Estimation of Density Function Parameters with Censored Data from Product Life Tests

Transcription

1 VŠB TU Ostrava Faulty of Mehanial Engineering Department of Control Systems and Instrumentation Estimation of Density Funtion Parameters with Censored Data from Produt Life Tests JiříTůma 7. listopadu 5, Ostrava Poruba Czeh Republi

2 Outline Suspension spring lifetime test Graphi solution Analyti solution Results See Engineering statisti Handbook

3 Lifetime Test for Suspension Spring Mileage Truk-suspension steel-spring speimens are an objet to be analyzed Mileage 0 3 km Broken speimen The spring lifetime is assessed by mileage (number of kilometres up to the time of spring failure) Results of the lifetime test

4 Eperimental Cumulative Distribution Funtion vs. Mileage Graphi solution The diagram sale orresponds to the oordinate transformation resulting in a straight line for the Weibull,5,0 0,5 0,0-0,5 -,0 -,5-2,0-2,5-3,0-3,5 umulative distribution funtion ln(-ln(-p/00)), P probability in % Tests terminated prior to breaking Mileage [0 3 km]

5 Basi Definitions for Analyti Solution f(, probability density funtion d F F(, umulative distribution funtion d... mileage [km], Θ funtion parameters ( ; f ( ; Probability funtions for mileage as a random variable ξ P { ξ < + } f ( ; { ξ < } F( ; f ( ; P d P + { ξ > } F( ; f ( d ;

6 , 2, 3,, n n+!, 2, 3,, n+m P P Σ f Likelihood funtion Likelihood Funtion + ( ) ( ) ( ), Θ... f n, Θ f ; Θ d... f ( ; ( )( ) n n Θ n+ + n+ m Σ L, 2,..., + m; n n+ m + ( ) ( ),..., n+ m; Θ f i; Θ f ( ; d i i i n+ n n+ m + L,..., n+ m; Θ ln f i; Θ + ln f ; Θ i i n+ i L ln... mileage up to speimens breaking.. mileage without speimens breaking Probability of obtaining that partiular set of the eperimental data L (, 2,..., ) n m + ;Θ ( ) ( ) ( ) d d

7 Let λ P where ( ; ) Θ P Failure Rate λ(; { ξ < + } λ( ; P{ ξ > } + O( ) lim O( ) 0 0 for the mileage interval The onditional probability that ξ > ξ < + { ξ < + } O( ) P{ ξ > } f ( ) ( ; λ ; Θ F( ; ( ; ( ; F After rearrangement of F the first-order linear differential equation in the standard form is obtained as F ( ; + λ( ; F( ; λ( ; whos solution is as follows F( ; ep λ( ; d 0 an be epressed as

8 Weibull probability funtion (Θ{,}) { } ( ), ; λ { } ( ) f ep., ; { } ( ) F ep, ; {} + Γ E ξ ( ) + Γ + Γ 2 var 2 2 ξ ( ) ( ) + Γ 0 ep d m m Failure rate Probability density funtion Cumulative distribution funtion where Epeted value and variane of ξ for the Weibull probability funtion is the Gama funtion

9 Maimum-likelihood Method for Estimate the Weibull Probability Funtion Parameters Likelihood funtion an be rewritten into a form n n+ m + ( ) ln L n.ln ln i i i i Conditions for the first derivative of ln L with respet to the unknown parameters ( ln L ) ( ln L ) 0, 0 Relationship between and at the maimum of ln L. n + m i n Numeri solution of minimizing problem onsists in finding a value of that minimizes likelihood funtion ln L

10 Iterative refinement of probability funtion parameters Mileage 0 3 km Broken speimen , km Mean value 0 3 km Standard deviation

11 Weibull Probability Funtion Parameters vs. Number of Broken Speimens [0 3 km] Number of broken speimens [-] 8,0 7,0 6,0 5,0 4,0 3,0 2,0,0 0,0

12 Weibull Probability Funtion Parameters vs. Mileage [0 3 km] Mileage [0 3 km] [-] 8,0 7,0 6,0 5,0 4,0 3,0 2,0,0 0,0

13 Observed Failure Rate vs. Mileage

14 Conlusion The paper deals with a statistial method for the evaluation of life test results. It is assumed that only some of the test speimens are observed until breaking while observation of others is terminated prior to breaking. That is, some observations are ensored. The estimation of probability density funtion parameters is based on the maimumlikelihood method. The Weibull distribution funtion is preferably used to desribe the omplete life distribution. It is suggested that estimation be performed sequentially after eah observed breakage, treating speimens while on test as ensored until the parameter estimates beome stable. Some results of simulation are presented using an eample of truk suspension tests.