# Hypothesis Designs for Three-Hypothesis  Test Problems

Authors: Yan Li; Xiaolong Pu
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

As a helpful guide for applications, the alternative hypotheses
of the three-hypothesis test problems are designed under the required error probabilities and
average sample number in this paper. The asymptotic formulas and the proposed
numerical quadrature formulas are adopted, respectively, to obtain the hypothesis
designs and the corresponding sequential test schemes under the Koopman-Darmois
distributions. The example of the normal mean test shows that our methods are
quite efficient and satisfactory for practical uses.


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## Body

## 1. Introduction

In practice, the multihypothesis test problems are of considerable interest in the areas of engineering, agriculture, clinical medicine, psychology, and so on. For instance, the multihypothesis tests are involved in pattern recognition [1–4], multiple-resolution radar detection [5–7], products' comparisons [8, 9], and information detection [10]. Before the inspections, the hypotheses must be determined according to such practical needs as the balance of risks and costs. As Wetherill and Glazebrook [11] pointed out, combinations of hypotheses, risks, and costs may need to be tried iteratively until an acceptable design is attained. This bothers and burdens the practitioners.To avoid too many troublesome trials and to produce the hypotheses directly, we discuss the hypothesis designs under the controlled risks and expected costs in this paper. As an initial exploration, only the three-hypothesis test problems are considered here. Indeed, our methods may extend to the multihypothesis cases.In practice, test costs are mainly determined by sample sizes. Therefore, the sample size becomes an issue relating to the statistical analysis of problems in many aspects; see for example, Chen et al. [12], Oliveira et al. [13], Li and Zhao [14], Li et al. [15], Bakhoum and Toma [16], Cattani [17], as well as Cattani and Kudreyko [18]. Accordingly, we consider the Average Sample Number (ASN), which is one of the most important values in evaluating the expected costs of sequential test schemes.In the three-hypothesis test problem, the null hypothesis is always set as a standard and medium status. For example, Anderson [8] discussed the three-hypothesis test problem to decide whether the difference of two yarns' strength is zero (the null hypothesis), positive or negative. Realistically, the standard and medium status (denoted as θ0) is definite, while the two alternatives beside it need to be designed to balance the risks and costs. Thus, in this paper, we try to design the alternatives θ-1 and θ1(θ-1<θ0<θ1) under the required error probabilities and ASN for testing the parameter θ of the Koopman-Darmois distribution(1.1)fθ(x)=exp{l(x)+θx-b(θ)},whereb(θ)isaconvexfunction,θ∈Θ.To simplify the discussion, we only consider the designs of the two alternative hypotheses symmetric with the null hypothesis, that is,θ1-θ0=θ0-θ-1=k(>0). Actually, the asymmetric designs may be obtained by extending our methods slightly.Then, the test problem here is(1.2)H-1:θ=θ-1=θ0-kvs.H0:θ=θ0vs.H1:θ=θ1=θ0+k.For the multihypothesis test problems, Armitage [19] provided a classical test scheme by simultaneously applying the method of Sequential Probability Ratio Test (SPRT) on each pair of the hypotheses. This test scheme pattern is simple and easy to implement. When testing the three hypotheses for the Koopman-Darmois distribution (1.1), Armitage's scheme may be illustrated as in Figure 1, where AL//CM are boundaries for “θ=θ1 versus θ=θ0" and CP//DQ are for “θ=θ0 versus θ=θ-1" when the boundaries for “θ=θ1 versus θ=θ-1" are encircled by AL and DQ and thus are neglected. According to Figure 1, the decision rule should be(1.3)AcceptH1ifTn≥a+ntanψ,AcceptH0ifc+(n-n0)tanφ≤Tn≤c+(n-n0)tanψ,AcceptH-1ifTn≤d+ntanφ,Continuesamplingwithoutanydecision,otherwise,
where Tn=∑i=1nXi and X1,X2,… are independent sequential observations from a Koopman-Darmois distribution.Figure 1For the givenθ-1,θ0, and θ1, the test scheme in Figure 1 is decided by 6 parameters (n0,a,c,d,ψ,φ). ψ and φ may be determined according to Armitage [19], that is, tanψ=[b(θ1)-b(θ0)]/(θ1-θ0), tanφ=[b(θ0)-b(θ-1)]/(θ0-θ-1) under the Koopman-Darmois distribution (1.1), then the remaining 4 parameters (n0,a,c,d) form the scheme. Altogether with the hypothesis design value k in the test problem (1.2), the 5 underdetermined values are (k,n0,a,c,d).In the three-hypothesis test problems, the error probabilitiesα and β should be assigned to the error probabilities (γ1,γ2,γ3,γ4) in correspondence with the requirements(1.4)P(AcceptH0∣H1)≤γ1,P(AcceptH0∣H-1)≤γ2,P(AcceptH1∣H0)≤γ3,P(AcceptH-1∣H0)≤γ4.
Commonly, we set γ1=γ2=β, γ3=γ4=α/2, as Payton and Young [20, 21] indicated.And the request on the ASN should be(1.5)ASN(θASN)≤N,
where N(>0) is a provided integer and θASN(∈Θ) is the point at which the ASN needs to be controlled. θASN may take values of θ-1, θ0, θ1, and so on according to practical needs.Then, under the constraints (1.4) and (1.5), we may find the proper (k,n0,a,c,d) by virtue of their relationships with the error probabilities and ASN.Unfortunately, however, to the best knowledge of the authors, the accurate formulas for the performances of the three-hypothesis test scheme are still unavailable possibly because of its sequential feature and anomalistic continuing sampling area. In the following, the hypothesis designs and the test scheme parameters are determined under the required error probabilities and ASN in terms of some approximate expressions, that is, the asymptotic formulas and the proposed numerical quadrature formulas.


## 2. Designs under Asymptotic Formulas

In this section, we try to find the hypothesis designs and test schemes under the required error probabilities and ASN by virtue of the asymptotic formulas of the multihypothesis sequential test scheme by Dragalin et al. in [22, 23].Firstly, we discuss how to control the error probabilities. LetCi be the critical value of the logarithmic likelihood ratio function for accepting θi, and let Ri be the probability limit of incorrectly accepting θi, i=-1,0,1. According to Dragalin et al. [22], under the condition of equal prior probabilities for the three hypotheses, the probability of wrongly accepting θi for the Armitage [19] scheme may be controlled by Ri if the critical value Ci is set as(2.1)Ci=ln{23Ri},i=-1,0,1.
Thus, the error probabilities (γ1,γ2,γ3,γ4) are in control if we follow the critical values in (2.1), where R-1=γ4, R0=min{γ1,γ2}, and R1=γ3. Setting the critical values (C-1,C0,C1) equal to the corresponding logarithmic likelihood ratio functions, we have the following expressions for the test scheme parameters under the Koopman-Darmois distribution (1.1):(2.2)n0=C0(1/(θ0-θ-1)-1/(θ0-θ1))(b(θ0)-b(θ1))/(θ0-θ1)-(b(θ0)-b(θ-1))/(θ0-θ-1)=2C0b(θ0+k)+b(θ0-k)-2b(θ0),a=C1θ1-θ0=C1k,c=C0θ0-θ1+n0tanψ=-C0k+n0b(θ0+k)-b(θ0)k,d=C-1θ-1-θ0=-C-1k.Note that the expressions in (2.2) define the relations between the hypothesis design parameter k and the test scheme parameters (n0,a,c,d), while k has not been determined so far.In the following, the hypothesis design parameterk is found with the help of Dragalin et al.'s asymptotic ASN formulas [23].Based on the nonlinear renewal theory, Dragalin et al. [23] summarized and developed the asymptotic ASN formulas under max{α,β}→0. Specifically, when θ1-θ0=θ0-θ-1, the asymptotic ASN formulas under the two alternatives θ-1 and θ1 are(2.3)ASN(θi)≈Ci+OθiDθi,i=-1,1,
where Dθi=minj≠iEθi(ln{fθi(x)/fθj(x)}) and Oθi is the expected limiting overshoot under θi, i=-1,0,1.And for the null hypothesisθ0 under θ1-θ0=θ0-θ-1, the asymptotic ASN formula is(2.4)ASN(θ0)≈F2(C0,Dθ0,v)+Oθ0Dθ0,
where F2(x,q,u)=x+uh2*x/q+u2(h2*)2/(4q2)+u2(h2*)2/(2q) (h2*=0.5641895835 here), and v is the value related to the covariance of the logarithmic likelihood ratio functions.Notice that the approximate ASN formulas (2.3) and (2.4) only depend on the hypothesis design parameter k when θ0 is given. Therefore, to find the proper hypothesis design under the desired number N, we set up an equation about k to meet the ASN requirement on one of the three hypothesis values, that is,(2.5)ASN(θASN)=N,
where θASN may be θ-1,θ0, or θ1.Then, the hypothesis design parameterk is the solution to (2.5) and the test scheme with (n0,a,c,d) may be obtained correspondingly according to (2.2). Illustrations are provided in Example 1 for testing the normal mean with the variance known.Example 1.
Suppose that the sequential observationsX1,X2,… are independent and identically distributed (i.i.d.) with N(μ,1). Let μ0=0, γ1=γ2=β, and γ3=γ4=α/2. Small values (≤30) are set on N as practical sequential inspections always require.
Accordingly, we haveC0=ln{2/(3γ1)}, C1=C-1=ln{2/(3γ3)}. In this example, the test scheme parameters should be
(2.6)n0=2C0k2,a=C1k,c=0,d=-a,tanψ=k2,tanφ=-tanψ.
And for the normal distribution N(μ,1), there are
(2.7)Dμi=minj≠i(μi-μj)22=k22,i=-1,0,1,Oμi=1+k24-k∑l=1∞1l[ϕ(k2l)-k2lΦ(-k2l)],i=-1,0,1,v=2k,
where ϕ(·) and Φ(·) are the probability density function (p.d.f.) and cumulative distribution function (c.d.f.) of the standard normal distribution, respectively.Consider the following 4 cases, respectively:(2.8)μASN=μ0,α=β=0.002,μASN=μ0,α=β=0.05,μASN=μ1,α=β=0.002,μASN=μ1,α=β=0.05.
Then, solving (2.5), we obtain the hypothesis designs k as shown in Column 2 of Tables 1, 2, 3, and 4. The corresponding test scheme parameters (n0,a,tanψ) from (2.6) are listed in Columns 3–5 of Tables 1–4. To evaluate the method's efficiency, we record the Monte Carlo simulation study results with 1,000,000 replicates in Tables 5, 6, 7, and 8, where ASN′(μASN) is the simulated value of ASN(μASN) and ɛ is the relative difference between ASN′(μASN) and N. Note that the simulated probabilities under μ-1 are neglected here since they are nearly equivalent to their counterparts under μ1 in terms of the schemes' symmetry.Table 1
Hypothesis designs and test schemes forμASN=μ0, α=β=0.002.
NUnder Asymptotic formulasUnder Gaussian quadrature formulasɛk(%)kn0atanψkn0atanψ52.10902.61213.08311.05451.89532.68763.07250.947711.28101.44005.60304.51550.72001.31935.88334.65760.65979.15151.16048.62795.60340.58021.06749.02665.89100.53378.71200.997711.67096.51700.49890.922812.33546.90390.46148.12250.888214.72577.32040.44410.824615.65927.79440.41237.71300.808217.78888.04580.40410.752318.99498.59840.37627.43Table 2
Hypothesis designs and test schemes forμASN=μ0, α=β=0.05.
NUnder Asymptotic formulasUnder Gaussian quadrature formulasεk(%)kn0atanψkn0atanψ51.60662.00722.04380.80331.30252.19532.21750.651323.35101.09084.35423.01020.54540.90875.00283.42330.454420.04150.87676.73953.74500.43840.74018.00014.22420.370118.46200.75279.14484.36240.37640.639211.00005.25250.319617.76250.669311.56304.90540.33470.570814.00005.90860.285417.26300.608513.99035.39570.30430.520417.00006.52440.260216.93Table 3
Hypothesis designs and test schemes forμASN=μ1, α=β=0.002.
NUnder Asymptotic formulasUnder Gaussian quadrature formulasεk(%)kn0atanψkn0atanψ51.78933.62903.63400.89471.71053.28643.46440.85534.61101.21797.83345.33910.60901.19007.18755.22550.59502.34150.980012.09736.63500.49000.966711.15606.56390.48341.38200.841916.39267.72360.42100.835215.15947.68820.41760.80250.749020.70808.68090.37450.746019.18518.67700.37300.40300.681225.03799.54540.34060.680323.22669.56980.34020.13Table 4
Hypothesis designs and test schemes forμASN=μ1, α=β=0.05.
NUnder Asymptotic formulasUnder Gaussian quadrature formulasɛk(%)kn0atanψkn0atanψ51.30893.02372.50850.65451.18752.91242.48580.593810.22100.88356.63693.71640.44180.82646.25453.82130.41326.91150.708310.32684.63580.35420.671510.00004.83510.33585.48200.607114.05665.40850.30360.580313.49935.68720.29024.62250.539317.81186.08830.26970.518317.00616.43680.25924.05300.489921.58536.70220.24500.472720.88427.11310.23643.64Table 5
Simulated performances for the schemes under asymptotic formulas in Table1.
NWhenμ1 is trueWhenμ0 is trueASN′(μ0)ε(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.00070.99930.00050.99900.00054.70536.261000.00090.99910.00070.99860.00079.36016.841500.00110.99890.00070.99850.000813.98457.262000.00110.99890.00080.99840.000818.59107.582500.00120.99880.00090.99820.000923.18857.813000.00130.99870.00100.99800.001027.80467.90Table 6
Simulated performances for the schemes under asymptotic formulas in Table2.
NWhenμ1 is trueWhenμ0 is trueASN′(μ0)ɛ(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.01860.98140.01480.97030.01494.440912.591000.02730.97270.01950.96100.01958.422918.721500.03170.96820.02190.95620.021912.368721.272000.03230.96770.02360.95290.023516.476721.382500.03440.96560.02460.95080.024520.353518.593000.03610.96380.02520.94940.025524.223319.26Table 7
Simulated performances for the schemes under asymptotic formulas in Table3.
NWhenμ1 is trueWhenμ0 is trueASN′(μ1)ɛ(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.00080.99920.00050.99900.00054.99070.191000.00100.99900.00070.99850.00089.98180.181500.00100.99900.00090.99830.000814.97050.202000.00120.99880.00100.99810.000919.95250.242500.00130.99870.00090.99810.001024.94940.203000.00120.99880.00100.99800.001029.90590.31Table 8
Simulated performances for the schemes under asymptotic formulas in Table4.
NWhenμ1 is trueWhenμ0 is trueASN′(μ1)ɛ(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.02260.97740.01730.96550.01724.79764.051000.03130.96870.02210.95600.02199.43225.681500.03360.96640.02410.95180.024114.12215.852000.03460.96540.02550.94920.025318.76656.172500.03670.96320.02650.94710.026423.33166.673000.03710.96280.02700.94590.027027.99896.67Obviously, the accuracy of the solutionk to (2.5) is decided by the efficiency of the ASN formulas (2.3) and (2.4). On one hand, from Dragalin et al. [23] and the ε’s in Tables 5–8, we conclude that the formulas in (2.3) for ASN(θ-1) and ASN(θ1) are more efficient than the one in (2.4) for ASN(θ0) when testing the normal mean. On the other hand, the asymptotic ASN formulas perform better under smaller error probabilities since the asymptotic limit is taken as max{α,β}→0. For applications, with such a simple computation, the efficiency of the design is quite satisfactory for small error probabilities conditions.However, this method may only serve to control the ASN on the three hypothesis values since the asymptotic ASN formulas out of these points are absent so far. And the quantitiesDθi,Oθi(i=-1,0,1), and v should be deduced according to specific distributions (see [23]). Besides, the discrepancies between the real performances and the required ones show the method's conservativeness. In the next section, an improved method is proposed and more efficient formulas are developed through the numerical quadrature.


## 3. Designs under Numerical Quadrature Formulas

This section proposes a method to obtain more efficient hypothesis designs and test schemes through a system of equations based on the numerical quadrature formulas of the error probabilities and ASN.In studies by Payton and Young in [20, 21], for the provided hypotheses, the error probabilities (γ1,γ2,γ3,γ4) are approximately attained by solving a system of equations about the 4 scheme parameters (n0,a,c,d). This method is hoped to fully make use of the required error probabilities and to obtain efficient designs. Enlightened by Payton and Young, we propose to find the hypothesis design and test scheme by solving the following system of equations:(3.1)P(AcceptH1∣H0)=γ3,P(AcceptH-1∣H0)=γ4,P(AcceptH0∣H1)=γ1,P(AcceptH0∣H-1)=γ2,ASN(θASN)=N.Obviously, the key is to find the formulas of the error probabilities and ASN on the left side of the equations in (3.1). Unfortunately, the available approximate formulas cannot meet applicable needs well. For example, Payton and Young [20, 21] adopted the formulas under the continuous-time process and the required minimum sample size before decisions, and obtained some inefficient results. Also, as mentioned in Section 2, Dragalin et al.'s results are restricted to the conditions of small error probabilities and θASN=θi(i=-1,0,1) [22, 23].To find efficient and applicable designs, we develop the approximate formulas through the numerical quadrature for the three-hypothesis test scheme's performances on the error probabilities and ASN.To deduce the formulas for the realistic discrete-time situation, we denotent as the minimum integer that is not less than n0. Let Lj and Uj be the values on the two boundaries DQ and AL in Figure 1 at n=j, that is, Lj=d+jtanφ, Uj=a+jtanψ, j=1,…,nt. Denote cL=c+(nt-n0)tanφ, cU=c+(nt-n0)tanψ, a′=a+nttanψ, and d′=d+nttanφ. With the decision rule (1.3), we rewrite the system of (3.1) as(3.2)R1(θ0)+S1(θ0)-L1(θ0)=γ3,R-1(θ0)+S-1(θ0)-L-1(θ0)=γ4,L1(θ1)+L-1(θ1)+L0(θ1)=γ1,L1(θ-1)+L-1(θ-1)+L0(θ-1)=γ2,N0(θASN)+N1(θASN)+N-1(θASN)+ntJ(θASN)=N,
where R1(θ)=Pθ(AcceptH1throughALwhenn≤nt);R-1(θ) = Pθ(AcceptH-1 through DQwhenn≤nt); S1(θ)=Pθ(cU<Tnt<a′, Lj<Tj<Uj, j=1,…,nt-1); S-1(θ)=Pθ(d′<Tnt<cL, Lj<Tj<Uj, j=1,…,nt-1); L1(θ)=Pθ(AcceptH0throughCMwhenn>nt); L-1(θ) = Pθ(AcceptH0throughCPwhenn>nt); L0(θ)=Pθ(AcceptH0atn=nt); J(θ)=S1(θ)+S-1(θ)+L0(θ);N0(θ) is the average sample number from a point in (d,a) at n=0 to the point of accepting H1 or H-1 when n≤nt. N1(θ) is the average sample number from a point in (cU,a′) at n=nt to the point of making a decision when n>nt. And N-1(θ) is the average sample number from a point in (d′,cL) at n=nt to the point of making a decision when n>nt.The following theorem provides the approximate formulas through the numerical quadrature for the quantities in (3.2). In fact, these formulas are developed by a stepwise dealing for the continuing sampling area before n0 and the results by Li and Pu in [24] for the parallel lines areas inside AL//CM and inside CP//DQ, respectively. With such an idea, the proof of Theorem 3.1 is trivial and is neglected here.Theorem 3.1.
Assume thatX1,X2,… are i.i.d. observations. Let fθ(x) and Fθ(x) be the p.d.f. and c.d.f. of X, respectively. Assume that Fθ-(x)=Pθ(X<x). Denote g̃1θ(x)=fθ(x), and g̃j+1θ(x) = ∑i=1mω(ui(j))g̃jθ(ui(j))fθ(x-ui(j)), where ui(j) is the ith numerical quadrature root for [Lj,Uj], i=1,…,m, j=1,…,nt-1, and ω(u) is the corresponding weight for the numerical quadrature root u. Let ui(nt) and ui(nt)' be the ith numerical quadrature root for [cU,a′] and for [d′,cL], respectively, i=1,…,m.
Then, the approximate valuesR̃1(θ), R̃-1(θ), S̃1(θ), S̃-1(θ), L̃1(θ), L̃-1(θ), L̃0(θ), Ñ0(θ), Ñ1(θ), and Ñ-1(θ) for the quantities in (3.2) are the following.(1)
(3.3)R̃1(θ)=1-Fθ-(U1)+∑j=2nt∑i=1mω(ui(j-1))g̃j-1θ(ui(j-1))[1-Fθ-(Uj-ui(j-1))].(2)
(3.4)R̃-1(θ)=Fθ(L1)+∑j=2nt∑i=1mω(ui(j-1))g̃j-1θ(ui(j-1))Fθ(Lj-ui(j-1)).(3)
(3.5)S̃1(θ)=∑i=1mω(ui(nt-1))g̃nt-1θ(ui(nt-1))[Fθ-(a′-ui(nt-1))-Fθ(cU-ui(nt-1))].(4)
(3.6)S̃-1(θ)=∑i=1mω(ui(nt-1))g̃nt-1θ(ui(nt-1))[Fθ-(cL-ui(nt-1))-Fθ(d′-ui(nt-1))].(5)
Denoteqθ(1)(x)=[qθj(1)(x)]1×m, where qθj(1)(x)=ω(uj(nt))fθ(uj(nt)+tanψ-x), j=1,…,m; pθ(1)=[pθi(1)]m×1, where pθi(1)=Fθ(cU+tanψ-ui(nt)), i=1,…,m; Qθ(1)=[qθij(1)]m×m, where qθij(1)=ω(uj(nt))fθ(uj(nt)+tanψ-ui(nt)), i,j=1,…,m. Let I be the m×m identity matrix. Then, there is
(3.7)L̃1(θ)=∑i=1mω(ui(nt))g̃ntθ(ui(nt))OC̃1θ(ui(nt)),
where OC̃1θ(x)=Fθ(cU+tanψ-x)+qθ(1)(x)(I-Qθ(1))-1pθ(1).(6)
Denoteqθ(2)(x)=[qθj(2)(x)]1×m, where qθj(2)(x)=ω(uj(nt)')fθ(uj(nt)'+tanφ-x), j=1,…,m; pθ(2)=[pθi(2)]m×1, where pθi(2)=Fθ(d′+tanφ-ui(nt)'), i=1,…,m; Qθ(2)=[qθij(2)]m×m, where qθij(2)=ω(uj(nt)')fθ(uj(nt)'+tanφ-ui(nt)'), i,j=1,…,m. Then, we have
(3.8)L̃-1(θ)=S̃-1(θ)-∑i=1mω(ui(nt)′)g̃ntθ(ui(nt)′)OC̃-1θ(ui(nt)′),
where OC̃-1θ(x)=Fθ(d′+tanφ-x)+qθ(2)(x)(I-Qθ(2))-1pθ(2).(7)
(3.9)L̃0(θ)=∑i=1mω(ui(nt-1))g̃nt-1θ(ui(nt-1))[Fθ(cU-ui(nt-1))-Fθ-(cL-ui(nt-1))].(8)
(3.10)Ñ0(θ)=1+∑j=2ntj∑i=1mω(ui(j-1))g̃j-1θ(ui(j-1))[1-Fθ-(Uj-ui(j-1))+Fθ(Lj-ui(j-1))].(9)
(3.11)Ñ1(θ)=∑i=1mω(ui(nt))g̃ntθ(ui(nt))ñ1θ(ui(nt)),
where ñ1θ(x)=1+qθ(1)(x)(I-Qθ(1))-11 with 1 being the m×1 vector of 1's.(10)
(3.12)Ñ-1(θ)=∑i=1mω(ui(nt)′)g̃ntθ(ui(nt)′)ñ-1θ(ui(nt)′),
where ñ-1θ(x)=1+qθ(2)(x)(I-Qθ(2))-11.Notice that the values on the left side of the equations in (3.2) must be obtained through a computer program with much iterative work, which reveals the method's complexity in computation and impairs the speed of solving the system of (3.2). Nevertheless, the time it costs is tolerable when the accuracy of solving the equations is not too demanded.Example 1 (Continued).
Consider the same problems as those in Example1 in Section 2. By applying the formulas (3.3)–(3.12) and the 64 Gaussian quadrature roots, we solve the system of (3.2) in a computer program. The hypothesis designs and the corresponding test schemes are listed in Columns 6–9 of Tables 1–4. As a comparison with the method under the asymptotic formulas in Section 2, ɛk in Column 10 of Tables 1–4 records the relative difference between the two hypothesis designs of the two methods. The Monte Carlo simulation study with 1,000,000 replicates in Tables 9, 10, 11, and 12 reveal the schemes' real performances.Table 9
Simulated performances for the schemes under Gaussian quadrature formulas in Table1.
NWhenμ1 is trueWhenμ0 is trueASN′(μ0)ɛ(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.00210.99790.00100.99800.00105.00030.011000.00200.99800.00100.99800.001010.00780.081500.00190.99810.00100.99800.000914.99490.032000.00200.99800.00100.99800.001019.98880.062500.00200.99800.00110.99790.001024.99800.013000.00200.99800.00100.99800.001029.99720.01Table 10
Simulated performances for the schemes under Gaussian quadrature formulas in Table2.
NWhenμ1 is trueWhenμ0 is trueASN′(μ0)ɛ(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.05000.95000.02500.95010.02494.99980.001000.04990.95010.02490.94990.025110.00320.031500.04860.95140.02710.94560.027415.00380.032000.05200.94800.02280.95430.022920.00610.032500.05100.94900.02360.95300.023424.97950.083000.05120.94880.02350.95320.023329.99260.02Table 11
Simulated performances for the schemes under Gaussian quadrature formulas in Table3.
NWhenμ1 is trueWhenμ0 is trueASN′(μ1)ɛ(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.00190.99810.00110.99790.00104.99940.011000.00200.99800.00100.99800.00109.99170.081500.00200.99800.00100.99800.001015.01540.102000.00200.99800.00100.99790.001020.01290.062500.00200.99800.00100.99800.001024.98800.053000.00200.99800.00100.99800.001030.01510.05Table 12
Simulated performances for the schemes under Gaussian quadrature formulas in Table4.
NWhenμ1 is trueWhenμ0 is trueASN′(μ1)ɛ(%)Probability of acceptingProbability of acceptingμ-1μ0μ1μ-1μ0μ1500.04990.95010.02520.95000.02484.99730.051000.04990.95010.02540.94960.02509.98760.121500.05050.94950.02500.94990.025215.01810.122000.05020.94980.02490.95020.024920.01050.052500.04980.95020.02510.95000.024924.98960.043000.04990.95000.02500.95020.024830.01800.06The real performances in Tables9–12 show that the requirements on controlling the error probabilities and ASN may be fully made use of under this method and the numerical quadrature formulas are almost accurate. Therefore, the hypothesis designs and test schemes are highly efficient in terms of, for instance, more efficient designs with smaller k in Tables 1–4 under this method.To further explain the methods, an example of the airbag quality inspection is provided in the appendix.


## 4. Conclusions and Remarks

For the three-hypothesis test problems, the methods of designing the hypotheses, together with obtaining the corresponding test schemes, are proposed by adopting asymptotic formulas or numerical quadrature formulas in this paper. As a helpful guide for practitioners, they aid to directly find proper hypotheses under controlled risks and costs in preventing from too many iterative trials on combinations of hypotheses to meet practical needs.The asymptotic formulas and the numerical quadrature formulas are both alternative tools for the hypothesis designs. Several aspects should be considered when choosing between them in applications.(1)
The method with numerical quadrature formulas outperforms the one under the asymptotic formulas especially when the error probabilities are not very small, as the example shows. In reality, the required error probabilities always range from 0.05 to 0.30 in sequential inspections, which seems to suggest choosing the numerical quadrature formulas to obtain efficient designs.(2)
In computation, the asymptotic formulas provide great convenience for applications, while the numerical quadrature formulas demand much iterative computational work especially when the number of numerical quadrature roots is large. But from the computation with the 64 Gaussian quadrature roots in the example, the time it costs in a common computer is tolerable if the start values for the system of equations are proper. We recommend finding the designs under the asymptotic formulas first, and then apply them as starts to obtain more efficient hypotheses from the numerical quadrature formulas when needed.(3)
When adopting the asymptotic formulas, the expressions for the quantitiesDθi,Oθi(i=-1,0,1), and v should be developed for a specific distribution (see [23]). For the use of numerical quadrature formulas, the quadrature roots may be particularly arranged to fit the support points in the discrete distributions (e.g, see Reynolds and Stoumbos [25]). And for the θASN out of (θ-1,θ0,θ1), only the method with numerical quadrature formulas may take effect.Actually, the two methods may apply to any distribution out of the Koopman-Darmois family. However, the test schemes under these distributions may be different from that in Figure1, and the numerical quadrature formulas should be changed according to the test scheme patterns.For the hypothesis designs asymmetric with the null hypothesis or the multihypothesis test problems, the methods proposed in this paper are still applicable by some extensions of adding more constraints on the designs. The hypothesis design problems under other requests, for example, under the desire of stopping sampling before a limit guaranteed by a provided probability, are still open to scholars and practitioners.



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