Scholars' Mine Doctoral Dissertations Student Theses and Dissertations 1972 Statistical studies of various time-to-fail distributions James Addison Eastman Follow this and additional works at: https://scholarsmine.mst.edu/doctoral_dissertations Part of the Mathematics Commons Department: Mathematics and Statistics Recommended Citation Eastman, James Addison, "Statistical studies of various time-to-fail distributions" (1972). Doctoral Dissertations. 186. https://scholarsmine.mst.edu/doctoral_dissertations/186 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. STATISTICAL STUDIES OF VARIOUS TIME -TO-FAI L DISTRIBUTIONS by JAMES ADDISON EASTMAN, 1943- A DISSERTATION Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI-ROLLA In Partial Fulfil l ment of the Requirements for the Degree DOCTOR OF PHILOSOPHY in MATHEMATICS T2789 85 pages 1972 c .l ~ Adv isor f 6' cu,...., 23~269 i i ABSTRACT Three models are considered that have U-shaped hazard functions, and a fourth model is considered that has a linear hazard function. Several methods for esti111ating the parameters are given for each of these models. Also, various tests of hyrotheses are considet~ed in the cas e of t 11 e model \'I i t r1 the l i n e a r h a z a r d f u n c t i on . 0 n e o f the r:1o del s with a U-shaped hazard function has a location and a scale parameter, and it is proved in general that any other parameters in a distriLution of this type are distributed independently of the location and scale parameters. A nevJ method used to estimate the parameters in the preceding distributions is also employed to estir.wte the parameters in the Logistic distribution, and comrarisons based on ~,1onte Carlo methods are made betvJeen these estimators and the r1aximum Likelihood estimators for n = 10, 20, 40, 80 and for complete samples and censoring from the right for r/n = .1, .3, .5 and .7. The distributions of the pivotal quantities ln(C - JJ)/o, /n(3/o- 1), and (C- JJ)/o + ko-;0, \vhere the estimates are the i1aximum Likelihood estimates, are obtained Gy 11onte Carlo sir11ulation for the sample sizes and level of censo~~ing given above, so that confidence intervals and tolerance lir.1its can be found. The means and variances of the estirlotors of reliability are given. ; ; ; ACKNOWL EDGEr~ENT The author wishes to express his sincere appreciation to his advisor Dr. Lee J. Bain and to Dr. Maxwell E. Engelhardt both of the Department of Mathematics for their assistance and advice. The author also expresses his gratitude to his wife for her patience through the research and the graduate studies which preceded it. iv TABLE OF CONTENTS Paqe ABSTRACT ............................................................. i i ACKNOWLEDGEMENT ..................................................... iii TABLE OF CONTENTS .................................................... i v LIST OF TABLES ....................................................... vi I . INTRODUCTION .................................................... 1 I I . LI NEAR FA I LURE- RATE M0 DEL . 13 A. Range for the Parameters ................................... 13 B. Maximum Likelihood Estimates ............................... 14 c. Maximum Agreement Estimates ................................ 18 D. Cramer-Rae Lower Bound (CRLB) .............................. 21 E. Rep a rameteri za t ion of the Mode 1 ............................ 26 F. Tests of Hypotheses ........................................ 27 1. Tests on the Parameter a with b Known .................. 27 2. Tests on the Parameter a with b Unknown ................ 31 3. Tests on the Parameter b ............................... 35 4. Comparisons of the Various Test Statistics ............. 38 I I I. MODELS WITH U-SHAPED FAILURE-RATES ............................. 41 A. Quadratic Failure-Rate Model ............................... 41 1. Range for the Parameters ............................... 41 2. Maximum Likelihood Estimates ........................... 41 3. Maximum Agreement Estimates ............................ 43 B. Model with Failure-Rate h(x) = c cosh [b(x-a)] ............. 45 1. Maximum Likelihood Estimators .......................... 45 2. Agreement Estimates .................................... 46 v Table of Contents (continued) Paqe c. Model with Failure-Rate h(x) = ~ (~)a- 1 B B + aBy (i)a-1 exp (~)a ....................................... 48 1. Range for the Parameters ............................... 48 2. Maximum Likelihood Estimators .......................... 50 IV. ESTIMATION AND INFERENCES ON THE LOGISTIC DI STR I BUT I ON ................................................... 54 A. Estimation of the Parameters of the Logistic Distribution ...................................... 54 B. Statistical Inferences for the Logistic Distribution Based on Maximum Likelihood Estimators ................................................. 57 1. Confidence Intervals for 1.1 and o ....................... 57 2. Point Estimation of 1.1, o, and R(t) ..................... 59 3. Tolerance Limits ....................................... 59 V. NUMERICAL TECHNIOUES ........................................... 61 A. Generation of Random Samples ............................... 61 B. Numerical Solution of the Maximum Likelihood Equations for the Logistic Distribution .................... 62 VI. SUMMARY, CONCLUSIONS AND FURTHER PROBLEMS ...................... 66 REFERENCES .......................................................... 67 VITA ........................................ ········ ................ 69 APPENDIX A. TA8LES FOR INFtRENCES OF THE LOGISTIC DISTRIBUTION ........................................... 70 vi LIST OF TAiJLES Table Page 1. t,1onte Carlo study comparing various estimates for n = 20, 40 Hith CRLB for a= 5, b = 1 ...................... 25 2. Cumulative percentage points of 2 L x?1 and~L x?;(~x.)1 L 1 2 values of c _~,n such that P[Ix?;(Ix.) < c ] =a 1 ~ 1 1 - 1-a,n 2 and P[I.J Ix? < c ] = a....................................... 34 1 - 1 -a,n 3. Po\'Jer for various tests of H : a 0 versus 0 H : a> 0, a= .01, .1, 1.0 and b = .01, .1, 1.0 1 vJi th a = . 05 and n = 20 Lased on 500 r·1onte Carl ocd sa mp 1 e s ........................................................ 4 0 4. t1onte Carlo study comparing maximum agreement estir.1ators vJi th maxi mum 1 i l~e 1 i hood est i rna tors for the standard Logistic distribution ............................................ 58 5. iJumi.Jer of times NevJton-f<.aphson r.1ethod failed to converge .............................................. ~ ........... 65 Al. Values of my such that P[Jrl(G-p)/~ < r.1 y ] "( . .................. 71 A2. Values of sy such that P[Jrl(o/o-1) < sy] "( ••••••••••••••••.•• 7 2 A3. f·1eans and variances of the maximum 1 i kel i hood estimators of the parameters of the Logistic distribution.(E~G ) denotes the mean of G for the standard log1st1c,0 u = 0, o = 1) ..•............................ 73 A4. t1eans and variu.nces of R........................................ 74 A5. Tolerance factors t such that L(x) = 0- tyo' /'. A y 7h U ( X ) = lJ +to ................................................. :J y 1 I. INTRODUCTION There are a number of distributions that have proven useful in describing the distribution of the time-to-failure of an item. Typical­ ly, the random variable x, which is the time it takes the item to fail is of the continuous type and has a ranqe from zero to infinity. Given that the probability density function is f(x), 0 ~ x < =, the cumulative distribution function is given by and is the probability that the item has failed by time x. The pro­ bability of an item failinq in the interval (x, x + L1x) is oiven by F(x + L1x)- F(x)~ so the average rate of failure in this interval is F(x + 6x) - F(x) 6x and the average rate of failure qiven that the item has survived to time x is F ( X + L1x) - F ( X ) 6x[l - F(x)] The instantaneous rate of failure is 1 im F(x + 6x) - F(x) = f(x) ( 1) b.x-+0 b. X [ 1 - F ( X ) ] --:-1 ---'-==-'F(:.-x-.-) . The probability that an item survives to timex is called the reliability and is given by R(x) = 1 - F(x). Therefore, (1) can be rewritten as f(x) _ -R • (x) RTXT- RTXT · 2 If we let h(x) be the instantaneous rate of failure~ also referred to as the failure rate or the hazard function~ then h (x ) = -R ' (_x) R(x) R(x) exp[- ( h(u)du] or F(x) = 1 - exp [- ( h(u)dj. (2) and f (X) = h (X ) eX p t ( h ( U) d l} (3) The most popular time-to-failure models have either a constant failure rate, which is the exponential model, or a monotonically in­ creasing or monotonically decreasing failure rate, such as with the Gamma or Weibull models. In this paper, more general models are con- sidered so as to include models that have failure rates with U or bath tub shapes. The purpose of this shape is to better describe items which initially have a hiqh rate of failure caused by some phenomenon such as faulty manufacturing, then have a useful life period in which the failure rate is at a minimum, and then have an increasing failure rate caused by wear out. Some general methods for constructing models with bath tub shaped failure rates will first be considered. The first method produces a model of the type referred to by Kao [1] as a mixed model (n fold). If F.(x),i=l, ... ,n is a cumulative