Approximate interval estimation for EPMC for improved linear discriminant rule under high dimensional frame work

Transcription

1 Hiroshima Statistical Research Group: Technical Report Approximate interval estimation for PMC for improved linear discriminant rule under high dimensional frame work Masashi Hyodo, Tomohiro Mitani, Tetsuto Himeno and Takashi Seo Last Modified: March 9, 5 Abstract. An observation is to be classified into one of two multivariate normal populations with equal covariance matrix. In this paper, we consider the confidence intervals for expected probability of misclassification PMC for improved linear discriminant rule in two types of data:namely, large sample data and high dimensional data. Our approximate confidence interval is based on the asymptotic normality of consistent estimator of PMC. We obtain new results of stochastic expression for two bilinear forms and two quadratic forms which are important for our asymptotic evaluation of PMC. We prove asymptotic normality under two different frameworks which could be convenient in different situations based on these results. Through simulation study, it is observed that our approximate confidence interval has a good performance not only in high dimensional and large sample settings, but also in large sample settings. AMS Mathematics Subject Classification. 6H, 6. Key words and phrases. asymptotic approximations, expected probability of misclassification, high dimensional data, linear discriminant function.. Introduction We consider the problem of classifying a future observation vector into one of the two population groups Π and Π. For each i,, Π i denotes a population from a multivariate normal distribution N p µ i, Σ, and it is supposed that x ij, j,..., N i, are observed from the population Π i. Here, µ i i, and Σ are unknown parameters, and they are estimated by the sample mean vectors x i N S n i Ni i j x ij i, and the pooled sample covariance matrix Ni j x ij x i x ij x i for n N N.

2 M. HYODO, T. MITANI, T. HIMNO AND T. SO The linear discriminant function is defined as T x x x S {x x x }. Observe however that the linear discriminant function T x has a bias. In fact, T x x Π i ] n i n p nn N p n p N N, i,, where µ µ Σ µ µ. For this reason, we use the bias-corrected discriminant function defined as. T x x x S {x x x } nn N p n p N N, where the subtraction of nn N p/{n p N N } in. is to guarantee that T x x Π i ] n/{n p } i, i,. Now using T x, a new observation x is to be assigned to Π if T x >, and to Π otherwise. The performance of this discriminant rule is evaluated by its probabilities of misclassification. The probabilities of misclassification have been obtained with respect to the distribution of the linear discriminant function T x. There are different types of misclassification probability associated with T x. These are the conditional probabilities of misclassification CPMC and expected probabilities of misclassification PMC. The CPMC is defined by. L P T x x Π, X], L P T x > x Π, X], where X x,..., x N, x,..., x N. We note that the CPMC is the conditional probability of misclassifying an observation x from Π i into Π j, i, j,, i j. On the other hand, the PMC is defined by. R L ], R L ]. We note that the PMC is the unconditional probability of misclassifying an observation x from Π i into Π j, i, j,, i j. Since the exact expression for the PMC is very complicated, there are much works for the approximation of PMC. The asymptotic approximation of PMC under a framework such that N and N are large with p is fixed has been studied. This approximation is called large sample approximation. For a review of these results, see, e.g., Okamoto 96, 968 and Siotani 98. Further, asymptotic approximation of PMC under a framework that N, N and p are all large have also been studied see, e.g., Lachenbruch 968 and Fujikoshi and Seo 998.

3 APPROXIMAT INTRVAL STIMATION FOR PMC This approximation is called high dimensional and large sample approximation. In addition, Fujikoshi gave an explicit formula of error bounds for a high dimensional and large sample approximation of PMC proposed by Lachenbruch 968. However, as their approximations are functions of unknown parameters, it must be estimated in practice. Based on the large sample approximation, Lachenbruch and Mickey 968 proposed the asymptotic unbiased estimator of PMC. On the other hand, Kubokawa, Hyodo and Srivastava proposed the second order asymptotic unbiased estimator of the PMC in high dimensional and large sample framework. In this paper, we consider the interval estimations for the PMC. Since the exact interval estimations for the PMC are very difficult problem, there are some works for the approximate confidence interval. McLachlan 975 proposed an approximate confidence interval for the CPMC based on the large sample approximation. Recently, Chung and Han 9 proposed the jackknife confidence interval and the bootstrap confidence interval for the CPMC. The problems with these methods are listed below. A Since CPMC is conditional probability, it is more desirable to derive interval estimation of PMC. B Since these methods are based on large sample asymptotic results, these methods do not perform well in high dimensional settings. For the problems A and B, we derive the asymptotic distribution of the estimator of PMC under the high dimensional and large sample frame works, and propose the approximate confidence interval for the PMC. For that purpose, we derive explicit expression of stochastic for two bilinear forms and two quadratic forms. The method used in this paper is to express based on eight primitive random variables, namely four random variables having the standard normal distribution and four random variables having chi-square distributions. This approach not only makes it easier to derive asymptotic distribution of estimator of PMC, but also enables us to show the asymptotic normality of CPMC. As a by product, we show asymptotic normality of CPMC. The organization of this paper is as follows. In Section, we propose consistent estimator of PMC. In Section, we propose new approximate confidence interval of PMC and show the asymptotic normality of CPMC. In Section 4, we investigate the performances of our approximate confidence intervals through the numerical studies. The conclusion of our study is summarized in Section 5. Some preliminary results are given in Appendix.. The consistent estimator of PMC In this section, we propose the consistent estimator of the PMC. Since R can be obtained from R simply by interchanging N and N, we only deal

4 4 M. HYODO, T. MITANI, T. HIMNO AND T. SO with R. Let c p/n, γ N /n, γ N /n. We assume the following asymptotic frameworks, in order to derive limiting value of R. A n, p with n c c for some c,, A n, N, N with n γ γ, n γ γ for some γ, γ,, A n with n for some,. Suppose that x Π. Under these conditions, a conditional distribution of T x given x, x, S is distributed as N U, V, where U x x S x µ x x S x x nn N p n p N N, V x x S ΣS x x. Then, R can be expressed as R Φ UV /], where Φ denotes the cumulative distribution function of N,. We rewrite U and V by using τ N N /n Σ / N N µ µ, u n Σ / x x, u Σ / N x N x N µ N µ, W nσ / SΣ /. n It is seen that u, u and W are mutually independently and distributed as u N p τ, I p, u N p, I p and W W p n, I p, respectively. Using these variables, we can rewrite U and V as.. U N N n u N N W n u u N N W u n τ W u N nn N p n p N N, V n n N N u W u. Applying Lemma A. to. and., we obtain the constants U and V as U lim U] n,p c, V lim V ] n,p c c γ γ.

5 APPROXIMAT INTRVAL STIMATION FOR PMC 5 Also, the expectations U U ] and V V ] can be evaluated as U U ] { n c 4 c. γ γ cγ γ } γ on, γ V V ] c n c { c c }.4 γ γ c { c c } ] γ on. γ under the asymptotic frameworks A-A. See details in Appendix B and C. Thus, using.,.4 and Chebyshev s inequality, we have that U p U and V p V. Furthermore, using continuous mapping theorem, we obtain that.5 Φ UV / Φ U V / p under the asymptotic frameworks A-A. On the other hand, it holds that Φ UV / Φ U V /.6 < a.s. Combining.5,.6 and dominated convergence theorem, we obtain the following lemma. Lemma.. Under the asymptotic frameworks A-A, it holds that R Φ c/. c/γ γ Since the limiting value of R is a function of, we begin by obtaining its consistent estimator. Lemma.. The estimator of is defined by n p x x S n p x x. n N N Under the asymptotic frameworks A-A, it holds that p. Proof We can rewrite the estimator.7 n p n N N u W u n p N N.

6 6 M. HYODO, T. MITANI, T. HIMNO AND T. SO Applying Lemma A. to.7, we have.8 ] n c 4 4 γ γ c γ γ on under the asymptotic frameworks A-A. See details in Appendix D. Thus, using.8 and Chebyshev s inequality, we have p under the asymptotic frameworks A-A. Substituting the consistent estimator into the limiting term ΦU V /, the consistent estimator of R is obtained by R Φ Û V, where Û c and V c { c/γ γ }. The following corollary is obtained from continuous mapping theorem and consistency of estimator. Corollary.. Under the asymptotic frameworks A-A, it holds that R p R.. Approximate interval estimation for PMC and asymptotic normality of CPMC In Section., we show the asymptotic normality of the estimator of PMC under two different frameworks, and propose the approximate confidence interval. In Section., we also show the asymptotic normality of CPMC... The asymptotic normality of the estimator of PMC At first, we derive the asymptotic distribution of the studentized statistics under the high dimensional frameworks A-A. We consider the following random variable n R Φ U V. To show the asymptotic normality of the above random variable, we consider the stochastic expansions of Û and V. Since the statistics Û and V are the functions of, it is essential to derive the stochastic expansion of. By using u and W, we rewrite as n p n N N u W u n p N N.

7 APPROXIMAT INTRVAL STIMATION FOR PMC 7 Define the variables where v ṽ p p, v ṽ n p n p, ṽ χ p, ṽ χ n p. Here, χ a a N means chi-square distribution with a degrees of freedom. The estimator is expanded as. D n o p n /, where D g v g v g u. Here, c c γ γ u N,, g, g, g γ γ cγ γ γ γ and v, v and u are mutually independent. From., it is noted that. Û U c D n o p n /, V V c D n o p n /, for c { c} and c c. expansion, it follows that where Here Û V U V V U V n Q o p n /, Using. and Taylor series c D n U V c D n o p n / Q q v q v q u. c c c γ γ q γ γ {cγ γ } /, q c c γ γ q c γ γ /. From the stochastic expansion of Û V, we have. R Φ c/ c/γ γ ϕ c γ γ γ γ cγ γ, c/ c/γ γ Q n o p n /,

8 8 M. HYODO, T. MITANI, T. HIMNO AND T. SO where ϕ is the p.d.f. of the standard normal distribution. Note that u is distributed as N,, v and v are asymptotically distributed as N, under the asymptotic framework A, and these variables are mutually independent. Hence, under the asymptotic frameworks A-A, it holds that.4 n R Φ c/ c/γ γ σ e where σ e ϕ c/ c/γ γ ϕ c/ c/γ γ q q q d N,, c γ γ c γ γ γ γ γ γ c γ γ /. Now turn to evaluate the difference of the limiting value of R and R. The remainder after using first term of the Taylor series of Φ at UV / U V / is given by Φ d! U V / U V / for some value d between UV / and U V /, and Φ d is equal or smaller than / πe uniformly in d,. Here, Φ is second derivative function of Φ. Hence, we have that ] R Φ U V / ϕ U V / U V / U.5 V / πe U V / U V /. We note that.6 U V / U U V / V / V U U U V / V V V V /V V V /V V V V V V / V U U V V. V

9 APPROXIMAT INTRVAL STIMATION FOR PMC 9 From A.8 and A. ].7 U U V ].8 U V / V V V c n on { U 4 n c V / c c } 4 γ γ c on. Since V /V > and V.9 V /V > V, U V V / V /V V V /V V V V < U V V ] V 4V / By using Lemma A., we obtain that V V ]. On V under the asymptotic frameworks A-A. From.9 and., See details in Appendix.. U V / V /V V V V /V V V On. V By using V V / V > V > and Cauchy Schwarz inequality, ] V V / U U V V. V V ] U U < V V U U ] V V ]. V V V V V By using Lemma A., we obtain that U U ]. On V

10 M. HYODO, T. MITANI, T. HIMNO AND T. SO under the asymptotic frameworks A-A. See details in Appendix F. From. and., ] V V / U U V V.4 On. V V Combining.7,.8,. and.4, under the asymptotic frameworks A-A, it holds that ] U V / U.5 V / On. Since V V V V >, U V / U V / U U V U U V U V V V V V V U V V V V V V U U U V V V V V V U U V U V V V U U U V V V V V By using Cauchy Schwarz inequality, we obtain that U V / U U U ] U.6 V V U V V / U U ] / V V ] /. V V From.,. and.6, we obtain that U V / U.7 V / On V V ] V under the asymptotic frameworks A-A. Combining.5,.5 and.7, under the asymptotic frameworks A-A, it holds that R Φ c/.8 c/γ γ On. By using.4 and.8, we obtain the following theorem..

11 APPROXIMAT INTRVAL STIMATION FOR PMC Theorem.. Under the asymptotic frameworks A-A, it holds that n R R T e σ e d N,. To propose the interval estimation of the PMC, we need to estimate σ e. We use truncated estimator ˆ max,, so that the estimator of σ e may be negative. Then it holds that.9 max, a.s. By using Markov s inequality,.8 and.9, we obtain ˆ p under the asymptotic frameworks A-A. Hence, ˆ is a consistent estimator of. Assigning the truncated estimator to the portion of σ e which may be negative, we propose σ e ˆ ϕ c/ ˆ ˆ c/γ γ c ˆ γ γ c ˆ γ γ ˆ γ γ γ γ c ˆ γ γ /. By using the consistent estimator σ e ˆ, we obtain the following statistics of T e n R R T e σ e ˆ Therefore we can obtain the following corollary. Corollary.. Under the asymptotic frameworks A-A, it holds that T e d N,.. Next, we show that asymptotic normality of Te the large sample framework is also established under A : p is fixed and n or A : n, p with p/ n.

12 M. HYODO, T. MITANI, T. HIMNO AND T. SO Under the frameworks A and A or the frameworks A, A and A, it holds that R Φ on / U, Φ Φ. on /, V σ e ϕ /γ γ. o, σ e ˆ p ϕ /γ γ,. T e ϕ σ e From.-., we have that Te 8 4γ γ v γ γ / u v Therefore we can obtain the following corollary. o p. γ γ / u o p. Corollary.. Assume the conditions A and A or the conditions A, A and A. Then, it holds that T e d N,. Remark.. From Corollary. and., T e has a asymptotic normality not only under high dimensional and large sample frame work, but also under the large sample framework. Based on Corollary. and., we propose an approximate α percentile confidence interval for PMC as following: C T R σ e y α, R σ e ]. y α, n n where y α denotes upper α percentile of standard normal distribution... Asymptotic normality of CPMC In this section, we show asymptotic normality of CPMC. The CPMC can be expressed as L Φ UV.

13 APPROXIMAT INTRVAL STIMATION FOR PMC Applying Lemma A. to. and., we obtain where U { n npn N ṽ N N n p nn N nu N N ṽ N N n u u ṽ nu 4 ṽ N N ṽ N N ṽ 4 n u V ṽ } u N N ṽ u ṽ u ṽ n ṽ N u N u, ṽ n N N ṽ 4 { n ṽ n n ṽ ṽ 4 ṽ 4 n n ṽ ṽ N N ṽ 4 N N ṽ u, u u ṽ } u i N, i,,, 4, ṽ χ p, ṽ χ n p, ṽ χ p, ṽ 4 χ n p, and these variables are mutually independent. Define the variables v ṽ p p, v ṽ n p n p, v ṽ p p, v 4 ṽ n p n p. Note that ṽ p p v, ṽ n p n p v, ṽ p p v, ṽ 4 n p n p v 4, and v, v, v and v 4 are asymptotically distributed as N, under the asymptotic framework A. By using Taylor series expansion based on these variables, we can expand U stochastically,.4 U U n U o p n /,

14 4 M. HYODO, T. MITANI, T. HIMNO AND T. SO where U c, cγ γ cγ γ γ γ U v v cγ γ c / γ γ cγ c / u γ γ c γ u c γ γ c c c u γ γ γ γ c / u 4. γ γ Using similar arguments, we can expand V stochastically,.5 V V V n o p n /, where V V c c, γ γ c c v c γ γ c c γ γ γ γ c 7/ v γ γ c v γ γ c c γ γ c 7/ v 4 γ γ c u. γ γ By using.4,.5 and Taylor series expansion, it follows that UV U V { U U } V / op n / nv V where Here, U V W n o p n /, W w v w v w v w 4 v 4 w 5 u w 6 u w 7 u w 8 u 4. w w c c c γ γ {cγ γ } / cγ γ γ γ cγ γ, γ c γ γ cγ γ, c c w cγ γ, w c 4 cγ γ, c w 5 γ γ {cγ γ } / c γ γ γ cγ γ, c γ w 6 γ γ cγ γ, w 7 c, w 8 c.

15 APPROXIMAT INTRVAL STIMATION FOR PMC 5 Using the Taylor series expansion, L is expressed as L ΦU V ϕu V W n o p n /. Since the random variables v, v, v, v 4, u, u, u and u 4 in W are mutually independent and asymptotically or exactly distributed as N,, we obtain the following theorem. Theorem.. Under the asymptotic frameworks A-A, it holds that nl R d N, σ, where σ {ϕu V } 8 i w i. Next, we evaluate asymptotic property of L under the large sample framework. We assume the conditions A and A or the conditions A, A and A. Then it holds that L Φ ϕ Thus, we obtain the following corollary. γ γ n u u o p n /. γ γ Corollary.. Assume the conditions A and A or the conditions A, A and A. Then, it holds that n L Φ d N, ϕ. 4γ γ Remark.. We consider the relation between the optimal rule.6 T opt x >resp. x Π resp.π, and our suggested rule.7 T x>resp. x Π resp.π, where T opt x µ µ Σ {x µ µ }, T x x x S {x x x }. From Corollary., we note that the distribution of the CPMC of the rule.7 under the condition A or A approaches a normal distribution with standard deviation shrinking in proportion to / n around the error rate of the optimal rule.6.

16 6 M. HYODO, T. MITANI, T. HIMNO AND T. SO 4. Simulation study In this section, we investigate the performance of proposed approximate confidence intervals.. In order to evaluate coverage probabilities of the approximate confidence intervals and the expected lengths, a Monte Carlo study is conducted. Without loss of generality, multivariate normal random samples are generated from Π : N p, I p and Π : N p 5, p, I p. The values of N, N and p are chosen as follows: n CaseA p,,,, 4, N : N :, :, :, p CaseB p 5, n,, 5, N : N :, :, :. In above configuration, we calculate the following coverage probabilities CP { R, R R n / σ e y α/, R n / σ e y α/ ]}, sim and the following expected lengths of approximate confidence interval L n / σ e y α/ y α/ ], where { } denotes number of element of set { }, sim denotes replication number of simulation. We also estimate the exact expected length by using Monte Carlo simulation as follows: L R α/ sim R α/ sim, where R i denotes i-th largest value among the sim values. Tables - give the coverage probabilities when p, and 5, respectively. Tables 4-6 give the expected lengths of approximate confidence interval and exact expected length when p, and 5, respectively. As can be seen from the Tables -, when the sample size or dimension is increased, probability for approximate confidence interval is close to confidence level. In addition, we observe that our approximations have a high level of accuracy in different situations: large sample settings Table, high dimensional and large sample settings Table -. From Tables 4-6, when the sample sizes increase, the expected lengths become narrower for each case. Through these simulation results, we can see that our approximate confidence interval has a good performance not only in high dimensional and large sample settings, but also in large sample settings. The asymptotic normality obtained by Corollary. is also demonstrated. Let B N,N N N /nl Φ / ϕ /, H p,n,n nl R σ.

17 APPROXIMAT INTRVAL STIMATION FOR PMC 7 Then Corollary. Theorem. show that B N,N H p,n,n converges in distribution to standard normal distribution as n n, p. To check for asymptotic normality make B N,N H p,n,n vs standard normal Q-Q plot in Case A. The straight line y x represents where asymptotic normality holds. Figure display the Q-Q plots of B N,N in Case B, and Figure, display the Q-Q plots of H p,n,n in Case A. From figures, it is confirmed that CPMC has normality when sample size is large enough compared with the dimension. 5. Conclusion The performance of classification procedure is evaluated by its error probability which usually depends on unknown parameters. In practice, we considered the interval estimation for PMC of improved linear discriminant rule. To derive an approximate confidence interval, we obtained the explicit expression of stochastic for two bilinear forms and two quadratic forms, and derived the asymptotic distribution for the studentized statistics of estimator of PMC under the high dimensional and large sample frame work. Our approximate confidence interval not only has been established in high dimensional and large sample settings, but also has been established in large sample settings. Also, we confirmed that the superiority of our approximate confidence intervals have been verified in the sense of the coverage probability and expected length by using Monte Carlo simulation. Appendix A. Stochastic expression quadratic form We present here the preliminary results about the quadratic form. Lemma A.. Let z N p ν, I p, g N p, I p, W W p n, I p and ν ν ν. Assume that n p > and p >. Then, it holds that i z W z u ν u ṽ, ṽ ii z W z u ν u ṽ ṽ { iii ν W z ν ṽ iv z W g ν u u ṽ ṽ 4 u ν u ṽ ṽ ṽ } ṽ 4,, { } ṽ u u 4, ṽ 4

18 8 M. HYODO, T. MITANI, T. HIMNO AND T. SO where u i N, i,,, 4, ṽ χ p, ṽ χ n p, ṽ χ p, ṽ 4 χ n p, and these variables are mutually independent. Here, χ p means chi-square distribution with p degrees of freedom. Proof The proof of assertions i-iv follows directly by applying the technique derived in Lemma, in Yamada et al. 5. B. Derivation of. By using Lemma A., U can be rewritten as A. { npn N N N n p nn N nu N N ṽ N N n u u ṽ nu 4 ṽ N N ṽ N N ṽ 4 n u u ṽ n ṽ N u N u. ṽ n N N ṽ 4 U n ṽ } u N N ṽ u ṽ By using above expression, we calculate the expectation of U as A. U] n n p c c n c on. n c The expectation of U is obtained by calculating the second moment of each term in A.. The second moment of each term in A. is calculated as

19 APPROXIMAT INTRVAL STIMATION FOR PMC 9 follows: { n ṽ npn N N N n p nn N N N ṽ u u ṽ n 4n p n p 4 n pn N N N n p n p n n pn N N N n p n p c c γ γ n c n c cγ γ γ γ n c γ γ nu N N ṽ N N n u u ṽ n n n p n p n c c n c γ γ on, } on, n p N N n p n p nu 4 ṽ N N ṽ N N ṽ 4 n u u ṽ n p n n p n p n p n p p N N n p n p n p c c n c n c on, γ γ { } n ṽ N u N u ṽ n N N ṽ 4 n { N n p N p } n N N n p n p n p γ γ c γ c nγ γ c on.

20 M. HYODO, T. MITANI, T. HIMNO AND T. SO Summarizing these results, we obtain that A. U ] From A. and A., we obtain that c n c n c γ c γ γ c n c γ γ n c on. γ γ U U ] U ] U U] U { n c 4 c γ γ cγ γ γ γ } on. C. Derivation of.4 By using Lemma A., V can be rewritten as A. 4 V { n ṽ ṽ n n ṽ 4 N N ṽ n n ṽ ṽ N N ṽ 4 u. ṽ u u } ṽ ṽ 4 By using above expression, we calculate the expectation of V as A. 5 V ] n ṽ ṽ n n ṽ 4 N N ṽ ṽ u u ] ṽ ṽ 4 n n n p n p n p n n n p N N n p n p n p { c } 4 n c { c c } 4 γ γ n c on. The expectation of V is obtained by calculating the second moment of each term in A.4. The second moment of each term in A.4 is calculated as

21 APPROXIMAT INTRVAL STIMATION FOR PMC follows: A. 6 { n ṽ ṽ n n ṽ 4 N N ṽ ṽ u ṽ 4 } { n 4 N N n p n p N N A. 7 { n } n ṽ ṽ u N N ṽ 4 p 4 n 4 n N N We note that Thus we obtain that ṽ 4 ṽ 4 ] ṽ 4 ] ṽ4 ] ṽ 4 p ṽ 4 ṽ4 ṽ 4 ]. u } ṽ c 4 n 4 6 c 5 n 5 on 5, c n c n on, cn. p ṽ ṽ 4 p 4 ṽṽ4 4 ] ṽ 4, A. 8 p ṽ ṽ 4 p 4 ṽṽ4 4 ] ṽ 4 c 7 c 6 n4 c 7 n 5 on 5. Substitute A.8 into A.6 and A.7, we obtain that A. 9 V ] c c 6 4 c 6 γ γ c 6 γ γ c 7 4 4{cc 7 } n c 7 c 7 γ γ c8c c 7 γ on. γ c

22 M. HYODO, T. MITANI, T. HIMNO AND T. SO From A.5 and A.9, we obtain that V V ] V ] V V ] V c n c { c c } γ γ c { c c } ] γ on. γ D. Derivation of.8 By using Lemma A., ˆ can be rewritten as A. ˆ n p n u τ u ṽ n p N N ṽ N N By using above expression, we calculate the expectation of ˆ as A. ˆ ] n p n u τ u ṽ ] n p N N ṽ N N n p n N N n p n p N N n n p N N. Also, we calculate the second moment of ˆ as A. ˆ 4 ] n p u n τ u N N ṽ ] ṽ n n p p u τ u N N ṽ ] n p ṽ N N.

23 APPROXIMAT INTRVAL STIMATION FOR PMC The expected term in A. can be calculated as A. u τ u ṽ ] ṽ A. 4 u τ u ṽ ] ṽ N N n p n n p, n p n p n {N N 4 n p N N n pp }. Substitute A. and A.4 into A., we obtain that A. 5 ˆ 4 ] 4 4n n n p n p N N n pn N N. n p From A. and A.5, we obtain that ] 4 ] ] n c γ γ c γ γ on.. Derivation of. From Lemma A., we note that < V V V < N N ṽ n n ṽ V V a.s. So, we consider to evaluate A. 6 N N ṽ ] n V V n ṽ N N ṽ n V n ṽ N N ṽ V n n ṽ ] N N ṽ V ]. ] n V n ṽ

24 4 M. HYODO, T. MITANI, T. HIMNO AND T. SO The each term on right hand side in A.6 is evaluated as A. 7 N N ṽ ] n V n ṽ { nṽ ṽ 4 N N n u u ṽ } n N N ṽ ṽ ṽ 4 n u ṽ ṽ 4 ṽ ṽ ṽ 4 n n n N N 4 n n p n p n p n pp 4 n n n p n p n p n p n pp 4 n n n n p p N N n p n p n p n pp 4 { } γ γ c 4 c c c γ γ c 5 c 4 n { } c 4 4c c c 5 cn c c 4 c c γ γ c 5 on, nγ γ and N N ṽ ] A. 8 n V n ṽ { ṽ ṽ 4 N N n u u ṽ } n ṽ ṽ 4 ] N N u ṽ ṽ 4 n ṽ ṽ 4 n N N n n pp 4 n p n pp 4 γ γ cc 4 cγ γ cc n c c ccn on.

25 APPROXIMAT INTRVAL STIMATION FOR PMC 5 Combining A.6-A.8, we obtain that N N ṽ ] n V V c 4γ γ {cc } n ṽ n c 5 c 4 c 5 c cc c 5 on. γ γ F. Derivation of. From Lemma A., it holds that < U U V So, we consider to evaluate < N N ṽ n n ṽ U U a.s. A. 9 N N ṽ ] n U U n ṽ N N ṽ ] n U N N ṽ ] U n ṽ n U n ṽ U N N ṽ ] n. n ṽ We evaluate the first term on right hand side of A.9. The random variable N N ṽ /{n n ṽ }U can be rewritten as A. N N ṽ n n ṽ / U { N N ṽ p n N N ṽ n p u u ṽ ṽ } N N n / ṽ u N N n ṽ n u u ṽ u 4 ṽ N N n ṽ ṽ 4 n u u ṽ ṽ n N u N u. ṽ ṽ 4 The expectation of N N ṽ /{n ṽ }U is obtained by calculating the second moment of each term on right hand side of A.. These second

26 6 M. HYODO, T. MITANI, T. HIMNO AND T. SO moments can be calculated as follows: N N ṽ p n p u u ṽ ṽ N N n N N ṽ n / A. ṽ A. N N 4n p 4 4 n N N n p 4n p n n p N N p n n p N N p 4, u N N n ṽ n u u ṽ N N n p 4 p n p 4 n N N n p 4 p n p 4, u 4ṽ N N A. n ṽ ṽ 4 n u u ṽ N N p n p 4n p p p n p 4n p {n N N }p p p n p 4n p n p 4n p, ṽ n N u N u N n p N p A. 4 ṽ ṽ 4 n p 4n p. From A.-A.4, we can obtain that N N ṽ ] A. 5 n U n ṽ N N 4n p 4 4 N { pp n N n p p } n p 4n p n p { n pn p N N } p n p 4 N N n p n p 4 γ γ c 4 γ γ 4c n c {cγ c γ } cc γ ] γ on. cγ γ

27 APPROXIMAT INTRVAL STIMATION FOR PMC 7 Also, we have that N N ṽ ] A. 6 n U n ṽ N N n p nn p 4 N N n p n p nn n p p 4 cγ γ {5 cc 4} γ γ c n c c γ ] γ on, c and A. 7 N N ṽ ] n n ṽ N N n p n p n n p 4 c γ γ c { cc} cγ γ c n on. Combining A.5-A.7, we obtain that N N ṽ ] n U U { 4 γ n ṽ n ccγ γ γ γγ cγ γ } on. Acknowledgments The authors would like to express their gratitude to Professor Yasunori Fujikoshi for many valuable comments and discussions. References ] Chung, H.-C. and Han, C.-P. 9. Conditional confidence intervals for classification error rate. Computational Statistics and Data Analysis, 5, ] Fujikoshi, Y.. rror bounds for asymptotic approximations of the linear discriminant function when the sample sizes and dimensionality are large. Journal of Multivariate Analysis, 7, -7. ] Fujikoshi, Y. and Seo, T Asymptotic approximations for PMC s of the linear and the quadratic discriminant functions when the sample sizes and the dimension are large. Random Operators and Stochastic quations, 6, 69-8.

28 8 M. HYODO, T. MITANI, T. HIMNO AND T. SO 4] Kubokawa, T., Hyodo, M. and Srivastava, M. S.. Asymptotic expansion and estimation of PMC for linear classification rules in high dimension. Journal of Multivariate Analysis, 5, ] Lachenbruch, P. A On expected probabilities of misclassification in discriminant analysis, necessary sample size, and a relation with the multiple correlation coefficient. Biometrics., 4, ] Lachenbruch, P. A. and Mickey, M. R stimation of error rates in discriminant analysis. Technometrics,, -. 7] McLachlan, G. J Confidence intervals for the conditional probability of misallocation in discriminant analysis. Biometrics,, ] Okamoto, M. 96. An asymptotic expansion for the distribution of the linear discriminant function. Annals of Mathematical Statistics, 4, ] Okamoto, M Correction to An asymptotic expansion for the distribution of the linear discriminant function. Annals of Mathematical Statistics, 9, ] Siotani, M. 98. Large sample approximations and asymptotic expansions of classification statistic. Handbook of Statistics P. R. Krishnaiah and L. N. Kanal, ds., North-Holland Publishing Company, 6-. ] Yamada, T., Himeno, T. and Sakurai, T. 5. Cut-off point of linear discriminant rule for large dimension. Technical Report, No.5-4, Hiroshima statistical research group, Hiroshima University.

29 APPROXIMAT INTRVAL STIMATION FOR PMC 9 Table. The coverage probabilities p α \ n N : N 4 : : : : : : : : : Table. The coverage probabilities p α \ n N : N : : : : : : : : : Table. The coverage probabilities p 5 α \ n N : N 5 : : : : : : : : :

30 M. HYODO, T. MITANI, T. HIMNO AND T. SO Table 4. The expected lengths p n 4 α N : N L L L L L L : : : : : : : : : Table 5. The expected lengths p n α N : N L L L L L L : : : : : : : : : Table 6. The expected lengths p 5 n 5 α N : N L L L L L L : : : : : : : : :

31 APPROXIMAT INTRVAL STIMATION FOR PMC N, N 5, 75 N, N 5, 5 N, N 75, 5 4 N, N 75, 5 N, N 5, 5 N, N 5, 75 4 N, N 5, 75 N, N 5, 5 N, N 75, 5 Figure. Q-Q plots of B N,N for Case B

32 M. HYODO, T. MITANI, T. HIMNO AND T. SO N, N 5, 5 N, N, N, N 5, 5 N, N 75, 5 N, N 5, 5 N, N 5, 75 N, N, N, N, N, N, Figure. Q-Q plots of H p,n,n in Case A p

33 APPROXIMAT INTRVAL STIMATION FOR PMC 4 4 N, N, N, N, N, N, N, N 5, 45 N, N, N, N 45, 5 N, N, 6 N, N 4, 4 N, N 6, Figure. Q-Q plots of H p,n,n in Case A p Masashi Hyodo Department of Mathematical Information Science, Tokyo University of Science - Kagurazaka, Shinjuku-ku, Tokyo 6-86, Japan -mail: caicmhy@gmail.com Tomohiro Mitani Graduate School of Science, Tokyo University of Science - Kagurazaka, Shinjuku-ku, Tokyo 6-86, Japan Tetsuto Himeno Faculty of Science and Technology, Seikei University -- Kichijoji-Kitamachi, Musashino-shi, Tokyo 8-86, Japan

34 4 M. HYODO, T. MITANI, T. HIMNO AND T. SO Takashi Seo Department of Mathematical Information Science Tokyo University of Science, - Kagurazaka, Shinjuku-ku, Tokyo 6-86, Japan