International Journal of Statistical Sciences

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1 International Journal of Statistical Sciences Editor-in-Chief Dr. Shahariar Huda, Professor, Dept. of Statistics and Operations Research Faculty of Science, Kuwait University, P.O. Box-5969, Safat-13060, Kuwait Executive Editor Dr. Mohammed Nasser, Professor, Dept. of Statistics, University of Rajshahi, Rajshahi-605, Bangladesh Associate Editors Dr. Md. Golam Hossain, Professor, Dept. of Statistics, University of Rajshahi, Rajshahi-605, Bangladesh Dr. Md. Sabiruzzaman, Dept. of Statistics, University of Rajshahi, Rajshahi-605, Bangladesh Dr. Md. Mostafizur Rahman, Dept. of Statistics, University of Rajshahi, Rajshahi-605, Bangladesh Secretary Dr. Md. Nasim Mahmud, Dept. of Statistics, University of Rajshahi, Rajshahi-605, Bangladesh Editorial Board Anwar H. Joarder, Dr., Prof., School of Business University of Liberals Arts Bangladesh, Dhanmondi, Dhaka 109, Bangladesh Biswas, S., Dr., Senior Prof., School of Insurance and Actuarial Science (ASIAS), Amity University Noida (U.P.), India Islam, M. Nurul., Dr., Prof., Dept. of Statistics University of Rajshahi Rajshahi-605, Bangladesh Kibria, B.M.G., Dr., Prof., Dept. of Statistics Florida International University FL 33199, USA Rahman, M., Dr., Prof., Dept. of Statistics Minnesota State University Mankato, MN 56001, USA Shah, M.A., Dr., Prof., Dept. of Statistics University of Rajshahi Rajshahi-605, Bangladesh Verma, Med Ram, Dr., Senior Scientist (Agri. Stat.) ICAR- Indian Veterinary Research Institute (IVRI) Izatnagar, Bareilly Uttar Pradesh, India 43 1 Beg, A.B.M. Rabiul Alam, Dr., School of Business James Cook University Townsville, QLD 4811, Australia El-Shaarawi, A.H., Dr., Prof., Dept. of Statistics McMaster University Hamilton, Ontario, Canada Karim, M. Rezaul, Dr., Prof., Dept. of Statistics University of Rajshahi Rajshahi-605, Bangladesh Latif, Mahbub, Dr., Prof., ISRT, University of Dhaka Dhaka-1000, Bangladesh Roy, M.K., Dr., Prof., Dept. of Statistics M. Bhashani Sci. & Tech. Univ. Santosh, Tangail-190, Bangladesh Sinha, B.K., Dr., Prof., Retired Faculty Indian Statistical Institute Kolkata , India Biswas, Atanu, Dr., Prof., Applied Statistics Unit Indian Statistical Institute Kolkata , India Imon, A.H.M.R., Dr., Prof., Dept. of Mathematical Sciences Ball State University Muncie, IN 47306, USA Khan, S., Dr., Assoc. Prof., Dept. of Math. & Computing University of Southern Queensland Qld 4350, Australia Rahman, M., Dr., Prof., Dept. of Statistics University of Jahangirnagar Savar, Dhaka-134, Bangladesh Sen, K.P., Dr., Prof., Dept. of Statistics University of Dhaka Dhaka-1000, Bangladesh Suzuki, K., Dr., Prof., Dept. of Systems Eng. The University of Electro-Communications Chofu, Tokyo, , Japan

2 International Journal of Statistical Sciences ISSN Vol. 14, 014 c 014 Dept. of Statistics, Univ. of Rajshahi, Bangladesh Editorial note It is a great pleasure for me to see the 014 Volume (Volume 14) of the International Journal of Statistical Sciences (IJSS) go into press. This volume contains five research articles entitled A New Class of Optimal Designs in the Presence of a Quantitative Covariate, Concomitants of Dual Generalized Order Statistics from Farlie Gumbel Morgenstern Type Bivariate Gumbel Distribution, Constructions of Nested Group Divisible Designs, Minimax Estimation of the Variance of a Normal Distribution for an Asymmetric Loss Function and Minimal Sufficient Statistics Emerge from the Observed Likelihood Functions. All the articles were peer reviewed and the referees are to be thanked for their help. The Editor and his team are to be congratulated for doing a great job under difficult circumstances. However, I hope that the quality of submission to the journal will be further improved in the future with help from well-wishers from all over the world. In particular, a more diverse collection of research areas covered by the articles would be most welcome. Prof. Shahariar Huda Editor-in-Chief, IJSS

3 International Journal of Statistical Sciences Publisher: Department of Statistics University of Rajshahi Rajshahi 605 Bangladesh. Published: Volume 14, December, 014. Copyright c 014 by the Dept. of Statistics, Univ. of Rajshahi. All rights reserved. Complementary Price: Individual: Tk (US $100.00) Organizations: Tk (US $00.00) All communication should be addressed to the Executive Editor, IJSS Department of Statistics, University of Rajshahi, Rajshahi 605 Bangladesh. Web Site: ijss ru@yahoo.com Phone: Fax:

4 International Journal of Statistical Sciences ISSN Vol. 14, 014, pp 1-15 c 014 Dept. of Statistics, Univ. of Rajshahi, Bangladesh A New Class of Optimal Designs in the Presence of a Quantitative Covariate Bikas K. Sinha 1 & P. S. S. N. V. P. Rao Retired Professor, Indian Statistical Institute 03 B. T. Road, Kolkata, India bikassinha1946@gmail.com Thomas Mathew Baltimore, MD 150, USA Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, MD 150, USA mathew@umbc.edu S. B. Rao CRRao AIMSCS University of Hyderabad Campus Hyderabad, India siddanib@yahoo.co.in [Received October 8, 014; Revised December 18, 014; Accepted April 1, 015] Abstract We propose to discuss at length the problem of placement of one controllable covariate in the context of an experiment involving several treatments. We do this while extracting maximum information on the unknown parameter attached to the covariate s values in the mean model for the observations. The experimental set-up is a bit different, and this calls for an interesting non-trivial study on optimality in the context of a single-covariate linear regression model. Keywords and Phrases: Treatment allocation experiments, models with covariates, optimal placement of covariate values, optimal allocations. AMS Classification: Primary 6K05, Secondary 6K10. 1 Corresponding Author

5 International Journal of Statistical Sciences, Vol. 14, Introduction There are two parallel developments in the construction, analysis and optimality of varietal designs. One of these is associated with the Analysis of Variance (ANOVA) set-up and in particular, the block design set-up. The pioneering work is due to Bose and his co-authors on the construction and analysis of Balanced Incomplete Block Designs (BIBDs) and other families of designs, and many authors contributed in this area. In this connection one may refer to Raghavarao (1971) for detailed references on initial development towards constructions of designs. The optimality problems have been formulated in various ANOVA models arising in experimental designs. For an exhaustive review of the work done in the area prior to 90 s, one is to referred to the monograph by Shah and Sinha (1989). The other development is associated with the analysis and optimality of regression designs where the response depends on the levels of a number of controllable quantitative factors, called covariates. Again, excellent text books are available in this area of research viz. Fedorov (197), Silvey (1980) and Pukelsheim (1993). A relatively recent monograph on this topic is by Liski et al (00). In the analysis of covariance models where both qualitative and quantitative factors are present, the problem of inference on varietal contrasts corresponding to qualitative factors were studied by Harville (1975), Wu (1981) and Nachtsheim (1989). The problem of determining optimum designs for the estimation of regression parameters corresponding to controllable covariates was first considered by Lopes Troya (198a, 198b). She restricted investigations in the set-up of Completely Randomized Designs (CRDs). Das et al. (003) extended it to the block design set-up viz. Randomised Block Designs (RBDs) and some series of Balanced Incomplete Block Designs (BIBDs) and constructed Optimum Covariate Designs (OCDs) for optimal and simultaneous estimation of covariates parameters. The literature on optimum designs is so vast and is developing so fast in different directions that it would be impossible to cover the area comprehensibly in such a short article. In the above we have tried to cite only those references which have direct link with one or other aspect of the problem to be considered here. In this paper, an attempt has been made to construct OCDs for the estimation of the covariate parameter in a specific experimental design set-up. The model formulation closely follows the usual covariates model as in Lopes Troya and others. But, it is not quite the same. We will consider two formulations of the specific experimental setup (and, consequently, of the underlying allocation design). In both the cases, we will search for optimal design(s) for most efficient estimation of the covariate parameter. Formulation I There is a non-stochastic feature, quantified as X, associated with every experimental unit in a finite population of n experimental units (eu s). Assume X lies in the closed

6 Sinha, Rao, Mathew and Rao: A New Class of Optimal Designs 3 interval [ 1,1] and that there are altogether t(> 1) distinct values of X covering all the n units according to the following scheme : [(x i,f i );1 i t; i f i = n; 1 x 1 <... < x t 1] There are v experimental treatments to be allocated, one to each of these eu s and with τ j as the effect of the j th treatment; j = 1,,...,v. The following Incidence Matrix describes the layout of the design: X values T reatment Allocations T otals x 1 f 11 f 1 f 1j f 1v f 1 x f 1 f f j f v f x 3 f 31 f 3 f 3j f 3v f 3 x i f i1 f i f ij f iv f i f 1v x t f t1 f t f tj f tv f t Totals r 1 r r j r v n Clearly, F = ((f ij )) is the treatment allocation matrix with f ij 0 and it is arbitrary subject to the pre-assigned row totals f i s and with arbitrary column totals r j s except that each r j > 0. We contemplate a fixed effects additive model given by Y iju = µ+βx i +τ j +e iju whenever f ij > 0, for u = 1,,...,f ij ;1 j v;1 i t. Usual assumptions on the error distributions apply. The given design parameters are n and v. The quantities to be ascertained are (i) t > 1;(ii) 1 x 1 < x <... < x t 1;(iii) f i s, andhence((f ij )) sandr j ssuchthatinformationonβ parameterismaximized, subject to estimability of the τ-contrasts. Joint Information Matrix for the τ-vector and the β-parameter for the above design layout is given by r i x if i1 = S 1,say 0 r 0 0 i x if i = S,say r v i x if iv = S v,say S 1 S S 3 S v i x i f i = SS,say

7 4 International Journal of Statistical Sciences, Vol. 14, 014 Note that ij x if ij = i x if i = j S j;s j = i x if ij. Let I(β) = SS j S j /r j;ss = ij x i f ij = i x i f i. Our purpose is to maximize I(β) for variations in (t,x i,f i,r j ) s, finally leading to the ((f ij )) matrix. This formulation is quite easy to sort out. We re-write I(β) as j [ i x i f ij S j /r j] = j [Q j], say. Lemma 1. For fixed r j,q j r j I(odd)/r j, where I(odd) = 1 if r j is odd; = 0 if r j is even. Proof. Easy since 1 x i 1 for each i. Therefore, I(β) = j Q j n j [I(odd)/r j] n. Henceforth, we assume n to be even and choose r j s also to be even integers so that I(β) may attain the upper bound n. In order to attain the bound, we set t =,x 1 = 1 and x = +1 and further, each r j as an even integer. One choice is : n = b = (v 1) + (b v + 1);r 1 =... = r (v 1) = ;r v = (b v +1). However, as is well known, this allocation design will stay away from a treatmentoptimal design which calls for equal or nearly equal treatment replications. Hence, we consider the representation n = vr + s where 0 < s < v. In that case, the treatment allocations are : r (with replication (v s)) and r + 1 (with replication s). At this stage, maximization rests on whether r is odd or even. It follows that I(β) = n (v s)i(r odd)/r si(r even)/(r +1). For r = q, I(β) = n s/(r +1) while n = qv +s and for r = q +1,I(β) = n (v s)/r while n = v(q +1) +s. Further, all treatment contrasts are estimable. With the formulation given above, β-parameter is estimated with maximum precision for the above allocations, though there are many choices of the underlying designs. 3 Formulation II We now confine to a situation wherein the observations are generated in pairs - based on pairs of distinct treatments chosen from the set of all v treatments. Thus, as before, we have t > 1 distinct choices of x-values inside [ 1,+1] and for each x i, there are pairs of treatments chosen and utilized to ( produce ) pairs of observations. We v assume that in the process we generate all b = pairs of observations so that ( ) v n = = v(v 1). Naturally, the choice of a design corresponds to a partition of b pairs into t classes. Note that this formulation trivially leads to estimability of all treatment effects contrasts. Our purpose is to characterize an optimal design for most efficient estimation of the β-parameter underlying the same model as stipulated above. It may be noted that this formulation is a bit different from usual linear regression set-up involving a number of treatments. Herein we are contemplating a situation

8 Sinha, Rao, Mathew and Rao: A New Class of Optimal Designs 5 involving a production process. Every run of the experiment using any covariate value utilizes enough input material to accommodate two distinct treatments which constitute a block, so to say. An immediate generalization would call for accommodating triplets of treatments and so on. We will discuss this aspect later. We start with a very general design specification, with k i treatment pairs generated from the i th block; i = 1,,,.,t, as outlined in Table 1. Table 1 Blocks x values T reatment P airs T reatment Replications T otal 1 x 1 [{1i p,1i q };1 <= p < q <= v;] f 11 f 1 f 1v k 1 x [{i p,i q };1 <= p < q <= v;] f 1 f f v k t x t [{ti p,ti q };1 <= p < q <= v;] f t1 f t f tv k t In the above, it is understood that all allocations (of pairs of treatments) within and across different blocks are distinct. Naturally, the above allocation parameters satisfy (i) f ij = k i ;(ii) k i = b. j i Further, we assume, without loss of generality, 1 x 1 < x <.. < x t 1. Define F astheblock treatment incidencematrix asindicated above. Thatis, F = ((f ij ));1 i t;1 j v. It follows that I(β) = x Qx/(v 1) where x =(x 1,x,...,x t ) and Q ii = (v 1)k i j f ij ;Q ii = j f ijf i j. In other words, Q = (v 1)Diag.(k i ) FF The decision variables are : t(> 1), distinct x-values, k-values subject to (ii) and elements of the matrix F subject to (i) above. The problem is that of maximization of I(β). Note that the treatment pairs can be listed according to the dictionary style, viz., [(1,),...,(1,v);(,3),...,(,v);...;(v 1,v)]. For any given specification of the decision variables listed above, starting with the vector x of order t 1, let us naturally extend it to a vector x of order b 1, by repeating x i in exactly k i positions as per the specification of the treatment pairs in the above allocation matrix and the dictionary style representation. Note that (ii) ensures i k i = b. Further to this, it also follows that x Diag.(k i )x = x x. Likewise, let us convert the F-matrix of order t v to an Incidence matrix N of order b v in an obvious manner so that F x = Nx is a vector of order v 1. With these two suggested conversions, I(β) = x [(v 1)I N N]x where x is a vector of b components (x 1,x,...,x b ) with elements not necessarily all distinct.

9 6 International Journal of Statistical Sciences, Vol. 14, 014 Naturally, the class of choices of the design parameters encompasses all choices of x subject to 1 x min < x max 1. Here, N is the treatment block incidence matrix involving elements ( 0) and 1 where the b blocks are arranged in dictionary style. Note v that since b =, each treatment is replicated exactly (v 1) times in the entire design. Theproblem is thus to make an optimal choice of thex -vector with elements not necessarily distinct. We have resolved this problem - rather non-trivially. We do not know if a simpler approach exists. For the rest of the paper, we will revert back to the notation of x from x so that x-values are not necessarily distinct and there are b = v c pairs of treatments attached to the x-values. Not to obscure the essential steps of reasoning, we will go through the following steps. Essentially, we claim that for every v >, there is an optimal choice of the x-values across all the b pairs of treatments, and that these are located at the extreme points, viz., 1 and 1. The specific allocations depend on the nature of v and we distinguish among Case 1. v = 4t Case. v = 4t+1 Case 3. v = 4t+ Case 4. v = 4t+3. First note that for t =, I(β) reduces to Q(x)/r where Q(x) = [x (ri N N)x] and r = v 1. Towards maximization of I(β), we may and will ignore the divisor r and work only with Q(x). The following lemmas are easy to establish and we defer the proofs to Appendix A. Lemma. Let Q(x) = x (ki N N)x be a quadratic form where k is a positive constant and all the diagonal elements of the matrix N N are equal to a constant d k. Then maximum of Q(x) with restriction on elements x i, that 1 x i 1, is attained at a vector x 0 with each element ±1. Lemma 3. For any vector x with elements ±1, Q(x) is a multiple of 8. Lemma 4. Let v be even. Then for every vector x with each element ±1, the value of x N Nx v. In view of Lemma, there exists a vector x 0 having each element ±1 that maximizes Q(x). Therefore Q(x 0 ) = rb x 0 N Nx 0 = PRU x 0 N Nx 0, where PRU = rb. Notice that Q(x 0 ) PRU and equality occurs if and only if x 0 N Nx 0 = 0. In view of Lemma 4, when v is even x 0 N Nx 0 v. Therefore, in this case Q(x 0 ) PRU v. And in view of Lemma 3, when v is of the form 4t + 3, PRU is not a multiple of 8 but a multiple of 4. Therefore, in this case Q(x 0 ) PRU 4. Thus we have, Theorem 1. Q(x 0 ) PRU v for the even values of v and Q(x 0 ) PRU 4 for the values of v of the form 4t+3. Further, Q(x 0 ) PRU, for the values of v of the form 4t+1.

10 Sinha, Rao, Mathew and Rao: A New Class of Optimal Designs 7 It is shown below by construction, that for each value of v 4 there exists a vector x 0 such that equality holds in the above theorem. This vector is constructed in two stages. First we get an eigen vector x corresponding to the eigenvalue r of the matrix (ri N N) that may contain some zero elements in addition to +1 and 1. Then we replace the zero elements of x with +1 or 1 to get x 0. Towards this, note that Q(x) being a quadratic form its maximum value is M(x x) where M is maximum eigenvalue of (ri N N) and x is an eigen vector (with the restriction) corresponding to M such that (x x) is maximum. As N N is positive semi-definite matrix its minimum eigenvalue is zero and hence maximum eigenvalue of (ri N N) is r. That is, M = r. It is easy to show that there is always an eigen vector corresponding to M with elements in the set 1,0,1. Clearly, the maximum of Q(x) is rb = v(v 1), attained when x x = b, that is, each element of x is ±1 and x N Nx = 0. This maximum value v(v 1) is denoted by PRU. In general, there may not be any such eigen vector x with each element as ±1, in which case, PRU is not attained. Notice that x is an eigen vector of (ri N N) corresponding to r if and only if Nx is null vector and this implies e x = 0. Now, let x be the eigen vector of (ri N N) corresponding to r. Construction of a vector x such that Nx = φ for the values of v of the form 4t,4t+,4t+1 and 4t+3 is given below. Theorem. Let N v b be as above. There exists a vector x b 1 such that Nx = φ, that is, x is an eigen vector of (ri N N) corresponding to the eigenvalue r such that 1. when v is even, that is of the form 4t or 4t+, the vector x has v/ zero elements.. when v is of the form 4t+1, the vector x has no zero elements. 3. when v is of the form 4t+3, the vector x has 3 zero elements. Proof. We construct a vector x of b elements as follows. Consider x as a vector of v 1 partitioned blocks of sizes v 1,v,,,1 such that the i th partitioned block contains i elements in the natural order, following dictionary style. Block No. and size v 1 v 1 x = Case 1. When v is even, that is, v is of the form 4t or 4t+. Step 1: For each odd sized block, put zero as the first element and 1 and +1 alternately till the end of block. Notice that there are v/ odd sized blocks. Step : For each even sized block, put +1 and 1 alternately till the end of block. Example 1. Let v = 8. Then b = 8. So x is written as 7 blocks. Block No x =

11 8 International Journal of Statistical Sciences, Vol. 14, 014 Example. Let v = 6. Then b = 15. So x is written as 5 blocks. Block N o x = Case. When v is of the form 4t+1. In this case v 1 is even and (v 1)/ is also even. Let x be the vector constructed as above for the even case of 4t. Starting from block 1 replace each zero element of x alternately with +1 and 1. Let the resulting vector be y which is of order 4t 1. Construct the new vector x as x = ( y N : y ) where N is the incidence matrix corresponding to the case of v = 4t. Example 3. Let v = 9, that is of the form 4t+1. Then b = 36. So x is written as 8 blocks. Here we use the vector constructed in example 1. Block No x = Case 3. When v is of the form 4t+3. In this case v 1 is even and (v 1)/ is odd. Let x be the vector constructed as above for the case of 4t +. Starting from block 1 replace each zero element of x alternately with +1 and 1 except the (v 1) th block. Let the resulting vector be y. Construct the new vector x as x = ( y N : y ). It is easy to check that 3 elements (first, second and v th elements) of x are zeros and the rest of the elements are ±1. Example 4. Let v = 7, that is of the form 4t+3. Then b = 1. So x is written as 6 blocks. Here we use the vector constructed in example. Block N o x = In all the above cases it is easy to check that Nx = φ. For these constructed vectors the following table gives the values of Q(x). Table v x x Q(x) 4t b v/ (v 1)(b v/) PRU v(v 1) 4t+1 b (v 1)b PRU 4t+ b v/ (v 1)(b v/) PRU v(v 1) 4t+3 b 3 (v 1)(b 3) PRU 6(v 1) It is readily seen that only in the case of v = 4t + 1, the constructed x-vector has all elements ±1. Except for this case, in all other cases, we will suggest appropriate conversion of x by suitably replacing 0 s by ±1 s. For the case v = 4t + 3 we take

12 Sinha, Rao, Mathew and Rao: A New Class of Optimal Designs 9 the vector x constructed above, replace the 3 zero elements (first, second and v th elements) of x, with +1, +1 and 1. Denoting the resulting vector by x 0, it is easy to see that only the first element of Nx 0 is and the rest of the elements are zeros. Hence x 0 N Nx 0 = 4. When v is even, starting from block 1 replace all the v/ zero elements of x with +1 and 1 alternately. Denoting the resulting vector by x 0, it is easy to check that each element of Nx 0 is ±1. Hence x 0 N Nx 0 = v. Therefore, we have established that Q(x 0 ) = PRU v, if v is even, that is, v is of the form 4t or 4t+. Q(x 0 ) = PRU, if v is of the form 4t+1. Q(x 0 ) = PRU 4, if v is of the form 4t+3. As per Theorem 1, x 0 constructed above dependingon the value of v, maximizes Q(x). A Table showing the values of Q(x 0 ) covering v = 4 to 17 is given later. Explicit solutions to the optimal allocation designs for the cases of v = 6,7,8,9 are given Appendix B. 4 Generalization In this section we consider the same problem but ( with distinct ) triplets ( of) treatments v 1 v instead of pairs of treatments. We write p = and c =. Further 3 ( ) v we denote by N v the incidence matrix of order v for the case of pairs of treatments ( ) considered in Section and denote by M v the incidence matrix of order v v for the case of triplets of treatments. Similarly, we denote by x v the vector constructed in Section that maximizes Q(x) of Section. It is easy to verify the function to be maximized in the case of triplets is T(y v ) = y v (3pI M v M v)y v, where y v denotes the vector of order c 1 of covariate levels with each element in the interval [ 1,1]. Notice that the maximum value T(y v ) can attain is 3pc, attained if and only if M v y v = φ. The following table gives the values of the scalar e v x v and the vector N v x v of order v 1 for different values of v. Table 3 v e v x v (N v x v ) Description 4t 0 ( e : e : e : : e ) e and e occur alternately 4t+1 0 ( 0 : 0 : 0 : : 0) All elements are zero 4t+ 1 ( e : e : e : : e ) e and e occur alternately 4t+3 1 ( : 0 : 0 : : 0) All elements are zero except the first which is

13 10 International Journal of Statistical Sciences, Vol. 14, 014 The proofs of the following two lemmas are on the same lines of proofs of Lemma and Lemma 4. Lemma 5. Maximum of T(y) is attained at a vector y with each element ±1. Lemma 6. Let p be odd, that is, v = 4t or 4t+3. Then for every vector y v with each element ±1, the value of y M My v. Now we are ready to state the main result of this section. Theorem 3. For v > 4, the vector that maximizes T(y v ) is given by y v = (x v 1 : y v 1 ) when v = 4t+3 and y v = ( x v 1 : y v 1 ) in other cases with y 4 = ( ). Proof. Notice that M v and hence M v y v are given by [ e M v = v 1 φ N v 1 M v 1) ] [ and M v y v = ±e v 1 x v 1 ±N v 1 x v 1 +M v 1 y v 1 ] Using the above expression, it is easy to check that the values of M v y v are as in the following table. Table 4 v (M v y v ) description 4t ( e : e : e : : e ) e and e occur alternately 4t+1 ( 0 : 0 : 0 : : 0) All elements are zero 4t+ ( 0 : 0 : 0 : : 0) All elements are zero 4t+3 (1 : e : e : e : : e ) First element is 1and then e and e alternate This results in y v M v M vy v = 0 in case of v = 4t + 1 or 4t + and y v M v M vy v = v in case of v = 4t or 4t + 3. Hence MaxT(y) = 3pc, for t = 4t + 1 or 4t + and MaxT(y) = 3pc v for t = 4t or 4t+3 at the above constructed y v. Remark. In general the vector x v that maximizes Q(x) of section is not unique and so in the construction of vector y v only x v constructed should be used. The following table gives the maximum values of Q(x v ) and T(y v ) for the values of v = 4 to 17. Herein PRU and PRU3 refer to the maximum possible values of Q(x) and T(y) respectively.

14 Sinha, Rao, Mathew and Rao: A New Class of Optimal Designs 11 Table 5 v r b rb = (PRU) MaxQ(x v ) p c 3pc = (PRU3) MaxT(y v ) Concluding Observations The formulation of the problem considered here is a bit different from that usually adopted in covariates studies involving treatment designs. Usually, the experimenter has the flexibility of choice of the quantitative covariate X for every single application of any of the v treatments under consideration. That gives rise to [(y ij ;x ij )] type of data. This seems to raise a concern for increase in the cost of experimentation since the labeling of the covariate-values may undergo constant and uncontrolled changes over the entire operation of the treatment-replicated experiments! In order to avoid such scenarios, the experimenter is constrained to generate observations in pairs as far as possible. We also considered briefly the generalization in the sense of generating data in triplets of the treatments under consideration. Even in the context of paired treatment scenario, consider a related problem. As before, we start with( v treatments ) but there is a restriction of generating paired data v only on a subset of treatment pairs. Given the subset, what would be the optimal allocation of x-values for most efficient estimation of the β-parameter? And, further to this, what would be the optimal subset selection and related optimal allocation of x-values for a given number of distinct treatment pairs to be covered in the experiment? These seem to be difficult issues. We believe Lemma has the potential for necessary generalization to address such issues. Acknowledgement. S B Rao s research was supported by DST-CMS Project No. SR/S4/MS : 516/07 Dated 1/04/008. The authors are thankful to the adminis-

15 1 International Journal of Statistical Sciences, Vol. 14, 014 trative authorities in their respective organizations for providing excellent research atmosphere in pursuing this work. An anonymous referee pointed out several typing mistakes in an earlier version. We are thankful to him for a critical reading. Appendix A. Proofs of Lemmas 4 Proof of Lemma. Writing x as (y + x i e i ) and N N as (di + M) where d is the common diagonal element of N N, we have Q(x) = x (ki N N)x = (y+x i e i ) (ki N N)(y +x i e i ) = y (ki N N)y+x i (k d) x i(y N Ne i ) = y (ki N N)y+x i (k d) x i(y (di +M)e i ) = y (ki N N)y+x i (k d) x i(y Me i ) = y (ki N N)y+x i (k d) x i((x e i x i ) Me i ) = y (ki N N)y+x i (k d) x i(x Me i ) = y (ki N N)y+[x i (k d) x i(mx) i ] Now, thefirsttermisindependentof x i anditis easytoseethat thevalueofthesecond term will increase if we choose x i as ±1 with the sign opposite to that of (Mx) i. It must be noted that (Mx) i dependents on other x j s and is indeed independent of x i. Hence the claim. Proof of Lemma 3. As each element of x is ±1, permute it so that all +1 elements are at the top and denote the transpose of this vector as (e u : e w ), where u is the number of positive elements and ( w is the number ) of negative elements. Permute and partition N N1 N N accordingly as 3 so that N N 3 ( x N Nx = (e u : e N1 N 3 w) N 3 N )( eu e w ) = e un 1 e u +e wn e w e un 3 e w = t 1 +t t 3,say. Also note that t 1 +t +t 3 = rb. Therefore, from the above corollary, we have, Q(x) = x (ri N N)x = r(x x) x N Nx = rb x N Nx = 4t 3. Further, we also have, t 1 +t 3 = (v 1)u which is even. Since t 1 is even, this implies t 3 is even. Thus Q(x) is a multiple of 8. Proof of Lemma 4. Let y = Nx. Then x N Nx = y i. So minimum of x N Nx is attained when each y i is minimum. Notice that when v is even, every y i effectively reduces to an odd (v 1) combination of +1s and 1s. Therefore minimum of y i = 1 and hence of x N Nx v.

16 Sinha, Rao, Mathew and Rao: A New Class of Optimal Designs 13 Appendix B. Explicit solutions for Optimal Allocations Designs The following two tables give the vectors x v for v = 4 to 9 for the case of pairs of treatments and y v for v = 5 to 8 for the case of triplets of treatments. Table 6. x v for v = 4 to 9 for the case of pairs of treatments v=4 x 4 v=5 x 5 v=6 x 6 v=7 x 7 v=8 x 8 v=9 x

17 14 International Journal of Statistical Sciences, Vol. 14, 014 Table 7. y v for v = 5 to 8 for the case of triplets of treatments v=5 y 5 v=6 y 6 v=7 y 7 v=8 y

18 Sinha, Rao, Mathew and Rao: A New Class of Optimal Designs 15 6 References Das.K., Mandal, N.K., Sinha, B.K. (003). Optimal experimental designs for models with covariates. Journal of Statistical Planning and Inference, 115, Fedorov, V.V. (197). Theory of Optimal Experiments, Academic Press, New York. Harville, D.A (1975). Computing optimum designs for covariate model. In A Survey of Statistical Designs and Linear Models. J. N. Srivastava Ed., North Holland Liski, E.P., Mandal, N.K., Shah, K.R., Sinha, B.K. (00). Topics in Optimal Design (Vol. 163), Lecture Notes in Statistics, Springer, New York. Lopes Troya, J. (198a). Optimal designs for covariate models. Journal of the Statistical Planning and Inference 6, Lopes Troya, J. (198b). Cyclic designs for a covariate model. Journal of the Statistical Planning and Inference 7, Nachtsheim, C.J. (1989). On the design of experiments in the presence of fixed covariates. Journal of Statistial Planning and Inference,, Pukelsheim, F. (1993). Optimal Design of Experiments, John Wiley, New York. Raghavarao, D. (1971). Constructions and Combinatorial Problems in Design of Experiments, John Wiley, New York. Shah, K.R., Sinha, B.K. (1989). Theory of Optimal Designs (Vol. 54), Lecture Notes in Statistics, Springer, New York. Silvey, S.D. (1980). Optimal Designs, Chapman and Hall, London. Wu, C.F.J. (1981). Iterative construction of nearly balanced assignments I: categorical covariates. Technometrics, 3,

19 International Journal of Statistical Sciences ISSN Vol. 14, 014 c 014 Dept. of Statistics, Univ. of Rajshahi, Bangladesh CALL FOR PAPERS IJSS shall welcome papers both theoretical and applied from all branches of Statistics. It shall also welcome papers from other disciplines where high quality of statistical applications are made. IJSS is a referred journal and abstracted in the Statistical Theory and Method Abstract published by the International Statistical Institute, Netherlands and Bangladesh Science and Technological Abstracts, Dhaka, Bangladesh. The authors are requested to submit a soft copy of the manuscript in MS Word to the following address. Prof. Mohammed Nasser, Ph.D Executive Editor, IJSS Department of Statistics University of Rajshahi Rajshahi Bangladesh ijss ru@yahoo.com

20 International Journal of Statistical Sciences ISSN Vol. 14, 014, pp 17-8 c 014 Dept. of Statistics, Univ. of Rajshahi, Bangladesh Concomitants of Dual Generalized Order Statistics from Farlie Gumbel Morgenstern Type Bivariate Gumbel Distribution Haseeb Athar and Nayabuddin Department of Statistics & Operations Research Aligarh Muslim University Aligarh - 000, India. M. Almech Ali Department of Statistics, Faculty of Sciences King Abdulaziz University Kingdom of Saudi Arabia. [Received September 9, 013; Revised April 15, 015; Accepted August 3, 015] Abstract Burkschat et al. (003) introduced the concept of dual generalized order statistics to enable a common approach to descendingly ordered random variables like reverse order statistics and lower record values. In this paper probability density function of single and the joint probability density function of two concomitants of dual generalized order statistics from Farlie Gumbel Morgenstern type bivariate Gumbel distribution have been obtained and expression for marginal and joint moments generating functions are derived. Further, the results are deduced for reverse order statistics and lower record values. Keywords and Phrases: Dual Generalized Order Statistics, Farlie Gumbel Morgenstern Type Bivariate Gumbel Distribution, Concomitants, Marginal and Joint Moments Generating Functions. AMS Classification: Primary 6E15; Secondary 6G30. 1 Introduction The Farlie Gumbel Morgenstern (FGM) family of bivariate distributions has found extensive use in practice. This family is characterized by the specified marginal dis-

21 18 International Journal of Statistical Sciences, Vol. 14, 014 tribution functions F X (x) and F Y (y) of random variables X and Y respectively and a parameter α, resulting in the bivariate distribution function (df) is given by F X,Y (x,y) = F X (x)f Y (y)[1+α(1 F X (x))(1 F Y (y))]. (1) with the corresponding probability density function (pdf) f X,Y (x,y) = f X (x)f Y (y)[1+α(f X (x) 1)(1 F Y (y))]. () Here, f X (x) and f Y (y) are the marginals of f X,Y (x,y). The parameter α is known as the association parameter. The two random variables X and Y are independent when α is zero. Such a model was originally introduced by Morgenstern (1956) and investigated by Gumbel (1960) for exponential marginals. The general form in (1) is due to Farlie (1960) and Johnson and Kotz (1975). The admissible range of association parameter α is 1 α 1 and the Pearson correlation coefficient ρ between X and Y can never exceed 1/3. The conditional df and pdf of Y given X, are given by and F Y X (y x) = F Y (y)[1+α(1 F X (x))(1 F Y (y))] (3) f Y X (y x) = f Y (y)[1+α(1 F X (x))(1 F Y (y))] (4) [c.f. Beg and Ahsanullah, 007] In this paper, we have considered FGM type bivariate Gumbel distribution with pdf f(x,y) = e x e y e e x e e y [1+α(1 e e x )(1 e e y )], and corresponding df < x,y <, 1 α 1 (5) F(x,y) = e e x e e y [1+α(1 e e x )(1 e e y )], < x,y <, 1 α 1 (6) Thus, the conditional pdf of Y given X is f(y x) = e y e e y [1+α(1 e e x )(1 e e y )], and the marginal pdf and df of X are < x,y <, 1 α 1 (7) f(x) = e x e e x, < x <. (8) F(x) = e e x, < x <. (9)

22 Athar, Nayabuddin and Ali: Concomitants of Dual Generalized Order Statistics 19 respectively. Kamps (1995) introduced the concept of generalized order statistics (gos). Using the concept of gos, Burkschat et al. (003) introduced the concept of dual generalized order statistics (dgos) as follows: Let X be a continuous random variables with df F(x) and pdf f(x), x (α,β). Further, Let n N, k 1, m = (m 1,m,...,m ) R n 1, M r = n 1 j=r m j such that γ r = k+(n r)+m r 1 for all r 1,,...,n 1. Then X d (r,n, m,k) r = 1,,...,n are called dgos if their pdf is given by (n 1 k j=1 )(n 1 γ j on the cone F 1 (1) > x 1 x... x n > F 1 (0). [ F(xi ) ] ) m i [F(xn f(x i ) ) ] k 1 f(xn ) (10) If m i = m = 0, i = 1,,...,n 1, k = 1, then X d (r,n,m,k) reduces to the (n r + 1) th reverse order statistics, X n r+1:n from the sample X 1,X,...,X n and when m = 1, then X d (r,n,m,k) reduces to k th lower record values. The pdf of X d (r,n,m,k) is f Xd (r,n,m,k) = C r 1 (r 1)! [F(x)]γr 1 f(x)g r 1 m (F(x)) (11) and joint pdf of X d (r,n,m,k) and X d (s,n,m,k), is 1 r < s n. where f Xd (r,s,n,m,k)(x,y) = C r 1 = C s 1 (r 1)!(s r 1)! [F(x)]m f(x) gm r 1 ( ) F(x) [h m ( F(y) ) hm ( F(x) ) ] s r 1 [F(y)] γs 1 f(y), α x < y β (1) r γ i, γ i = k+(n i)(m+1) 1 h m (x) = m+1 xm+1, m 1 logx, m = 1 and g m (x) = h m (x) h m (1), x (0,1). Let (X i,y i ),i = 1,,...,n, be n pairs of independent random variables from some bivariate population with df F(x,y). If we arrange the X variates in descending

23 0 International Journal of Statistical Sciences, Vol. 14, 014 order as X d (1,n,m,k) X d (,n,m,k)... X d (n,n,m,k), then Y variates paired ( not necessarily in descending order ) with these dgos are called the concomitants of dgos and are denoted by Y [1,n,m,k],Y [,n,m,k],...,y [n,n,m,k]. The pdf of Y [r,n,m,k], the r th concomitant dgos, is given as g [r,n,m,k] (y) = f Y X (y x) f Xd (r,n,m,k)(x) dx (13) and the joint pdf of Y [r,n,m,k] and Y [s,n,m,k] 1 r < s n is g [r,s,n,m,k] (y 1,y ) = x1 f Y X (y 1 x 1 ) f Y X (y x ) f Xd (r,s,n,m,k)(x 1,x ) dx 1 dx (14) where f Xd (r,s,n,m,k)(x) is the joint pdf of X d (r,n,m,k) and X d (s,n,m,k), 1 r < s n. The most important use of concomitants arises in selection procedures when k < n individuals are chosen on the basis of their X values. Then the corresponding Y values represent performance on an associated characteristic. For example, X might be the score of a candidate on a screening test and Y the score on a later test. There are vast literature which deals with concomitants. An excellent review on concomitants of order statistics is given Bhattacharya (1984) and David and Nagaraja (1998). Balasubramanian and Beg (1996, 1997, 1998) studied the concomitants for bivariate exponential distribution of Marshall- Olkin, Morgesnstern type bivariate exponential distribution and Gumbel s bivariate exponential distribution and gave the recurrence relation between single and product moment of order statistics. Begum and Khan (1997, 1998, 000) studied the concomitants for Gumbel s bivariate Weibull distribution, bivariate Burr distribution, Marshall and Olkin bivariate Weibull distribution and gave expression for single and product moment of order statistics. Ahsanullah and Beg (006) studied the concomitants for Gumbel s bivariate exponential distribution and derived the recurrence relation between single and product moment of generalized order statistics. Here in this paper we have considered FGM type bivariate Gumbel distribution and obtained pdf for r th, 1 r n and the joint pdf of r th and s th, concomitants of dgos. Also, moment generating function(mgf) and cumulant generating function(cgf) are obtained and expressions for means, variances and covariances are derived.

24 Athar, Nayabuddin and Ali: Concomitants of Dual Generalized Order Statistics 1 Probability Density Function of Y [r,n,m,k] For the FGM type bivariate Gumbel distribution as given in (5), using (7) and (11) in (13), the pdf of r th concomitants of dgos Y [r,n,m,k] for m 1 is given as g [r,n,m,k] (y) = C r 1 r 1 ( ) r 1 (r 1)!(m+1) r 1 ( 1) i e y e e y i i=0 Setting e x = t in (15), we get = = C r 1e y e e y r 1 (r 1)!(m+1) r 1 e x[ e e x ] γr i [1+α(1 e e x )(1 e e y )]dx. (15) i=0 ( ) r 1 [ 1 ( 1 ( 1) i +α i γ r i γ r i C r 1 r 1 ( ) r 1 (r 1)!(m+1) re y e e y ( 1) i i [ 1 ( +α k m+1 +n r +i i=0 1 k m+1 +n r +i k+1 m+1 +n r +i ) ] (1 e e y ). γ r i +1 (16) ) ] (1 e e y ). (17) = C r 1 r 1 ( ) r 1 [ ( k ) (r 1)!(m+1) re y e e y ( 1) i B i m+1 +n r +i,1 i=0 { ( k ) ( k+1 )} ] +α B m+1 +n r +i,1 B m+1 +n r +i,1 (1 e e y ). (18) For real positive p, c and a positive integer b, we have b ( ) b ( 1) a B(a+p,c) = B(p,c+b). (19) a a=0 Thus, using (19) in (18), we get g [r,n,m,k] (y) = C r 1 (r 1)!(m+1) re y e e y [ B ( k m+1 +n r,r )

25 International Journal of Statistical Sciences, Vol. 14, 014 { ( k ) ( k+1 )} ] +α B m+1 +n r,r B m+1 +n r,r (1 e e y ). (0) After simplification, we get g [r,n,m,k] (y) = e y e e y[ { 1+α(1 e e y ) 1 r (1+ 1 γ i ) 1 }]. (1) Remark.1: Set m = 0,k = 1 in (1), to get the pdf of r th concomitants of reverse order statistics from FGM type bivariate Gumbel distribution as g [n r+1:n] (y) = e y e e y[ { 1+α(1 e e y ) 1 (n r +1) }]. n+1 By replacing (n r +1) by r, we get the pdf of ordinary order statistics. g [r:n] (y) = e y e e y[ { 1+α 1 r } ] (1 e e y ). n+1 Remark.: At m = 1 in (1), we get the pdf of r th concomitants of k th lower record values from Farlie FGM type bivariate Gumbel distribution as g [r,n, 1,k] (y) = e y e e y[ { ( k ) r }] 1+α(1 e e y ) 1. k +1 3 Moment Generating Function of Y [r,n,m,k] In this section,we derive the moment generating function (mgf) of Y [r,n,m,k] for FGM type bivariate Gumbel distribution by using the results of the previous section. In view of (1), the mgf of Y [r,n,m,k] is given as M [r,n,m,k] (t) = Let e y = z in (), then we have = 0 e ty e y e e y[ { 1+α(1 e e y ) 1 z (1 t) 1 e z[ { 1+α(1 e z ) 1 r (1+ 1 ) 1 }] dy. () γ i r (1+ 1 ) 1 }] dz (3) γ i

26 Athar, Nayabuddin and Ali: Concomitants of Dual Generalized Order Statistics 3 [ { = Γ(1 t) 1+α(1 t ) 1 r (1+ 1 γ i ) 1 }]. (4) Remark 3.1: Set m = 0,k = 1 in (4), to get the mgf of concomitants of reverse order statistics from FGM type bivariate Gumbel distribution as [ { M [n r+1:n] (t) = Γ(1 t) 1+α(1 t ) 1 (n r+1) n+1 Further, by replacing (n r + 1) by r, we get the mgf of concomitants of ordinary order statistics [ { M [r:n] (t) = Γ(1 t) 1 α(1 t ) 1 r }]. n+1 Remark 3.: Setting m = 1 in (4), we get the mgf of concomitants of k th lower record values from FGM type bivariate Gumbel distribution as [ { ( k ) r }] M [r,n, 1,k] (t) = Γ(1 t) 1+α(1 t ) 1. k +1 Remark 3.3: The cumulative generating function (cgf) of r th dgos Y [r,n,m,k] for FGM type bivariate Gumbel distribution is K [r,n,m,k] (t) = ln Γ(1 t)+ln [ { 1+α(1 t ) 1 }]. ) Since, mean= E (Y [r,n,m,k] = µ 1 [r,n,m,k] = d dt K [r,n,m,k](t) at t = 0 and variance= µ [r,n,m,k] = d dt K [r,n,m,k] (t) at t = 0. Thus, [ µ 1 [r,n,m,k] = α r µ [r,n,m,k] = Π 6 α(ln)[ 1 r (1+ 1 γ i ) 1 }]. (1+ 1 γ i ) 1 1 ] ln ψ(1). (5) r (1+ 1 ) 1 ] [ {. 1+α 1 γ i r (1+ 1 γ i ) 1 }]. (6) where, ψ(x) = d dx lnγ(x) = Γ (x) Γ(x), ψ(1) = γ is known as Eulers constant.

27 4 International Journal of Statistical Sciences, Vol. 14, Joint Probability Density Functionof Y [r,n,m,k] andy [s,n,m,k] For the FGM type bivariate Gumbel distribution as given in (5), using (7) and (1) in (14), the joint pdf of r th and s th concomitants of dgos Y [r,n,m,k] and Y [s,n,m,k] for m 1 is given as g [r,s,n,m,k] (y 1,y ) = where, r 1 I(x 1,y ) = i=0 C s 1 (r 1)!(s r 1)!(m+1) s e y 1 e y e e y1 e e y s r 1 j=0 ( 1) i+j ( r 1 i )( ) s r 1 j e x 1 ( ) e e x 1 (s r+i j)(m+1) [1+α(1 e e x1 )(1 e e y1 )]I(x 1,y )dx 1, (7) x1 By setting e x = t in (8), we get ( e x ) γs j[1+α(1 e e e x e x )(1 e e y )]dx. (8) [ e e (γ s j )x 1 ( e e (γ s j )x 1 I(x 1,y ) = +α γ s j γ s j e e (γ s j +1)x 1 ) ] (1 e e y ). (9) γ s j +1 Now putting the value of I(x 1,y ) from (9) in (7), we obtain [ ( )( ) g [r,s,n,m,k] (y 1,y ) = e y 1 e y e e y1 e e y 1+α 1 e e y 1 1 e e y { 1 r (1+ 1 ) 1 s (1+ 1 ) 1 r + 4 (1+ ) 1 } γ i γ i γ i {( ) ( )}{ +α 1 e e y e e y 1 s (1+ 1 γ i ) 1 }]. (30) 5 Joint Moment Generating Function of two concomitants Y [r,n,m,k] and Y [s,n,m,k] Joint moment generating function of two concomitant Y[r,n,m,k] and Y[s,n,m,k] is given by M [r,s,n,m,k] (t 1,t ) = e t 1y 1 e t y g [r,s,n,m,k] (y 1,y ) dy 1 dy. (31)

28 Athar, Nayabuddin and Ali: Concomitants of Dual Generalized Order Statistics 5 In view of (30) and (31), we have M [r,s,n,m,k] (t 1,t ) = { 1 { 1 e t 1y 1 e t y e y 1 e y e e y1 e e y s Let z 1 = e y 1, then M [r,s,n,m,k] (t 1,t ) = Γ(1 t 1 ) { 1 { 1 (1+ 1 ) 1 } ( )( ) +α 1 e e y 1 1 e e y γ i [ {( ) ( )} 1+α 1 e e y e e y r (1+ 1 ) 1 s (1+ 1 ) 1 r + 4 (1+ ) 1 }] dy 1 dy. (3) γ i γ i γ i s [ {( ) ( )} e t y e y e e y 1+α 1 t e e y (1+ 1 ) 1 } ( )( ) +α 1 t 1 1 e e y γ i r (1+ 1 ) 1 s (1+ 1 ) 1 r + 4 (1+ ) 1 }] dy. (33) γ i γ i γ i Setting z = e y in (33) and simplifying, we have [ {( ) ( )} M [r,s,n,m,k] (t 1,t ) = Γ(1 t 1 )Γ(1 t ) 1+α 1 t t { 1 { 1 s (1+ 1 ) 1 } ( )( ) +α 1 t 1 1 t γ i r (1+ 1 ) 1 s (1+ 1 ) 1+4 r (1+ ) 1 }]. (34) γ i γ i γ i Remark 5.1: Set m = 0,k = 1 in (34), to get the joint mgf of two concomitants of reverse order statistics from FGM type bivariate Gumbel distribution as [ {( ) ( )}{ M [n s+1,n r+1:n] (t 1,t ) = Γ(1 t 1 )Γ(1 t ) 1+α 1 t t 1 (n s+1) } n+1 ( )( ){ +α 1 t 1 1 t 1 (n r+1) n+1 (n s+1) + n+1 4(n r +1)(n r +) }]. (n+1)(n+)

29 6 International Journal of Statistical Sciences, Vol. 14, 014 Now replacing (n r + 1) by s and (n s + 1) by r, we get the joint mgf of two concomitants of ordinary order statistics [ {( ) ( )}{ M [r,s:n] (t 1,t ) = Γ(1 t 1 )Γ(1 t ) 1 α 1 t t 1 r n+1 ( )( ){ +α 1 t 1 1 t 1 s n+1 r n+1 + 4s(s+1) (n+1)(n+) Remark 5.: Setting m = 1 in (34), we get the joint mgf of two concomitants of k th lower record values from FGM type bivariate Gumbel distribution as [ {( ) ( )}{ ( M [r,s,n, 1,k] (t 1,t ) = Γ(1 t 1 )Γ(1 t ) 1+α 1 t t k 1 ( )( ){ ( +α 1 t 1 1 t k ) r ( k 1 k+1 k+1 k+1 ) s+ 4 ( k k + } }]. ) s } ) r }]. Remark 5.3: Joint cgf of two concomitant Y[r,n,m,k] and Y[s,n,m,k] for FGM type bivariate Gumbel distribution is given as K [r,s,n,m,k] (t 1,t ) = ln Γ(1 t 1 )+ ln Γ(1 t ) + ln { 1 [ {( ) ( )}{ 1+α 1 t t 1 Noting that, s (1+ 1 ) 1 } ( )( ) +α 1 t 1 1 t γ i r (1+ 1 ) 1 s (1+ 1 ) 1 r + 4 (1+ ) 1 }]. (35) γ i γ i γ i ] Cov [Y [r,n,m,k],y [s,n,m,k] = d K dt 1 st [r,s,n,m,k] (t 1,t ) at t = 0, t = 0 Then we get ] ( ) [{ Cov [Y [r,n,m,k],y [s,n,m,k] = αln r r (1+ 1 ) 1 s (1+ 1 ) 1 γ i γ i (1+ ) 1 } { 1 γ i s (1+ 1 γ i ) 1 } ]. (36)

30 Athar, Nayabuddin and Ali: Concomitants of Dual Generalized Order Statistics 7 Acknowledgement Authors are thankful to Editor in Chief, IJSS and learned referee who spent their valuable time to review this manuscript. References [1] Ahsanullah, M. and Beg, M.I. (006). Concomitants of generalized order statistics from Gumbel s bivariate exponential distribution, J. Statist. Theory and Application. 6(), [] Beg, M.I. and Ahsanullah, M. (007). Concomitants of generalized order statistics from Farlie Gumbel Morgenstern type bivariate Gumbel distributio, Statistical Methodology [3] Begum, A.A. and Khan, A.H. (1997). Concomitants of order statistics from Gumbel s bivariate Weibull distribution, Cal. Statist. Assoc. Bull. 47, , [4] Begum, A.A. and Khan, A.H.(1998). Concomitants of order statistics from bivariate Burr distribution, J. Appl. Statist. Sci. 7 (4), [5] Begum, A.A. and Khan, A.H. (000). Concomitants of order statistics from Marshall and Olkin bivariate Weibull distribution, Cal. Statist. Assoc. Bull. 50, [6] Balasubramnian, K. and Beg, M.I. (1996). Concomitants of order statistics in bivariate exponential distribution of Marshall and Olkin, Cal. Statist. Assoc. Bull. 46, [7] Balasubramnian, K. and Beg, M.I. (1997). Concomitants of order statistics in Morgenstern type bivariate exponential distribution, J. App. Statist. Sci. 54 (4), [8] Balasubramnian, K. and Beg, M.I. (1998). Concomitants of order statistics in Gumbel s bivariate exponential distribution, Sankhya B, 60, [9] Bhattacharya, P.K. (1984): Induced order statistics: Theory and Applications. In: Krishnaiah, P.R. and Sen, P.K. (Eds.), Handbook of Statistics. Elsevier Science. 4, [10] Burkschat, M., Cramer, E. and Kamps, U. (003). Dual generalized order statistics. Metron, LXI(1), [11] David, H. A. and Nagaraja H.N. (1998). Concomitants of order statistics In: N. Balakrishnan and C.R. Rao (eds), ( Handbook of Statistics.), 16,