A Note on Coverage Probability of Confidence Interval for the Difference between Two Normal Variances
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1 pplied Mathematical Sciences, Vol 6, 01, no 67, Note on Coverage Probability of Confidence Interval for the Difference between Two Normal Variances Sa-aat Niwitpong Department of pplied Statistics, Faculty of pplied Science King Mongkut s University of Technology North Bangkok, Thailand snw@kmutnbacth bstract We investigate, in this paper, the new confidence interval for the difference between two normal population variances based on the closed form method of variance estimation of Zou, Huo and Taleban [Simple confidence intervals for lognormal means and their differences with environmental applications Environmetrics 0: , 009] We derive analytic expressions of coverage probability and expected length of our proposed confidence interval compared to the existing confidence interval Mathematics Subject Classification: 6F5 Keywords: Coverage probability; Expected length; Variances 1 Introduction Recent paper of Herbert et al [9] have argued that analytical methods for interval estimation of the difference between variances have not been described They proposed a simple analytical method to construct confidence interval for the difference between variances of two independent samples It is shown that their proposed confidence interval generated coverage probability close to the nominal level 1 α even when sample sizes are small and unequal Their method also works well when observations are highly skewed and leptokurtic Cojbasica and Tomovica [11] has shown that the method based on t-statistics combined with bootstrap techniques for determination of nonparametric confidence interval of the population variance of two-sample problems works very well when data are from exponential families Related works of the confidence
2 331 Sa-aat Niwitpong interval for the difference between two variances, the reader is referred to the references cited in the above two papers In this paper, we emphasize only data is from a normal distribution and we propose the new confidence interval for the difference of two normal population variances based on the closed form method of variance estimation(cmve), Zou et al [5] The closed form method of variance estimation has been used to construct the confidence interval for the parameters in the forms of θ 1 + θ and θ 1 /θ, see eg Zou et al [5], Zou and Donner [7], Zou et al [8] and the latest paper of Donner and Zou [1] Zou et al [6] have argued that, in terms of the coverage probability, the confidence interval constructed from the method of variance recovery estimation method performs as well as the confidence interval from the generalized confidence interval (GCI), Weerahandi [10], but in some situations the average length width from this method is shorter than that of the confidence interval constructed from GCI method, see eg Niwitpong et al [3]and [] Niwitpong [3] shown that, by means of simulation, the confidence interval for the difference between normal variances based on CMVE method is preferable compared to the GCI method and the standard method Niwitpong [3], however, has not derived the expressions of the coverage probability and expected length of each interval In this paper, it is therefore of interest to derive mathematically the expressions of the coverage probability and the expected length of the confidence interval for the difference of normal population variances based on the CMVE compared to the standard method, see eg Cojbasica and Tomovica [11] Confidence intervals for the difference of two normal population variances Let X 1,,X n and Y 1,,Y m be random samples from two independent normal distributions with means μ x, μ y and standard deviations σ x and σ y, respectively The sample means and variances for X and Y are, respectively, denoted as X, Ȳ, S x and S y when X = n 1 n i=1 X i, Ȳ = m 1 m i=1 Y i, S x = (n 1) 1 n i=1 (X i X) and S y = (m 1) 1 m i=1 (Y i Ȳ ) We are interested in 100(1-α)% confidence interval for θ = σ x σ y 1 Standard confidence interval for θ It is well known that statistic (n 1)Sx/σ x has Chi-square distribution (χ n 1) with n 1 degrees of freedom and hence we have the following results, E(χ n 1 )= n 1 and Var(χ n 1 )=(n 1) Define Z = (n 1)S x/σx (n 1) (n 1) = S x σ x Var(S x )
3 Confidence interval for the difference between normal variances 3315 when Var(Sx )=σ x /(n 1) and the test statistic Z is the standard normal distribution s a results, we have Z = (Sx S y ) θ Var(S x )+Var(Sy ) = (S x S y ) θ σ x (n 1) + σ y (m 1) In general, σx and σ y are unknown parameters and the unbiased estimators for these parameters are respectively Sx and S y It is easily seen that, using Z as a pivotal statistic, the approximate 100(1-α)% confidence interval for θ is [ ] CI s = (Sx S y ) z S x 1 α/ (n 1) + S y (m 1) where z 1 α/ is an upper 1 α/ quantile of the standard normal distribution Coverage probability of CI s The following Lamma 1 is useful to prove Theorem 1 Lemma 1 Suppose f(v, u) = r 1 v + r u, then f(v, u) is a convex function in (v, u) when r 1 and r are constants and v 0 and u 0 Proof It is easy to see that f (v) 0 and f (u) 0 Hence f(v) and f(u) are convex functions in v and u respectively s a result, f(v, u), which is function of v and u, is a convex function in (v, u) Theorem 1 Suppose = σ x + σ y, B = S x + S y and K = c B n 1 m 1 n 1 m 1, the coverage probability of CI s and its expected length are respectively E [ Φ[K] Φ[ K] ] and ( (n +1)σx c + (m ) +1)σ y (n 1) (m 1) where Φ[ ] is a cdf normal distribution function and c is an upper percentile of the standard normal distribution function Proof This Theorem is proved similarly to Niwitpong and Niwitpong [], Pr(θ CI s ) = Pr(Sx S y c B<θ<Sx S y + c B) = Pr( c B<θ (Sx Sy) <c B) = Pr( c B<(Sx S y ) θ<c B) = Pr( c B < (S x S y ) θ < c B )
4 3316 Sa-aat Niwitpong = Pr( c B <Z< c B ) where Z N(0, 1) length of CI s is { 1, if ξ { K<Z<K} = E[I { K<Z<K} (ξ)],i { K<Z<K} (ξ) = 0, otherwise = E[E[I { K<Z<K} (ξ)] S],S =(μ x,μ y ) = E[Φ(K) Φ( K)] Using Lemma 1 and Jensen s inequality, the expected E ( c B ) ( ) S = ce x n 1 + S y m 1 ( σ = ce x(n 1) Sx + σ y (m 1) S ) y (n 1) 3 σx (m 1) 3 σy ( ) σx = ce (n 1) V + σ y 3 (m 1) U,V χ 3 n 1,U χ m 1 ( ) σx c (n 1) E(V )+ σ y 3 (m 1) E(U ), by Jensen s inequality 3 ( ) σx = c (n 1) (Var(V)+(E(V 3 )) )+ σ y (m 1) 3 (Var(U)+(E(U)) ) ( ) σx = c (n 1) ((n 1) + (n 3 1) )+ σ y (m 1) ((m 1) + (m 3 1) ) ( (n +1)σx = c + (m ) +1)σ y (n 1) (m 1) 3 The closed form method of variance estimation for θ The idea of estimating confidence limits for lognormal data by the closed form method of variance estimation was presented by Zou et al [5] Their strategy is to recover variance estimates from these limits and then to form approximate confidence intervals for functions of the parameters Using the central limit theorem, a general approach to setting two-sided (1 α)100% confidence limits for θ 1 + θ is given by (ˆθ 1 + ˆθ ) z (1 α/) var( ˆθ 1 )+var( ˆθ )
5 Confidence interval for the difference between normal variances 3317 where the estimators ˆθ i (i =1, ) and var(ˆθ i ) are statistically independent of each other Since var(ˆθ i ) is unknown, Zou et al [5] argued that the method to improve of estimating variance of (ˆθ 1 + ˆθ ) in the neighborhood of the limits (L, U) for θ 1 + θ was estimated in the setting of min(θ 1 + θ ) for L and that of max(θ 1 + θ ) for U Suppose that a (1 α)100% two-sided confidence interval for θ i is given by (l i,u i ) Using the central limit theorem, we have (ˆθ i l i ) var(ˆθ i ) z 1 α/ which outcomes in the estimated variance var(ˆθ i ) under the condition θ i = l i of var l (ˆθ i ) (ˆθ i l i ) z 1 α/ Similarly, the estimated variance var(ˆθ i ) under the condition θ i = u i of var u (ˆθ i ) (u i ˆθ i ) z1 α/ Therefore, the confidence limit [L, U] is given by [L, U] = [ (ˆθ 1 + ˆθ ) z 1 α/ var( ˆθ 1 )+ var( ˆθ ) ] = [ (ˆθ 1 + ˆθ ) z ( ˆθ 1 l 1 ) 1 α/ z 1 α/ + ( ˆθ l ) z 1 α/ (ˆθ 1 + ˆθ )+z (u 1 ˆθ 1 ) 1 α/ z1 α/ = [ (ˆθ 1 + ˆθ ) ( ˆθ 1 l 1 ) +(ˆθ l ), (ˆθ 1 + ˆθ )+ (u 1 ˆθ 1 ) +(u ˆθ ] ) + (u ˆθ ) z 1 α/ Confidence limits for θ 1 θ can be constructed by replacing θ 1 θ in the form θ 1 +( θ ) and replacing the confidence limits for θ which is ( u, l ) We therefore construct the confidence interval for θ as [ ] L1,U 1 = [ (ˆθ 1 ˆθ ) ( ˆθ 1 l 1 ) +(u ˆθ ), (1) (ˆθ 1 ˆθ )+ (u 1 ˆθ ] 1 ) +(ˆθ l ) () We now propose the confidence interval for θ Define θ 1 = σx, θ = σy,ˆθ 1 = Sx and ˆθ = Sy lso, confidence limits for θ i,i=1,, is (l i,u i ) where [ (n 1)S [l 1,u 1 ]= x, (n ] 1)S x χ 1 α/,n 1 χ α/,n 1, ]
6 3318 Sa-aat Niwitpong and [ (n 1)S [l,u ]= y, (n ] 1)S y χ 1 α/,m 1 χ α/,m 1 Replacing ˆθ 1 = S x,ˆθ = S y, l i and u i in the above equations into (1) and (), we then obtain a new confidence interval for θ which is called CI r, CI r =[L 1,U 1 ] where and L 1 = Sx Sy ( c1 n +1 c 1 ) S x + ( m 1 c c ) S y ( ) ( ) n 1 U 1 = Sx S y + c Sx c + c3 m +1 Sy c 3 Coverage probability of CI r Theorem The coverage probability and expected length of CI r are respectively E[Φ(K ) Φ(K 1 )] and (n +1)σx r 3 (n 1) + r (m +1)σ y (m 1) + r 1 (n +1)σ x (n 1) where r 1 = ( c 1 n+1 c 1 ), r = ( m 1 c c ), r3 = ( n 1 c c + r (m +1)σ y (m 1) ), r = ( ) c 3 m+1, c1 c 3 is a 1 α/ percentile of the χ n 1 with (n-1) degrees of freedom, c is a α/ percentile of the χ n 1 with (n-1) degrees of freedom, c 3 is a 1 α/ percentile of the χ m 1 with (m-1) degrees of freedom, is a α/ percentile of the χ m 1 with (m-1) degrees of freedom, K 1 = r 3 S x +r S y and K = r1 S x +r S y Proof Pr(θ CI r ) = Pr(Sx S y r 1 Sx + r Sy <θ<s x S y + r 3 Sx + r Sy ) = Pr( r 1 Sx + r Sy <θ (S x S y ) < r 3 Sx + r Sy ) = Pr( r 1 Sx + r Sy > (S x S y ) θ> r 3 Sx + r Sy ) = ( r3 Sx Pr Sy < (S x Sy) θ r1 Sx < S ) y ( r3 Sx = Pr + r Sy <Z< r1 Sx + r Sy )
7 Confidence interval for the difference between normal variances 3319 { 1, if ξ {K1 <Z<K = E[I {K1 <Z<K }(ξ)],i {K1 <Z<K }(ξ) = } 0, otherwise = E[E[I { K1 <Z<K }(ξ)] S],S =(μ x,μ y ) = E[Φ(K ) Φ(K 1 )] Similarly to Theorem 1, using the Jensen s inequality and Lemma 1, the expected length of CI r is E( r 3 Sx + r Sy r 1 Sx + r Sy) ( σx = E r 3 (n 1) V σ y + r (m 1) U σx r 1 (n 1) V σ ) y + r (m 1) U σx r 3 (n 1) E(V σ y )+r (m 1) E(U ) σx r 1 (n 1) E(V σ y )+r (m 1) E(U ) = (n +1)σx r (m +1)σ y 3 + r + (n +1)σx r (m +1)σy 1 + r (n 1) (m 1) (n 1) (m 1) 3 Discussion and Conclusion We proposed, in this paper, the confidence interval for the difference between two normal population variances based on CMVE method We derived analytic expressions of the coverage probability and the expected length of our proposed confidence interval compared to the existing confidence interval, see eg Niwitpong [3] Niwitpong [3] has shown via Monte carlo simulation that the coverage probability of the confidence interval CI s is far below the nominal value of 095 and its expected value is very narrow Niwitpong [3] also found that confidence interval of the difference between normal variances constructed by the CMVE method, CI r, performs better than that of the confidence interval based on the generalize confidence interval (Weerahandi [10]) but she did not derived analytic expressions of the coverage probabilities and their expected lengths s a result, all results (Theorems 1-) in this paper are very useful to compare the new confidence interval for the difference between normal variances via their expected lengths, see eg Niwitpong and Niwitpong [] References [1] Donner, GY Zou, Closed-form confidence intervals for function of the normal mean and standard deviation, Statistical Methods in Medical Research, (010) 1 1
8 330 Sa-aat Niwitpong [] S Niwitpong and S Niwitpong, Confidence interval for the difference of two normal population means with a known ratio of variances, pplied Mathematical Sciences, (010), [3] S Niwitpong, Confidence interval for the difference of two normal population varainces, Prooceedings of World cademy of Sciences, Engineering and technology, Vol 80, (011), [] S Niwitpong, S Koonprasert and S Niwitpong, Confidence intervals for the difference between normal population means with known coefficients of variation, pplied Mathematical Sciences, Vol6, no 1, 7-5 [5] GY Zou, CY Huo, J Taleban, Simple confidence intervals for lognormal means and their differences with environmental applications, Environmetrics 0 (009) [6] GY Zou, J Taleban, C Huo, Confidence interval estima tion for lognormal data with application to health economics, Computational Statistics and Data nalysis, 53 (009) [7] GY Zou, Donner, simple alternative confidence interval for the difference between two proportions, Controlled Clinical Trials 5 (00) 3 1 [8] GY Zou, W Huang, X Zhang, note on confidence interval estimation for a linear function of binomial proportions, Computational Statistics and Data nalysis, 53 (009) [9] RD Herbert, Hayen, P Macaskill, SD Walter, Interval estimation for the difference of two independent variances, Communications in Statistics- Simulation and Computation, 0 (011) [10] S Weerahandi, Generalized confidence intervals, The Journal of merican Statistical ssociation, 88 (1993) [11] V Cojbasica, Tomovica, Nonparametric confidence intervals for population variance of one sample and the diffeence of variances of two samples, Computational Statistics & Data nalysis, 51 (007) Received: February, 01
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