MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2021/99175759917575Research ArticleApproximate Calculation Method for Noncentral t-Distribution Quantilehttps://orcid.org/0000-0002-8116-9283GaoJun12https://orcid.org/0000-0001-7299-2115YaoJitao12Gómez-DénizEmilio1College of Civil EngineeringXi’an University of Architecture and TechnologyXi’an 710055Chinaxauat.edu.cn2Key Lab of Structural Engineering and Earthquake ResistanceMinistry of Education (XAUA T)Xi’an 710055China202125520212021133202175202125520212021Copyright © 2021 Jun Gao and Jitao Yao.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the process of structural design and structural performance evaluation, the inference of the reliable life of the structure and the representative value of the material strength is a necessary work. The determination of material strength is the presumption of the quantile of the normal distribution, and the determination of the confidence level of the quantile of the normal distribution involves the noncentral t-distribution function. However, the calculation of the quantile is very complicated and is often provided in the form of a numerical table, which often involves multiparameter interpolation calculation, so it is not convenient to apply. The existing approximate calculation methods for noncentral t-distribution quantiles have strict application conditions, and the calculation process is relatively cumbersome. It is still difficult to meet actual needs in terms of fitting accuracy, application range, and convenience. In this paper, a new calculation method for noncentral t-distribution quantiles is proposed by introducing new probability expressions and related approximate distributions, based on theoretical derivation and numerical fitting. The comparative analysis results show that the method not only is convenient for calculation but also has the advantages of higher accuracy and wider application range, and it is more in line with the actual needs of engineering.

National Natural Science Foundation of China51278401
1. Introduction

The structural reliability design covers the entire civil engineering and is the basic guarantee for the normal operation of the building structures [1, 2]. There are various reliability modeling methods. In practical engineering, uncertain failure modes can result in the system failure . The noncentral t-distribution is mainly used for small sample inference of normal distribution quantile, and it is widely used in the estimation of structural reliability life, representative value of material strength, and so on [6, 7]. The probability density function of the noncentral t-distribution is complex, and its quantile is usually calculated by numerical methods and provided in the form of a numerical table. However, in the current numerical table, only some typical values are provided, and in other cases, interpolation calculation is required, and multiple parameters are involved, which is inconvenient to apply . Therefore, a variety of approximate calculation methods have been proposed, but they are still difficult to meet the actual needs in terms of fitting accuracy, scope of application, and convenience [12, 13].

In this paper, a new calculation method for noncentral t-distribution quantiles is proposed by introducing new probability expressions and related approximate distributions, based on theoretical derivation and numerical fitting. The comparative analysis results show that the method not only is convenient for calculation, but also has the advantages of higher accuracy and wider application range, and it is more in line with the actual needs of engineering.

2. Current Approximate Calculation Method

The random variable X obeys the normal distribution N(μ, σ2), and the distribution parameters μ and σ2 are unknown. Then, the upper p and 1 − p quantile values of X can be expressed as(1)xp=μz1pσ,x1p=μ+z1pσ,where z1p is the upper 1 − p quantile of the standard normal distribution; p is the assurance rate of X ≥ xp or X ≤ x1−p, and its value is generally more than 0.5, and xp and x1−p are small and large quantile values, respectively.

Let X1, X2, …, Xn be the n samples of the random variable X. According to the statistical method, after obtaining the sample realization value x1, x2, …, xn, the lower limit estimate of xp and the upper limit estimate of x1−p are, respectively,(2)x^p=x¯ks,x^1p=x¯+ks,where(3)x¯=1ni=1nxi,s=1n1i=1nxix¯2,k=tn1,z1pn,αn,where tn1,z1pn,α is the upper α quantile value of the noncentral t-distribution with a degree of freedom of X and a parameter of z1pn, and α is the significance level.

The coefficient k is related to the sample size n, the distribution parameter z1pn and the significance level α, etc., and is mainly determined by the quantile value tn1,z1pn,α of the noncentral t-distribution. According to the approximate statistics of the F distribution , it is recommended to approximate as(4)tn1,λ,α=λCnFγ1,γ1,α,λ>0,λCnFγ1,γ1,1α,λ<0,where(5)Cn=2n1Γn/2Γn1/2,γ1=2λ2,γ2=2Cn21Cn2,where tn1,γ,α is the upper α quantile value of the noncentral t-distribution with the degree of freedom n − 1 and the parameter γ. Fγ1,γ1,α and Fγ1,γ1,1α are the upper α and 1 − α quantile values of the F distribution with parameters γ1 and γ2, respectively. When γ1 and γ2 are nonpositive integers, the values of Fγ1,γ1,α and Fγ1,γ1,1α can be determined by linear interpolation method. This method is mainly applicable to the case of n > 9, and, in general, it is still inconvenient to check the table.

When(6)T1=X¯xpS/n,X¯=1ni=1nXi,S=1n1i=1nXiX¯2.

Then, T1 obeys a noncentral t-distribution with a degree of freedom of n − 1 and a parameter of z1pn. Jitao  approximates that the probability distribution of S is a normal distribution Nσ,σ2/2n1. By numerical fitting, it is recommended to approximate the noncentral t-distribution of the quantile value as(7)tn1,γ,α=γ+zα1zα2γ2/2nm1zα2/2nmzα22nm,(8)m=1.51α2,where zα is the upper α quantile value of the standard normal distribution.

This method is mainly applicable to the case of n ≥ 5, has a wider scope of application, and is simpler than the method proposed by Huirong . But it can be known from the comparison analysis of the accuracy of the following, the approximation methods proposed by Huirong  and Jitao and Yaokui  are difficult to meet the actual needs in terms of accuracy, and the scope of application is relatively limited.

3. New Approximate Calculation Method

If(9)T2=U+γV/n1,where U obeys the standard normal distribution and V obeys the chi-square distribution with degrees of freedom of n – 1, then, T2 obeys a noncentral t-distribution with a degree of freedom of n − 1 and a parameter of γ. When(10)PT2tn1,γ,α=PU+γV/n1tn1,γ,α=1α,W=Vn1.

Then,(11)PU+γtn1,γ,αW0=1α

The probability density function and mean value of V are(12)fVv=12n1/2Γn1/2ev/2vn1/21,v0,μV=n1.

So, the probability density function of W is(13)fWw=2n1wfVn1w2,w0.

Therefore,(14)μW=0wfWwdw=0vn1fVvdv=2n1Γn/2Γn1/2,EW2=0w2fWwdw=0vn1fVvdv=1,σW=EW2μW2=12n1Γ2n/2Γ2n1/2.

Referring to the method proposed by Shisong et al.  and Jitao , approximating that W obeys the normal distribution NμW,σW2, then(15)PU+γtn1,γ,αW0Φγtn1,γ,αμW1+tn1,γ,α2σW2=1α,tn1,γ,αμWγ1+tn1,γ,α2σW2zα.

It can be obtained that(16)tn1,γ,α=γ+zα1zα2γ2δW2μW1zα2δW2,(17)δW=σWμW=n12Γ2n1/2Γ2n/2110.25n1.

After the overall numerical fit, it can be found in equation (16):(18)μW=10.25n1p+0.60.1α,p=Φγn.

If(19)γ=z1pn,then(20)k=z1pn+zα1zα2z1p2nδW2μWn1zα2δW2.

4. Comparative Analysis

Compare and analyze the calculation accuracy of the four methods: method 1: the method proposed by Huirong  calculates k according to (4); method 2: the method proposed by Jitao  calculates k according to (7); method 3: the method in this paper calculates k according to (20). Tables 15 list the corollary results of the coefficient k according to the different values of C.

Corollary results of coefficient k (C = 0.5).

 p 0.7 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 0.597986 0.415034 −30.59 0.524401 −12.31 0.608001 1.67 4 0.572487 0.435142 −23.99 0.524401 −8.40 0.577322 0.84 5 0.560064 0.449987 −19.65 0.524401 −6.37 0.563115 0.54 6 0.552664 0.460839 −16.61 — — 0.554921 0.41 7 0.547827 0.469046 −14.38 0.524401 −4.28 0.54959 0.32 8 0.544407 0.475423 −12.67 0.524401 −3.68 0.545844 0.26 9 0.541868 0.480519 −11.32 0.524401 −3.22 0.543068 0.22 10 0.539877 0.484634 −10.23 0.524401 −2.87 0.585111 0.19 20 0.53166 0.503979 −5.21 0.524401 −1.37 0.562683 0.08 30 0.529147 0.510691 −3.49 0.524401 −0.90 0.554224 0.05 40 0.527907 0.514103 −2.61 0.524401 −0.66 0.549534 0.04 Error range −30.59∼−2.61 −12.31∼−0.66 0.04∼1.67 p 0.8 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 0.970715 0.88509 −8.82 0.841621 −13.3 0.990143 2 4 0.924547 0.859236 −7.06 0.841621 −9.0 0.935135 1.15 5 0.902606 0.849782 −5.85 — — 0.909861 0.8 6 0.889844 0.845484 −4.99 — — 0.895342 0.62 7 0.881479 0.843315 −4.33 0.841621 −4.52 0.885917 0.5 8 0.875564 0.842031 −3.83 0.841621 −3.88 0.879306 0.43 9 0.871202 0.841351 −3.43 0.841621 −3.4 0.874412 0.37 10 0.867798 0.840878 −3.1 0.841621 −3.02 0.870643 0.33 20 0.853853 0.840228 −1.6 0.841621 −1.43 0.855123 0.15 30 0.849585 0.840502 −1.07 0.841621 −0.94 0.850419 0.1 40 0.847534 0.84068 −0.81 0.841621 −0.7 0.848145 0.07 Error range −8.82∼−0.81 −13.3∼−0.7 0.07∼2 p 0.9 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 1.498545 1.481939 −1.11 1.281552 −14.5 1.530211 2.11 4 1.418883 1.403378 −1.09 — — 1.437254 1.29 5 1.381884 1.368009 −1 — — 1.394886 0.94 6 1.360419 1.348175 −0.9 — — 1.370643 0.75 7 1.346566 1.335561 −0.82 1.281552 −4.83 1.354944 0.62 8 1.336729 1.326928 −0.73 1.281552 −4.13 1.343949 0.54 9 1.32965 1.320672 −0.68 1.281552 −3.62 1.335819 0.46 10 1.324126 1.315862 −0.62 1.281552 −3.22 1.329564 0.41 20 1.301272 1.29683 −0.34 1.281552 −1.52 1.303854 0.2 30 1.294449 1.291352 −0.24 1.281552 −1 1.296077 0.13 40 1.291037 1.28876 −0.18 1.281552 −0.73 1.292321 0.1 Error range −1.11∼−0.18 −14.5∼−0.73 0.1∼2.11 p 0.95 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 1.938445 1.954945 0.85 — — 1.978771 2.08 4 1.829525 1.83853 0.49 — — 1.853356 1.3 5 1.77926 1.784694 0.31 — — 1.796427 0.96 6 1.75043 1.753874 0.20 — — 1.763918 0.77 7 1.731816 1.734234 0.14 1.644854 −5.02 1.742891 0.64 8 1.718722 1.72031 0.09 1.644854 −4.3 1.728176 0.55 9 1.709037 1.710337 0.08 1.644854 −3.76 1.717302 0.48 10 1.701662 1.702589 0.05 1.644854 −3.34 1.708939 0.43 20 1.671163 1.67113 0 1.644854 −1.57 1.6746 0.21 30 1.661982 1.661913 0 1.644854 −1.03 1.664222 0.13 40 1.657536 1.657417 0 1.644854 −0.77 1.659212 0.1 Error range 0∼0.85 −5.02∼−0.77 0.1∼2.08

Corollary results of coefficient k (C = 0.6).

 p 0.7 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 0.799895 0.58474 −26.9 0.627209 −21.6 0.801897 0.25 4 0.733265 0.578577 −21.1 — — 0.733647 0.05 5 0.698054 0.577128 −17.32 0.702867 0.69 0.69811 0.01 6 0.675672 0.576407 −14.69 0.659712 −2.36 0.675599 −0.01 7 0.659758 0.575744 −12.73 0.642013 −2.69 0.659767 0 8 0.647848 0.574975 −11.25 0.630945 −2.61 0.647881 0.01 9 0.63852 0.574221 −10.07 0.622911 −2.44 0.63855 0 10 0.630996 0.57343 −9.12 0.616635 −2.28 0.630981 0 20 0.594181 0.566305 −4.69 0.586898 −1.23 0.594198 0 30 0.579687 0.561364 −3.16 0.574889 −0.83 0.57971 0 40 0.571485 0.557856 −2.38 0.567906 −0.63 0.571498 0 Error range −26.9∼−2.38 −21.6∼0.69 0∼0.25 p 0.8 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 1.211267 1.111396 −8.25 0.867739 −28.36 1.216148 0.4 4 1.110931 1.038738 −6.5 — — 1.113718 0.25 5 1.060245 1.003604 −5.34 1.082299 2.08 1.062497 0.21 6 1.02914 0.982315 −4.55 1.010636 −1.8 1.030933 0.17 7 1.007569 0.967799 −3.95 0.983455 −2.39 1.009184 0.16 8 0.991641 0.957063 −3.49 0.967535 −2.43 0.993113 0.15 9 0.97935 0.948756 −3.12 0.95652 −2.33 0.980658 0.13 10 0.969502 0.942029 −2.83 0.948208 −2.2 0.970662 0.12 20 0.922891 0.909472 −1.45 0.911562 −1.23 0.923535 0.07 30 0.905238 0.89636 −0.98 0.897651 −0.84 0.905674 0.05 40 0.895406 0.888792 −0.74 0.889712 −0.64 0.895735 0.04 Error range −8.25∼−0.74 −28.36∼2.08 0.04∼0.4 p 0.9 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 1.805399 1.77636 −1.61 — — 1.812208 0.38 4 1.6497 1.628998 −1.25 — — 1.655357 0.34 5 1.574205 1.557654 −1.05 1.618757 2.83 1.579205 0.32 6 1.528963 1.514778 −0.93 1.506009 −1.5 1.533232 0.28 7 1.498267 1.485975 −0.82 1.464439 −2.26 1.502045 0.25 8 1.475927 1.465039 −0.74 1.440912 −2.37 1.479285 0.23 9 1.458642 1.449051 −0.66 1.42511 −2.3 1.461825 0.22 10 1.445112 1.436289 −0.61 1.413474 −2.19 1.447934 0.2 20 1.382494 1.377987 −0.33 1.365121 −1.26 1.384065 0.11 30 1.359598 1.356631 −0.22 1.347844 −0.86 1.360658 0.08 40 1.347075 1.344737 −0.17 1.338184 −0.66 1.347853 0.06 Error range −1.61∼−0.17 −2.37∼2.83 0.06∼0.38 p 0.95 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 2.30676 2.308331 0.07 — — 2.312866 0.26 4 2.102118 2.104901 0.13 — — 2.108769 0.32 5 2.004726 2.006578 0.09 2.065685 3.04 2.010919 0.31 6 1.946573 1.948 0.07 1.918951 −1.42 1.952369 0.3 7 1.90781 1.9086 0.04 1.86521 −2.23 1.912919 0.27 8 1.879692 1.880188 0.03 1.835126 −2.37 1.884289 0.24 9 1.858126 1.85848 0.02 1.815139 −2.31 1.862426 0.23 10 1.841239 1.841258 0 1.800564 −2.21 1.8451 0.21 20 1.764225 1.764128 −0.01 1.741567 −1.28 1.766358 0.12 30 1.736488 1.736406 0 1.721125 −0.88 1.737963 0.08 40 1.721437 1.721317 −0.01 1.709825 −0.67 1.72256 0.07 Error range −0.01∼0.07 −2.37∼3.04 0.07∼0.26

Corollary results of coefficient k (C = 0.7).

 p 0.7 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 1.056856 0.823382 −22.09 — — 1.041203 −1.48 4 0.92804 0.770563 −16.97 0.982941 5.92 0.919138 −0.96 5 0.861097 0.742317 −13.79 0.850421 −1.24 0.855142 −0.69 6 0.818649 0.723444 −11.63 0.799625 −2.32 0.814274 −0.53 7 0.788632 0.709445 −10.04 0.769229 −2.46 0.785322 −0.42 8 0.766156 0.698382 −8.85 0.747888 −2.38 0.763447 −0.35 9 0.748413 0.689293 −7.9 0.731641 −2.24 0.74618 −0.3 10 0.73401 0.681582 −7.14 0.718652 −2.09 0.732107 −0.26 20 0.663438 0.639206 −3.65 0.655701 −1.17 0.662707 −0.11 30 0.635261 0.619653 −2.46 0.6302 −0.8 0.634827 −0.07 40 0.619144 0.607677 −1.85 0.615421 −0.6 0.61886 −0.05 Error range −22.09∼−1.85 −2.46∼5.92 −1.48∼−0.05 p 0.8 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 1.528369 1.412346 −7.59 — — 1.507871 −1.34 4 1.343264 1.267934 −5.61 1.458218 8.56 1.333216 −0.75 5 1.251379 1.1953 −4.48 1.248493 −0.23 1.245424 −0.48 6 1.19489 1.150102 −3.75 1.173717 −1.77 1.190894 −0.33 7 1.15592 1.118601 −3.23 1.131398 −2.12 1.153027 −0.25 8 1.1271 1.09512 −2.84 1.102841 −2.15 1.12485 −0.2 9 1.104595 1.076739 −2.52 1.081717 −2.07 1.102872 −0.16 10 1.0865 1.061783 −2.27 1.06519 −1.96 1.085136 −0.13 20 1.00039 0.988779 −1.16 0.988889 −1.15 0.999985 −0.04 30 0.967035 0.959519 −0.78 0.959368 −0.79 0.966858 −0.02 40 0.948261 0.942748 −0.58 0.942525 −0.6 0.948161 −0.01 Error range −7.59∼−0.58 −2.15∼8.56 −1.34∼−0.01 p 0.9 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 2.219999 2.162173 −2.6 — — 2.190339 −1.34 4 1.943907 1.912762 −1.6 2.14235 10.21 1.931962 −0.61 5 1.81224 1.790519 −1.2 1.819984 0.43 1.806181 −0.33 6 1.733163 1.716398 −0.97 1.708671 −1.41 1.729754 −0.2 7 1.679532 1.665911 −0.81 1.647691 −1.9 1.67754 −0.12 8 1.640374 1.628914 −0.7 1.60763 −2 1.63918 −0.07 9 1.610364 1.600302 −0.62 1.578625 −1.97 1.609567 −0.05 10 1.586351 1.577401 −0.56 1.556317 −1.89 1.585873 −0.03 20 1.474504 1.470441 −0.28 1.457608 −1.15 1.474811 0.02 30 1.432615 1.429924 −0.19 1.421102 −0.8 1.432884 0.02 40 1.409306 1.407292 −0.14 1.400619 −0.62 1.409556 0.02 Error range −2.6∼−0.14 −2∼10.21 −1.34∼0.02 p 0.95 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 2.8089821 2.769737 −1.4 — — 2.769095 −1.42 4 2.4530226 2.438936 −0.57 2.717347 10.8 2.437778 −0.62 5 2.2858326 2.278508 −0.32 2.300963 0.66 2.278716 −0.31 6 2.186787 2.181884 −0.22 2.158486 −1.29 2.183009 −0.17 7 2.120093 2.116571 −0.17 2.081324 −1.83 2.118108 −0.09 8 2.071657 2.069042 −0.13 2.031161 −1.95 2.070709 −0.05 9 2.034752 2.032585 −0.11 1.995165 −1.95 2.034297 −0.02 10 2.0052348 2.003434 −0.09 1.967689 −1.87 2.00528 0 20 1.8702006 1.869459 −0.04 1.848538 −1.16 1.870855 0.03 30 1.820288 1.819858 −0.02 1.805492 −0.81 1.820876 0.03 40 1.7927441 1.7925 −0.01 1.781563 −0.62 1.793272 0.03 Error range −1.4∼−0.01 −1.95∼10.8 −1.42∼0.03

Corollary results of coefficient k (C = 0.8).

 p 0.7 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 1.441459 1.198475 −16.86 2.240555 55.44 1.392952 −3.37 4 1.199003 1.056408 −11.89 1.233037 2.84 1.171917 −2.26 5 1.079699 0.980532 −9.18 1.065791 −1.29 1.06182 −1.66 6 1.006262 0.931047 −7.47 0.984826 −2.13 0.993137 −1.3 7 0.955358 0.895292 −6.29 0.933649 −2.27 0.945115 −1.07 8 0.917479 0.867706 −5.42 0.897132 −2.22 0.909135 −0.91 9 0.887813 0.845589 −4.76 0.869215 −2.09 0.880892 −0.78 10 0.863964 0.827366 −4.24 0.846899 −1.98 0.857968 −0.69 20 0.748168 0.733359 −1.98 0.739788 −1.12 0.745878 −0.31 30 0.70254 0.693546 −1.28 0.697122 −0.77 0.70116 −0.2 40 0.676569 0.670218 −0.94 0.672596 −0.59 0.675594 −0.14 Error range −16.86∼−0.94 −2.27∼55.44 −3.37∼−0.14 p 0.8 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 2.015963 1.864198 −7.53 3.350253 66.19 1.959245 −2.81 4 1.67476 1.595514 −4.73 1.756294 4.87 1.644344 −1.82 5 1.513834 1.461578 −3.45 1.508846 −0.33 1.494208 −1.3 6 1.417233 1.378831 −2.71 1.395374 −1.54 1.403193 −0.99 7 1.35164 1.321513 −2.23 1.326381 −1.87 1.340847 −0.8 8 1.303523 1.278875 −1.89 1.278522 −1.92 1.294852 −0.67 9 1.266389 1.245617 −1.64 1.242699 −1.87 1.259186 −0.57 10 1.236572 1.218831 −1.43 1.214529 −1.78 1.230522 −0.49 20 1.096332 1.089234 −0.65 1.084534 −1.08 1.094046 −0.21 30 1.042614 1.038382 −0.41 1.03481 −0.75 1.041283 −0.13 40 1.012475 1.009494 −0.29 1.00666 −0.57 1.011538 −0.09 Error range −7.53∼−0.29 −1.92∼66.19 −2.81∼−0.09 p 0.9 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 2.870411 2.734096 −4.75 4.95068 72.47 2.800163 −2.45 4 2.372481 2.313091 −2.5 2.520845 6.25 2.337578 −1.47 5 2.145211 2.11023 −1.63 2.153216 0.37 2.124036 −0.99 6 2.011888 1.988188 −1.18 1.989734 −1.1 1.997447 −0.72 7 1.922926 1.905291 −0.92 1.892856 −1.56 1.912164 −0.56 8 1.858496 1.844579 −0.75 1.827017 −1.69 1.850058 −0.45 9 1.809182 1.797828 −0.63 1.778539 −1.69 1.802401 −0.37 10 1.770182 1.760544 −0.54 1.740922 −1.65 1.764428 −0.33 20 1.58978 1.586399 −0.21 1.573196 −1.04 1.587908 −0.12 30 1.522643 1.520797 −0.12 1.511473 −0.73 1.521655 −0.06 40 1.485504 1.484164 −0.09 1.477065 −0.57 1.484813 −0.05 Error range −4.75∼−0.09 −1.69∼72.47 −2.45∼−0.05 p 0.95 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 3.603836 3.451824 −4.22 6.293786 74.64 3.519022 −2.35 4 2.968097 2.908834 −2 3.169819 6.8 2.928243 −1.34 5 2.682675 2.65053 −1.2 2.700191 0.65 2.659239 −0.87 6 2.5169789 2.496323 −0.82 2.493566 −0.93 2.501342 −0.62 7 2.4069464 2.392608 −0.6 2.372342 −1.44 2.395769 −0.46 8 2.3278784 2.31704 −0.47 2.29066 −1.6 2.319351 −0.37 9 2.2677779 2.259295 −0.37 2.230948 −1.62 2.261001 −0.3 10 2.220237 2.213298 −0.31 2.184893 −1.59 2.214701 −0.25 20 2.003516 2.001622 −0.09 1.982907 −1.03 2.001987 −0.08 30 1.924118 1.92322 −0.05 1.910046 −0.73 1.923348 −0.04 40 1.880421 1.879741 −0.04 1.869768 −0.57 1.879932 −0.03 Error range −4.22∼−0.04 −1.62∼74.64 −2.35∼−0.03

Corollary results of coefficient k (C = 0.9).

 p 0.7 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 2.227727 1.95304 −12.33 3.927345 76.29 2.172989 −2.46 4 1.692664 1.592182 −5.94 1.818662 7.44 1.627362 −3.86 5 1.455446 1.409432 −3.16 1.468263 0.88 1.406016 −3.4 6 1.317233 1.294781 −1.7 1.30651 −0.81 1.279076 −2.9 7 1.225071 1.214403 −0.87 1.208123 −1.38 1.19427 −2.51 8 1.15805 1.154134 −0.34 1.139942 −1.56 1.132505 −2.21 9 1.106482 1.106785 0.03 1.088974 −1.58 1.084946 −1.95 10 1.065516 1.068362 0.27 1.048945 −1.56 1.046877 −1.75 20 0.873674 0.88102 0.84 0.86496 −1 0.866203 −0.86 30 0.800592 0.806951 0.79 0.794931 −0.71 0.79604 −0.57 40 0.759473 0.764901 0.71 0.755368 −0.54 0.756278 −0.42 Error range −12.33∼0.84 −1.58∼76.29 −3.86∼−0.42 p 0.8 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 3.039117 2.738367 −9.9 5.814814 91.33 3.03893 −0.01 4 2.294540 2.186456 −4.71 2.534914 10.48 2.236538 −2.53 5 1.975691 1.92394 −2.62 2.023345 2.41 1.927019 −2.46 6 1.794452 1.766061 −1.58 1.797266 0.16 1.755281 −2.18 7 1.675329 1.658734 −0.99 1.664035 −0.67 1.643227 −1.92 8 1.589881 1.580105 −0.61 1.573872 −1.01 1.563058 −1.69 9 1.525238 1.519469 −0.38 1.507702 −1.15 1.502188 −1.51 10 1.474096 1.470974 −0.21 1.456488 −1.19 1.454012 −1.36 20 1.240591 1.243243 0.21 1.229623 −0.88 1.232126 −0.68 30 1.154152 1.15689 0.24 1.146775 −0.64 1.148946 −0.45 40 1.106253 1.108681 0.22 1.100748 −0.5 1.102528 −0.34 Error range −9.9∼0.24 −1.19∼91.33 −2.53∼−0.01 p 0.9 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 4.258037 3.827763 −10.11 8.552616 101.86 4.340567 1.94 4 3.187642 3.026395 −5.06 3.594739 12.77 3.144705 −1.35 5 2.742235 2.658306 −3.06 2.842316 3.65 2.698114 −1.61 6 2.493491 2.442339 −2.05 2.518136 0.99 2.456109 −1.5 7 2.332315 2.298158 −1.46 2.331059 −0.05 2.301023 −1.34 8 2.218358 2.193841 −1.11 2.206595 −0.53 2.191628 −1.2 9 2.132705 2.114345 −0.86 2.116515 −0.76 2.109516 −1.09 10 2.065574 2.051333 −0.69 2.047597 −0.87 2.045142 −0.99 20 1.765161 1.762608 −0.14 1.751624 −0.77 1.756306 −0.5 30 1.656991 1.656266 −0.04 1.647536 −0.57 1.651498 −0.33 40 1.597842 1.597666 −0.01 1.590631 −0.45 1.593871 −0.25 Error range −10.11∼−0.01 −0.87∼101.86 −1.61∼1.94 p 0.95 n Exact solution Method 1 Relative error (%) Method 2 Relative error (%) Method 3 Relative error (%) 3 5.311158 4.75345 −10.5 10.85606 104.4 5.457418 2.75 4 3.955887 3.743654 −5.36 4.5002 13.76 3.924381 −0.8 5 3.399631 3.286259 −3.33 3.543087 4.22 3.35901 −1.19 6 3.0916004 3.020273 −2.31 3.134426 1.39 3.055638 −1.16 7 2.8934486 2.844041 −1.71 2.90053 0.24 2.862743 −1.06 8 2.7540745 2.717418 −1.33 2.746018 −0.29 2.727541 −0.96 9 2.6498059 2.62142 −1.07 2.634866 −0.56 2.626593 −0.88 10 2.568089 2.545597 −0.88 2.550266 −0.69 2.547805 −0.79 20 2.207616 2.201785 −0.26 2.19221 −0.7 2.19873 −0.4 30 2.079723 2.077018 −0.13 2.06865 −0.53 2.074132 −0.27 40 2.010191 2.008534 −0.08 2.00167 −0.42 2.006152 −0.2 Error range −10.5∼−0.08 −0.7∼104.4 −1.19∼2.75

In order to better research the trend of the coefficient k corollary results, the relative error scatter plots of the coefficient k corollary results can be drawn when C = 0.5, C = 0.6, C = 0.7, C = 0.8, and C = 0.9, respectively, as shown in Figures 15. Since the relative error of Method 2 when n < 5 is too large, this part of the data is deleted when the figure is made.

Relative error (C = 0.5).

Relative error (C = 0.6).

Relative error (C = 0.7).

Relative error (C = 0.8).

Relative error (C = 0.9).

From the results of comparative analysis, the error range of method 1 is −30.59%∼0.84%, the error range of method 2 is −28.36%∼104.4%, and the error range of method 3 is −3.86%∼2.75%. Separately, when p = 0.7, the relative error ranges of these three methods are −30.59%∼0.84%, −21.6%∼76.29%, and −3.86%∼1.67%; when p = 0.8, the relative error ranges of these three methods are −9.9%∼0.24%, −28.36%∼91.33%, and −2.81%∼2.0%; when p = 0.9, the relative error ranges of these three methods are −10.11%∼−0.01%, −14.5∼101.86%, and −2.45%∼2.11%; and when p = 0.95, the relative error ranges of these three methods are −10.5%∼0.85%, −5.02%∼104.4%, and −2.35%∼2.75%. It can be obtained that the error range calculated by method 2 is the largest, method 1 is in the middle, and method 3 is the smallest, indicating that the relative error calculated by method 3 is the smallest with the highest precision.

When C = 0.9, p = 0.7∼0.95, and n = 3∼40, the absolute values of the average and maximum relative errors of method 1 are 2.28%, 12.33%, the absolute values of the average and maximum relative errors of method 2 are 10.33%, 104.4%, and the absolute values of the average and maximum relative errors of method 3 are 1.38%, 3.9%. It can be seen that the precision of method 2 is the lowest, method 1 is in the middle, and method 3 has the highest precision.

With an absolute value of relative error not exceeding 5%, the applicable range of the method 1 is n > 10, and the applicable range of method 2 is n > 7, and method 3 satisfies all the values of n in the above study. When n < 5, the difference of the relative error calculated by method 2 is the largest, indicating that method 2 is more suitable for the approximate solution of the coefficient k when n ≥ 5. It can be seen from the figure that when p is increased from 0.7 to 0.95, the absolute value of relative error calculated by these three methods has a decreasing trend, indicating that the precision of these three methods is improved with the increase of p. With the increase of n, the absolute value of the relative error calculated by these three methods has a decreasing trend, indicating that the precision of these three methods is improved with the increase of n. When n > 10, the relative error trend curve of the coefficient k calculated by these three methods tends to be gentle, indicating that the accuracy of these three methods is improved with the increase of n. And the relative error calculated by method 3 fluctuates around 0, and there is no large variability. In summary, method 3 has a wider range of applications than the first two methods.

5. Concluding Remarks

Through the study of the inference method of the noncentral t-distribution value, the following main conclusions can be obtained:

By comparing and analyzing the error range of coefficient k under different conditions, it is shown that the quantile value corollary method proposed in this paper is more accurate than the current approximation method.

By comparing the influence of different number of samples n on the coefficient k, it indicates that the proposed method in this paper has a wider scope of application.

The method of inferring the quantile value proposed in this paper does not need to repeatedly check the value table and does not need to perform linear interpolation calculation, which simplifies the calculation process and has better convenience.

Data Availability

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Jun Gao and Jitao Yao contributed the central idea, and Jun Gao analysed most of the data and wrote the initial draft of the paper. The remaining authors contributed to refining the ideas, carrying out additional analyses, and finalizing this paper.

Acknowledgments

The authors would like to gratefully acknowledge the financial support from the National Natural Science Foundations of China (51278401).

MengD.XieT.WuP.ZhuS.-P.HuZ.LiY.Uncertainty-based design and optimization using first order saddle point approximation method for multidisciplinary engineering systemsASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering2020630402002810.1061/AJRUA6.0001076MengD.HuZ.WuP.Reliability-based optimisation for offshore structures using saddlepoint approximationProceedings of the Institution of Civil Engineers—Maritime Engineering20201732334210.1680/jmaen.2020.2LiH.YuanR.FuJ.A reliability modeling for multi-component systems considering random shocks and multi-state degradationIEEE Access2019716880510.1109/access.2019.2953483YuanR.LiH.WangQ.Simulation-based design and optimization and fatigue characteristics for high-speed backplane connectorAdvances in Mechanical Engineering201911611010.1177/16878140198567522-s2.0-85067338929YuanR.TangM.WangH.LiH.A reliability analysis method of accelerated performance degradation based on bayesian strategyIEEE Access2019716904710.1109/access.2019.2952337GairongZ.Probability Theory and Mathematical Statistics2006Beijing, ChinaChina Commercial Publishing House PressRustP. F.Noncentral T Distribution: Introduction. Wiley StatsRef: Statistics Reference Online2014New Yark, NY, USAJohn Wiley & Sons, LtdHarveyA.LangeR.-J.Volatility modeling with a generalized t distributionJournal of Time Series Analysis201738217519010.1111/jtsa.122242-s2.0-85006056946Fatma Zehra DoğruY.Murat bulut,olcay arslan. doubly reweighted estimators for the parameters of the multivariate t-distributionCommunications in Statistics—Theory and Methods2018474751477110.1080/03610926.2018.14458612-s2.0-85044088689KimJ.HayterA. J.Efficient confidence interval methodologies for the non centrality parameter of a non central t distributionCommunications in Statistics—Simulation and Computation200837466067810.1080/036109107017396052-s2.0-41049083322YuanM.HuangJ. Z.HuangRegularized parameter estimation of high dimensional t distributionJournal of Statistical Planning and Inference200913972284229210.1016/j.jspi.2008.10.0142-s2.0-62049084054HuirongC.On an approximate computational method for fractile of non-central t distributionJournal of Agricultural Sciences200046567JitaoY.YaokuiX.Statistical inference for coefficient of variation in reliability assessment of existing structureJournal of Building Structure201031101105JitaoY.Reliability Assessment of Existing Structures Based on Uncertainty Reasoning2011Beijing, ChinaScience PressShisongM.JinglongW.DinghuaS.Statistics Handbook2003Beijing, ChinaScience press