Statistical Parametric And Non-parametric Methods Of Determining The .
EUR4282eCOMMISSION OF THE EUROPEAN COMMUNITIESSTATISTICAL PARAMETRIC ANDNON-PARAMETRIC METHODSOF DETERMINING THE RELIABILITYOF MECHANICAL COMPONENTSbyD. BASILE and G. VOLTA1970Joint Nuclear Research CenterIspra Establishment - ItalyEngineering DepartmentTechnologyEUR/20. 7. mf?. Τ,ftiSw
« åmLEGAL NOTICEThis document was prepared under the sponsorship of theCommission of the European Communities.Neither the Commission of the European Communities, itscontractors nor any person acting on their behalf :Make any warranty or representation, express or implied, withrespect to the accuracy, completeness, or usefulness of theinformation contained in this document, or that the use of anyinformation, apparatus, method, or process disclosed in thisdocument may not infringe privately owned rights ; orAssume any liability with respect to the use of, or for damagesresulting from the use of any information, apparatus, methodor process disclosed in this document.This report is on sale at the addresses listed on cover page 4at the price of F F 11,— FB 100,—DM 7.30Lit. 1,250,—Fl. 7.25,-When ordering, please quote the EUR number and the title,which are indicated on the cover of each report.Printed by Vanmelle, GhentLuxembourg, June 1970This document was reproduced on the basis of the best available copy.
EUR 4282 eSTATISTICAL PARAMETRIC AND NON-PARAMETRIC METHODS OF DETERMINING THE RELIABILITY OF MECHANICAL COMPONENTS by D. BASILE and G. VOLTACommission of the European CommunitiesJoint Nuclear Research Center — Ispra Establishment (Italy)Engineering Department — TechnologyLuxembourg, June 1970 — 70 Pages — 6 Figures — FB 100,—Various statistical methods were studied permitting determinationof the distribution of the failure probability of a mechanical componentusing the experimental data obtained by tests on the component itself.Parametrical methods were considered (applying the maximumlikelihood principle) and non-parametrical methods (order statistics) ;particular emphasis was also given to the use of probability papers.EUR 4282 eSTATISTICAL PARAMETRIC AND NON-PARAMETRIC METHODS OF DETERMINING THE RELIABILITY OF MECHANICAL COMPONENTS by D. BASILE and G. VOLTACommission of the European CommunitiesJoint Nuclear Research Center — Ispra Establishment (Italy)Engineering Department — TechnologyLuxembourg, June 1970 — 70 Pages — 6 Figures — FB 100,—Various statistical methods were studied permitting determinationof the distribution of the failure probability of a mechanical componentusing the experimental data obtained by tests on the component itself.Parametrical methods were considered (applying the maximumlikelihood principle) and non-parametrical methods (order statistics) ;particular emphasis was also given to the use of probability papers.EUR 4282 eSTATISTICAL PARAMETRIC AND NON-PARAMETRIC METHODS OF DETERMINING THE RELIABILITY OF MECHANICAL COMPONENTS by D. BASILE and G. VOLTACommission of the European CommunitiesJoint Nuclear Research Center — Ispra Establishment (Italy)Engineering Department — TechnologyLuxembourg, June 1970 — 70 Pages — 6 Figures — FB 100 —Various statistical methods were studied permitting determinationof the distribution of the failure probability of a mechanical componentusing the experimental data obtained by tests on the component itself.Parametrical methods were considered (applying the maximumlikelihood principle) and non-parametrical methods (order statistics) ;particular emphasis was also given to the use of probability papers.
These methods for the Weibull distribution are applied to somesamples concerning rupture resistance of intermetallic joints and thelifetime of mechanical components.Also presented are three digital computer programs (IBM 360/65)to determine the Weibull parameters, the ranks with a fixed confidencelevel and application of the Kolmogorov test.These methods for the Weibull distribution are applied to somesamples concerning rupture resistance of intermetallic joints and thelifetime of mechanical components.Also presented are three digital computer programs (IBM 360/65)to determine the Weibull parameters, the ranks with a fixed confidencelevel and application of the Kolmogorov test.These methods for the Weibull distribution are applied to somesamples concerning rupture resistance of intermetallic joints and thelifetime of mechanical components.Also presented are three digital computer programs (IBM 360/65)to determine the Weibull parameters, the ranks with a fixed confidencelevel and application of the Kolmogorov test.
EUR4282eCOMMISSION OF THE EUROPEAN COMMUNITIESSTATISTICAL PARAMETRIC ANDNON-PARAMETRIC METHODSOF DETERMINING THE RELIABILITYOF MECHANICAL COMPONENTSbyD. BASILE and G. VOLTA1970Joint Nuclear Research CenterIspra Establishment - ItalyEngineering DepartmentTechnology
ABSTRACTVarious statistical methods were studied permitting detei-minationof the distribution of the failure probability of a mechanical componentusing the experimental data obtained by tests on the component itself.Parametrical methods were considered (applying the maximumlikelihood principle) and non-parametrical methods (order statistics) ;particular emphasis was also given to the use of probability papers.These methods for the Weibull distribution are applied to somesamples concerning rupture resistance of intermetallic joints and thelifetime of mechanical components.Also presented are three digital computer programs (IBM 360/65)to determine the Weibull parameters, the ranks with a fixed confidencelevel and application of the Kolmogorov test.KEYWORDSELEMENTARY MECHANICAL STRUCTURESPLANTS
3 -CONTENTS1. INTRODUCTION52. PARAMETRIC METHODSH3. NON-PARAMETRIC METHODS25k.28METHOD OF PROBABILITY PAPERS5. APPLICATIONS¿9BIBLIOGRAPHY45APPENDIX*f6
- 5STATISTICAL PARAMETRIC AND NON-PARAMETRIC METHODS OF DETERMININGTHE RELIABILITY OF MECHANICAL COMPONENTS1.*)INTRODUCTION1.1 Subject MatterThe theory of reliability can be divided into two main sections.The first deals with the ways of handling the availableexperimental material so as to discover a posteriori the statistical law of behaviour of a component. (The notion of a ' component"or ' system·' is not to be associated with any image of a physicalcomplex.The component is the elementary unit under consideration,for which the statistical law of behaviour is to be defined. Thesystem is the result of the functional connexion of a number ofcomponents.)The second section starts from the assumption of knowledgeof the statistical properties of the components to deduce, by meansof appropriate probabilistic models that simulate the functionalrelations between components, the properties of a system.This report is a contribution to the first section.To processthe experimental material, which consists of data (lifetime, breaking stresses, etc.) corresponding to events considered as random,one uses statistical methods already developed to a large extentfor an immense variety of applications.The specific applicationof these mathematical methods to reliability problems depends onthe type of component in question, the context and the purpose ofthe application.The method that can and must be employed to assess thereliability of mass-produced electronic components in a designstudy for a data bank, for instance, is of little use to someonewho wants to evaluate the reliability of mechanical components ofa plant in operation so that the management can be duly adjustedat once.*) Manuscript received on 2 March 1970
In this report we adopted the position of someone concernedwith the reliability of mechanical and electromechanical components,i.e., components for which:- the dimensions of the available sample are always fairly small;- the deterioration of the properties (through wear, corrosion,fatigue, etc.) with time is significant with respect to thelifetimes regarded as useful;- the reliability analysis effected during operation, taking intoaccount the damage that has occurred on only a fraction of aseries of functioning components, can be of more immediate interest than the reliability analysis that can be obtained whenthe sampling procedure is completed in full.Adopting this point of view, to which is not yet givenenough consideration in the literature on reliability, we haveset out the typical and suitable methods of analysis, developingfor each the appropriate digital programmes.1.2 Plan of the ReportSection 2 briefly describes the main outlines of what arecalled parametric methods for the statistical analysis of samples,i.e., the methods most commonly used in the case of large numbersof samples. We have dwelt more particularly on the application ofthese methods to cases of exponential and Weibull distributionsof failure.The range of reference works available for this matter isenormous as far as the general principles are concerned, but isfar more limited when it comes to specific application to Weibulldistributions. We referred chiefly to the excellent book by Lloydand lipow (Ref. 1).Section 3 shows a non-parametric method which can be regardedas a direct application of a general property of the statisticalvariables associated with ordered events (order statistics).
- 7-This method has been insistently advocated and illustratedby L.G. Johnson (Refe. 2 and 3) of General Motors, precisely inthe context of its application to mechanical components.The method is extremely simple when suitable tabulatedvalues are available; for small samples it is better than theparametrio methods and, unlike them, enables one to take intoconsideration incomplete samples, such as occur in the case ofa set of in-service components only a fraction of which is damaged· We describe the method and have also developed a digitalprogramme by which the tabulated values can be obtained forsamples composed of 1-30 elements and for various degrees ofconfidence.Section k contains a critical analysis of the method of"probability papers", a method which combines the advantages ofthe non-parametrio method with the potentialities inherent tothe parametrio methods. For this method we referred principallyto the works by Gumbel (Ref. k) and Weibull (Ref. 5 ) .Lastly, in Section 3, the various methods mentioned areapplied to some real cases and the results are compared withreference to the extreme values.1.3 Some General Concepts1.3.1 Definition of reliabilityOut of the various definitions of reliability we quote theone adopted by the IECt "The characteristic of an item expressedby the probability that it will perform a required function understated conditions for a stated period of time". The probabilityindicated, a function of time, is the complement to 1 of theprobability of non-function or probability of failure.Considerations on reliability are based on the considerationson the failure distribution, since the failure is the physicallyobserved event.
-81·3·2 Failure distribution function and failure rate functionThe functions of failure distribution versus time are alsoindioated as life characteristics of the given component. We shalltake F(t) to be the failure distribution, i.e., the probabilitythat the component will fail before time t, and f(t) the correspond ing density. It is also expedient to introduce a "failure rate"v(t) defined as,.v(t) f(t)1 - F(t)This function is also known as the "force of mortality","mills ratio", "intensity function" or "hazard rate". The failurerate function is useful because amongst other things, it allowsof dividing the distribution functions into two main categories the failure rate functions that increase with time, and thosethat decrease with time.The fact of belonging to one or other of these categories hasan immediate physical significance: an increasing f.r.f. correspondsto the existence of wear or fatigue phenomena, a decreasing f.r.f.to the running-in situation, for instance; but the subdivisionalso has an important formal significance: one need only knowthat a distribution belongs to one or the other category to beable to deduce limit statistical properties of the componentconcerned or of the system consisting of a number of components(Ref. 6 ) .1·3·3 Most commonly used continuous failure distributionsExponential distributionF(t) l-e"Xtf(t) Xe Xtv(t) λMean ι/χ τt 0 , λ 0
Weibull distributionF(t) 1 - e " X ( t " e )t O , e * 0 , X 0 , a 0f(t) λ «(fe)01'1 e"A(t'e)aait-Qf1v(t) λMean λ 1 / α Γ ( 1 1 / α )Normaldistribution-i—'Γ 7 Æ7F(t) f(t) -JL e2e2σdt*Æ7 i ( 31)2v(t) —Ξ 2Ce2vσ 'σdtMean uLog-normaldistributionI f y » I n t i s a normal v a r i a t e w i t h mean ,\x and v a r i a n c e 6* t h ed i s t r i b u t i o n o f t i s known a s l o g - n o r m a l :1fmteσ /2T J --A-e- Ì(ZIi2 (v σ ) 22σ2σ¿2ΪΓ ' of(t) . I ¿* σ /δϊ"Mean2μ σ / 21e2/lnt-ux2(-T-)dy*
10The exponential distribution is characterized by a constantrate of failure function (A.). The reciprocal of is the mean timebetween two failures (MTBF).This law interprets failure phenomena corresponding topurely random events and it also interprets phenomena of failureof complex systems, when the number of components tends to becomevery large, independently of the law of failure of the individualcomponents. Furthermore it takes advantage of the fact that asystem consisting of components characterized by an exponentiallaw will likewise have an exponential failure law.The normal and log normal distributions are used mainly tointerpret failure phenomena due to wear. They are characterizedby failure rate functions that increase with time.The Weilbull distribution, with three parameters, is moreflexible than the foregoing ones. Its limit case, for oC 1, isthe exponential distribution, and it too can be used to interpretfailure due to wear. Moreover it is suitable for a linear repre sentation on log log paper, so that it does not require specialprobability papers. Lastly it is an asymptotic distribution ofthe extreme values of a wide class of distributions (Ref. ¿0,for which reason it appears in particular to be inherently suitedto represent the phenomena of material failure, interpreted asthe failure of the weakest link in a chain.For these reasons this distribution, proposed originallyby Weibull to interpret data on tensile and fatigue failure ofmaterials, has been increasingly used in the field of electro mechanical components which we shall be considering in particular.
111.3.1* The reliability functionThe reliability function R is defined as the differencebetween the failure distribution values corresponding to theextremes of the event (period of time intended and operatingconditions encountered).R F(t2) - F(t1)In general one assumes forthe time interval(V V '. ."» ' *lsothat:R(T) 1 - F(T)The time Τ is often indicated as "mission time".On the basis of this definition R(t) is to be deducedstraightaway in the cases F(t) indicated.2. PARAMETRIC METHODS2.1 General SchemeThe term "parametric" applied to these methods is due tothe fact that, starting from the sample, they evaluate the para meters of the distribution of failure and hence of reliability,a distribution hypothesized a priori. Roughly speaking, theirstages of use are as follows:i) availability of a complete set (*) of values (sample) referring(*)An incomplete set is one of defined dimensions but only partiallydefined values. Take, for instance, a fixed number of componentsbeing tested simultaneously. The set of lifetimes will be completewhen the last surviving component fails; it will be incomplete forall the preceding times.
- 12 -to the component's characteristic used for the reliabilityestimate (lifetime, breaking stress, etc.). These valuesobviously have to be obtained from tests or operating expe rience on components belonging to the same statistical popu lation.ii) Assumption of one or more forms of statistical distribution towhich the sample is assumed to belong.iii) Estimate of the distribution parameters, based on the samplevalues.The most practical and suitable procedure for thispurpose is the one based on the principle of maximum likelihood.iv) Test for goodness of fit on the various assumed distributionsto see which one fits the interpretation of the sample bestfor a given significance level.v) Determination of the variances of the estimated parametersand, if appropriate, of their confidence intervale.vi) Calculation of the reliability value, by means of the distri bution adopted and the estimated parameters. This reliabilityvalue will likewise be an estimated value. Hence a confidenceinterval will have to be established for it.2.2 Estimate of Parameters2.2.1 The maximum-likelihood methodThis very general method is mentioned in all text books on statistics. Considering, for simplicity's sake, adistribution with a single parameter of, f(t,e(), of which themathematical form is assumed to be known, we form from the sample(t , t ,., t ) the functionL(t1,t2,.,tn;a) Π. fit a)(1)
13known as the function of likelihood of the sample. It correspondsto the compound probability of n random independent variables,each with the same probability distribution, i.e., it correspondsto the probability of obtaining the sample under study out ofall the possible samples of the same size. The method consistsin determining which value of the parameter oc renders it mostprobable that the sample under study will turn up. Thus, ifwe call that value oc it must satisfy the equationflii(orfj with 0(2)λ log L)known as the likelihood equation.Under very general conditions, the maximum likelihoodestimate has a normal distribution when the sample dimensionstend to o4. This asymptotic property of the maximum likelihoodestimates is most useful, because it means that the propertiescharacteristic of a normal distribution can be attributed tothose estimates. At the same time, inasmuch as it is an asymp totic property, it is the chief limitation of the method, sincesmall samples cannot be taken into consideration (accordingto Ref. 1, page 172, the correct use of the normal approximationoalls for sample sizes of not less than 50).2.2.1 Determination of the variances of the estimated parametersA distribution dependent on two parameters oC , X is con sidered. Let of and λ. be the values of these parameters esti mated by the maximum likelihood method from the sample values.It has been shown (Ref. 7) that by using the asymptotic propertyAof the estimated parameters, approximated values of the ô and Xvariances are obtained by constructing the matrix
14 ¿a3a23α3λA s(3)23 43α3λ3λ'Var αC ov (α,λ)Between A and matrixΒ sA A(«OACov(a,X;Var λthere is the simple relation:B *A(5)It will be noted that A is a function of the real para meters of, X ; approximated values are obtained by substitutingAfor the real, unknown values the estimated valuesfit,Λ.In the oase where the distribution depends on a single parameterat, we obtain from the foregoing formulae:Var o 32«CV 1(—j)3α(6)2.3 Goodness of FitThe choice of the form of distribution to which the dataare assumed to belong is, a priori, arbitrary. Hence, thedistributions adopted, whose parameters have been estimated onthe basis of the sample, must be tested to decide which fitsbeet with the sample. Let us briefly describe two widely usedtests, namely the chi squared test and the Kolmogoroff test·The first applies to the density of distribution, the secondto the distribution. The efficiency of both methods is limitedby the size of the sample. The first method is not applicable
15 -to small samples because it calls for division of the sampleinto classes and calculation of the frequency for each class.The second method does not have this drawback. But being basedon asymptotic properties, neither is very significant when itcomes to small samples.2.3.1 The chi-squared test (Ref. 8)The data for the sample of size n are classified in kintervaleAtand the values νiare considered, corresponding to the numberof sample data comprised in the i-th generic interval.If f(t) is the density function of the assumed distribution*i Ati/2w*i -Atf(t)dt(7)i/2will represent the probability that the statistical variablein question belongs to the i-th interval.If the assumption concerning the distribution is valid,thenlim P( V i - npj e) 1(8)Hence a measurement of the data's goodness of fit withthe hypothesis is related to the complex of differences * - n p ) .
- 16With a choice owed to Pearson, we can establish thefollowing magnitude as the measurement of this goodness offit:kΔ2(v. - nPl s i » .xι(9)"Piand it can be shown that Δis a random variable distributed,2with n*""* , according to a γ law with k-1 degrees of freedom,in the event that the parameters of the assumed distributionare known.If, on the other hand, the parameters are estimated fromthe sample, the number of degrees of liberty will be lowerthan k-1 by as many units as there are estimated parameters.2For practical application of the test, having calculatedand set a level of significance γ,,2value X r such that:Awe find from the tables ar(x 2 x* x(10)The assumed distribution satisfies the test if.22Δ χYFor a valid application of the test the sample dimensionsmust be such thatnp. 10i 1» — » k2.3.2 Kolmogoroff test (Ref. 9)This is a test which examines the cumulative distri bution. Let F(t) be this distribution assumed to be continuousand let S (t) be the empirical distribution of the sample ofndimensions n, arranged in ascending order of values.
17Furthermore let:Dsmx F(t) S (t) (11) » X eo»Q(X) Σ2 2k( l) e"2k λλ 0(12)—«.The test is based on Kolmogoroff's theorem which states:lim P(Dη·*· — ) * QU)n(13)JãFor application purposes, once Dhas been calculated and a levelof significance of has been chosen, we find in the tables value Λ for whichQUa) 1 aThe distribution in question will satisfy the test ifDη(IH) λ /ÆαIn Appendix 1 will be found the description of the KTESTcode, programmed in IBM 36O/65 to effect the Kolmogoroff test onvarious distributions. The normal, log normal, Weibull and expo nential distributions are considered.2»h Reliability EstimateWhen the failure distribution parameters have beenestimated, we can estimate the reliability value correspondingto a time T.R(T) R(«, X , T)Now comes the problem of evaluating the confidence wecan have in this estimate.The general method, which is valid only for numeroussamples and does not require knowledge of the distribution ofAAthe parameter estimates, i.e., of ot, λ, etc., makes use of R's
18property of being asymptotically normal (Ref. 1, page 192). HenceAAit is necessary to know E(R) and Var R, i.e., the mean value andvariance of the estimate.It has been shown (Ref. 10, page 35Ό thatE[R(CX,X)1 R(a,X) 0(l/n)(15)Var[R(o,X)] Ä 2 Var α (¿ )2 Var λ 9α α9λ 2 Ä3α &\9λ(16)C ov(â, λ) Od/n1'5)λBoth 0(ΐ/η1 *·5) and 0(ΐ/η) are terms which tend towardszero as the sample dimensions increase. An estimate of E(R) andΛAAVar(R) can be obtained by substituting α, λ, for of, λ in (15) and(16).This general procedure is not necessary in cases wherethe reliability is a function of a single parameter (see exponen tial distribution). In such a case a reliability confidence in terval can be found directly from the parameter confidence inter val. For this purpose one must know the parameter distributionor else apply the property of normal asymptotic behaviour ofthe estimate using the variance calculated in Section 2.2.1.2.3 Applications2.5.1 Exponential distributionThe failure distribution density is given by:f(t,X) λ e Xtt2 0 ,λ 0Starting from the sample (t ,.,t ) the maximum-likelihoodfunction will be:η "λ V iL λ e(17)
19 -and from the maximum-likelihood equation3 lp83λL owe obtain1 .Σ ΐ t iA(18) 'ληi.e., the mean of the sample is the inverse of the estimate ofparameter λ . An estimated value of the reliability at time Τ willbe given by:R R(T,X) e" XT 19 The calculation of the confidence interval of this esti mate can be done by two different routes as already mentioned inSection 2.,k.One procedure, which we might call general, entailscalculation of the variance of the distribution parameter, followedby calculation of the R variance and, using the normal approximation,the R confidence interval.From; s . (ifiogji) - 1var(6) weobtainhis3λ2nand from (15) and (16)E(R) ι s e-XT" 2Var R ( % , Var λ3λ ΑaRΤ.λ-λΤin λ2 π.2 - 2 Τ . λΤ eηR l o g l / R :: R l o g l / RÆÆThe v a r i a b l eη R - E(R)σR(20)
20is asymptotically a standardized normal variate. Having establisheda confidence level γ, we find the confidence interval for thereliability:Rì ; 'Rη(1 γ)/2A second procedure, valid only in the case of exponentialdistribution, allows one to avoid the repeated use of the asymp totic approximations employed in the previous procedure.This second route is based on two characteristics of theexponential distribution:Λ the distribution of the estimated parameter λ is known; the reliability is a monotonie function of the parameter.It has been shown (Ref. 11, page 190) that the estimatedparameterτ — has a gamma distribution:λt(21)by putting2τΧ 2η —this distribution can be reducedto a chi squared distribution with 2n degrees of freedom.Thus (21) is equivalent to:Γ tn l e t/2 d tP(x y) (22)2ηΓ(η)By using (2.2)Awe can obtain an exact evaluation of theAconfidence limit on R R( ) at a given confidence level γ .For having fixed a value for V, we obtain from (22) :P( 2nL τ) γ(23)Χχ γThus T'. 2ητi " 2Xl Yrepresents a lower limit of t with confidence level f .
21 τ/τAs the reliability R » eis an increasing function ofΤ , it follows that a lower limit for the reliability at time T,with confidence level f, will be given by:T*l yRi ) e *;i(24)It is important to note that we have been able to transferAto the reliability the confidence limit calculated for Τ onlyinasmuch as the distribution has only one parameter. In generalthis is not possible where there is more than one parameter.2.5.2 Weibull distributionThe most general form of this distribution has threeparameters:F(t) 1 β λ * θ)βfor the sake of simplicity, we shall assume » 0; the probabilitydensity function therefore is:f(t) -- αλ t""1 e" Att* 0,a 0,\ 0The log of the likelihood function, for a sample (t1i«««»t ) isgiven by:ç¿jηη η loga t η logX (a 1) E . l o g t, Α Σ . t?i1 *1i iBy imposing the maximum likelihood conditions on Åtè-»ir o(25)
22we get two equations for the determination of the estimatedvalues of, X , of the two parameters:(26)λ Σ t.(27)α λ Σ t. log t. Σ log t¿AAThe calculation of oc and A from these equations is donewith an iteration process programmed on IBM 36O/65· To obtain areasonable initial value for α the following relation is used,which expresses equality between the sample mean and the distri bution mean: Σ t. λ" 1/α Γ(1ηιί)α (28)Το determine the variance and covariance of the two parametersit is necessary to invert the matrixÛL92cC9a9α9λ2 9JU3α9λ9X29(29)Having calculated Var ot, V a r A , C ov(«A.) in this way,we can find the variance of the estimated reliability value.If Τ is the mission time for the component for which the relia bility is to be ascertained, then the estimated reliabilityvalue is.AT"R eAlso, by reference to (I5) and (16)Λ-λ τ αE(R) » eT(30)
23*2α -2λΤβ 22*"- Var R Τ e(λ* log ' Τ Var α Var λ 2λ log Τ Οον(α,λ))(31)Estimated values for E(R) and Var R can be obtained byreplacing of and Λ in the previous equations with their estimatesot and A.Using the normal approximation we can then find a confidence interval for R.The foregoing calculations have been programmed onIBM 36Ο/65. Appendix 2 gives a description of the VITA oodeemployed.3. NON-PARAMETRIC METHODS3.1 GeneralGiven a fairly small sample (with fewer than, say, 20values) the mathematically laborious method described in theprevious chapter yields results whose significance is not pro portionate to the effort required.The method we shall give here, however, enables thereliability corresponding to the measured values to be easilyand directly evaluated, even with very small samples.It also permits of evaluating a confidence interval,likewise in respect of the measured values.Lastly, it allows the sample size to be taken intoaccount in cases where the sample is incomplete: from thisstandpoint it offers a possibility not allowed by the methoddescribed in the previous chapter.On the other hand, as it does not aim to evaluate thedistribution but confines itself to evaluation of a series ofdiscrete values, it does not provide indications for interpola tion or extrapolation.
- 243.2 Statistical Properties of Ordered Samples3.2.1 Distribution of the m-th valueLet (t , — , t , — , t ) be a sample of size n with1mnvalues in order of increase. The distribution (t) from which thesample was taken is unknown. The problem is to estimate thecumulative probability 5(t ) , using for the purpose the sample'smproperty of being ordered. If the population is sampled again,the value t', arrayed in the m-th position, will in general bemdifferent from t and one can say that the sample order positionm characterizes, by means of all the samples extractable fromthe population, a set of values, the t values, which will bemdistributed according to their own law of probability, whosedensity is:n mmm mn" m '(t )m(1)This law can be determined at once by using the poly nomial distribution and the sample's property of order. It isa known fact that, given three events with probabilities p 1 , pand ρat the instant of a test, the probability that in η teststhe event with probability ρability p. ηl Vand ttimes, that of prob times, and that of probability ρnn !will occur ηlnn2ηtimes is:3n ! Pl ?2 Pa3If we now let the event "value of t lying between tm dt " correspond to p„, thenrmmρχ ζ(t )dtm mwhere ξ (t) Φ'(t) i*
25 Similarly let the event "value of t t" correspond toρ
STATISTICAL PARAMETRIC AND NON-PARAMETRIC ME THODS OF DETERMINING THE RELIABILITY OF MECHANI CAL COMPONENTS by D. BASILE and G. VOLTA Commission of the European Communities Joint Nuclear Research Center — Ispra Establishment (Italy) Engineering Department — Technology Luxembourg, June 1970 — 70 Pages — 6 Figures — FB 100,—
parametric models of the system in terms of their input- output transformational properties. Furthermore, the non-parametric model may suggest specific modifications in the structure of the respective parametric model. This combined utility of parametric and non-parametric modeling methods is presented in the companion paper (part II).
parametric and non-parametric EWS suggest that monetary expansions, which may reflect rapid increases in credit growth, are expected to increase crisis incidence. Finally, government instability plays is significant in the parametric EWS, but does not play an important role not in the non-parametric EWS.
Surface is partitioned into parametric patches: Watt Figure 6.25 Same ideas as parametric splines! Parametric Patches Each patch is defined by blending control points Same ideas as parametric curves! FvDFH Figure 11.44 Parametric Patches Point Q(u,v) on the patch is the tensor product of parametric curves defined by the control points
Non-parametric, Abstract: Checking the normality assumption is necessary to decide whether a parametric or non-parametric test needs to be used. Different ways are suggested in literature to use for checking normality. Skewness and kurtosis values are one of them. However, there is no consensus which values indicated a normal distribution.
that the parametric methods are superior to the semi-parametric approaches. In particular, the likelihood and Two-Step estimators are preferred as they are found to be more robust and consistent for practical application. Keywords Extreme rainfall·Extreme value index·Semi-parametric and parametric estimators·Generalized Pareto Distribution
Learning Goals Parametric Surfaces Tangent Planes Surface Area Review Parametric Curves and Parametric Surfaces Parametric Curve A parametric curve in R3 is given by r(t) x(t)i y(t)j z(t)k where a t b There is one parameter, because a curve is a one-dimensional object There are three component functions, because the curve lives in three .
Some continuity: template tting !ILC !non-parametric SMICA. Non-parametric foreground modeling with SMICA. All the more useful for CMB cleaning as long as polarized foreground models remain uncertain. The parametric / non- parametric also is a tradeo between statistical e ciency and robustness. Need to learn from forthcoming Planck data and .
An Alphabetical List of Diocesan and Religious Priests of the United States REPORTED TO THE PUBLISHERS FOR THIS ISSUE (Cardinals, Archbishops, Bishops, Archabbots and Abbots are listed in previous section)
