A SURPRISING RESULT IN COMPARING ORTHOGONAL AND NONORTHOGONAL LINEAR EXPERIMENTS

Comparing Linear Experiments Czesław St ˛epniak vol. 10, iss. 2, art. 31, 2009

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A SURPRISING RESULT IN COMPARING

ORTHOGONAL AND NONORTHOGONAL LINEAR EXPERIMENTS

CZESŁAW ST ˛EPNIAK

Institute of Mathematics University of Rzeszów Rejtana 16 A

35-359 Rzeszów, Poland EMail:cees@univ.rzeszow.pl

Received: 09 February, 2009

Accepted: 24 March, 2009

Communicated by: Terry M. Mills

2000 AMS Sub. Class.: Primary 62K05, 62B15; Secondary 15A39, 15A45.

Key words: Linear experiment, orthogonal/nonorthogonal experiment, single parameters, comparison of experiments.

Abstract: We demonstrate by example that within nonorthogonal linear experiments, a use- ful condition derived for comparing of the orthogonal ones not only fails but it may also lead to the reverse order.

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Contents

1 Preliminaries 3

2 Estimation and Comparison of Linear Experiments for Single

Parameters 5

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1. Preliminaries

Any linear experiment is determined by the expectation E(y) and the variance- covariance matrixV(y)of the observation vector y. In the standard case these two moments have the following representation:

(1.1) E(y) = Xβ and V(y) =σIn,

where X is a known n ×p design matrix while β = (β₁, ..., β_p)⁰ and σ are un- known parameters. To secure the identifiability of the parameters β_i’s we assume that rank(X) =p. Any standard linear experiment, being formally a structure of the form(y,Xβ,σI_n), will be denoted by L(X) and may be identified with its design matrix.

Now let us consider two linear experimentsL₁ =L(X₁)andL₂ =L(X₂)with design matricesX₁ andX₂, respectively, and with common parametersβandσ. In St˛epniak [7], St˛epniak and Torgersen [8] and St˛epniak, Wang and Wu [9] the exper- imentL₁ is said to be at least as good asL₂ if for any parametric functionϕ =c⁰β the variance of its Best Linear Unbiased Estimator (BLUE) inL₁ is not greater than inL₂. It was shown in the above papers that this relation among linear experiments reduces to the Loewner ordering for their information matrices M1=X⁰₁X1 and M2=X⁰₂X2. It appears that this ordering is very strong.

Many authors, among others Kiefer [1] , Pukelsheim [4] Liski et al. [3], suggest some weaker criteria, among others of typeA, DandE, based on some scalar func- tions of the information matrices. In this paper we focus on a reasonable criterion considered by Rao ([5, p. 236]).

Denote byC_p the class of all linear experiments with the same parametersβand σ, and by O_p its subclass containing orthogonal experiments only. Inspired by Rao we introduce the following definition.

Comparing Linear Experiments Czesław St ˛epniak vol. 10, iss. 2, art. 31, 2009

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Definition 1.1. We shall say that an experimentL₁ belonging toC_pis better than L₂ with respect to the estimation of single parameters (and write:L₁ L₂) if for any β_i, i= 1, ..., p, its BLUE inL₁ does not have greater variance than inL₂ and less for somei.

One can easily state an algebraic criterion for comparing experiments withinO_p. The aim of this note is to reveal the fact that this criterion may lead to a reverse order outside this class.

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2. Estimation and Comparison of Linear Experiments for Single Parameters

In this section we focus on estimation and comparison of linear experiments with respect to the estimation of single parametersβ_ifor alli= 1, ..., p. In this context a simple result provided by Scheffé ([6, Problem 1.5, p. 24]) will be useful. We shall state it in the form of a lemma.

Let L = L(X) be a linear experiment of the form (1.1), whereX is an n ×p design matrix of rank p and let x₁, ...,x_p be the columns of X. For a given x_i, i= 1, ..., pdenote byP_ithe orthogonal projector onto the linear space generated by the remaining columnsx_j,j 6=i.

Lemma 2.1. Under the above assumptions each parameter β_i in the experiment (1.1) is unbiasedly estimable and the variance of its BLUE may be presented in the formσ(a⁰_iai)⁻¹, whereai= (I−P_i)x_i.

In fact this lemma is a consequence of the well known Lehmann-Scheffé theorem on minimum variance unbiased estimation (cf. Lehmann and Scheffé [2]).

Now let us consider the classO_pof all orthogonal experiments, i.e. satisfying the conditionx⁰_ix_j = 0 fori 6= j, with the same parameters β andσ. LetX₁ andX₂ be matrices with columnsx_1,1, ...,x_1,p andx_2,1, ...,x_2,p, respectively. The following theorem is a direct consequence of Lemma2.1.

Theorem 2.2. For any orthogonal experiments L₁ = L(X₁) and L₂ = L(X₂) belonging to the classO_p the first one is better than the second one for estimation of single parameters, i.e. L₁ L₂, if and only if,

(2.1) x⁰_1,ix_1,i≥x⁰_2,ix_2,i fori= 1, ..., pwith strict inequality for somei.

Now we shall demonstrate by example that the ordering rule (2.1) may lead to unexpected results outside the classOp.

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Example 2.1. Let x be an arbitrary n-column such that x⁰1_n 6= 0 and x6=λ1_n for any scalar λ. Consider two linear experiments L₁ = L([1_n,x]) and L₂ = L([1_n,(I_n−P)x]) where P = _n¹1_n1⁰_n is the orthogonal projector onto the one- dimensional linear space generated by 1_n. Since x⁰(I−P)x<x⁰x, the condition (2.1) holds for X₁=[1_n,x]andX₂= [1_n,(I_n−P)x]. This may suggest that the ex- perimentL₁ is at least as good asL₂ for estimation of the single parametersβ₁ and β₂, i.e. thatL(X₁) L(X₂).However, by Lemma2.1, the variances of the BLUE’s forβ₂in these two experiments are the same, while forβ₁the corresponding variance inL(X₂)is less than inL(X₁).

Conclusion. In this example the condition (2.1) is met whileL(X₂) L(X₁).

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References

[1] J. KIEFER, Optimum Experimental Designs, J. Roy. Statist. Soc. B, 21 (1959), 272–304.

[2] E.L. LEHMANNANDH. SCHEFFÉ, Completeness, similar regions, and unbi- ased estimation - Part I, Sankhy¯a, 10 (1950), 305–340.

[3] E.P. LISKI, K.R. MANDAL, K.R. SHAHANDB.K. SINHA, Topics in Optimal Designs, Lecture Notes in Statistics, Springer-Verlag, New York- (2002)

[4] F. PUKELSHEIM, Optimal Designs of Experiments, Wiley, New York, 1993.

[5] C.R. RAO, Linear Statistical Inference and its Applications, 2nd. Ed., Wiley, New York, 1973.

[6] H. SCHEFFÉ, The Analysis of Variance, Wiley, New York, 1959.

[7] C. ST ˛EPNIAK, Optimal allocation of units in experimental designs with hierar- chical and cross classification, Ann. Inst. Statist. Math. A, 35 (1983), 461–473.

[8] C. ST ˛EPNIAK ANDE. TORGERSEN, Comparison of linear models with par- tially known covariances with respect to unbiased estimation, Scand. J. Statist., 8 (1981), 183–184.

[9] C. ST ˛EPNIAK, S.G. WANGANDC.F.J. WU, Comparison of linear experiments with known covariances, Ann. Statist., 12 (1984), 358–365.