios. It follows that the approximate test is more accurate for a small design and lower variance ratios. It is not detected any signi…cant rise or descend of the power from null to non null kurtosis. Thus, departing from null kurtosis does not have an impact to the power of the approximate test.