Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

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1 Journal of Computational and Applied Mathematics 216 (2008) Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution Yildiz E. Yilmaz a, Aysen D. Akkaya a,b, a Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 b Department of Statistics, Middle East Technical University, Ankara, Turkey Received 12 December 2006; received in revised form 29 May 2007 Abstract We consider one-way classification model in experimental design when the errors have generalized secant hyperbolic distribution. We obtain efficient and robust estimators for block effects by using the modified maximum likelihood estimation (MML) methodology. A test statistic analogous to the normal-theory F statistic is defined to test block effects. We also define a test statistic for testing linear contrasts. It is shown that test statistics based on MML estimators are efficient and robust. The methodology readily extends to unbalanced designs Elsevier B.V. All rights reserved. MSC: 62K10; 62F10; 62F12; 62F03; 62F35 Keywords: Experimental design; Non-normality; Generalized secant hyperbolic; Modified maximum likelihood; Linear contrast; Robustness 1. Introduction Analysis of variance procedures have traditionally been based on the assumption of normality. In practice, however, non-normal distributions occur so frequently. A number of studies have been made to investigate the effect of nonnormality on the test statistics used in analysis of variance. The effect of non-normality on Type I error was studied in [11,7,6,2,9,3,15]. The effect of non-normality on power of the test was studied in [4,13,5,18]. They concluded that the effect of moderate non-normality on Type I error is not very serious but the power is adversely affected; the values of the power are, in fact, considerably smaller than under normality if, particularly, the kurtosis of the underlying distribution is different than 3 (kurtosis of normal). The above investigations for symmetric non-normal distributions have been carried out when the mathematical forms of short-tailed distributions (kurtosis less than 3) and long-tailed distributions (kurtosis greater than 3) are quite distinct from one another, e.g., the former is uniform and the latter is Student s t [22, Chapter 1]. The purpose of this paper is to present unified results by considering the family of generalized secant hyperbolic (GSH) distributions. The properties of GSH distributions have been studied by Vaughan [24]. They represent both short-tailed and long-tailed symmetric distributions with kurtosis ranging from 1.8 to 9 and include logistic as a particular case, uniform as a limiting case, and closely approximate the normal and Student s t distributions. Maximum Corresponding author. Department of Statistics, Middle East Technical University, Ankara, Turkey. Tel.: ; fax: address: akkay@metu.edu.tr (A.D. Akkaya) /$ - see front matter 2007 Elsevier B.V. All rights reserved. doi: /j.cam

2 546 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) likelihood estimators being intractable [12,24], we derive modified maximum likelihood (MML) estimators of block effects (and scale parameter) in the framework of one-way classification experimental design and show that they are asymptotically fully efficient. We also study their properties for small sample sizes n and show that they are in general considerably more efficient than the normal theory (i.e., least squares) estimators. In fact the least squares estimators have a disconcerting feature, namely, their efficiencies relative to the MML estimators decrease as the sample size in a block increases. A test statistic analogous to the normal-theory F statistic is defined to test block effects. We also define a test statistic for testing linear contrasts. It is shown that test statistics based on MML estimators are efficient and robust. The methodology obtained readily extends to unbalanced designs. 2. One-way classification model Consider the one-way classification fixed-effects model y ij = μ + τ i + e ij (i = 1, 2,...,k; j = 1, 2,...,n), (2.1) having k blocks with n observations in each block; y ij is the jth observation in the ith block, μ is the overall mean, τ i is the ith block effect, and e ij are iid random errors. In the fixed effects model, τ i are defined as deviations from their overall mean. Thus, k τ i = 0. In (2.1), suppose that e ij are iid and have GSH distribution [24] GSH(0, ; t): f(e)= c 1 exp(c 2 e/) exp(2c 2 e/) + 2a exp(c 2 e/) + 1 where for π <t 0: a = cos(t), c 2 = (π 2 t 2 )/3 and c 1 = sin(t) t c 2 ( <e< ), (2.2) and for t>0: a = cosh(t), c 2 = (π 2 + t 2 )/3 and c 1 = sinh(t) c 2. t For t > π, t < π and t = π, GSH(0, ; t) represents short-tailed, long-tailed and approximately normal distributions, respectively; is the scale parameter and t is a nuisance parameter. The coefficient of kurtosis, β 2 = μ 4 /μ 2 2, is given below for a few representative values of the shape parameter, t: t= π 2/3 π/2 0 π π 11 β 2 = The likelihood function is L = cn 1 N k n exp(c 2 z ij ) exp(2c 2 z ij ) + 2a exp(c 2 z ij ) + 1, (2.3) where N = nk, z ij = y ij μ τ i (1 i k, 1 j n). Let y i(1) y i(2) y i(n) (1 i k) be the order statistics of the n observations y ij (1 j n) in the ith block. Then z i(j) = (y i(j) μ τ i )/ (1 i k) are the ordered z ij (1 j n) variates. Since complete sums are invariant to ordering, the likelihood equations for estimating μ, τ i (1 i k) and can be expressed in terms of z i(j) as follows: ln L μ = N c c 2 ln L = n c 2 τ i + 2 c 2 g(z i(j) ) = 0, g(z i(j) ) = 0 (2.4) j=1

3 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) and ln L = N 1 c 2 z i(j) + 2 c 2 z i(j) g(z i(j) ) = 0. where g(z) = (exp(2c 2 z) + a exp(c 2 z))/(exp(2c 2 z) + 2a exp(c 2 z) + 1). Eqs. (2.4) do not admit explicit solutions because of the terms involving the non-linear function g(z). Solving these equations by iteration is difficult and time consuming since there are k + 1 equations to solve iteratively. To alleviate these difficulties, we use the method of modified maximum likelihood due to Tiku [16,17] and Tiku and Suresh [21]. This method gives explicit and highly efficient estimators [14,20,22,23]. 3. The MML estimators Let t i(j) = E(z i(j) ) be the expected value of the jth order statistic z i(j) in the ith block. Since the function g(z) is almost linear in small intervals a<z<b[16,17] and z i(j) is located in the vicinity of t i(j) = E(z i(j) ) at any rate for large n, an appropriate linear approximation for g(z i(j) )(1 i k) is obtained by using the first two terms of a Taylor series expansion, namely g(z i(j) ) g(t i(j) ) + g (t i(j) )(z i(j) t i(j) ) = α i(j) + β i(j) z i(j) (1 j n), (3.1) where α i(j) = g(t i(j) ) β i(j) t i(j) and β i(j) = g (t i(j) ). Here, t 1(j) = t 2(j) = =t k(j) = t (j), α 1(j) = α 2(j) = =α k(j) = α j, β 1(j) = β 2(j) = =β k(j) = β j for all j = 1, 2,...,n (3.2) and α j = exp(2c 2t (j) ) + a exp(c 2 t (j) ) exp(2c 2 t (j) ) + 2a exp(c 2 t (j) ) + 1 β j t (j), (3.3) β j = ac 2 exp(3c 2 t (j) ) + 2c 2 exp(2c 2 t (j) ) + ac 2 exp(c 2 t (j) ) [exp(2c 2 t (j) ) + 2a exp(c 2 t (j) ) + 1] 2. (3.4) If β j < 0 then we set β j = 0 [24]. Thus, ˆ in (3.6) is always real and positive. It may be noted that n j=1 α j = n/2 and nj=1 β j t (j) = 0. Remark. It may be noted that the coefficients β j have inverted umbrella ordering (i.e., they increase until the middle value and then decrease in a symmetric fashion) when the GSH(0, ; t) represents short-tailed distributions. The coefficients β j have umbrella ordering when the GSH(0, ; t) represents long-tailed and approximately normal distributions. This gives MML estimators excellent robustness properties. Although the formulation to find the exact values of expected values t (j),1 j n, is available [24], it is difficult to implement. Therefore, we use their following approximate values which is often done in practice [12,20,25]: t (j) = 1 c 2 ln[sin(tq j )/ sin(t(1 q j ))], π <t<0 3 = π ln(q j /(1 q j )), t = 0 = 1 ln[sinh(tq c j )/ sinh(t(1 q j ))], t >0, (3.5) 2 where q j = j/(n + 1).

4 548 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) The use of these approximate values in place of the exact values does not affect the efficiency of the MML estimators in any substantial way. Incorporating (3.1) in (2.4), the modified maximum likelihood equations are obtained. These equations are asymptotically equivalent to the corresponding likelihood equations (2.4) under very general regularity conditions [25] and their solutions are the following MML estimators: ˆμ =ˆμ.., ˆτ i =ˆμ i ˆμ and ˆ = B + B 2 + 4NC 2 N(N k) (bias corrected), (3.6) where ˆμ.. = 1 km β j y i(j), ˆμ i = 1 m β j y i(j), m= β j, B = Nc 2 (ȳ.. ȳ a ), j=1 C = 2c 2 β j (y i(j) ˆμ ˆτ i ) 2, ȳ.. = 1 N y i(j), ȳ a = 2 N α j y i(j). The estimators are explicit functions of sample observations and therefore easy to compute. 4. Efficiency The estimator ˆμ i is unbiased for all n; follows from symmetry. Moreover, ˆμ i is the best asymptotically normal (BAN) estimator; see Appendix A. Given in Table 1 are the simulated variances of the MML estimator ˆμ i of μ i (1 i k), relative efficiency (RE) of the LS estimator μ i =ȳ i, the minimum variance bound (MVB) V 11 (Appendix A) for estimating μ i and the efficiency (Eff) of ˆμ i. From the values of the relative efficiencies RE( μ i ), it is concluded that the MML estimator ˆμ i is considerably more efficient than the LS estimator μ i for all n other than the GSH distribution with t =π. This distribution is indistinguishable from normal and that is the reason for μ i to be highly efficient; however, ˆμ i being asymptotically fully efficient, it takes a lead as n increases and is a little more efficient for n 20. From the values of the efficiencies Eff(ˆμ i ), it is concluded that for all the distributions the variances of ˆμ i steadily decrease as n increases and attain subsequently the minimum variance bounds V 11 (Appendix A). As a result, ˆμ i is 100% efficient for large n. For some of the distributions, in fact, ˆμ i is almost 100% efficient even for moderate sample sizes (say, n 20). Table 1 Variances of the MML estimator and the relative efficiencies of the LS estimator: (1) V(ˆμ i )/ 2 ; (2) RE( μ i )=[V(ˆμ i )/V ( μ i )] 100; (3) MVB(μ i )/ 2 ; (4) Eff( ˆμ i ) =[MVB(μ i )/V ( ˆμ i )] 100 k = 4 β 2 = n = 6 (1) (2) (3) (4) n = 10 (1) (2) (3) (4) n = 20 (1) (2) (3) (4)

5 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) Table 2 Variances of the LS and M-estimators of μ: (1) V( μ)/ 2 ; (2) V(ˆμ w )/ 2 k = 1 β 2 = n = 6 (1) (2) n = 10 (1) (2) n = 20 (1) (2) M-estimators A class of robust estimators of the location parameter μ that are known to be considerably more efficient [20, p. 181] than sample median based estimators (e.g., Tukey s median polish estimator) are called M-estimators and are due to Huber [10]. The most prominent from this class is [10] the wave estimator ˆμ w. The equation for calculating ˆμ w is given in [19, p. 130] reproduced from [8]. Like the MML estimator ˆμ and the LS estimator μ, ˆμ w is unbiased for all symmetric distributions. To compare the efficiency of ˆμ w with ˆμ and μ for the family (2.2), we simulated their variances. The simulated values are given in Table 2 for ˆμ w and μ and in Table 1 for ˆμ. All the simulated values in this paper are based on [100, 000/n] (integer value) Monte Carlo runs. It can be seen that ˆμ has the smallest variances and is, therefore, the most efficient. The M-estimator ˆμ w is more efficient than the LS estimator μ only for long-tailed distributions (kurtosis β 2 > 3). Since ˆμ w is not predominantly more efficient than μ for the family (2.2), we do not consider it any further. Besides, the distribution of ˆμ w is intractable for small n [22, p. 138]. That is due to the fact that the method implicitly censors a number (which is not the same for every sample of size n) of observations [20,22]. 5. Testing block effects To test equality of block effects, that is, to test the null hypothesis H 0 : τ 1 = τ 2 = =τ k = 0 against H 1 : τ i = 0 for some i = 1, 2,...,k, the following decomposition of total sum of squares S T is used: S T = S b + S e, S T = 2c 2 β j (y i(j) ˆμ) 2, S b = 2c 2 m (ˆμ i ˆμ) 2 and S e = 2c 2 β j (y i(j) ˆμ i ) 2. (5.1) S T / 2 and S b / 2 are for large n distributed as chi-squares with N 1 and k 1 degrees of freedom, respectively, and irrespective of H 0 and H 1, S e / 2 is distributed as chi-square with N k degrees of freedom (Appendix A). Since S b / 2 and S e / 2 are independently distributed chi-square random variables, for large n, the null distribution of the ratio kˆτ 2 i W = S b/(k 1) S e /(N k) 2c 2m (k 1)ˆ 2 (5.2) is central F with (k 1, N k) degrees of freedom. The distribution of W under H 1 is non-central F with (k 1, N k) degrees of freedom and non-centrality parameter λ 2 W = 2c 2m k (τ i /) 2 for large n. Large values of W lead to the rejection of H 0 in favour of H 1. The normal-theory test statistic for testing H 0 is F = ((N k)/(k 1)) k τ 2 i / k nj=1 (y ij μ i ) 2 having F distribution with k 1 and N k degrees of freedom. Under H 1, it is distributed as non-central F with k 1 and N k degrees of freedom and non-centrality parameter λ 2 F = k (τ i /) 2. Since λ 2 W > λ2 F, the W-test is more powerful than the F-test. This was to be expected since more efficient estimators are used in the W-test.

6 550 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) Testing linear contrasts The W-test gives an overall assessment whether block differences exist or not. If the W statistic is not significantly large, that does not necessarily imply that no block differences exist. It is, therefore, always advisable to construct linear contrasts to assesss the block differences [20]. A linear function L = k l i τ i = k l i μ i is called linear contrast if k l i = 0. A linear contrast represents a comparison between μ i (1 i k). The MML estimator of the linear contrast L = k l i μ i is k l i ˆμ i with variance 2mc 2 k 2 li 2, for large n. Since ˆμ i are asymptotically normally distributed and ˆ converges to as n becomes large, the distribution of the statistic k 2mc2 l i ˆμ T = i k ˆ li 2 (6.1) is asymptotically normal N(0, 1), under the null hypothesis H 0 : k l i μ i = 0. Large values of T lead to the rejection of H 0. The asymptotic power function of the test is (with Type I error α) 1 β P( Z z α/2 λ T ), (6.2) where Z is a standard normal variate and λ 2 T = 2mc 2( k l i μ i ) 2 / 2 k li 2 is the non-centrality parameter. Note that the normal-theory statistic for testing H 0 is t = n L/. The null distribution of this statistic for large n is Student s t with k(n 1) degrees of freedom and its distribution under the alternative hypothesis is non-central t with k(n 1) degrees of freedom and non-centrality parameter λ 2 t = nl 2 / 2. Since λ 2 T > λ2 t, the T-test is more powerful than the t-test, for large n. 7. Robustness of estimators and tests Since deviations from an assumed model are very common, the issue of robustness becomes important. An estimator is called robust if it is fully efficient (or nearly so) for an assumed distribution and maintains high efficiency for plausible alternatives. A test is said to have criterion robustness if its Type I error is not substantially higher than a pre-specified level and is said to have efficiency robustness if its power is high, at any rate for plausible alternatives to an assumed distribution [22, Preface, 21, Preface]. To illustrate the robustness of both the MML estimators and the tests based on them we consider, for illustration, the following plausible alternatives (1) (5) to the assumed distribution GSH in (2.2) with t = π/2: Misspecification of the distribution: (1) GSH(μ,, π/4). Dixon s outlier model: (2) (n 1) observations come from GSH(μ,, π/2) but one observation (we do not know which one) comes from GSH(μ, 4, π/2). Mixture model: (3) 0.90 GSH(μ,, π,/2) GSH(μ, 4, π/2). Contamination model: (4) 0.90 GSH(μ,, π/2) Uniform( 1/2, 1/2). (5) Normal distribution with mean μ and standard deviation. Note that the coefficients α j and β j in (1) (5) are computed from (2.3) with t = π/2, the assumed distribution. The simulated variances of μ i and ˆμ i are given in Table 3. Also given are the values of the relative efficiency of the LS estimator of μ i. It can be seen that the MML estimator ˆμ i is remarkably efficient and robust. For the normal Table 3 Variances and relative efficiencies: n = 10, = 1 Model Variance RE μ i ˆμ i μ i (1) (2) (3) (4) (5)

7 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) Table 4 Values of Type I error and power for the W and F-tests: d = τ 1 /, τ 2 = d, τ 3 = τ 4 = 0; = 1, k = 4, n = 10 Alternative models (1) (2) (3) (4) d F W d F W d F W d F W distribution, of course, ˆμ i is somewhat less efficient as expected since μ i is the MVB estimator. To show the robustness property of the W-test, the simulated values of the Type I error and power are given in Table 4. It may be noted that the W-test has a double advantage: it has not only smaller Type I error but has also higher power than the F-test. The T-test based on the MMLE for testing linear contrasts has efficiency and robustness properties exactly similar to those in Table 4. We do not reproduce details for conciseness. For the normal distribution, the W-test is a little less powerful as expected since the F-test is known to be UMP (uniformly most powerful). 8. Unbalanced design The methodology readily extends to unbalanced designs, the number of observations in the ith block being n i (1 i k). Without loss of generality, assume k m i τ i = 0 where m i = n i j=1 β i(j) and β i(j) is the coefficient β j in (3.4) with n = n i (1 i k). The MML estimators in (3.6) are exactly similar to those for balanced designs with obvious changes such as ˆμ.. = 1 M m i ˆμ i, ˆμ i = 1 n i n i β m i(j) y i(j), M = m i, C = 2c 2 β i(j) (y i(j) ˆμ ˆτ i ) 2, i j=1 ȳ = 1 n i y i(j), ȳ a = 2 n i α i(j) y i(j), N = n i and N N α i(j) is the coefficient α j in (3.3) with n = n i (1 i k). (8.1) To test equality of block effects, the variance ratio statistic is W = 2c k 2 m i ˆτ 2 i (k 1)ˆ 2. (8.2) The null distribution of W for large n i (1 i k) is central F with (k 1,N k) degrees of freedom and the distribution of W under H 1 is non-central F with (k 1,N k) degrees of freedom and non-centrality parameter λ 2 W = 2c k 2 (m i τ i /) 2. The W-test has exactly similar robustness properties as the test statistic (5.2). In testing a linear contrast, the T-test statistic is k 2c2 l i ˆμ T = i ˆ k li 2 m i (8.3)

8 552 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) and is asymptotically normal N(0, 1), under the null hypothesis H 0 : k l i μ i = 0. The non-centrality parameter in (6.2) becomes λ 2 T = 2c 2( k l i μ i ) 2 / 2 k (l 2 i /m i). Acknowledgment This research work was supported by the Scientific and Technical Research Council of Turkey under grant: TBAG (103T033). Thanks are due to the referees for helpful comments. Appendix A. A.1. Information matrix The Fisher information matrix for one-way classification model is ( ) 2 ( ln L 2 ) ln L E μ 2 E μ I = i i ( 2 ) ( ln L 2 ) ln L, E E μ i 2 where E( 2 ln L/ μ i ) = 0, ( ) 2 ln L for π <t<0, E μ 2 i ( 2 ) ln L E 2 = N ( π 2 t sin 2 t ( ) 2 ln L for t 0, E μ 2 i = c2 2n(t sin t cos t) 2 2 tsin 2 t (π2 3t 2 ) ) cos t t sin t = c2 2n(sinh t cosh t t) 2 2 tsinh 2 t ( 2 ) ln L E 2 = N ( (π 2 + 3t 2 ) cosh t 6 2 π2 + t 2 ) t sinh t sinh 2. t The variance covariance matrix is V = I 1 = (V ij ), where 1 V 11 = E( 2 ln L/ μ 2 i ), V 1 12 = V 21 = 0 and V 22 = E( 2 ln L/ 2 ). (A.1) A.2. Asymptotic properties A.2.1. One-way clasification model Lemma 1. Asymptotically, the estimator ˆμ i =ˆμ i. is BAN with variance V(ˆμ i ) 2 2mc 2. (A.2) Proof. Since ln L / μ i is asymptotically equivalent to ln L/ μ i and assumes the form ln L = 2mc 2 μ i 2 (ˆμ i μ i ), (A.3) [ ] and E r ln L = 0 for all r 3, the result follows [1]. μ r i

9 Y.E. Yilmaz, A.D. Akkaya / Journal of Computational and Applied Mathematics 216 (2008) Corollary 1. The estimator ˆμ =ˆμ.. is BAN with variance 2 V(ˆμ). 2kmc 2 (A.4) Corollary 2. Since ˆμ i (1 i k) are independent of each other and ˆμ = (1/k) k ˆμ i, (k 1)2 V(ˆτ i ). (A.5) 2kmc 2 Remark. The estimators ˆτ i and ˆ are uncorrelated and since E( r+s ln L / τ r i s ) = 0 for all r 1 and s 1, they are independent of each other, asymptotically [1]. Lemma 2. Asymptotically, N ˆ 2 (μ i )/ 2 is conditionally (μ i = μ + τ i ) distributed as chi-square with N degrees of freedom. Proof. For large n, B/ nc 1 0 where C 1 = 2c k nj=1 2 β j (y i(j) μ i ) 2. Therefore, ln L / assumes the form ln L N ( ) C1 3 N 2. (A.6) Evaluation of the cumulants of ln L / in terms of the expected values of the derivatives of ln L / immediately leads to the result that N ˆ 2 (μ i )/ 2 is distributed as chi-square with N degrees of freedom [1,20,22]. Corollary 3. Asymptotically, N ˆ 2 / 2 is distributed as chi-square with N k degrees of freedom. References [1] M.S. Bartlett, Approximate confidence intervals, Biometrika 40 (1953) [2] G.E.P. Box, S.L. Andersen, Permutation theory in the derivation of robust criteria and the study of departures from assumption, J. Roy. Statist. Soc. B 17 (1955) [3] G.E.P. Box, G.S. Watson, Robustness to non-normality of regression tests, Biometrika 49 (1962) [4] F.N. David, N.L. Johnson, The effect of non-normality on the power function of the F-test in the analysis of variance, Biometrika 58 (1951) [5] T.S. Donaldson, Robustness of the F-test to errors of both kinds and the correlation between the numerator and denominator of the F-ratio, J. Amer. Statist. Assoc. 63 (1968) [6] A.K. Gayen, The distribution of the variance ratio in random samples of any size drawn from non-normal universes, Biometrika 37 (1950) [7] R.C. Geary, Testing for normality, Biometrika 34 (1947) [8] A.M. Gross, Confidence interval robustness with long-tailed symmetric distributions, J. Amer. Statist. Assoc. 71 (1976) [9] H.R.B. Hack, An empirical investigation into the distribution of the F-ratio in samples from two non-normal populations, Biometrika 45 (1958) [10] P.J. Huber, Robust estimation of a location parameter, Ann. Math. Statist. 35 (1964) [11] E.S. Pearson, The analysis of variance in cases of non-normal variation, Biometrika 23 (1931) [12] B. Senoglu, M.L. Tiku, Analysis of variance in experimental design with non-normal error distributions, Comm. Statist. Theory Methods 30 (2001) [13] A.B.L. Srivastava, Effect of non-normality on the power of the analysis of variance test, Biometrika 46 (1959) [14] W.Y. Tan, On Tiku s robust procedure-a Bayesian insight, J. Statist. Plann. Inference 11 (1985) [15] M.L. Tiku, Approximating the general non-normal variance ratio sampling distributions, Biometrika 51 (1964) [16] M.L. Tiku, Estimating the mean and standard deviation from a censored normal sample, Biometrika 54 (1967) [17] M.L. Tiku, Estimating the parameters of log-normal distribution from censored samples, J. Amer. Statist. Assoc. 63 (1968) [18] M.L. Tiku, Power function of the F -test under non-normal situations, J. Amer. Statist. Assoc. 66 (1971) [19] M.L. Tiku, Robustness of MML estimators based on censored samples and robust test statistics, J. Statist. Plann. Inference 4 (1980) [20] M.L. Tiku, A.D. Akkaya, Robust Estimation and Hypothesis Testing, New Age International Publishers (Wiley Eastern), New Delhi, [21] M.L. Tiku, R.P. Suresh, A new method of estimation for location and scale parameters, J. Statist. Plann. Inference 30 (1992) [22] M.L. Tiku, W.Y. Tan, N. Balakrishnan, Robust Inference, Marcel Dekker, New York, [23] D.C. Vaughan, On the Tiku-Suresh method of estimation, Comm. Statist. Theory Methods 21 (1992) [24] D.C. Vaughan, The generalized secant hyperbolic distribution and its properties, Comm. Statist. Theory Methods 31 (2002) [25] D.C. Vaughan, M.L. Tiku, Estimation and hypothesis testing for a non-normal bivariate distribution with applications, J. Math. Comput. Modeling 32 (2000)