Computing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions

Transcription

1 Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin

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3 Working Paper 2013:5 May 2013 Department of Statistics Uppsala University Box 513 SE UPPSALA SWEDEN Working papers can be ownloae from Title: Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Author: Shaobo Jin

4 Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an Their Functions Shaobo Jin May 28, 2013 Abstract There are many methos to calculate simultaneous confience intervals for multinomial proportions with categories. They are asymptotic 1 α simultaneous confience intervals. Wang (J. Multivar. Anal. (2008) ) propose a metho to fin the exact confience coefficient for a fixe sample size n instea of using simulation results. However, her proceure is only applicable to 1 intervals instea of intervals. Most of the methos for such intervals focus on intervals simultaneously. In this paper, we generalize her iea to fin the exact confience coefficient for simultaneous confience intervals for multinomial proportions. Our metho has to consier lots of new points in orer to correctly ientify the point which attains the confience coefficient. Generalizations to the SCIs for the contrasts an ratios are briefly iscusse. Keywors: Simultaneous confience intervals, Multinomial proportions, Confience coefficient Department of Statistics, Uppsala University, Box 513, SE-75120, Sween shaobo.jin@statistik.uu.se 1

5 1 Introuction The multinomial istribution plays an important role in statistical practice. Contingency tables in the survey analysis or experimental esigns can be regare as multinomial istributions. In an election with more than two caniates, the counts of votes also construct a multinomial istribution. The problem of estimating the cell probabilities arises, for example, when the meia wants to preict the result of an election. Two generic methos are the point estimation an the interval estimation. For a multinomial istribution, X = (X 1, X 2,, X ), the probability mass function is P (X 1 = x 1, X 2 = x 2,, X = x ) = with n = i=1 x i. The parameter space is efine as Θ = { p = (p 1,, p ); x 1!x 2! x! px 1 1 p x 2 2 p x, (1) } p i = 1, p i > 0, i = 1,,. (2) i=1 For some α (0, 1), we want to construct simultaneous confience intervals (SCIs) for p 1, p 2,, p, enote as I 1, I 2,, I, such that the joint coverage probability is larger than or equal to 1 α. Such methos are fruitful in the literature. To our best knowlege, Quesenberry an Hurst (1964) propose the first SCIs, where a = χ 2 1 α( 1). p i = a + 2X i ± a[a + 4X i (n X i )/n], i = 1,,, (3) 2(n + a) Gooman (1965) suggeste that this estimation can be improve by replacing a = χ 2 1 α( 1) by b = χ 2 1 α/ (1) using an argument base on the Bonferroni inequality. This technique is suppose to obtain shorter interval estimations. Hou et al. (2003) stuie a family of simultaneous confience intervals base on the power-ivergence type statistics propose by Cressie an Rea (1984). 2

6 For each proportion p i, the lower boun an the upper boun are the minimum an maximum solutions of the function (1 ˆP i ) λ+1 P λ i + λ+1 ˆP i (1 P i ) λ = A λ Pi λ (1 P i ) λ ; i = 1, 2,,, (4) where A λ = λ(λ + 1)χ 2 1 α( 1)/(2n) + 1 an ˆP i = X i /n. However, there are usually no explicit forms of the lower an upper limits. The SCIs mentione above are all obtaine from the χ 2 approximation. There are also some SCIs which are not base on the χ 2 approximations. Gooman (1965) also introuce simpler simultaneous confience intervals given by X i n a X i n (1 X i n )/n < p i < X i n + a X i n (1 X i )/n, i = 1, 2,,. (5) n Sison an Glaz (1995) propose two intervals. One was base on the approximation for multinomial proportions introuce by Levin (1981). The other metho was base on an inequality for a multinomial istribution. The propose simultaneous confience intervals can be formulate as ˆP i c n < p i < ˆP i + c + 2γ, i = 1,,, (6) n for some c, r an ˆP i = X i /n, i = 1, 2,,. Fitzpatrick an Scott (1987) stuie the conventional simultaneous confience interval ˆP i p i < 1/ n. By aiing of the Bonferroni s inequality, they propose quick simultaneous intervals, given by ˆP i Φ 1 (1 α 2 ) 2 n < p i < ˆP i + Φ 1 (1 α) 2 2, (7) n where Φ 1 was the inverse of the istribution function of a stanar normal istribution. One common measure to assess the properties of SCIs is the confience coefficient 3

7 which is efine as the infimum of the coverage probability over the parameter space Θ. This represents the most angerous case which of course we woul like to control. Fitzpatrick an Scott (1987) gave a lower boun of the confience coefficient of their quick SCIs. Wang (2008) was the first one who consiere the metho to calculate the exact confience coefficients of SCIs for multinomial proportions instea of using simulations. The parameter space was efine as Ω = { (p 1,, p 1 ); } 1 p i 1, 0 p i 1, i = 1,, 1, (8) i=1 in Wang (2008), which was slightly ifferent from Θ by allowing the appearance of parameters with value 0. An only the coverage probability of SCIs for p 1, p 2,, p 1 were use (the form of coverage probability will be given in the next section). However, all the SCIs mentione above are constructe for parameters simultaneously. So the metho evelope for the 1 SCIs will only give a upper boun to the coverage coefficients sometimes if it is use to assess the methos in the literature. In Wang (2008), she argue that in multinomial istribution there were only 1 parameters, so it was natural to consier the 1 SCIs. She use the binomial istribution as an example. But the reason to consier SCIs instea of 1 SCIs is also obvious. In binomial case, we have two parameters an one restriction about the parameters. If we construct a confience interval for one parameter p, say (L, U), we can easily obtain the confience interval for another parameter q = 1 p as (1 U, 1 L) which is equivariant in the sense of Blyth an Still (1983). However this is not the case in multinomial cases. In the multinomial problems, if we have the SCIs for p 1,, p 1, there is no easy way to obtain the confience interval for the last parameter p = 1 1 i=1 p i. We can not use one minus the sum of the upper (lower) boun as the lower (upper) boun for p because most likely both of them will be negative. So it is necessary to evelop a metho to fin the exact confience coefficients for the SCIs. 4

8 In this paper, we exten the work of Wang (2008) to obtain the exact confience coefficients of SCIs which is mostly focuse on in the literature. The paper is organize as follows. In Section 2, we introuce some main results from Wang (2008) an present some results uner our setting. The iea is generalize to other types of SCIs in Section 3. In Section 4, some numerical results are illustrate. A iscussion ens the paper. 2 General Methos For the i-th cell probability p i, the lower boun an upper boun are enote as L i (X) an U i (X). They can be reformulate as functions of ˆPi, for i = 1,, which are the MLE of the population proportions. Assumption 1 contains some keys assumptions in the paper. Assumption L i (X) have the same form for all i an U i (X) have the same form for all i as well. 2. For any fixe n, L i (X) an U i (X) only epen on x i, not x j where j i. Hence we can write L i (X) = L(X i ) an U i (X) = U(X i ). 3. L(X i ) an U(X i ) are increasing functions of ˆP i. 4. When X i = 0, then L(X i ) 0 U(X i ). 5. For any fixe p in the parameter space, there exists an x 0 such that p I(x 0 ) an P p (X = x 0 ) > 0. The Assumption 1(2)-(5) are just the Assumption 1 in Wang (2008). 5

9 2.1-1 SCIs For a fixe p in the parameter space, the coverage probability for p 1, p 2,, p 1 is efine as CP 1 (p 1,, p 1 ) = n x 1 =0 n x 1 =0 [ 1 x 1!x 2! x! px 1 1 p x 2 2 p x i=1 I {pi I i } ]. (9) Collect all the values of L(X) an U(X) by increasing orer into a set v = {v i } which has at most 2n + 2 elements. Define the set v = {v i / {0, 1}}, hence v v = {v i }. Let g = #v an g = #v. The parameter space can be ivie into several subsets following Wang (2008) s metho. For each p i, i = 1,,, there exists v j an v j+1 such that v j p i v j+1, so each subset can be formulate as ((v i1, v i 1 ) (v i 1, v i 1 )) Ω, (10) where v ij v an i j {i j, i j +1}. If i j = i j, then the corresponing interval (v ij, v i j ) is efine as the hyperplane p j = v ij. Figure 1(a) gives an example of the subsets when n = 5 an = 3. In fact, the parameter space is separate by the hyperplanes p i = v j, i = 1,, 1, j = 1,, g. Let s be the number of total subsets an ω t collects all (x 1,, x ) such that for all (p 1,, p ) in the subset τ t with the form (10), we have p i (L(x i ), U(x i )) for all i = 1,, 1. Lemma 1. If the SCIs satisfy the Assumption 1 (1)-(3), within each subset (10) without any other elements of v = {v i } in the interior of the subset, the coverage probability (9) can be expresse as CP 1 (p 1,, p ) = b 1 x 1 =a 1 b 2 (x 1 ) x 2 =a 2 b 1 (x 1,,x 2 ) x x 1 =a 1!x 2! x! px 1 1 p x 2 2 p x, (11) 1 for some constants a 1,, a 1, b 1 an some functions b 2 (x 1 ),, b 1 (x 1,, x 2 ). (Lemma 3, Wang (2008)). 6

10 p lower limit upper limit p lower limit upper limit p1 p1 (a) Wang (2008) s metho (b) Our metho Figure 1: An example of partioning the parameter space using the lower limits an upper limits of the SCIs when n = 5 an = 3. Rewrite the formula (11) as CP 1 (p 1 p 1,, p 2 ) representing the conitional function given p 1,, p 2. Then we have the following lemma. Lemma 2. If b i (x 1,, x i 1 ) > a i, for some i = 1,, 2, then 1. CP 1 (p 1 p 1,, p 2 ) is a unimoal function or an increasing function of p 1 if a 1 > 0 an x b 1 (x 1,, x 2 ) < n. 2. CP 1 (p 1 p 1,, p 2 ) is a ecreasing function of p 1 if a 1 = CP 1 (p 1 p 1,, p 2 ) is an increasing function of p 1 if x b 1 (x 1,, x 2 ) = n. (Lemma 2, Wang (2008)). 7

11 2.2 SCIs The coverage probability of SCIs is CP (p 1,, p ) = n x 1 =0 [ n x 1!x 2! x! px 1 1 p x 2 2 p x x =0 i=1 I {pi I i } ]. (12) Here we nee a new metho to separate the parameter space. Consier the following subset ((v i1, v i 1 ) (v i, v i )) Θ, (13) where v ij v an i j {i j, i j + 1}. If i j = i j, then the corresponing open interval (v ij, v i j ) is efine as the hyperplane p j = v ij. Figure 1(b) gives an example of the subsets when n = 5 an = 3. In fact, the parameter space is separate by the hyperplanes p i = v j, i = 1,,, j = 1,, g. So each subset (10) efine before is partitione into several smaller sets. Let s be the total number of subsets an ω t collects all (x 1,, x ) such that for all (p 1,, p ) in the subset τ t with the form of (13), we have p i (L(x i ), U(x i )) for all i = 1,,. If we consier the coverage probability (12) an the subset (13), we o not have a nice result as equation (11). The best we can o is to eal with CP (p 1,, p ) = x ω t = b 1 Assume we have a subset [ ] x 1!x 2! x! px 1 1 p x 2 2 p x b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2 p x. (14) τ t =((v i 1, v i 1 +1) (v i, v i +1)) Θ, 8

12 with no hyperplane involve, the corresponing set for X is ω t. First, consier the case that we ecrease p 1 from p 1 = v i 1 +ε to p 1 = v i 1 while p j, j = 2,,, are still in τ t. When v i 1 is L(x) for some x, v i 1 +ε (L(x), U(x)) but v i 1 the observation X 1 = x from ω t. When v i 1 / (L(x), U(x)). We lose is U(x) for some x, v i 1 + ε / (L(x), U(x)) an v i 1 / (L(x), U(x)) still cannot cover p 1 = v i 1. ω t stays the same. Secon, consier the case that we increase p 1 from p 1 = v i 1 +1 ε to p 1 = v i 1 +1 while p j, j = 2,,, still in τ t. When v i 1 +1 is L(x) for some x, (L(x), U(x)) still cannot cover p 1 = v i 1. ω t stays the same. When v i 1 +1 is U(x) for some x, (L(x), U(x)) cannot cover p 1 = v i 1. We lose the observation X 1 = x from ω t. Thus within each subset, the values that X can take are fixe but it may or may not vary from one subset to the other. Since Equation (14) has 1 free arguments, we can stuy CP (p 1,, p ) conitioning on 2 free arguments. Define two functions first B t (p i ) = C t (p i ) = b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) x 1!x 2! x! px 1 1 p x 1 1 x ip x 1 x 1, x 1!x 2! x! px 1 1 p x 1 1 x p i p x 1 x 1 1, where the subscript t means x ω t an the argument p i means that p k, k i an are fixe. First we stuy CP (p 1,, p ) conition on p 1,, p 2, enote by CP (p 1 p 1,, p 2 ). Equation (14), B t (p 1 ) an C t (p 1 ) are continuous an ifferentiable functions in p 1. Although they are restricte in one or some specific ( subsets τ t, we temporarily exten the omains to 0, 1 ) 2 j=1 p j. Lemma 3. If the SCIs satisfying the Assumption 1 (1)-(3), within each subset (13) without any other elements of v = {v i} in the interior of the subset, then 1. If a 1 (x 1,, x 2 ) = b 1 (x 1,, x 2 ) = 0 for all x 1,, x 2 an x 1 + 9

13 +b 1 (x 1,, x 2 ) < n for some x 1,, x 2, then CP (p 1 p 1,, p 2 ) is a ecreasing function in p If a 1 (x 1,, x 2 ) = b 1 (x 1,, x 2 ) 0 an, for all x 1,, x 2, x b 1 (x 1,, x 2 ) = n, CP (p 1 p 1,, p 2 ) is an increasing function in p If a 1 (x 1,, x 2 ) = b 1 (x 1,, x 2 ) = 0 an, for all x 1,, x 2, x b 1 (x 1,, x 2 ) = n, CP (p 1 p 1,, p 2 ) is a constant function in p 1. Further if B t (p 1 ) C t (p 1 ) = 0 has no more than two solutions, then 4. CP (p 1 p 1,, p 2 ) is an unimoal function for other values of a 1 an b 1. This lemma is the extension of Lemma 2 from -1 SCIs to SCIs. We impose one assumption about B t (p 1 ) an C t (p 1 ) to have unimoality. Such an assumption oes not appear in Wang (2008). When she showe the unimoality property, the argument that two convex functions have at most two intersections are use. that is the case, we o not nee the assumption regaring B t (p 1 ) an C t (p 1 ) anymore. However, in general this argument is not true an we have to check this assumption. It can be checke numerically. If B t (p 1 ) C t (p 1 ) = 0 has more than two solutions, CP (p 1 p 1,, p 2 ) may have several local minima. We nee to pay more attention to these local minima. However, CP (p 1 p 1,, p 2 ) is ( actually efine on some subset ω t instea of 0, 1 ) 2 j=1 p j. If in the subset ω t, CP (p 1 p 1,, p 2 ) is unimoal or monotone, we on t nee to worry about these local minima. Another remark regaring Lemma 3 is that we can switch the fixe cell probabilities. For example, if we consier CP (p 1 p 2,, p 1 ) where p 1 is the free parameter an p 2,, p 1 are fixe, similar results as Lemma 3 can be obtaine easily. If 10

14 The main iea is that if we fixe 2 cell probabilities, the (conitional) coverage probability is a ecreasing/increasing/constant/unimoal function uner ifferent conitions. Using Lemma 3, we can stuy the coverage probability within each subset (13). If the coverage probability is a ecreasing/increasing/constant/unimoal function within a subset, the minimum is taken on the bounaries. This establishes Theorem 1 following. Theorem 1. Assume Assumption 1 is fulfille, an let E 0 = {(π 1, π 2,, π ); π i = 0 for some i = 1,,, an π j v for j i} E 1 = {(π 1, π 2,, π ); π i = v j, i = 2,,, j = 1,, g, π m < 1} Θ, m=2 E k = {(π 1, π 2,, π ); π i = v j, i k, j = 1,, g, m k π m < 1} Θ, k = 2,, 1, E = {(π 1, π 2,, π ); π i = v j, i = 1,, 1, j = 1,, g, M 1,t = {(π 1, π 2,, π ); B t (π i ) = C t (π i ), i = 2,,, 1 m=1 π m < 1} τ t, m=2 π m < 1} Θ, M k,t = {(π 1, π 2,, π ); B t (π i ) = C t (π i ), i k, m k π m < 1} τ t, k = 2,, 1, M,t = {(π 1, π 2,, π ); B t (π i ) = C t (π i ), i = 1,, 1, t = 1,, s. 1 m=1 π m < 1} τ t, Then 11

15 1. If L(x) = 0 if an only if x = 0, the confience coefficient is obtaine at some of the points in E 1 E ( s ) (M 1,t M,t ). t=1 2. If L(x) 0 for some nonzero x, the confience coefficient is obtaine at some of the points in E 0 E 1 E ( s ) (M 1,t M,t ). t=1 In Theorem 1, E 0 is the collection of bounary points, E 1 E is the collection of the cross points an s t=1 (M 1,t M,t ) is the collection of local minima. If L(0) < 0 an L(x) = 0 for a nonzero x, CP (p 1 p 1,, p 2 ) > CP (0 p 1,, p 2 ), if p 1 is sufficiently close to 0. The value itself is smaller than the right limit. Thus CP 0 (p 1 p 1,, p 2 ) is a minimum value locally. We fail to prove the nonexistence of the local minima which makes the theorem no so easy to use. The coverage probability (14) is a sum of unimoal (or monotone) functions. But the sum of unimoal functions are not guarantee to be unimoal, so we have to consier the possibility of local minima as well. Lemma 3 says that when B t (p 1 ) C t (p 1 ) = 0 has no more than two solutions, CP (p 1 p 1,, p 2 ) is a unimoal function uner some conitions. In some subsets, B t (p 1 ) C t (p 1 ) = 0 has more than two solutions. This happens when x i an x j are exchangeable in the sense that if (x 1,, x i,, x j,, x ) ω t 12

16 in equation (14) then (x 1,, x j,, x i,, x ) ω t as well. For instance, if n = 5, = 3 an ω t = {(4, 0, 1), (4, 1, 0), (5, 0, 0)}, then B = C for all fixe p 1 (U(3), U(4)) an p 2 (L(1), 1 2L(1)). An within this subset CP 3 (p 1, p 2, p 3 ) = x ω t [ ] x 1!x 2!x 3! px 1 1 p x 2 2 p x 3 3 = 5p 4 1 4p 5 1, which is a constant when p 1 is fixe. Actually, the constant coverage can happen in many subsets. However, if we exclue such constant coverage case, we argue that it is unlikely that we face the problem of local minima. We are ealing with the sum of x 1!x 2! x! px 1 1 p x 2 2 p x over some ω t, the possible values of x 1 in ω t are continous in the sense that all the values between a 1 (x 1,, x 2 ) an b 1 (x 1,, x 2 ) belong to ω t if x 1,, x 2 are fixe. Thus b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2 p x is unimoal if a 1 (x 1,, x 2 ) > 0 an x b 1 (x 1,, x 2 ) < n when we fix p 1,, p 2. The moe takes place at p 1 = ( ) 1 b 1!(n x 1 x 2 b 1 1)! b 1 a 1 +1 a 1!(n x 1 x 2 a 1 1)! ( b 1!(n x 1 x 2 b 1 1)! a 1!(n x 1 x 2 a 1 1)! ) 1 b 1 a (1 p 1 p 2 ). 13

17 Hence unless we have iscrete x i in ω t, CP (p 1 p 1,, p 2 ) is unimoal or monotone or a constant over ω t. 3 Simultaneous Confience Intervals for Functions of Proportions Theorem 1 is only vali to ientify the exact confience coefficient of SCIs for multinomial proportions. It cannot be irectly applie to other types of SCIs. In practice, we are also intereste in the SCIs for the contrasts an the ratios. From the proof of Theorem 1, we can see that the theorem can actually be moifie to some other cases as well. As long as the coverage probability can be expresse as (14), the iea of Theorem 1 can ientify the point attaining the coverage coefficient. We only nee to examine the intersections, local minima an the bounary points. The only ifference is the partition of the parameter space. 3.1 Contrast Consier the SCIs for the ( 1)/2 ifferences p i p j with i < j. The confience interval for p j p i can be obtaine by inverting the one for p i p j. Gol (1963) propose the SCIs of the form p i p j = X i X j n ± a n ( X i + X j n ( ) ) 2 Xi X j, (15) n where a was efine in the introuction. Gooman (1965) generalize the iea to the SCIs for the all possible linear combinations of p i. If we are intereste in the 14

18 SCIs for δ i p i for some constants δ 1,, δ, then the SCIs are δi p i = δ i X i n ± ( a δ 2 i n X i n ( ) ) 2 X i δi. (16) n We can also erive the SCIs from the confience regions which are calle Pearson SCIs in this paper. For example if we have a confience region CR α, an we want a confience interval for p i p j, we can consier the line p i p j = z. The lower boun an upper boun are the values of z such that p i p j = z an CR α have only one intersection respectively. Figure 2(a) shows the partition of the parameter space base on the SCIs for the contrasts from the Pearson χ 2 confience regions. p lower limit upper limit p lower limit upper limit p1 p1 (a) Contrasts p i p j (b) Ratios, p i /p j Figure 2: An example of the partition of the parameter space base on the SCIs erive from the Pearson χ 2 confience regions. n = 5, = 3 an α =

19 3.2 Ratio We are intereste in the SCIs for the ratio p i /p j or equivalently the natural logarithm of p i /p j. Gooman (1965) provie a metho to construct SCIs for p i /p j as { ( X i 1 /exp a + 1 X j X i X j ) } p i p j X i X j exp { ( 1 a + 1 ) }. (17) X i X j However, these SCIs face the problem of 0 observations. SCIs for the ratio can also be erive from confience regions. Assume we have a confience region CR α, an we want simultaneous confience intervals for p i /p j. We can consier the line p i /p j = z, the lower boun an upper boun are the values of z such that p i /p j = z an CR α have only one intersection respectively or when z = 0 or. Especially when z = 0 or, the line p i /p j = z coincies with the coorinate axis. Figure 2(b) shows the partition of the parameter space base on the SCIs for the ratios from the Pearson χ 2 confience regions. 4 Numerical Examples In this part, we first stuy the confience coefficient of the Q-H SCIs (3) assuming n = 5 an = 3. We set α = 0.05 throughout this section. It is easy to verify that the Q-H SCIs (3) satisfies Assumption 1 an the conition in Theorem 1(1). Accoring to Theorem 1(1), the confience coefficient will be taken at the ot points shown in Figure 3 an all possible local minima within each subset. But no local minimum is foun. The confience coefficient is , which is attaine at (p 1, p 2, p 3 ) = (L(1), L(1), 1 L(1) L(1)) = (0.0260, , ) (p 1, p 2, p 3 ) = (L(1), 1 L(1) L(1), L(1)) = (0.0260, , ) (p 1, p 2, p 3 ) = (1 L(1) L(1), L(1), L(1)) = (0.9480, , ). 16

20 The confience coefficient is invariant uner permutations of p 1, p 2 an p 3 if we consier SCIs. Thus from now on, we only present one element within each invariant class. p lower limit upper limit p1 Figure 3: Fining exact confience coefficient of Q-H SCIs when n=5 an =3. We tabulate the exact confience coefficients of Q-H SCIs (3) assuming =3 for ifferent sample sizes n in Table 1. Only limite sample sizes are use here just for illustration purposes. We fin that for Q-H SCIs (3), the exact confience coefficient consiering -1 SCIs an SCIs are exactly the same. The points that we fin for 1-SCIs case are also inclue in -SCIs case. All the confience coefficients are achieve at the points when two proportions equal L(1). No local minimum is foun. The Q-H SCIs are special cases of the power-ivergence SCIs (4). Such SCIs are in fact functions of λ. Hou et al. (2003) propose that we shoul choose the λ which minimize the volume of the SCIs while the coverage probability was at least 1 α. 17

21 Table 1: The exact confience coefficients for Q-H SCIs (3) when =3 an ifferent n. α = For both -1 SCIs an SCIs, there are several points achieving the confience coefficient as coverage probabilities. We only list one example in the table. The other points can be obtaine by permutating the coorinates. n Confience coefficient (p 1, p 2, p 3 ) 1 SCIs SCIs (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0260,0.0260,0.9480) (0.0260,0.0260,0.9480) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0128,0.0128,0.9744) (0.0128,0.0128,0.9744) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0085,0.0085,0.9830) (0.0085,0.0085,0.9830) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0064,0.0064, ) (0.0064,0.0064, ) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0051,0.0051,0.9898) (0.0051,0.0051,0.9898) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0043,0.0043,0.9914) (0.0043,0.0043,0.9914) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0025,0.0025,0.9950) (0.0025,0.0025,0.9950) But no systematic stuy has been one in the literature about the choice of λ in the case of confience interval estimation. Rea an Cressie (1988) recommene the use of λ = 2 3 as a compromise of the Pearson χ2 an the likelihoo ratio if we use the power-ivergence statistic as a test statistic. Meak an Cressie (1991) stuie the confience region base on the power-ivergence statistic, they conclue that λ = 2 3 gave an improvement in small samples. So we stuie the SCIs by fixing λ = 2. The requirements for Theorem 1(1) are fulfille. From Table 2 we can see 3 that all the confience coefficients are larger than those in Table 1. All the points are still the cross points. Another example is the F-S SCIs (7). It is an example where L(x) < 0 for some x. Theorem 1(2) is applie. From Table 3 we can see that the confience coefficients of the SCIs are all smaller than those of the -1 SCIs when =3. As the sample size increases, the points which achieve the confience coefficients as coverage probabilities become ifferent. An for the -1 SCIs, uner some sample sizes, the 18

22 Table 2: The exact confience coefficients for power-ivergence SCIs (4) with λ = 2/3 when =3 an ifferent n. α = For both -1 SCIs an SCIs, there are several points achieving the confience coefficient as coverage probabilities. We only list one example in the table. The other points can be obtaine by permutating the coorinates. n Confience coefficient (p 1, p 2, p 3 ) 1 SCIs SCIs (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0188,0.0188,0.9624) (0.0188,0.0188,0.9624) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0093,0.0093,0.9814) (0.0093,0.0093,0.9814) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0061,0.0061,0.9878) (0.0061,0.0061,0.9878) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0046,0.0046, ) (0.0046,0.0046, ) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0037,0.0037,0.9926) (0.0037,0.0037,0.9926) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0031,0.0031,0.9938) (0.0031,0.0031,0.9938) (L(1),L(1),1-L(1)-L(1))= (L(1),L(1),1-L(1)-L(1))= (0.0018,0.0018,0.9964) (0.0018,0.0018,0.9964) 19

23 points achieving the confience coefficient are not necessary to have the equality p 1 = p 2 which is always the case in Table 1 an Table 2. Again, all the confience coefficients are attaine at cross points an no local minimum exists. Further, the non-consistency of the confience coefficient can be observe here. Even we increase the sample size, the confience coefficient ecreases sometimes. Table 3: The exact confience coefficients for F-S SCIs (7) when =3 an ifferent n. α = For both -1 SCIs an SCIs, there are several points achieving the confience coefficient as coverage probabilities. We only list one example in the table. The other points can be obtaine by permutating the coorinates. n -1 SCIs SCIs Confience coefficient (p 1, p 2, p 3 ) Confience coefficient (p 1, p 2, p 3 ) (L(4),L(4),1-L(4)-L(4))= (L(4),L(4),1-L(4)-L(4))= (0.3617, , ) (0.3617, , ) (L(7),L(7),1-L(7)-L(7))= (L(6),L(7),1-L(6)-L(7))= (0.3901, , ) (0.2901,0.3901,0.3198) (L(10),L(10),1-L(10)-L(10))= (L(9),U(1),1-L(9)-U(1))= (0.4136, , ) (0.3470, ,0.3333) (L(12),L(13),1-L(12)-L(13))= (U(3),U(3),1-U(3)-U(3))= (0.3809,0.4309,0.1883) (0.3691,0.3691,0.2617) (L(15),L(15),1-L(15)-L(15))= (U(4),L(13),1-L(13)-U(4))= (0.4040,0.4040,0.1920) (0.3560,0.3240,0.3200) (L(17),L(18),1-L(17)-L(18))= (L(16),L(16),1-L(16)-L(16))= (0.3877,0.4211,0.1912) (0.3544,0.3544,0.2912) (L(27),L(28),1-L(27)-L(28))= (L(23),U(10),1-L(23)-U(10))= (0.4014,0.4214,0.1772) (0.3214,0.3386,0.3400) At last we stuy the SCIs for the contrasts. The Gol SCIs (15) have problems when we observe x i = x j = 0 or x i = n an x j = 0. In the former case, both lower boun an upper boun is 0. In the latter case, both the lower boun an upper boun is 1. Thus it is natural to expect a low confience coefficient. From Table 4 we see that the confience coefficients are zero even though asymptotic 95% SCIs are esire. As a comparison, we stuie the confience coefficients of the Pearson SCIs. They are superior over the Gol SCIs (15) an therefore are preferre to the Gol SCIs (15). 20

24 Table 4: The exact confience coefficients for Gol SCIs (15) for contrasts an SCIs for contrasts erive from Pearson χ 2 confience regions when =3 an ifferent n. α = n Gol SCIs (15) Pearson SCIs Confience coefficient Confience coefficient (p 1, p 2, p 3 ) (0.0173,0.0173,0.9654) (0.0086,0.0086,0.9828) (0.0057,0.0057,0.9886) (0.0043,0.0043,0.9914) (0.0034,0.0034,0.9932) (0.0028,0.0028,0.9944) 5 Conclusions In this paper, we exten the work by Wang (2008) by consiering the confience coefficient of SCIs instea of -1 SCIs. The confience coefficients obtaine using our metho are still true values, not estimates. If the metho in Wang (2008) is use to fin the confience coefficient, the result is only an upper boun. orer to ientify the correct confience coefficient, extra points shoul be taken into consieration. However, for certain power-ivergence SCIs both methos give the same results. The examples within the power-ivergence family show that the confience coefficients are often attaine at p i = L(1), i = 1,, 1. This is not always the case, see Jin (2013) for the binomial case. The metho can be generalize to SCIs for ifference in the proportions p i p j or ratios p i /p j. We briefly iscusse these in this paper. Simulation stuies suggest that the confience coefficients are always achieve at the cross points even though we have to eal with the existence of possible local minima within some subsets. Within some subsets, the coverage probability is constant. If we exclue this case from the local minima problem, it is still an open question to prove the non-existence of local minima. If we only look at the confience coefficient, the F-S SCIs (7) for the multinomial proportions are preferre to the power-ivergence SCIs with λ = 2/3, 1 since they have higher confience In 21

25 coefficients. References Blyth, C. R. an Still, H. A. (1983). Binomial confience intervals. Journal of the American Statistical Association, 78: Cressie, N. an Rea, T. (1984). Multinomial gooness-of-fit tests. Journal of the Royal Statistical Society. Series B (Methoological), 46: Fitzpatrick, S. an Scott, A. (1987). Quick simultaneous confience intervals for multinomial proportions. Journal of the American Statistical Association, 82: , 1 Gol, R. Z. (1963). Tests auxiliary to χ 2 tests in a markov chain. The Annals of Mathematical Statistics, 34: Gooman, L. A. (1965). On simultaneous confience intervals for multinomial proportions. Technometrics, 7: , 1, 3.1, 3.2 Hou, C., Chiang, J., an Tai, J. J. (2003). A family of simultaneous confience intervals for multinomial proportions. Computational Statistics & Data Analysis, 43: , 4 Jin, S. (2013). Best power-ivergence confience interval for a binomial proportion. Unpublishe manuscript. 5 Levin, B. (1981). A representation for multinomial cumulative istribution functions. The Annals of Statistics, 9: Meak, F. an Cressie, N. (1991). Confience regions in ternary iagrams base on the power-ivergence statistics. Mathematical Geology, 23:

26 Quesenberry, C. P. an Hurst, D. C. (1964). Large sample simultaneous confience intervals for multinomial proportions. Technometrics, 6: Rea, T. an Cressie, N. (1988). Gooness-of-fit statistics for iscrete multivariate ata. Springer-Verlag New York. 4 Sison, C. P. an Glaz, J. (1995). Simultaneous confience intervals an sample size etermination for multinomial proportions. Journal of the American Statistical Association, 90: Wang, H. (2008). Exact confience coefficients of simultaneous confience intervals for multinomial proportions. Journal of Multivariate Analysis, 99: , 1, 2, 2.1, 1, 2, 2.2, 5 A Mathematical Appenix Proof of Lemma If a 1 (x 1,, x 2 ) = 0 an b 1 (x 1,, x 2 ) = 0 for all x 1,, x 2 = CP (p 1 p 1,, p 2 ) b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 2 (x 1,,x 2 ) x 2 =a 2 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2 px. Consier the first erivative with respect to p 1 = p 1 CP (p 1 p 1,, p 2 ) b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 2 (x 1,,x 2 ) x 2 =a 2 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2 ( x )p x 1, 23

27 since x b 1 (x 1,, x 2 ) < n for some x 1,, x 2. Thus p 1 CP (p 1 p 1,, p 2 ) < 0, an subsequently CP (p 1 p 1,, p 2 ) is a ecreasing function in p If a 1 (x 1,, x 2 ) = b 1 (x 1,, x 2 ) 0 an, for all x 1,, x 2, x b 1 (x 1,, x 2 ) = n, then x is always 0. = CP (p 1 p 1,, p 2 ) b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2 px 1 1. Consier the first erivative with respect to p 1 = p 1 CP (p 1 p 1,, p 2 ) b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 2 (x 1,,x 2 ) x 2 =a 2 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2 (x 1)p x 1 1 1, which is positive. Subsequently CP (p 1 p 1,, p 2 ) is a increasing function in p If a 1 (x 1,, x 2 ) = b 1 (x 1,, x 2 ) = 0 an, for all x 1,, x 2, x b 1 (x 1,, x 2 ) = n = CP (p 1 p 1,, p 2 ) b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 2 (x 1,,x 2 ) x 2 =a 2 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2. Thus no p 1 is involve in the function any more. CP (p 1 p 1,, p 2 ) is fixe when p 1,, p 2 are fixe. 24

28 4. For other values of a 1 an b 1 p 1 CP (p 1 p 1,, p 2 ) = p 1 = = b 1 b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) x p x 1 1 px 1 ) b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) (x 1 p x 1 1 p x 1 x 1 b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) [ ] x 1!x 2! x! px 1 1 p x 2 2 p x x 1!x 2! x! px 1 1 p x 2 2 p x 2 x 1!x 2! x! px 1 1 p x 2 2 p x 2 2 x p x p x 1 x 1 1 ). 2 (x 1p x p n p 1 p x We can o this even when x 1 or x is 0, since in that case x 1 p x p x = 0 an x p x 1 1 px 1 = 0. Thus p 1 CP (p 1 p 1,, p 2 ) = pn p 1 (B t (p 1 ) C t (p 1 )) an p 1 CP (p 1 p 1,, p 2 ) = 0 if an only if B t (p 1 ) C t (p 1 ) = 0. Note that p x 1 1 p x 1 x 1 functions in p 1. p 1 B = b 1 x 1 =a 1 x 2 =a 2 (x 1 ) is an increasing function in p 1, thus B an C are increasing b 2 (x 1 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) x 1!x 2! x! px 1 1 p x 2 2 p x 2 2 x 1 (x 1 p x p x 1 x 1 + (x x 1 )p x 1 1 p x 1 x 1 1 ), p 1 C = b 1 b 2 (x 1 ) x 1 =a 1 x 2 =a 2 (x 1 ) b 1 (x 1,,x 2 ) x 1 =a 1 (x 1,,x 2 ) ((x 1 + 1)p x 1 1 p x 1 x 1 1 x 1!x 2! x! px 1 1 p x 2 2 p x 2 2 x + (x x 1 + 1)p x p x 1 x 1 2 ), which are both positive an increasing in p 1. Thus 2 p 2 1 B > 0 an 2 C > 0. p

29 This means B an C are strictly increasing convex functions. If B C = 0 has no solution, then CP (p 1 p 1,, p 2 ) is either an increasing function or a ecreasing function on the whole support. However which is a contraiction. lim CP (p 1 p 1,, p 2 ) = 0, p 1 0 lim p 1 1 CP (p 1 p 1,, p 2 ) = 0. 2 i=1 p i If B C = 0 has exactly one solution, say p 1 = p 1, then CP (p 1 p 1,, p 2 ) is a unimoal function attaining the maximum at p 1 = p 1. If B C = 0 has two solutions, CP (p 1 p 1,, p 2 ) is still a unimoal function. Because at the bounary lim CP (p 1 p 1,, p 2 ) = 0, p 1 0 lim p 1 1 CP (p 1 p 1,, p 2 ) = 0, 2 i=1 p i so one of the solution is not a maximum or minimum point. Proof of Theorem 1. The proof consists of two parts. In the first part, we assume that L(0) = 0 an U(n) = 1 which implies L(x) (0, 1) an U(x) (0, 1) for all x 0, n. The secon part consiers a more general case, where L(x) < 0 for some nonzero x. Subsequently, U(x) > 1 for some x n. We start with the first part. (1). Without loss of generality, we assume we have p 1 (0, v 1 ) first. So unless we observe x 1 = 0 an subsequently a 1 = 0 an b 1 (x 1,, x 2 ) = 0 for all x 1,, x 2, the SCIs cannot cover p 1. From Lemma 3 we know that within those subsets CP (p 1 p 1,, p 2 ) is a ecreasing function of p 1 if p / (0, v 1 ). This means the infimum of CP (p 1 p 1,, p 2 ) within this subset is attaine at p 1 v 1. Assumption 1(5) means we shall have U(x) > L(x + 1) for all 26

30 x = 0,, n 1. Together with Assumption 1(3), we know that the infimum is actually attaine at another subset nearby where p 1 = v 1 an p 1,, p 2 stay the same. This applies to all CP (p i p 1,, p i 1, p i+1,, p 1 ). Note that we can treat any p j as a linear combination of the others. For example, if we let p 1 be a linear combination of p 2,, p an consier CP (p p 2,, p 1 ), then the infimum occurs at p = v 1. If p (0, v 1 ) as well, then Lemma 3 shows CP (p 1 p 1,, p 2 ) is a constant function in p 1. In this case, there shoul be some p i / (0, v 1 ). We choose such p i to be the linear epenent argument an fix p 1,, p i 1, p i+1,, p 2, p, so CP (p 1 p 1,, p i 1, p i+1,, p 2, p ) is a ecreasing function in p 1. This means the infimum is attaine at p 1 v 1. Thus we o not nee to consier the subsets where some enpoints are 0. Secon, we consier the subsets where no enpoints are 0 an v i j v i for all j j = 1,,. We can ivie those subsets into two kins: with local minima an without local minima. Case I, assume CP (p 1 p 1,, p 2 ) has no local minimum in the subsets. Given p 1,, p 2, CP (p 1 p 1,, p 2 ) is unimoal or monotone. So the infimum occurs when p 1 goes to the infimum or supremum that it can take. Here because we introuce (v i, v i +1) in (13), p 1 may not attain v i 1 or v i 1 +1 for all values of p 1,, p 2. Thus when p 1 v i 1 or v i 1 +1 or on the hyperplane p v i or p v i 1, the infimum is attaine. By inuction, the infimum of CP (p j p 1,, p j 1, p j+1,, p 1 ) is attaine when p j v i j or v i j +1 or on the hyperplane p v i or p v i 1 if p 1,, p j 1, p j+1,, p 1 are fixe for j = 1,, 1. An further let p 1 be a linear combination of p 2,, p, then the infimum occurs at p 2 v i 2 or v i 2 +1,,p v i or v i +1 or on the hyperplane p 1 v i1 or p 1 v i1 1. Above all, the infimum occurs at when p j v i j or v i j +1 for j k where p k is a linear combination of the others k = 1,,. Case II, assume there is a local minimum in the subset, perhaps many local minima, Given p 1,, p 2, the infimum occurs when p 1 goes to the infimum or 27

31 supremum that it can take or when B(p 1 ) = C(p 1 ). By inuction an the symmetric property, the infimum occurs when p j v i j or v i j +1 or when B(p j ) = C(p j ) for j k where p k is a linear combination of the others k = 1,,. Use the arguments below Equation (14), we know can be replace by =. Combine all the arguments above, we conclue that the confience coefficient is obtaine at some of the points in E 1 E ( s ) (M 1,t M,t ). t=1 (2). Assume L(x) < 0 for some nonzero x. For p 1 (0, v 1 ), there coul be multiple x where the resulting confience interval covers p 1. When it happens, the conitional coverage probability CP (p 1 p 1,, p 2 ) is not a ecreasing function in (0, v 1 ) any more. So we have to take the possible unimoality into account. Therefore, the infimum of CP (p 1 p 1,, p 2 ) is attaine at either p 1 = v 1 or p 1 0. Although zero parameters are not allowe in the parameter space Θ, it is still vali to calculate the coverage probability in case of zero cell probabilities. Such points construct E 0. The remaining part of the proof is the same as the first part, so the confience coefficient is attaine at E 0 E 1 E ( s ) (M 1,t M,t ). t=1 28