# One-Stage Multiple Comparisons of the Mean Lifetimes of k Treatments with the Average for Exponential Distributions under Heteroscedasticity

## Abstract

**:**

## 1. Introduction

## 2. One-Stage Multiple Comparisons with the Average for Exponential Mean Lifetimes of k Treatments

_{th}population ${\pi}_{i}$ following the exponential distribution denoted by $E({\theta}_{i},{\sigma}_{i})$, the mean lifetime is regarded as ${\mu}_{i}={\theta}_{i}+{\sigma}_{i}$ for the i

_{th}population, i = 1, …, k. Take a one-stage random sample ${X}_{i1},\dots ,{X}_{im}$ of size $m(\ge 2)$ from the i

_{th}population. Let ${Y}_{i}=\mathrm{min}({X}_{i1},\dots ,{X}_{im})$ be the smallest order statistic and ${S}_{i}={\displaystyle {\sum}_{j=1}^{m}({X}_{ij}-{Y}_{i})}/(m-1)$. Then Y

_{i}and S

_{i}are the uniformly minimum-variance unbiased estimators (UMVUEs) of ${\theta}_{i}$ and ${\sigma}_{i}$, respectively. Furthermore, Y

_{i}+ S

_{i}is the UMVUE of the mean lifetime ${\mu}_{i}={\theta}_{i}+{\sigma}_{i}$ for the i

_{th}population, i = 1, …, k.

_{th}mean lifetime with the average denoted by ${\mu}_{i}-\overline{\mu}$, i = 1, …, k. It appears that ${Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}$ is the UMVUE of ${\mu}_{i}-\overline{\mu}$, i = 1, …, k, where $\overline{Y}={\displaystyle \sum _{i=1}^{k}{Y}_{i}}/k$ and $\overline{S}={\displaystyle \sum _{i=1}^{k}{S}_{i}}/k$. The pivotal quantities for building our confidence intervals are given by ${G}_{i}^{}=\frac{-m{S}_{i}/{\sigma}_{i}+m-m({Y}_{i}-{\theta}_{i})/{\sigma}_{i}}{{S}_{i}/{\sigma}_{i}}$, i = 1, …, k. Making use of the UMVUE of ${\mu}_{i}-\overline{\mu}$ and these pivotal quantities, the one-sided and two-sided confidence intervals for ${\mu}_{i}-\overline{\mu}$, i = 1, …, k are proposed in the following theorem:

**Theorem 1.**

- (a)
- $\mathrm{P}({\mu}_{i}-\overline{\mu}\le {Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}+{c}_{i}^{*}{s}_{U}^{*},i=1,\dots ,k)\ge {P}^{*}$, where ${s}_{U}^{*}$ is the $100{P}_{th}$ percentile of the distribution of $\mathrm{max}(-{\tilde{W}}_{i},{G}_{i},{G}_{i}-{\tilde{W}}_{i},i=1,\dots ,k)$ multiplied by (k − 1)/k, with ${\tilde{W}}_{i}={\mathrm{min}}_{l\ne i}{G}_{l}^{}$.Thus, the upper confidence bound for ${\mu}_{i}-\overline{\mu}$ with confidence coefficient ${P}^{*}$ is $({Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}+{c}_{i}^{*}{s}_{U}^{*})$, $i=1,\dots ,k$.
- (b)
- $\mathrm{P}({\mu}_{i}-\overline{\mu}\ge {Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}-{c}_{i}^{*}{s}_{L}^{*},i=1,\dots ,k)\ge {P}^{*}$, where ${s}_{L}^{*}$ is the $100{P}_{th}$ percentile of the distribution of $\mathrm{max}(\text{}{W}_{i},-{G}_{i},{W}_{i}-{G}_{i},i=1,\dots ,k)$ multiplied by (k − 1)/k, with ${W}_{i}={\mathrm{max}}_{l\ne i}{G}_{l}^{}$.Thus, the lower confidence bound for ${\mu}_{i}-\overline{\mu}$ with confidence coefficient ${P}^{*}$ is $({Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}-{c}_{i}^{*}{s}_{L}^{*})$, $i=1,\dots ,k$.
- (c)
- $\mathrm{P}({Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}-{c}_{i}^{*}{s}_{t}^{*}\le {\mu}_{i}-\overline{\mu}\le {Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}+{c}_{i}^{*}{s}_{t}^{*},i=1,\dots ,k)\ge {P}^{*}$ where ${s}_{t}^{*}$ is the $100{P}_{th}$ percentile of the distribution of $\mathrm{max}(|{G}_{i}|,{W}_{i},{W}_{i}-{G}_{i},$ $-{\tilde{W}}_{i},{G}_{i}-{\tilde{W}}_{i},i=1,\dots ,k)$ multiplied by (k − 1)/k with ${\tilde{W}}_{i}={\mathrm{min}}_{l\ne i}{G}_{l}^{}$ and ${W}_{i}={\mathrm{max}}_{l\ne i}{G}_{l}^{}$.Thus, $({Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}\pm {c}_{i}^{*}{s}_{t}^{*})$ is the two-sided simultaneous confidence interval for ${\mu}_{i}-\overline{\mu}$ with confidence coefficient ${P}^{*}$,$i=1,\dots ,k$.

**Lemma 1.**

**Proof of Theorem 1**:

_{i}, we require three distributional results from Roussas [21] as follows:

- (D1) $2(m-1){S}_{i}/{\sigma}_{i}={Q}_{i},$ i = 1, …, k follows a chi-squared distribution with 2m − 2 df denoted by ${\chi}_{2m-2}^{2}$.
- (D2) $\text{}m({Y}_{i}-{\theta}_{i})/{\sigma}_{i}={E}_{i},$ i = 1, …, k follows a standard exponential distribution denoted by $Exp(1)$.
- (D3) E
_{i}and Q_{i}are two independent variables.

_{i}is presented in the following theorem:

**Theorem 2.**

**Proof of Theorem 2:**

_{i}, where U

_{i}~U(0,1). Solving this equation for x, we have ${G}_{i}^{}=\left(\nu -2m-\nu {({e}^{m}{U}_{i})}^{\frac{1}{1-m}}\right)/2$ coming from the distribution of ${G}_{i}^{}$. The random variable ${G}_{i}^{}$ can be generated using the equation above.

^{*}=0.875, 0.90, and 0.925 are listed in Table A2. The critical values for P

^{*}= 0.95 and 0.975 are listed in Table A3. The software we use to find the critical values is Fortran 90 and the programming manual refers to Mourik [22]. In Table A1, Table A2 and Table A3, it can be seen that the approximate critical values ${s}_{U}^{*}$, ${s}_{L}^{*}$, and ${s}_{t}^{*}$ are increasing while ${P}^{*}$ is increasing for any given k and m or while k is increasing for any given ${P}^{*}$ and m. Let L

_{1}be the length of the two-sided confidence intervals for ${\mu}_{i}-\overline{\mu}$, and we have the average length L

_{1}= 2$\overline{c}{s}_{t}^{*}$, where $\overline{c}={\displaystyle \sum _{i=1}^{k}{c}_{t}^{*}/k}$. From the equation of L

_{1}= 2$\overline{c}{s}_{t}^{*}$, it is evident that, as ${P}^{*}$ increases, the value of ${s}_{t}^{*}$ increases and then the confidence length of L

_{1}increases for any given k and m. Furthermore, we can also see that, as the number of populations k increases, the value of ${s}_{t}^{*}$ increases, and then the confidence length of L

_{1}increases for any given m and ${P}^{*}$.

## 3. A Biometrical Example

_{th}category of lung cancer with the average survival days, the required statistics and critical values of ${s}_{t}^{*}$ for P* = 0.90, 0.95, and 0.975 are summarized in Table 2.

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The critical values of ${s}_{U}^{*}$ = ${s}_{L}^{*}$ = ${s}_{t}^{*}$ for P* = 0.75, 0.80, and 0.85.

k | ||||||||
---|---|---|---|---|---|---|---|---|

P* | m | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

0.75 | 2 | 8.92 | 13.82 | 18.69 | 23.74 | 28.85 | 33.40 | 38.26 |

3 | 5.11 | 7.22 | 9.02 | 10.69 | 12.14 | 13.46 | 14.67 | |

4 | 4.68 | 6.42 | 7.86 | 9.07 | 10.18 | 11.10 | 11.95 | |

5 | 4.66 | 6.31 | 7.63 | 8.76 | 9.68 | 10.51 | 11.20 | |

6 | 4.80 | 6.41 | 7.72 | 8.78 | 9.69 | 10.45 | 11.20 | |

7 | 4.96 | 6.62 | 7.91 | 8.97 | 9.84 | 10.61 | 11.25 | |

8 | 5.15 | 6.83 | 8.11 | 9.22 | 10.05 | 10.85 | 11.47 | |

9 | 5.32 | 7.04 | 8.35 | 9.46 | 10.38 | 11.13 | 11.75 | |

10 | 5.50 | 7.30 | 8.64 | 9.72 | 10.66 | 11.42 | 12.08 | |

15 | 6.42 | 8.44 | 9.96 | 11.16 | 12.16 | 13.00 | 13.71 | |

20 | 7.23 | 9.46 | 11.15 | 12.49 | 13.58 | 14.46 | 15.30 | |

25 | 7.99 | 10.42 | 12.25 | 13.70 | 14.87 | 15.88 | 16.77 | |

30 | 8.66 | 11.30 | 13.30 | 14.83 | 16.12 | 17.19 | 18.11 | |

0.80 | 2 | 11.46 | 17.68 | 24.24 | 30.26 | 36.81 | 43.17 | 49.02 |

3 | 5.99 | 8.46 | 10.46 | 12.27 | 13.84 | 15.50 | 16.87 | |

4 | 5.34 | 7.24 | 8.80 | 10.13 | 11.36 | 12.41 | 13.26 | |

5 | 5.26 | 7.05 | 8.45 | 9.63 | 10.66 | 11.58 | 12.32 | |

6 | 5.36 | 7.09 | 8.53 | 9.58 | 10.51 | 11.34 | 12.06 | |

7 | 5.55 | 7.26 | 8.64 | 9.75 | 10.66 | 11.44 | 12.22 | |

8 | 5.68 | 7.48 | 8.85 | 9.96 | 10.86 | 11.67 | 12.37 | |

9 | 5.86 | 7.68 | 9.07 | 10.24 | 11.16 | 11.94 | 12.67 | |

10 | 6.06 | 7.92 | 9.35 | 10.49 | 11.42 | 12.23 | 12.90 | |

15 | 7.01 | 9.12 | 10.70 | 11.93 | 12.97 | 13.78 | 14.52 | |

20 | 7.84 | 10.22 | 11.94 | 13.31 | 14.41 | 15.32 | 16.10 | |

25 | 8.68 | 11.23 | 13.08 | 14.51 | 15.73 | 16.75 | 17.67 | |

30 | 9.41 | 12.11 | 14.14 | 15.69 | 16.99 | 18.06 | 19.04 | |

0.85 | 2 | 15.71 | 24.43 | 32.57 | 41.18 | 50.24 | 59.25 | 66.50 |

3 | 7.36 | 10.12 | 12.59 | 14.59 | 16.42 | 18.22 | 19.88 | |

4 | 6.26 | 8.47 | 10.15 | 11.64 | 12.93 | 14.04 | 15.15 | |

5 | 6.07 | 8.05 | 9.55 | 10.85 | 11.94 | 12.94 | 13.79 | |

6 | 6.11 | 8.07 | 9.48 | 10.73 | 11.73 | 12.58 | 13.40 | |

7 | 6.26 | 8.15 | 9.61 | 10.72 | 11.73 | 12.60 | 13.34 | |

8 | 6.42 | 8.29 | 9.76 | 10.96 | 11.89 | 12.78 | 13.50 | |

9 | 6.59 | 8.51 | 10.02 | 11.14 | 12.19 | 12.99 | 13.67 | |

10 | 6.78 | 8.78 | 10.23 | 11.49 | 12.41 | 13.21 | 14.01 | |

15 | 7.76 | 9.95 | 11.58 | 12.86 | 13.94 | 14.84 | 15.57 | |

20 | 8.66 | 11.05 | 12.85 | 14.26 | 15.38 | 16.33 | 17.24 | |

25 | 9.53 | 12.12 | 14.12 | 15.57 | 16.83 | 17.83 | 18.72 | |

30 | 10.28 | 13.11 | 15.19 | 16.76 | 18.07 | 19.22 | 20.17 |

**Table A2.**The critical values of ${s}_{U}^{*}$ = ${s}_{L}^{*}$ = ${s}_{t}^{*}$ for P* = 0.875, 0.90, and 0.925.

k | ||||||||
---|---|---|---|---|---|---|---|---|

P* | m | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

0.875 | 2 | 19.47 | 29.50 | 40.22 | 49.24 | 59.63 | 70.68 | 80.54 |

3 | 8.34 | 11.39 | 13.95 | 16.26 | 18.38 | 20.52 | 21.96 | |

4 | 6.91 | 9.20 | 11.13 | 12.68 | 14.04 | 15.25 | 16.43 | |

5 | 6.63 | 8.71 | 10.26 | 11.63 | 12.80 | 13.78 | 14.67 | |

6 | 6.64 | 8.57 | 10.18 | 11.42 | 12.45 | 13.36 | 14.22 | |

7 | 6.70 | 8.70 | 10.21 | 11.43 | 12.48 | 13.31 | 14.15 | |

8 | 6.91 | 8.88 | 10.35 | 11.59 | 12.60 | 13.43 | 14.17 | |

9 | 7.04 | 8.99 | 10.59 | 11.80 | 12.76 | 13.69 | 14.39 | |

10 | 7.25 | 9.25 | 10.86 | 12.01 | 13.02 | 13.94 | 14.61 | |

15 | 8.18 | 10.47 | 12.14 | 13.46 | 14.55 | 15.45 | 16.23 | |

20 | 9.13 | 11.64 | 13.42 | 14.87 | 16.04 | 17.05 | 17.80 | |

25 | 10.05 | 12.67 | 14.68 | 16.23 | 17.49 | 18.48 | 19.35 | |

30 | 10.83 | 13.71 | 15.83 | 17.47 | 18.81 | 19.92 | 20.88 | |

0.90 | 2 | 24.30 | 37.05 | 48.67 | 62.96 | 75.91 | 88.74 | 100.46 |

3 | 9.43 | 12.91 | 15.90 | 18.55 | 20.81 | 23.05 | 24.94 | |

4 | 7.68 | 10.25 | 12.28 | 13.95 | 15.40 | 16.60 | 17.86 | |

5 | 7.30 | 9.52 | 11.25 | 12.66 | 13.94 | 14.92 | 15.96 | |

6 | 7.21 | 9.30 | 10.94 | 12.29 | 13.41 | 14.38 | 15.31 | |

7 | 7.27 | 9.38 | 10.94 | 12.24 | 13.30 | 14.23 | 15.13 | |

8 | 7.39 | 9.48 | 11.09 | 12.28 | 13.38 | 14.31 | 14.99 | |

9 | 7.61 | 9.77 | 11.33 | 12.50 | 13.62 | 14.40 | 15.19 | |

10 | 7.77 | 9.94 | 11.50 | 12.80 | 13.72 | 14.65 | 15.42 | |

15 | 8.81 | 11.12 | 12.81 | 14.19 | 15.25 | 16.24 | 16.98 | |

20 | 9.75 | 12.23 | 14.13 | 15.65 | 16.81 | 17.77 | 18.55 | |

25 | 10.64 | 13.43 | 15.36 | 16.96 | 18.23 | 19.31 | 20.22 | |

30 | 11.49 | 14.47 | 16.62 | 18.26 | 19.65 | 20.74 | 21.66 | |

0.925 | 2 | 33.10 | 50.61 | 67.37 | 85.87 | 100.34 | 120.16 | 137.23 |

3 | 11.40 | 15.11 | 18.74 | 21.76 | 24.39 | 26.96 | 29.38 | |

4 | 8.86 | 11.64 | 13.83 | 15.70 | 17.42 | 18.90 | 20.17 | |

5 | 8.24 | 10.61 | 12.47 | 14.07 | 15.37 | 16.55 | 17.52 | |

6 | 8.09 | 10.34 | 12.04 | 13.52 | 14.68 | 15.70 | 16.55 | |

7 | 8.08 | 10.33 | 11.97 | 13.36 | 14.45 | 15.43 | 16.23 | |

8 | 8.16 | 10.31 | 12.10 | 13.43 | 14.51 | 15.39 | 16.25 | |

9 | 8.34 | 10.59 | 12.25 | 13.59 | 14.61 | 15.50 | 16.31 | |

10 | 8.49 | 10.79 | 12.36 | 13.77 | 14.77 | 15.80 | 16.51 | |

15 | 9.48 | 11.93 | 13.66 | 15.13 | 16.24 | 17.16 | 18.00 | |

20 | 10.48 | 13.17 | 15.05 | 16.54 | 17.69 | 18.71 | 19.63 | |

25 | 11.43 | 14.28 | 16.38 | 17.93 | 19.30 | 20.27 | 21.19 | |

30 | 12.30 | 15.31 | 17.49 | 19.21 | 20.62 | 21.75 | 22.69 |

**Table A3.**The critical values of ${s}_{U}^{*}$ = ${s}_{L}^{*}$ = ${s}_{t}^{*}$ for P* = 0.95 and 0.975.

k | ||||||||
---|---|---|---|---|---|---|---|---|

P* | m | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

0.95 | 2 | 51.54 | 75.5 | 100.76 | 129.26 | 153.89 | 174.7 | 213.25 |

3 | 14.57 | 19.25 | 23.27 | 27.37 | 30.45 | 33.50 | 36.48 | |

4 | 10.70 | 13.96 | 16.63 | 18.79 | 20.45 | 22.19 | 23.80 | |

5 | 9.69 | 12.31 | 14.51 | 16.30 | 17.49 | 18.78 | 20.13 | |

6 | 9.30 | 11.78 | 13.71 | 15.23 | 16.60 | 17.58 | 18.57 | |

7 | 9.24 | 11.64 | 13.43 | 14.96 | 16.23 | 17.25 | 18.19 | |

8 | 9.22 | 11.65 | 13.45 | 14.93 | 16.04 | 17.04 | 17.94 | |

9 | 9.38 | 11.78 | 13.47 | 14.98 | 16.11 | 17.06 | 17.96 | |

10 | 9.56 | 11.94 | 13.67 | 15.10 | 16.27 | 17.30 | 17.98 | |

15 | 10.56 | 13.04 | 14.95 | 16.32 | 17.57 | 18.56 | 19.40 | |

20 | 11.55 | 14.29 | 16.32 | 17.85 | 19.12 | 20.07 | 20.92 | |

25 | 12.50 | 15.49 | 17.65 | 19.18 | 20.58 | 21.75 | 22.66 | |

30 | 13.46 | 16.57 | 18.81 | 20.52 | 22.06 | 23.14 | 24.15 | |

0.975 | 2 | 101.01 | 154.17 | 205.49 | 252.59 | 312.71 | 367.33 | 409.42 |

3 | 21.18 | 28.17 | 34.28 | 39.45 | 43.91 | 47.80 | 53.06 | |

4 | 14.46 | 18.52 | 21.83 | 24.50 | 26.34 | 28.85 | 30.63 | |

5 | 12.47 | 15.65 | 18.48 | 20.47 | 22.19 | 23.54 | 24.74 | |

6 | 11.67 | 14.60 | 16.86 | 18.66 | 20.11 | 21.52 | 22.51 | |

7 | 11.23 | 14.19 | 16.16 | 17.82 | 19.12 | 20.44 | 21.52 | |

8 | 11.37 | 13.89 | 15.85 | 17.59 | 18.81 | 20.14 | 20.92 | |

9 | 11.26 | 14.03 | 15.91 | 17.54 | 18.73 | 20.07 | 20.87 | |

10 | 11.38 | 14.05 | 16.08 | 17.50 | 18.79 | 19.82 | 20.78 | |

15 | 12.22 | 15.05 | 17.06 | 18.78 | 19.86 | 20.81 | 21.89 | |

20 | 13.19 | 16.19 | 18.38 | 20.07 | 21.35 | 22.44 | 23.35 | |

25 | 14.37 | 17.51 | 19.80 | 21.47 | 22.98 | 23.98 | 25.05 | |

30 | 15.30 | 18.61 | 21.00 | 22.86 | 24.36 | 25.59 | 26.54 |

## References

- Bechhofer, R.E. A single sample multiple decision procedure for ranking means of normal populations with known variances. Ann. Math. Stat.
**1954**, 25, 16–39. [Google Scholar] [CrossRef] - Gupta, S.S. On a Decision Rule for a Problem in Ranking Means. Ph.D. Thesis, Mimeograph Series No. 150. Institute of Statistics, University of North Carolina, Chapel Hill, NC, USA, 1956. [Google Scholar]
- Lawless, J.F. Statistical Models and Methods for Lifetime Data; Wiley: New York, NY, USA, 2003. [Google Scholar]
- Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions; Wiley: New York, NY, USA, 1994. [Google Scholar]
- Bain, L.J.; Engelhardt, M. Statistical Analysis of Reliability and Life Testing Models; Marcel Dekker: New York, NY, USA, 1991. [Google Scholar]
- Lawless, J.F.; Singhal, K. Analysis of data from life test experiments under an exponential model. Nav. Res. Logist. Q.
**1980**, 27, 323–334. [Google Scholar] [CrossRef] - Balakrishnan, K. Exponential Distribution Theory, Methods and Applications; Routledge: London, UK, 1996. [Google Scholar]
- Balakrishnan, N.; Joshi, P.C. Product moments of order statistics from the doubly truncated exponential distribution. Nav. Res. Logist. Q.
**1984**, 31, 27–31. [Google Scholar] [CrossRef] - Balakrishnan, N.; Sandhu, R.A. Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive type-II censored samples. Sankhyā Indian J. Stat. Ser. B.
**1996**, 58, 1–9. [Google Scholar] - Khan, M.J.S.; Iqrar, S.; Faizan, M. Characterization of exponential distribution through normalized spacing of generalized order statistics. J. Stat. Theory Appl.
**2019**, 18, 303–308. [Google Scholar] [CrossRef] [Green Version] - Ng, C.K.; Lam, K.; Chen, H.J. Multiple comparison of exponential location parameters with the best under type II censoring. Am. J. Math. Manag. Sci.
**1993**, 12, 383–402. [Google Scholar] [CrossRef] - Lam, K.; Ng, C.K. Two-stage procedures for comparing several exponential populations with a control when the scale parameters are unknown and unequal. Seq. Anal.
**1990**, 9, 151–164. [Google Scholar] [CrossRef] - Wu, S.F.; Lin, Y.P.; Yu, Y.R. One-stage multiple comparisons with the control for exponential location parameters under heteroscedasticity. Comput. Stat. Data Anal.
**2010**, 54, 1372–1380. [Google Scholar] [CrossRef] - Maurya, V.; Goyal, A.; Gill, A.N. Multiple comparisons with more than one control for exponential location parameters under heteroscedasticity. Commun. Stat.-Simul. Comput.
**2011**, 40, 621–644. [Google Scholar] [CrossRef] - Maurya, V.; Gill, A.N.; Singh, P. Multiple comparisons with a control for exponential location parameters under heteroscedasticity. J. Appl. Stat.
**2013**, 40, 1817–1830. [Google Scholar] [CrossRef] - Wu, S.F. One stage multiple comparisons of k-1 treatment mean lifetimes with the control for exponential distributions under heteroscedasticity. Commun. Stat.-Simul. Comput.
**2018**, 47, 2968–2978. [Google Scholar] [CrossRef] - Wu, S.F. One stage multiple comparisons with the control for exponential mean lifetimes based on doubly censored samples under heteroscedasticity. Commun. Stat.-Simul. Comput.
**2021**, 50, 1473–1483. [Google Scholar] [CrossRef] - Wu, S.F. Multiple comparison procedures for exponential mean lifetimes compared with several controls. Mathematics
**2022**, 10, 609. [Google Scholar] [CrossRef] - Wu, S.F. New one-stage multiple comparisons procedures with the average for exponential location parameters under heteroscedasticity. J. Stat. Comput. Simul.
**2016**, 86, 2740–2748. [Google Scholar] [CrossRef] - Lam, K. Subset selection of normal populations under heteroscedasticity. In Proceedings of the IPASRAS-II Proceedings and Discussions of the Second International Conference on Inference Procedures Associated with Statistal Ranking and Selection on The frontiers of Modern Statistical Inference Procedures, II, Sydney, Australia, August 1992; pp. 307–344. [Google Scholar]
- Roussas, G.G. A Course in Mathematical Statistics, 3rd ed.; Elsevier Science & Technology Books: Amsterdam, The Netherlands, 2014. [Google Scholar]
- Mourik, T.V. Fortran 90/95 Programming Manual; Chemistry Department, University College London: London, UK, 2005. [Google Scholar]

Category | m | Survival Times | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | Squamous | 9 | 72 | 10 | 81 | 110 | 100 | 42 | 8 | 25 | 11 |

2 | Small | 9 | 30 | 13 | 23 | 16 | 21 | 18 | 20 | 27 | 31 |

3 | Adeno | 9 | 8 | 92 | 35 | 117 | 132 | 12 | 162 | 3 | 95 |

4 | Large | 9 | 177 | 162 | 553 | 200 | 156 | 182 | 143 | 105 | 103 |

Statistics | Category 1 | Category 2 | Category 3 | Category 4 |
---|---|---|---|---|

Y_{i} | 8 | 13 | 3 | 103 |

S_{i} | 48.375 | 10.250 | 78.265 | 106.750 |

${c}_{i}^{*}$ | 7.232 | 8.644 | 8.696 | 11.861 |

${Y}_{i}-\overline{Y}+{S}_{i}-\overline{S}$ | −36.285 | −69.410 | −11.395 | 117.090 |

P* | ${s}_{U}^{*}$= ${s}_{L}^{*}$= ${s}_{t}^{*}$ | |||

0.900 | 9.77 | |||

0.950 | 11.78 | |||

0.975 | 14.03 |

**Table 3.**The 90%, 95% and 97.5% upper confidence bounds and lower confidence bounds for the mean survival times of four categories of lung cancer compared with the average.

Parameter | $({\mathit{Y}}_{\mathit{i}}-\overline{\mathit{Y}}+{\mathit{S}}_{\mathit{i}}-\overline{\mathit{S}}+{\mathit{c}}_{\mathit{i}}^{*}{\mathit{s}}_{\mathit{U}}^{*})$$,({\mathit{Y}}_{\mathit{i}}-\overline{\mathit{Y}}+{\mathit{S}}_{\mathit{i}}-\overline{\mathit{S}}-{\mathit{c}}_{\mathit{i}}^{*}{\mathit{s}}_{\mathit{L}}^{*})$ | ||
---|---|---|---|

90% | 95% | 97.5% | |

1. ${\mu}_{1}-{\overline{\mu}}_{}$ | (34.372), (−106.942) | (48.908), (−121.478) | (65.180), (−137.750) |

2. ${\mu}_{2}-{\overline{\mu}}_{}$ | (15.043), (−153.863) | (32.417), (−171.237) | (51.87), (−190.686) |

3. ${\mu}_{3}-{\overline{\mu}}_{}$ | (73.566), (−96.356) | (91.045), (−113.835) | (110.61), (−133.40) |

4. ${\mu}_{4}-{\overline{\mu}}_{}$ | (232.973), (1.207) | (256.81), (−22.634) | (283.50), (−49.321) |

**Table 4.**The 90%, 95% and 97.5% two-sided confidence intervals for the mean survival times of four categories of lung cancer compared with the average.

Parameter | $({\mathit{Y}}_{\mathit{i}}-\overline{\mathit{Y}}+{\mathit{S}}_{\mathit{i}}-\overline{\mathit{S}}-{\mathit{c}}_{\mathit{i}}^{*}{\mathit{s}}_{\mathit{t}}^{*},{\mathit{Y}}_{\mathit{i}}-\overline{\mathit{Y}}+{\mathit{S}}_{\mathit{i}}-\overline{\mathit{S}}+{\mathit{c}}_{\mathit{i}}^{*}{\mathit{s}}_{\mathit{t}}^{*})$ | ||
---|---|---|---|

90% | 95% | 97.5% | |

1. ${\mu}_{1}-{\overline{\mu}}_{}$ | (−106.942, 34.372) | (−121.478, 48.908) | (−137.750, 65.180) |

2. ${\mu}_{2}-{\overline{\mu}}_{}$ | (−153.863, 15.043) | (−171.237, 32.417) | (−190.686, 51.87) |

3. ${\mu}_{3}-{\overline{\mu}}_{}$ | (−96.356, 73.566) | (−113.835, 91.045) | (−133.40, 110.61) |

4. ${\mu}_{4}-{\overline{\mu}}_{}$ | (1.207, 232.973) | (−22.634, 256.81) | (−49.321, 283.50) |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, S.-F.
One-Stage Multiple Comparisons of the Mean Lifetimes of k Treatments with the Average for Exponential Distributions under Heteroscedasticity. *Axioms* **2023**, *12*, 312.
https://doi.org/10.3390/axioms12030312

**AMA Style**

Wu S-F.
One-Stage Multiple Comparisons of the Mean Lifetimes of k Treatments with the Average for Exponential Distributions under Heteroscedasticity. *Axioms*. 2023; 12(3):312.
https://doi.org/10.3390/axioms12030312

**Chicago/Turabian Style**

Wu, Shu-Fei.
2023. "One-Stage Multiple Comparisons of the Mean Lifetimes of k Treatments with the Average for Exponential Distributions under Heteroscedasticity" *Axioms* 12, no. 3: 312.
https://doi.org/10.3390/axioms12030312