ORDER RESTRICTED STATISTICAL INFERENCE ON LORENZ CURVES OF PARETO DISTRIBUTIONS. Myongsik Oh. 1. Introduction
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1 J. Appl. Math & Computing Vol. 13(2003), No. 1-2, pp ORDER RESTRICTED STATISTICAL INFERENCE ON LORENZ CURVES OF PARETO DISTRIBUTIONS Myongsik Oh Abstract. The comparison of two or more Lorenz curves of Pareto distributions of first kind under arbitrary order restriction is studied. The problem is turned out to be a statistical inference problem concerning scale parameters under order restriction. We assume that the location parameters of Parato distributions are completely unknown. In this paper the maximum likelihood estimation and likelihood ratio tests for and against order restriction are proposed. AMS Mathematics Subject Classification: 62H12, 62F03 Key words and Phrases : Isotonic regression, Lorenz curve, Lorenz order, maximum likelihood estimation, Pareto distribution. 1. Introduction A distribution of a positive random variable can be characterized by so-called Lorenz curve. The Lorenz curve of a distribution function of a positive random variable is defined as, for u [0, 1], u 0 L(u) = F 1 X (t)dt 1 0 F 1 X (t)dt, where F 1 X (t) = inf x{x : F X (x) t} and F X is the distribution function of random variable X. The typical Lorenz curve is concave down. The area between the line L(u) =u and the Lorenz curve may be regarded as a measure of inequality of income, or more generally, of variability in the distribution of X. This area is also called the area of concentration. The Gini s concentration index is twice this area. A concise account of properties of Lorenz curve is found in Dagum (1985). Received January 14, Revised July 14, c 2003 Korean Society for Computational & Applied Mathematics. 457
2 458 Myongsik Oh Let X and Y be positive random variables and the corresponding Lorenz curves be L X (u) and L Y (u), respectively. The Lorenz ordering is defined according to L X (u) ( )L Y (u) for all u (0, 1). If two curves cross at least one point of u, then X and Y are not comparable. The Lorenz ordering is widely used for comparing the amounts of inequality in two or more distributions. Arnold (1987) wrote an excellent research book about Lorenz curve and majorization. See also Shaked and Shanthikumar (1994) for brief review of Lorenz Order. The comparison of Lorenz curve may be quite complicated. This is true for general distributions but becomes quite simple for some well-known distributions, such as Pareto distribution of first kind. The probability density function of Pareto distribution of first kind is written as, f(x) =θδ θ x θ 1,θ>0,x>δ>0, and the Lorenz curve is L(u) =1 (1 u) (θ 1)/θ. Since the Lorenz curve is monotonic with respect to parameter θ, Lorenz ordering for Pareto distribution is equivalent to the ordering among parameters θ s. The Pareto distribution is widely used for many areas of science, such as astronomy, economics, and so on. Donnison (1990) and Donnison and Peers (1992) used Pareto distributions to model the brightness (same as the magnitude) of comets. They assumed a trend in the brightness of comets as a function of some factors such as perihelion distance or orbital period. The statistical method they employed is so-called order restricted statistical inference concerning Pareto distributions under trend, which is usually called the simple ordering. Pettit (1993) considered Bayesian inferential method for ordered parameter using the same data. They, however, did not considered the unknown nuisance parameter, which is lower bound for random variable X. In this paper we consider the statistical inference concerning scale parameters of Pareto distributions under arbitrary ordering when location parameters are assumed to be unknown. The ordering generated by Lorenz ordering becomes a partial order among k scale parameters of Pareto distributions. In Section 2, we discuss maximum likelihood (ML) estimation of Pareto parameters. We consider both common location parameter case and completely unknown case. In Section 3, likelihood ratio test statistics for and against partial order restriction are given and their null distributions are studied. Two types of approximation for deriving null distributions are discussed when underlying order is simple ordering. Critical values for both tests are given for simple order case. In Section 4, other types of testing procedures, such as ad hoc test, are briefly discussed and the further research topics are also discussed. 2. Maximum Likelihood Estimation Suppose we have k populations which follow Pareto distributions with probability distribution functions f(x; θ i,δ i )=θ i δ θi i x θi 1, θ i > 0,x > δ i > 0 for
3 Inference on Lorenz curves of Pareto distributions 459 i =1,...,k. The Lorenz curve for ith population is L i (u) =1 (1 u) (θi 1)/θi. As we mentioned in Sectio, the Lorenz ordering can be represented by a certain ordering among scale parameters θ i s. Let I = {1, 2,...,k} be an index set. We consider a partial order,, on I. For i, j I, i j whenever L i (u) L j (u) for all u (0, 1). In terms of parameters θ s, we have, for i, j I, i j whenever θ i θ j. Let C = {y = (y 1,...,y k ) R k : y i y j whenever i j} and A = {y R k : y C }.We note that C is a closed convex cone associated with partial order. Let Θ = {θ R k : θ i θ j whenever i j, θ i > 0,i =1,...,k} with θ =(θ 1,...,θ k ). The most frequently used orderings among others are simple ordering, simple tree ordering. For example, the simple order is 1 2 k and the simple tree order is 1 j for j =2,...,k. Now we discuss the ML estimation of θ i s under an arbitrary partial ordering. For each i, let X ij,j =1,...,n i be random sample of size n i from ith population. Then the likelihood function is k i=1 θ ni i δ niθi i ni j=1 xθi+1 ij. (1) It is straightforward to show that the unrestricted maximum likelihood estimates (MLEs, hereafter), ˆθ i, ˆδ i,ofθ i and δ i are n i / n i j=1 ln(x ij/ mi j ni {x ij }) and mi j ni {x ij }, for each i. Donnison(1990) and Donnison and Peers(1992) studied the ML estimation when δ i s are known with application to astronomical data. Transforming the data, which divided by δ i first and taken natural logarithm, enable us to use generalized isotonic regression for exponential family. They, however, assumed that the values of δ i. Here in this paper we do not assume that δ i s are known to us. We need to consider the two cases: (1) δ i s are the same but the common value is unknown, and (2) δ i s are completely unknown. First we consider the case that δ 1 = = δ k = δ. We assume that the common value of δ is known temporally. Focusing on the restriction, which does not relate the two types of parameters θ i s and δ s, it is not difficult to expect that the estimation for parameter θ is quite similar to those of Donnison and Peers(1992). Note that δ<x ij for j =1,...,n i, i =1,...,k. Now the log of the likelihood function is given by n i n i n i ln θ i θ i ln (x ij /δ) ln x ij. (2) i=1 j=1 i=1 j=1 The leading term is of form of exponential likelihood function. Hence it follows from Theorem of Robertson, Wright and Dykstra (1988) that the MLE of θ s under restriction can be obtained by an isotonic regression. Note that ˆθ i = n i / n i j=1 ln(x ij/δ) for i =1,...,k., i.e., ˆθ i is the reciprocal of the mean of log transformed data divided by δ. We need to be careful to use
4 460 Myongsik Oh the isotonic regression notation. We use the reciprocal of θ rather than θ. Let θ be the restricted MLE of θ. Now θ 1 = E n (ˆθ 1 A ), where n =(,...,n k ), x 1 =(x 1 1,...,x 1 k ). The projection operator E w( ) is defined as follows. For any collection of positive weights, w =(w 1,w 2,...,w k ), let < x, y > w be the inner product on R k defined by < x, y > w = k i=1 x iy i w i ; let w denote the induced norm, i.e., x 2 w = k i=1 x2 i w i; and for any subset A of R k let E w (x A) denote the projection (i.e., closest point under w )ofx onto A provided it exists and is unique. For details of projection theory, see Robertson et al. (1988). The Pool-Adjacent Violators Algorithm (PAVA) is a widely used computational method for finding isotonic regression. If violation occurs, the algorithm pools those estimators into a common estimator using appropriate weights. Assume, for example, i j but ˆθ i > ˆθ j. These two estimators do not satisfy the order restriction. The PAVA combines these two estimators into a common estimator which is n i + n j ni l=1 ln(x il/δ)+ n j l=1 ln(x jl/δ). (3) The PAVA is basically the repetition of this pooling process until all the violations are cleared up. Now by plugging θ i s into (2) we have [ ni ln θ ] n i i n i ln x ij. (4) i=1 Now let δ vary. Note that each of θ i is monotonic increasing function of δ s as we see in (3). To maximize the log-likelihood function (2) we need to choose maximum value as the MLE of δ. Hence the MLE, δ, ofδ becomes min { min {x ij}}. 1 i k 1 j n i, The equality assumption of δ i does not affect on estimation procedure. It, however, causes some difficulties in finding the null distributions of test statistics. We will discuss it later in Section 3. The above discussion is summarized in the following Theorem. i=1 j Theorem 2.1. Let X ij,j =1,...,n i,i =1,...,k be random sample from the populations with probability density functions f(x) =θ i δ θi x θi 1, θ i > 0,x > δ>0. The MLEs, θ and δ, ofθ and δ under the restriction θ Θ are given by θ = E n (ˆθ 1 A ) 1 and δ = min 1 i k { min 1 j n i {x ij }}.
5 Inference on Lorenz curves of Pareto distributions 461 Now we consider the case that δ i s are completely unknown. Note that δ i < mi j ni {x ij }. The log likelihood function is n i n i n i ln θ i θ i ln (x ij /δ i ) ln x ij. (5) i=1 j=1 i=1 j=1 Since it suffices to consider the case of k = 2, we assume k = 2. And we also assume θ 1 θ 2. Let ˆθ i ( θ i ) be the unrestricted MLEs of θ i when δ i are fixed to be ˆδ i ( δ i ). Assume δ i < ˆδ i. Suppose ˆθ 1 ˆθ 2, i.e. ˆθi do not violate the order restriction and may serve as the MLEs of θ i under the order restriction. For this case, the value of log likelihood function becomes ln n1 j=1 ln(x 1j/ˆδ 1 ) +n 2 ln n2 j=1 ln(x 2j/ˆδ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ). j=1 j=1 Suppose this is not true but θ i is the true unrestricted MLE of θ i. Then the corresponding MLEs are given by θ 1 = n1 j=1 ln(x 1j/ δ 1 ), θ2 = n2 j=1 ln(x 2j/ δ 2 ), if n1 j=1 ln(x 1j/ δ 1 ) n2 j=1 ln(x 2j/ δ 2 ), (6) θ 1 = θ + 2 = n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) if n1 j=1 ln(x 1j/ δ 1 ) > n2 j=1 ln(x 2j/ δ 2 ). (7) Then the log likelihood functions for each of (6) and (7) become ln n1 j=1 ln(x 1j/ δ 1 ) + ln n2 j=1 ln(x 2j/ δ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ), and + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ), j=1 j=1 respectively. First assume case (6). Since the log likelihood function is a monotonic increasing function with respect to each of δ i, we have ln n1 j=1 ln(x 1j/ˆδ 1 ) + ln ln n2 j=1 ln(x 2j/ˆδ 2 ) n1 j=1 ln(x 1j/ δ 1 ) + ln j=1 j=1 n2 j=1 ln(x 2j/ δ 2 ).
6 462 Myongsik Oh Hence for this case, θ i and δ i can not be a candidate for the MLE. Next assume (7) is the case. Since we assume θ 1 > θ 2, we have + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ n2 n1 j=1 ln(x 1j/ δ 1 ) = ( + )ln n1 j=1 ln(x 1j/ δ 1 ). Noting that ˆθ 1 ˆθ 2, we also have ln n1 j=1 ln(x 1j/ˆδ 1 ) + n 2 ln n2 j=1 ln(x 2j/ˆδ 2 ) (n 1 + )ln n1 j=1 ln(x 1j/ˆδ 1 ). From the above two inequalities, we have ln n1 j=1 ln(x 1j/ˆδ 1 ) + ln n2 j=1 ln(x 2j/ˆδ 2 ) + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ). This means that θ i and δ i can not be a candidate for the MLE for this case. Next suppose ˆθ 1 > ˆθ 2, i.e., ˆθ i violate the order restriction. If ˆθ i are the true unrestricted MLEs, then the value of log likelihood function is given by + ( + )ln n1 j=1 ln(x 1j/ˆδ 1 )+ j=1 ln(x 2j/ˆδ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ). j=1 j=1 Suppose this is not true but θ i are the true unrestricted MLEs of θ i. First suppose that no violations occur i.e., θ 1 < θ 2. Then we have ln n1 j=1 ln(x 1j/ δ 1 ) + ln n2 j=1 ln(x 2j/ δ 2 ) ( + )ln n2 j=1 ln(x 2j/ δ 2 ) ( + )ln n2 j=1 ln(x 2j/ˆδ 2 ) + ( + )ln n1 j=1 ln(x 1j/ˆδ 1 )+ j=1 ln(x 2j/ˆδ 2 ) The last inequality is due to ˆθ 1 > ˆθ 2. Hence θ i and δ i can not be the MLEs for this case. Next suppose that violation occurs and hence θ 1 and θ 2 need to be pooled. The common value is + ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ).
7 Inference on Lorenz curves of Pareto distributions 463 Since the likelihood function is monotonically increasing with respect to δ i s, we have + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) + ( + )ln n1 j=1 ln(x 1j/ˆδ 1 )+ j=1 ln(x 2j/ˆδ 2 ). This shows that the MLE for δ i s should be ˆδ i s. Now the following theorem summarizes the above arguments. Theorem 2.2. Let X ij,j =1,...,n i,i =1,...,k be random sample from the populations with probability density functions f(x) =θ i δ θi i x θi 1, θ i > 0,x > δ i > 0. The MLEs, θ and δ i,ofθ and δ i s under the restriction θ Θ are given by θ = E n (ˆθ 1 A ) 1 and δ i = min {x ij} for i =1,...,k. 1 j n i, Next we need to find the MLEs of θ i under the equality assumption of θ i s. We briefly state the MLEs, θ i s, when θ 1 = = θ k. Theorem 2.3. Let X ij,j =1,...,n i,i =1,...,k be random sample from the populations with probability density functions f(x) =θ i δ θi x θi 1, θ i > 0,x > δ>0. The MLEs, θ and δ,ofθ and δ s under the restriction θ 1 = = θ k are given by θ 1 = = θ k = + + n k k i=1 ni j=1 ln(x ij/δ ) and δ = ˆδ = min i i k { min 1 j n i, {x ij}}. Theorem 2.4. Let X, j =1,...,n i,i =1,...,k be random sample from the population with probability density functions f(x) =θ i δ θi i x θi 1, θ i > 0,x > δ i > 0. The MLEs, θ and δi,ofθ and δ i s under the restriction θ 1 = = θ k are given by θ1 = = θk + + n k = k ni i=1 j=1 ln(x ij/δi ) and δi = ˆδ i = min {x ij} for i =1,...,k. 1 j n i, To avoid notational confusion we use ˆδ i and ˆδ rather than δ i and δ. Finally we briefly mention the strong consistency of the estimators given above. It follows from the fact that the projection operators are continuous with
8 464 Myongsik Oh respect to weights and the arguments given polyhedral cone and the unrestricted estimators are strongly consistency. 3. Likelihood Ratio Tests Consider the following three hypotheses: H 0 : θ 1 = = θ k, H 1 : θ Θ, H 2 : No restriction on θ i s except θ i > 0. In this section we consider the likelihood ratio tests for H 0 versus H 1 H 0 and for H 1 versus H 2 H 1. As we discussed in previous section we need to consider two cases according to the assumption on δ s. First we consider the case that all the δ i s are unknown completely. For the case of equal δ i, we need some modifications to derive the null distribution of the test statistic. First consider the test for H 0 versus H 1 H 0. It is not difficult to show that the test reject H 0 for large value of T 01 = 2 ln(λ) = 2 n i ln θ i θ, (8) i=1 i where Λ is the likelihood ratio. Malik (1970) shows that mi j ni {x ij } and n i / n i j=1 ln(x ij/ mi j ni {x ij }) are jointly sufficient for (δ i,θ i ) and independent to each other, for each i = 1,...,k. It also has been shown that 2n i θ ˆθ 1 i i is distributed as chi-square with 2(n i 1) degrees of freedom. Hence n i j=1 ln(x ij/ˆδ i ) is distributed as gamma with shape parameter n i 1 and scale parameter θ 1 i. Noting that the shape parameter is known, this is just a testing problem for order restriction in the scale parameters of gamma distributions with known shape parameters. Appealing to Theorem of Robertson et al. (1988) or Robertson and Wegman (1978), we obtain the following theorem. Theorem 3.1. If H 0 is true, then, for all real t, lim P [T 01 t] = P (l, k; w, )P [χ 2 l 1 t], (9) N l=1 where N = k i=1 n i, w =(,...,n k ), and P (l, k; w, ) is the level probabilities. The level probability P (l, k; w, ) is the probability that E w (X A ) has exactly l distinct levels, where X =(X 1,...,X l ) and X 1,...,X l are independent normal variables with zero means and variances w1 1,...,w 1, respectively. k
9 Inference on Lorenz curves of Pareto distributions 465 Table 1. Critical values for T 01 and T 12 with k populations, simple order, equal sample sizes, and approximation based on Robertson and Wegman (1978) significance k test level T T T T T T Usually the exact values of level probabilities are intractable. But the level probabilities are known to be robust with respect to weights. So we may use critical values for T 01 with equal weights. Table 1 display critical values for T 01 for equal weights i.e., equal sample sizes. This approximation is generally acceptable because gamma distribution is pretty much same as normal distribution when shape parameters get large, for instance, n i > 10 for i =1,...,k. However, if is the usual simple order and the sample sizes are the same, then other type of approximation is also available. The details for this type of approximation can be found in Guffey and Wright (1986) and Robertson et al. (1988). Let τ = 1= = n k 1. Under H 0 we have P [T 01 t] P S (l, k)p [χ 2 b(τk/l,l)(l 1) c(τk/l, l)t], (10) l=2 where P S (l, k) is the level probability for simple ordering with equal weights, and c(τ,l) = lφ 1(τ) φ 1 (lτ) lφ 2 (τ) φ 2 (lτ), φ 1 (τ) φ 2 (τ) b(τ,l)= [lφ 1(τ) φ 1 (lτ)]c(τ,l), l 1. = τ + τ 6(1 + τ) 2 2τ ln(1 + 1 τ. = 1+ 2+τ 3(1 + τ) (1 + τ) 3. ), and
10 466 Myongsik Oh Table 2. Critical values for T 01 with k populations, simple order, equal sample sizes, and approximation based on Bain and Engelhardt (1975) Sample Size k Significance level = Significance level =
11 Inference on Lorenz curves of Pareto distributions 467 For the details of this approximation, see also Bain and Engelhardt(1975). For other ordering, such as simple tree or unimodal ordering, some modifications are required. The approximation for these types of ordering appears elsewhere. Table 2 shows the critical values for T 01 for k =3,...,20 and various sample sizes based on (10), i.e., approximation of Bain and Engelhardt (1975). For each k, the critical values converge to the corresponding value. The rate of convergence is rather slow. Even for moderate sample size, however, the critical value is quite close the limiting value which is given in Table 1. Therefore we recommend the reader to use Table 1 for moderate or large sample size cases and to use Table 2 for small sample sizes cases. Now consider the likelihood ratio test for H 1 against H 2 H 1. The test rejects H 1 for the large value of T 12 = 2 ln(λ) = 2 n i ln ˆθ i. (11) θ i Consider the partial order, θ induced by and θ on I which requires that, for i, j I, i θ j only when i j and θ i = θ j. Suppose i j and θ i <θ j. By the strong law of large numbers, for sufficiently large sample sizes, we have ˆθ i < ˆθ j with probability one. This means that there will be no violation and hence no amalgamation between these two θ s. Since this happens with probability one, no order needs to be defined between these two. For this reason, the partial ordering associated with the limiting distribution of T 12 depends upon and θ through θ. Now we have Theorem 3.2. If H 1 is true and θ is true value, then, for all real t, lim P [T 12 t] = P (l, k; w, N θ )P [χ 2 k l t]. (12) l=1 Moreover, lim P [T 12 t] P (l, k; w, )P [χ 2 k l t]. (13) N l=1 Since the true value of θ is unknown, (12) can not be used for finding critical value for the test. The equation (13) provides the critical value for conservative test. These critical values may be found in Table 1. The distribution associated to the right side of (13) is so called least favorable distribution which is the stochastically largest distribution with respect to parameter θ. The least favorable configuration is obtained when θ 1 = = θ k. For some values of θ, however, the test based on (13) will give substantially lower power. i=1
12 468 Myongsik Oh Table 3. Critical values for T 12 with k populations, simple order, equal sample sizes, and approximation based on Bain and Engelhardt (1975) Sample Size k Significance level = Significance level =
13 Inference on Lorenz curves of Pareto distributions 469 Oh(1994) studied the approximate test for resolving this difficulties. One might approximate θ using an estimate of θ. This will not be discussed here. Another approximation similar to (10) is also available. Under H 1 we have P [T 12 t] P S (l, k)p [χ 2 b(τ,k/l)(k l) c(τ,k/l)t]. (14) l=2 The critical values based on (14) are displayed in Table 3. As we have seen in T 01 the rate of convergence to the vales in Table 1 is rather slow. Even for moderate sample size, however, the critical value is quite close the limiting value which is given in Table 1. Therefore we also recommend the reader to use Table 1 for moderate or large sample size cases and to use Table 3 for small sample sizes cases. Next we consider the case that the location parameters are the same i.e., δ i are assumed to be the same. For this case we need a slight modification but use the same critical values as in the case of different location parameters. Let δ 1 = = δ k = δ. Note that n i ˆθ i = ni j=1 ln(x ij/ mi i k {mi j ni {x ij }}). That means 2n i θ ˆθ 1 i i is no longer distributed as chi-square with 2(n i 1) degrees of freedom. We also note that θ i s are not independent to each other. This fact makes us unable to find the null distributions of test statistics. However, we can find null distribution for the case of sufficiently large sample size. It is straightforward to show that mi i k {mi j ni {x ij }} converges to δ with probability one. And, for each i =1,...,k, mi j ni {x ij } also converges to δ with probability one. Then we have, for sufficiently large sample sizes, n i ˆθ i ni j=1 ln(x ij/ mi j ni {x ij }). Hence Theorem 3.1 and 3.2 are still valid for the case that equal value of δ. We, however, are not aware of whether (10) and (14) are still valid. We conjecture that those approximations are valid since the derivation of the fact does not depend heavily upon independence assumption. 4. Concluding Remarks It is generally known that a likelihood ratio test is a powerful test but under order restriction it has major drawback. That is, it is not easy to find level probabilities even for moderate value of k. Several approximation methods are suggested but it is still unsatisfactory. So some considered ad hoc tests and other type of tests. We will not pursue these tests in this paper. Interested reader refers to Robertson et al. (1988). Pettit (1993) studied Bayesian test for ordered
14 470 Myongsik Oh scale parameter of Pareto distributions when shape parameters are completely known. As we have seen in Donnison (1990) and Donnison and Peers (1992) it is not easy to expect that we have sufficiently large sample sizes. We need to find near-exact null distribution or some other approximation method for finding critical value. Bootstrapping method is likely to be a candidate for resolving this problem. References 1. Arnold. B. C. (1987). Majorization and Lorenz Order, Lecture Notes in Statistics No. 43, Springer-Verlag., New York and Berlin. 2. Bain, L. J., and Engelhardt, M. (1975). A Two-Moment Chi-Square Approximation for the Statistic log( x/ x), Journal of the American Statistical Association, 70, Dagum, C. (1985). Lorenz Curve, Encyclopedia of Statistical Sciences, 5, S. Kotz, N. L. Johnson, and C. B. Read (editors), , Wiley, New York. 4. Donnison, J. R. (1990). The Distribution of Cometary Magnitudes, Monthly Notice of Royal Astronomical Society, 245, Donnison, J. R., and Peers, H. W. (1992). An Approach to trend Analysis in Data with Special Reference to Cometary Magnitudes, Monthly Notice of Royal Astronomical Society, 256, Guffey, J. M., and Wright, F. T. (1986). Testig for trends in a nonhomogeneous Poisson process Technical report, Department of Mathematics and Statistics, University of Missori- Rolla. 7. Malik, H. J. (1970). Estimation of the Parameter of the Pareto Distribution, Metrika, 15, Oh, M. (1994). Statistical Tests Concerning a Set of Multinomial Parameters under Order Restrictions: Approximations to Null Hypotheses Distributions, Ph. D. Thesis, The University of Iowa. 9. Pettit, L. I. (1993). Inferences about Ordered Parameters - An Astronomical Problem. Journal of the Royal Statistical Society, Series D, The Statistician, Vol. 42, Robertson, T., and Wegman, E. J. (1978). Likelihood ratio tests for order restrictions in exponential families, The Annals of Statistics, 6, Robertson, T., F. T. Wright, and R. L. Dykstra (1988). Order Restricted Statistical Inference, Wiley, Chichester. 12. Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications, Academic Press, New York. Myongsik Oh received his Ph.D at the University of Iowa. He is an associate professor in Department of statistics, Pusan University of Foreign Studies. His research interest focuses on order restricted statistical inference and mathematical statistics. Department of Statistics, Pusan University of Foreign Studies, Pusan , Korea. moh@stat.pufs.ac.kr
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