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- Abstract
- 1 Introduction
- 2 Estimation of parameters under order restrictions
- 3 Hypothesis Testing
- 4 Application to the study of estrogen-like compounds
- References

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Sankhya Ser B. Author manuscript; available in PMC 2010 October 16.

Published in final edited form as:

Sankhya Ser B. 2009; 71(1): 79–96.

PMCID: PMC2955899

NIHMSID: NIHMS202681

See other articles in PMC that cite the published article.

In this article we introduce a new procedure for estimating population parameters under inequality constraints (known as *order restrictions*) when the unrestricted maximum liklelihood estimator (UMLE) is multivariate normally distributed with a known covariance matrix. Furthermore, a Dunnett-type test procedure along with the corresponding simultaneous confidence intervals are proposed for drawing inferences on elementary contrasts of population parameters under order restrictions. The proposed methodology is motivated by estimation and testing problems encountered in the analysis of covariance models. It is well-known that the restricted maximum likelihood estimator (RMLE) may perform poorly under certain conditions in terms of quadratic loss. For example, when the UMLE is distributed according to multivariate normal distribution with means satisfying simple tree order restriction and the dimension of the population mean vector is large. We investigate the performance of the proposed estimator analytically as well as using computer simulations and discover that the proposed method does not fail in the situations where RMLE fails. We illustrate the proposed methodology by re-analyzing a recently published rat uterotrophic bioassay data.

In many applications researchers are interested in testing for inequality constraints among population means *θ _{i}*,

Suppose * ^{UMLE}* is the unrestricted maximum likelihood estimator (UMLE) of

Let *Y _{ij}* denote the response of the

$${Y}_{ij}={\theta}_{i}+{x}_{ij}^{\prime}\beta +{\epsilon}_{ij},\phantom{\rule{0.16667em}{0ex}}i=1,2,\dots ,p,\phantom{\rule{0.16667em}{0ex}}j=1,2,\dots ,{n}_{i},$$

(1)

where *θ _{i}* are the parameters of interest and

$${\mathbf{Y}}_{N\times 1}=\mathit{Diag}({\mathbf{1}}_{1},\dots {\mathbf{1}}_{p}){\mathit{\theta}}_{p\times 1}+{\mathbf{X}}_{N\times k}{\mathit{\beta}}_{k\times 1}+{\mathit{\epsilon}}_{N\times 1},$$

(2)

where ** θ** = (

The basic idea of the proposed estimation procedure is the following: First, let us consider the special case when *p* = 2, and hence ** θ** = (

$${\widehat{\mathit{\theta}}}^{\mathit{RMLE}}=\{\begin{array}{ll}{\widehat{\mathit{\theta}}}^{\mathit{UMLE}}\hfill & \text{if}\phantom{\rule{0.16667em}{0ex}}{\widehat{\theta}}_{1}^{\mathit{UMLE}}\le {\widehat{\theta}}_{2}^{\mathit{UMLE}}\hfill \\ ({\mathbf{1}}^{\prime}{\mathbf{\sum}}^{-1}\widehat{\mathit{\theta}}){({\mathbf{1}}^{\prime}{\mathbf{\sum}}^{-1}\mathbf{1})}^{-1}(1,1)\hfill & \text{otherwise}.\hfill \end{array}$$

(3)

Recall that, this RMLE universally dominates the UMLE for this special case of *p* = 2 (see, Rudeda et. al. 1997). Our proposed method makes use of this result repeatedly. First, for each *θ _{i}*, reduce the

A pair of parameters *θ _{i}* and

For a parameter vector ** θ**, the set of order restrictions is said to be a

Let *ω*_{11}, *ω*_{22}, …, *ω _{pp}* denote the diagonal elements of

$${\widehat{\theta}}_{i}^{\mathit{ave}}=\frac{{\sum}_{j=1}^{r}{\alpha}_{j}{\widehat{\theta}}_{i}^{(i,j)\mathit{RMLE}}}{{\sum}_{j=1}^{r}{\alpha}_{j}},$$

where *α _{j}* is the inverse of

We now estimate all the parameters that are linked to *θ _{k}*. Identify all the maximally linked subgraphs which

$${\widehat{\theta}}_{{k}_{{i}_{j}}}=min({\widehat{\theta}}_{k({i}_{j-1})}^{\mathit{ave}},{\widehat{\theta}}_{{k}_{{i}_{j}}}),j=l,l=1,\dots ,2\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}{\widehat{\theta}}_{{k}_{{i}_{j}}}=max({\widehat{\theta}}_{{k}_{{i}_{j}}},{\widehat{\theta}}_{{k}_{{i}_{j+1}}}^{\mathit{ave}}),j=l,l+1,\dots ,s-1.$$

We repeat the above process by identifying the next lead parameter in the complement of
${\mathrm{\Lambda}}_{i=1}^{t}{\mathcal{M}}_{ki}$ and repeat the above process until all the components of ** θ** are estimated. Unlike the RMLE and the algorithm provided in Peddada et al. (2005), this procedure is not iterative.

We now simplify the estimators for some common order restrictions to illustrate the above methodology.

- Simple order (
*θ*_{1}≤*θ*_{2}≤ … ≤*θ*):_{p}Suppose the lead parameter is*θ*, where_{k}*θ*_{1}≤*θ*_{2}≤ …*θ*_{k}_{−1}≤*θ*≤_{k}*θ*_{k}_{+1}… ≤*θ*. Then ${\widehat{\theta}}_{k}\equiv {\widehat{\theta}}_{k}^{\mathit{ave}},\phantom{\rule{0.16667em}{0ex}}{\widehat{\theta}}_{i}=min({\widehat{\theta}}_{i}^{\mathit{ave}},{\widehat{\theta}}_{i+1})$,_{p}*i*=*k*− 1,*k*− 2, …,1 and ${\widehat{\theta}}_{i+1}=max({\widehat{\theta}}_{i},{\widehat{\theta}}_{i+1}^{\mathit{ave}})$,*i*=*k, k*+ 1, …,*p*− 1. - Simple tree order (
*θ*_{1}≤*θ*,_{i}*i*= 2, …,*p*):In this case the lead parameter is*θ*_{1}. Therefore ${\widehat{\theta}}_{1}\equiv {\widehat{\theta}}_{1}^{\mathit{ave}}$ and for all*i*≥ 2 ${\widehat{\theta}}_{i}=max({\widehat{\theta}}_{1},{\widehat{\theta}}_{i}^{\mathit{RMLE}})$. - Umbrella order (
*θ*_{1}≤*θ*_{2}≤ … ≤*θ*_{k}_{−1}≤*θ*≥_{k}*θ*_{k+1}≥ … ≥*θ*):_{p}In this case the lead parameter is*θ*. Therefore ${\widehat{\theta}}_{k}\equiv {\widehat{\theta}}_{k}^{\mathit{ave}},{\widehat{\theta}}_{i}=min({\widehat{\theta}}_{i}^{\mathit{ave}},{\widehat{\theta}}_{i+1})$ for_{k}*i*=*k*− 1,*k*− 2, …, 1, and ${\widehat{\theta}}_{i+1}=max({\widehat{\theta}}_{i},{\widehat{\theta}}_{i+1}^{\mathit{ave}})$ for*i*=*k*,*k*+ 1, …,*p*− 1.

Suppose * ^{UMLE} ~ N*(

$$E\psi (\mid {\widehat{\theta}}_{i}^{\mathit{ave}}-{\theta}_{i}\mid )\le E\psi (\mid {\widehat{\theta}}_{i}^{\mathit{UMLE}}-{\theta}_{i}\mid ),\phantom{\rule{0.38889em}{0ex}}i=1,2,\dots ,p,$$

(4)

with a strict inequality holding for at least one *i* (*i* = 1, 2, …, *p*).

Relative to
${\widehat{\theta}}_{i}^{\mathit{UMLE}}$, if an estimator * _{i}* satisfies (4), then we shall say

As a consequence of the above theorem we have the following important corollary which justifies the proposed estimation procedure for any arbitrary order restriction .

Under the conditions of Theorem 2.1, suppose *θ _{k}* is the lead parameter of

Under the conditions of Theorem 2.1, suppose *θ _{k}* is the only nodal parameter of

- For simple tree order
*θ*_{1}≤*θ*,_{i}*i*≥ 2,_{1}dominates ${\widehat{\theta}}_{1}^{\mathit{UMLE}}$ in monotonic convex risk. Recall from Lee (1988) and Hwang and Peddada (1994) that the RMLE of*θ*_{1}fails to dominate the UMLE as*p*increases, whereas the proposed methodology yields an estimator which actually dominates the UMLE. - For umbrella order
*θ*_{1}≤*θ*_{2}≤ … ≤*θ*_{k}_{−1}≤*θ*≥_{k}*θ*_{k}_{+1}≥ … ≥*θ*,_{p}dominates ${\widehat{\theta}}_{k}^{\mathit{UMLE}}$ in monotonic convex risk._{k}

We evaluted the performance of the proposed modified RMLE (MRMLE) with the following estimators: (a) UMLE, (b) RMLE, (c) estimator of Hwang & Peddada (1994), and (d) the iterative algorithm of Peddada et al. (2005). In the simulation study reported in this paper, we assumed *~ N*(*θ**,* **Σ**), where the components of ** θ** satisfy the simple tree order

- (P1) Independence:
**Σ**=*Diag*[1:*σ*^{2}: ···:*σ*^{2}]. - (P2) Intraclass among the
*p*− 1 treatments:**Σ**= (*σ*,_{ij}*i, j*= 1, 2, …,*p*), where*σ*_{11}= 1,*σ*_{1}=_{j}*σ, j*≥ 2,*σ*=_{ij}*σ*^{2}*, i*≠*j, i*≥ 2*, j*≥ 2,*σ*= 1 +_{ii}*σ*^{2}*, i*≥ 2. - (P3) Intraclass correlation:
**Σ**= (1 −*ρ*)**I**+*ρ***J**.

We considered two parameters, (1) the nodal parameter *θ*_{1}, and (2) vector of elementary contrasts ** η** = (

All our simulation results are based on 10,000 simulation runs, with a variety of patterns for *p*, *σ*, *ρ*. Although a wide range of patterns were considered in our simulation study, in this paper we provide a representative sample of our results. Table 1 summarizes the results for the nodal parameter *θ*_{1} and Table 2 summarizes the results for the elementary contrast vector ** η**. In all our simulations we considered the worst case scenario for

Mean Squared Error (MSE) and coverage rate comparisons of (a) UMLE, (b) RMLE, (c) estimator of Hwang & Peddada (1994), (d) the iterative algorithm of Peddada et al. (2005), and (e) the proposed modified RMLE (MRMLE). The parameter of interest **...**

Mean Squared Error (MSE) and coverage rate comparisons of (a) UMLE, (b) RMLE, (c) estimator of Hwang & Peddada (1994), (d) the iterative algorithm of Peddada et al. (2005), and (e) the proposed modified RMLE (MRMLE). The parameter of interest **...**

As expected, in each case as the dimension increased, the RMLE performed very poorly in terms of MSE and the coverage probability. In several cases the MSE of RMLE was substantially larger than all its competitors. It also had poor coverage probability compared to the nominal level of 0.95. As expected, the estimator of Hwang & Peddada (1994) performed extremely well when **Σ** was diagonal (pattern P1) or when **Σ** had an intraclass correlation structure (pattern P3). However, its performance can be very poor for other covariance structures (pattern P2). The proposed estimator MRMLE consistently performed very well for all patterns considered in our simulation studies. Relative to RMLE, in some cases, its MSE was almost one-third or even one-fourth that of RMLE. Also, its coverage rate always exceeded the nominal level. Furthermore, as expected, the performance of MRMLE and the estimator proposed in Peddada et al. (2005) performed similarly for diagonal and intraclass correlation matrices. However, for other non-diagonal covariance matrices, the MRMLE performed better than the estimator in Peddada et al. (2005) in terms of MSE. In conclusion, MRMLE never failed the way RMLE and the estimator of Hwang & Peddada (1994) did, and was always better than the UMLE.

The coverage probability results reported in Table 2 suggest that MRMLE may be used for constructing simultaneous confidence intervals for performing inferences on the elementary contrasts when the components of ** θ** are subject to order restrictions. Analogous to Williams (1977) and Marcus and Talpaz (1992), we begin with the following testing problem along the lines of Dunnett (1955).

$${H}_{0}:{\theta}_{1}={\theta}_{2}=\cdots ={\theta}_{p},\phantom{\rule{0.38889em}{0ex}}\text{versus}\phantom{\rule{0.38889em}{0ex}}{H}_{a}:\mathit{\theta}\in \mathrm{\Theta},$$

(5)

where Θ denotes the set of order restrictions on the components of ** θ** = (

For an elementary contrast *θ _{i}* −

$$G=\underset{i,j}{max}\frac{\mid {\widehat{\theta}}_{i}-{\widehat{\theta}}_{j}\mid}{\widehat{\sigma}\sqrt{{\mathbf{a}}_{i,j}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,j}}},$$

(6)

where max* _{i,j}* is the maximum over all possible linked pairs

*Simple Order*: If Θ is a simple order then (6) simplifies to$$G=\underset{1\le i\le p-1}{max}\frac{{\widehat{\theta}}_{i+1}-{\widehat{\theta}}_{i}}{\widehat{\sigma}\sqrt{{\mathbf{a}}_{i+1,i}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i+1,i}}},$$*Simple Tree Order*: If Θ is the simple tree order restriction then (6) simplifies to$$G=\underset{2\le i}{max}\frac{{\widehat{\theta}}_{i}-{\widehat{\theta}}_{1}}{\widehat{\sigma}\sqrt{{\mathbf{a}}_{i,1}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,1}}}.$$

Let **Z** *~ N*(**0***,* **Σ**) and let *U ~ χ*^{2} with *df* = *N* − *p* − *k* degrees of freedom. Under the order restriction specified in *H _{a}*, for each

Under the null hypothesis θ_{1} = θ_{2} = ··· = θ_{p} = θ and model (2)

$$\underset{i,j}{max}\frac{\mid {\widehat{\theta}}_{i}-{\widehat{\theta}}_{j}\mid}{s\sqrt{{\mathbf{a}}_{i,j}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,j}}}\stackrel{d}{=}\underset{i,j}{max}\frac{\sqrt{df}\mid {W}_{i}-{W}_{j}\mid}{U\sqrt{{\mathbf{a}}_{ij}^{\prime}\mathbf{\sum}{\mathbf{a}}_{ij}}}$$

(7)

In view of the above theorem, the null distribution of *G* can be approximated by simulating the following statistic:

$${G}^{\ast}=\underset{i,j}{max}\sqrt{df}\frac{\mid {W}_{i}-{W}_{j}\mid}{U\sqrt{{\mathbf{a}}_{i,j}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,j}}}.$$

Thus

$$P(G\ge {G}_{\alpha}^{\ast}\mid {H}_{0})=P(\underset{i,j}{max}\frac{\mid {\widehat{\theta}}_{i}-{\widehat{\theta}}_{j}\mid}{\widehat{\sigma}\sqrt{{\mathbf{a}}_{i,j}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,j}}}\ge {G}_{\alpha}^{\ast}\mid {H}_{0})=\alpha .$$

Hence we reject the global null *H*_{0} at a level of significance *α* if, for all linked parameters *θ _{i}* and

$$\frac{\mid {\widehat{\theta}}_{i}-{\widehat{\theta}}_{j}\mid}{\widehat{\sigma}\sqrt{{\mathbf{a}}_{i,j}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,j}}}\ge {G}_{\alpha}^{\ast}.$$

Hence, for example if *H _{a}* is the simple tree order as defined above, then we reject

$$\underset{i\ge 2}{max}\frac{{\widehat{\theta}}_{i}-{\widehat{\theta}}_{1}}{\widehat{\sigma}\sqrt{{\mathbf{a}}_{i,1}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,1}}}\ge {G}_{\alpha}^{\ast}.$$

As done in Dunnett (1955) and Marcus & Talpaz (1992) (see also Hsu, 1996), the above critical values can be used for constructing (1 − *α*) × 100% simultaneous confidence intervals for elementary contrasts *θ _{i}* −

$$\underset{i,j}{\cap}\{{\theta}_{i}-{\theta}_{j}\in {\widehat{\theta}}_{i}-{\widehat{\theta}}_{j}\pm {G}_{\alpha}^{\ast}\widehat{\sigma}\sqrt{{\mathbf{a}}_{i,j}^{\prime}\mathbf{\sum}{\mathbf{a}}_{ij}}\},\phantom{\rule{0.16667em}{0ex}}\forall {\theta}_{j}\le {\theta}_{i}.$$

(8)

The size of the above confidence hypercube is given by ${\prod}_{i,j}\{2\times {G}_{\alpha}^{\ast}\times \widehat{\sigma}\sqrt{{\mathbf{a}}_{i,j}^{\prime}\mathbf{\sum}{\mathbf{a}}_{i,j}}\}$.

We performed extensive simulation studies to compare the performance of the proposed procedure with the RMLE based procedure of Marcus & Talpaz (1992) when the parameters are subject to simple tree order restriction. We compared them in terms of (a) power and (b) size of the simultaneous confidence intervals for the *p* − 1 vector (*θ*_{2} − *θ*_{1}, *θ*_{3} − *θ*_{1}, …, *θ _{p}* −

Power comparison of Marcus & Talpaz test (MT) and the proposed test. Data are generated according *Y*_{ij} ~^{independent} N (*θ*_{i}, 1), *i* = 1, 2, …*, p*, *j* = 1, 2, …*, n*, *θ*_{1} ≤ *θ*_{i}. The powers are simulated using **...**

Size (log volumes) of simultaneous confidence intervals. The results are based on 10,000 simulation runs. The critical constants are chosen so that the coverage rate of the simultaneous confidence interval is 0.95.

From Table 3 it is apparent that the proposed procedure competes very well with MT procedure, for the extreme points such as ** θ** = (0, 0, 0, …, 1)′, the MT procedure has higher power than the proposed test. However, for more central values such as

Our simulation results reported in Table 4 suggest that the size of the simultaneous confidence intervals centered at MRMLE are smaller than those centered at RMLE. We observed this result for a variety of patterns. Hence the test based on the proposed simultaneous confidence intervals has larger power than the corresponding test based on RMLE.

Recently the Organization for Economic Cooperation and Development conducted a large uterotrophic bioassay to evaluate the effects of estrogen-like compounds on rat uterine weight (Kanno et al., 2002a, 2002b). One of the participating labs studied the effects of (1) Bisphenol A, (2) DBP, (3) DDT, (4) Genestein, (5) Ethinyl Estradiol (EE high dose), (6) Ethinyl Estradiol (EE low dose), (7) Methoxychlor.

Their study protocol consisted of 6 immature animals per test group, where the compound was administered using oral gavage for three consecutive days and the uterine weights were obtained on day 4 of the study. Immature animals were used so that the only source of estrogen was from the compounds used in the study.

The sample mean uterine weights and standard errors (in parentheses) along with mean body weight necropsy and the corresponding standard errors (in parentheses) are summarized in the following Table 5. Although the lab also studied the effects of Nonylphenol, we did not include this chemical in the present illustration since only 2 animals out of 6 survived in this group at the end of the study.

Since the uterine volume had a skewed distribution we log-transformed the data. Thus *θ _{i}*,

To illustrate the proposed methodology, we compute 95% simultaneous confidence intervals for *θ _{i}* −

Based on a two-way ANOVA, we found that the body weight was a significant covariate (*P* = 0.0238). Hence when comparing the above seven compounds with control group, we adjusted for the body weight of the animal. The unrestricted maximum likelihood estimates of mean uterine weights (in log scale) *θ _{i}, i* ≥ 0 (standard errors in parentheses) with body weight at necropsy as a covariate are as follows: 2.90 (0.12), 3.31 (0.12), 2.80 (0.10), 4.35 (0.12), 3.99 (0.11), 4.33 (0.10), 3.79 (0.10), 3.98 (0.10), respectively.

Under the simple tree order restriction, *θ*_{1} ≤ *θ _{i}*, ∀

Note that the 95% confidence interval for the mean change in uterine weights (relative to vehicle control) for the DBP group contains 0, suggesting no significant difference in the mean uterine weights between the DBP treated animals and the controls. None of the remaining confidence intervals contain 0. Thus the mean uterine volumes for all other test compounds differed from that of the control group. Except for Bisphenol A, our results regarding all other compounds agree with the results of Kanno et al. (2002b). They did not find Bisphenol A to be significantly different from the placebo group, but the proposed procedure did. Interestingly, our finding is in agreement with data obtained by some of the other labs that tested this compound (Kanno et al., 2002b).

The research of SDP was supported by the Intramural Research Program of the NIH, National Institute of Environmental Health Sciences[Z01 ES101744-04]. The authors thank Drs. Gregg Dinse and Anastasia Ivanova for carefully reading this manuscript and providing several useful comments that improved the presentation of the manuscript. We also thank the associate editor and the two referees for several insightful comments that led to a significant improvement in the presentation of the paper.

Using the notations in (2) let

$$\mathbf{U}={[\mathit{Diag}({\mathbf{1}}_{1},\dots {\mathbf{1}}_{p}):\mathbf{X}]}_{N\times (p+k)},$$

where the rank of **U** is *p* + *k*. Then the following lemma is useful for computing the covariance matrix of * ^{UMLE}*.

Suppose **J**_{i} is a n_{i} × n_{i} matrix of 1’s,
$\mathbf{M}=\mathit{Diag}({\scriptstyle \frac{{\mathbf{J}}_{1}}{{n}_{1}}},\dots ,{\scriptstyle \frac{{\mathbf{J}}_{p}}{{n}_{p}}}),\mathbf{L}=\mathit{Diag}({\scriptstyle \frac{1}{{\mathit{n}}_{\mathbf{1}}}},\dots ,{\scriptstyle \frac{1}{{\mathit{n}}_{\mathit{p}}}}),{\scriptstyle \frac{1}{{\mathit{n}}_{\mathit{i}}}}={(1/{n}_{i},1/{n}_{i},\dots 1/{n}_{i})}^{\prime}$ is a n_{i} × 1 vector, and **E** = (**X**′(**I** − **M**)**X**)_{k×k}, then

$${({\mathbf{U}}^{\prime}\mathbf{U})}^{-1}=\left(\begin{array}{cc}\mathit{Diag}({\scriptstyle \frac{1}{{n}_{1}}},\dots ,{\scriptstyle \frac{1}{{n}_{p}}})+{\mathbf{L}}^{\prime}{\mathbf{E}}^{-1}{\mathbf{X}}^{\prime}\mathbf{L}& -{\mathbf{L}}^{\prime}\mathbf{X}{\mathbf{E}}^{-1}\\ -{\mathbf{E}}^{-1}{\mathbf{X}}^{\prime}\mathbf{L}& {\mathbf{E}}^{-1}\end{array}\right)$$

Since *Diag*(*n*_{1}, …, *n _{p}*) and

$${\mathbf{U}}^{\prime}\mathbf{U}=\left(\begin{array}{cc}\mathit{Diag}({n}_{1},\dots ,{n}_{p})& \mathit{Diag}({\mathbf{1}}_{1}^{\prime},\dots ,{\mathbf{1}}_{p}^{\prime})\mathbf{X}\\ {\mathbf{X}}^{\prime}\mathit{Diag}({\mathbf{1}}_{1},\dots {\mathbf{1}}_{p})& {\mathbf{X}}^{\prime}\mathbf{X}\end{array}\right)$$

therefore from Rao (1973) we have

$${({\mathbf{U}}^{\prime}\mathbf{U})}^{-1}=\left(\begin{array}{cc}\mathit{Diag}({\scriptstyle \frac{1}{{n}_{1}}},\dots ,{\scriptstyle \frac{1}{{n}_{p}}})+{\mathbf{L}}^{\prime}{\mathbf{E}}^{-1}{\mathbf{X}}^{\prime}\mathbf{L}& -{\mathbf{L}}^{\prime}\mathbf{X}{\mathbf{E}}^{-1}\\ -{\mathbf{E}}^{-1}{\mathbf{X}}^{\prime}\mathbf{L}& {\mathbf{E}}^{-1}\end{array}\right)$$

Hence we have

$$\mathit{Cov}({\widehat{\theta}}^{\mathit{UMLE}})={\sigma}^{2}\mathbf{\sum}={\sigma}^{2}\left(\mathit{Diag}(\frac{1}{{n}_{1}},\dots ,\frac{1}{{n}_{1}})+{\mathbf{L}}^{\prime}\mathbf{X}{\mathbf{E}}^{-1}{\mathbf{X}}^{\prime}\mathbf{L}\right).$$

Accordingly, the variance of the UMLE of a contrast *θ _{i}* −

$$\mathit{Var}({\widehat{\theta}}_{i}^{\mathit{UMLE}}-{\widehat{\theta}}_{j}^{\mathit{UMLE}})={\sigma}^{2}{\mathbf{a}}_{ij}^{\prime}\mathbf{\sum}{\mathbf{a}}_{ij}.$$

(9)

From Rueda et al. (1997), we note that
${\widehat{\theta}}_{i}^{(i,j)\mathit{RMLE}}$ universally dominates
${\widehat{\theta}}_{i}^{\mathit{UMLE}}$. Thus for all monotonic functions (·),
$E(\phi (\mid {\widehat{\theta}}_{i}^{(i,j)\mathit{RMLE}}-{\theta}_{i}\mid ))\le E(\phi (\mid {\widehat{\theta}}_{i}^{\mathit{RMLE}}-{\theta}_{i}\mid ))$, with a strict inequality for at least one value of *θ _{i}*. Appealing to Dunbar et al. (2001) we therefore have
$E(\psi (\mid {\widehat{\theta}}_{i}^{\mathit{ave}}-{\theta}_{i}\mid ))\le E(\psi (\mid {\widehat{\theta}}_{i}^{\mathit{RMLE}}-{\theta}_{i}\mid ))$, for all monotonic convex functions

From (9) we have

$$\underset{i,j}{max}\frac{\mid {\widehat{\theta}}_{i}-{\widehat{\theta}}_{j}\mid}{s\sqrt{\mathit{Var}({\widehat{\theta}}_{i}^{\mathit{UMLE}}-{\widehat{\theta}}_{j}^{\mathit{UMLE}})}}=\underset{i,j}{max}\frac{\sigma \mid ({\widehat{\theta}}_{i}-\theta )-({\widehat{\theta}}_{j}-\theta )\mid}{s\sqrt{{\mathbf{a}}_{ij}^{\prime}\mathbf{\sum}{\mathbf{a}}_{ij}}}.$$

Under the null hypothesis *θ*_{1} = *θ*_{2} = ··· = *θ _{p}* =

$$\underset{i,j}{max}\frac{\sigma \mid ({\widehat{\theta}}_{i}-\theta )-({\widehat{\theta}}_{j}-\theta )\mid}{s\sigma \sqrt{{\mathbf{a}}_{ij}^{\prime}\mathbf{\sum}{\mathbf{a}}_{ij}}}\stackrel{d}{=}\underset{i,j}{max}\frac{\sigma \mid {W}_{i}-{W}_{j}\mid}{s\sqrt{{\mathbf{a}}_{ij}^{\prime}\mathbf{\sum}{\mathbf{a}}_{ij}}}\stackrel{d}{=}\underset{i,j}{max}\frac{\sqrt{df}\mid {W}_{i}-{W}_{j}\mid}{U\sqrt{{\mathbf{a}}_{ij}^{\prime}\mathbf{\sum}{\mathbf{a}}_{ij}}}$$

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