Homoscedastic controlled calibration model

Homoscedastic con trolled calibrati on mo del Betsab ´ e G. Blas A chic a , Mˆ onica C. Sando v al b and Olga Satomi Y os hida c a, b Departamen to d e Estat ´ ıstica, Univ ersid ade de S˜ ao P aulo, S˜ ao P au lo, Brasil c Cen tro de Metrologia de Fluidos, Instituto de P esquisas T ecnol´ ogicas, S˜ ao P aulo, Brasil Abstract In the con text of the usual calibration mo del, we consider the case in whic h the in d ep end ent v ariable is unobserv able, but a pre-fixed v alue on its surrogate is a v ailable. Th u s , consid ering controlle d v ari- ables and assum ing that the measur emen t errors hav e equal v ariances w e p rop ose a new calibration mo del. Lik eliho o d based metho dology is used to estimate the mo del parameters and the Fisher in formation matrix is used t o construct a co nfi dence in terv al f or the unkno wn v alue of the r egressor v ariable. A sim ulation study is carried out to asses the effect of the measuremen t error on the estimatio n of the parameter of in terest. T his n ew approac h is illustrated with an example. Keyw ords: Regression mo del, linear calibration mo del, measurement error mo del, Berkson mo d el. 1 1 In tro duction In the first stage of a calibration problem, a pair of data sample ( x i , Y i ) , i = 1 , 2 , · · · n is observ ed. In the second stage, it is o bserv ed one o r more v alues, whic h are the resp o nses corresp onding to a single unkno wn v alue of the re- gressor v ariable, X 0 . The first and second stage equations of the usual linear calibration mo del are defined, resp ectiv ely , as Y i = α + β x i + ǫ i , i = 1 , 2 · · · , n, (1.1) Y 0 i = α + β X 0 + ǫ i , i = n + 1 , n + 2 , · · · , n + k . (1.2) It is considered the fo llo wing assumptions: • x 1 , x 2 , · · · , x n tak e fixed v alues, whic h a re considered as true v alues. • ǫ 1 , ǫ 2 , · · · , ǫ n + k are indep enden t and normally distributed with mean 0 and v ariance σ 2 ǫ . The mo del parameters are α, β , X 0 and σ 2 ǫ and the main interest is to estimate the quantit y X 0 . The maxim un lik eliho o d estimators of the usual calibration mo del are giv en b y ˆ α = ¯ Y − ˆ β ¯ x, ˆ β = S xY S xx , ˆ X 0 = ¯ Y 0 − ˆ α ˆ β , (1.3) σ 2 ǫ = 1 n + k [ n X i =1 ( Y i − ˆ α − ˆ β x i ) 2 + n + k X i = n +1 ( Y 0 i − ¯ Y 0 ) 2 ] , (1.4) where ¯ x = 1 n n X i =1 x i , ¯ Y = 1 n n X i =1 Y i , S xY = 1 n n X i =1 ( x i − ¯ x )( Y i − ¯ Y ) , S xx = 1 n n X i =1 ( x i − ¯ x ) 2 , ¯ Y 0 = 1 n n + k X i = n +1 Y 0 i . In [10] an approxim ate express ion is deriv ed for the v aria nce of the esti- mator ˆ X 0 , whic h is deriv ed through the propagation error law. Another appro ximation for the v ariance of ˆ X 0 is giv en b y the Fisher information of θ = ( α, β , X 0 , σ 2 ǫ ) whic h, after some length a lg ebraic manipulations, it can b e sho wn to b e giv en b y I ( θ ) = 1 σ 2 ǫ       n + k k X 0 + n ¯ x k β 0 k X 0 + n ¯ x k X 2 0 + P n i =1 x 2 i κβ X 0 0 k β κβ X 0 k β 2 0 0 0 0 n + k 2 σ 2 ǫ       . (1.5) 2 The maximum likelihoo d estimator of ˆ θ = ( ˆ α, ˆ β , ˆ X 0 , ˆ σ 2 ǫ ) has approx i- mately normal distribution with mean θ and cov ariance matrix I ( θ ) − 1 , when k = q n, q ∈ Q + and n − → ∞ . Th us, the appro ximation of order n − 1 for the v ariance of ˆ X 0 is given by V 1 ( ˆ X 0 ) = σ 2 ǫ β 2 " 1 k + 1 n + ( ¯ X − X 0 ) 2 nS xx # . (1.6) On the other hand, in [4] the size k of the second stage is considered fixed, so that expanding ˆ X 0 in T aylor series aro und the p oin t ( α, β ) and ignoring terms of order less than n − 2 , w e can find the following approximations for the bias and v ariance of ˆ X 0 , r esp ective ly , B ias ( ˆ X 0 ) = σ 2 ǫ ( X 0 − ¯ x ) nβ 2 S xx , (1.7) V 2 ( ˆ X 0 ) = σ 2 ǫ β 2 " 1 k + 1 n + ( ¯ X − X 0 ) 2 nS xx + 3 σ 2 ǫ nk β 2 S xx # . (1.8) In o r der to construct a confidence in terv a l for X 0 , we consider that ˆ X 0 − X 0 q ˆ V ( ˆ X 0 ) D − → N (0 , 1) , (1.9) where ˆ V ( ˆ X 0 ) is the estimated v ariance computed according to (1.6) or (1.8). Hence, the appro ximated confidence in terv al fo r X 0 with a confidence lev el (1 − α ) , is giv en by  ˆ X 0 − z α 2 q ˆ V ( ˆ X 0 ) , ˆ X 0 + z α 2 q ˆ V ( ˆ X 0 )  , (1.10) where z α 2 is the quan tile of o rder (1 − α 2 ) of the standard normal distribution. The usual calibra tion problem has b een discussed in the literature fo r sev eral decades (see [1]- [6]). An illustration of this mo del is presen ted for example in [7]. W e can find a review of the litera t ur e on statistical calibration in [8 ], where some approac hes to the solution of the calibration problem are summarized. This model encoun ters applications in differen t areas, but it is not we ll suited in some instances a s, for example, in c hemical analysis, where the preparation pro cess of standa r d solutio ns are sub ject to measuremen t erro r ([10]). There exists some situations, as mentioned ab o v e, where the indep enden t v ariable, x i , is measured with error. In this case, [1 1] defines tw o types of observ ations: controlled and uncon trolled . In the uncontrolled situation, the usual pro cedure to obtain the true v alue of the indep enden t v ariable x i generates an error and the observ ed v alue is X i = x i + δ i , i = 1 , · · · , n. (1.11) 3 W e hav e tha t x i is a n unknown quantit y , δ i is a measuremen t error and X i is a random v ariable. Assuming that x i is a parameter the mo del defined b y (1.1) a nd (1 .1 1) is na med as functional mo del ([1 2]). In this case there exists correlation b etw een the mo del error and the v aria ble X i . Assuming that x i is a random v aria ble the mo del (1.1) and (1.11 ) is called a s structural mo del ([12]). On the other hand, the mo del defined by (1.1), (1.2) and (1.11) is called as the functional or structural calibration mo del if x i is assumed as a parameter or a ra ndom v ariable, resp ectiv ely ([13]). The con tro lled observ atio n is defined by a pre-fixed v alue X i according to the exp erimen ter con v enience and a pro cedure is established in or der to attain the pre-fixed v alue. The exp eriment give s the uno bserv ed x i and it is suc h that x i = X i − δ i , i = 1 , · · · , n. (1.12) In this case, the fixed quan tity is X i , the measureme nt error is δ i and x i is the random v ariable. The mo del (1.1) and (1.12) is kno wn a s Berkson regression mo del ([9]). Notice that the mo del error and the quan tity X i are indep enden t. The mo del defined b y (1.1), (1.2) and (1.12) has not b een considered b efore in t he measuremen t error literature and in this w ork it will b e called as the con trolled calibration mo del . In the calibration mo del defined b y (1.1), (1.2) and (1.11), the v alues of the regressor, X i , from the first stage are randomly g enerated, where as in the con trolled calibration mo del, (1.1), (1.2) and (1.12), they are assumed as pre-fixed b y the exp erimen ter. This w ork is organized a s follo ws. In Section 2, w e derive the maxim um lik eliho o d estimators o f the homoscedastic controlled calibration mo del by considering b oth cases: σ 2 δ unkno wn and kno wn . In Section 3, a sim ulation study is undertake n to inv estigate t he sensitivit y of parameter estimates of the prop o sed mo del. In Section 4, an example is presen ted to illustrate our new approac h. In Section 5 , the concluding r emark is presen ted. 2 P arameter estimation In this section w e study the controlled calibra tion mo del. F rom the equations (1.1), (1 .2) and (1.12) w e can write Y i = α + β X i + ( ǫ i − β δ i ) , i = 1 , 2 · · · , n, (2.1) Y 0 i = α + β X 0 + ǫ i , i = n + 1 , n + 2 , · · · , n + k . (2.2) with the following assumptions for the random errors • ǫ i are independen t N (0 , σ 2 ǫ ) random v aria bles. • E( δ i )=0, V( δ i )= σ 2 δ i . • co v( δ i , δ j )=0 fo r an y i 6 = j . 4 • co v( ǫ i , δ j )=0 fo r all i, j . Some commen ts are in order here. The v ariable X i in (2.1) is controlled and the error mo del ( ǫ i − β δ i ) is independen t of X i . The error mo del in (2.2) is only in function of erro r measure ǫ i related to Y 0 i , this mo del assume t hat there is not error in the prepara tion sample related to parameter X 0 . W e define the homoscedastic con trolled calibrat io n mo del by considering that the errors δ i are indep enden t and normally distributed with mean 0 a nd constan t v ariance, σ 2 δ . The study of this mo del is carried o ut f o llo wing similar ana lysis to the usual calibration mo del a s summarized ab ov e. The maxim um like liho o d estimator for the homoscedastic con trolled cal- ibration mo del is deriv ed in the following. The logarithm of the lik eliho o d function is giv en by : l ( α , β , X 0 , σ 2 ǫ , σ 2 δ ) ∝ − n 2 lo g ( σ 2 ǫ + β 2 σ 2 δ ) − k 2 lo g ( σ 2 ǫ ) − 1 2  1 σ 2 ǫ + β 2 σ 2 δ n X i =1 ( Y i − α − β X i ) 2 + 1 σ 2 ǫ n + k X i = n +1 ( Y 0 i − α − β X 0 ) 2  . (2.3) Solving ∂ l /∂ α = 0 a nd ∂ l /∂ X 0 = 0 w e hav e the maxim um lik eliho o d esti- mator of α and X 0 , whic h are giv en, resp ectiv ely , b y ˆ α = ¯ Y − ˆ β ¯ X and ˆ X 0 = ¯ Y 0 − ˆ α ˆ β . (2.4) F rom (2.3) and (2.4), it follows that the likelih o o d for ( β , σ 2 ǫ , σ 2 δ ) can b e written as l ( β , σ 2 ǫ , σ 2 δ ) ∝ − n 2 lo g ( σ 2 ǫ + β 2 σ 2 δ ) − k 2 lo g ( σ 2 ǫ ) − 1 2  1 σ 2 ǫ + β 2 σ 2 δ n X i =1 [( Y i − ¯ Y ) − β ( X i − ¯ X )] 2 + 1 σ 2 ǫ n + k X i = n +1 ( Y 0 i − ¯ Y 0 ) 2  . (2.5) Next, w e consider t w o cases for σ 2 δ . F ir stly , w e obtain the maxim um like- liho o d estimator of β , σ 2 ǫ and σ 2 δ from (2.5). In the second case w e assume that the v a riance σ 2 δ is know n and obtain the maxim um lik eliho o d estimators for β and σ 2 ǫ . Case 1: unknown v ar iance σ 2 δ T aking the partial deriv ativ e of (2.5) with resp ect to β , σ 2 ǫ and σ 2 δ and equating to zero w e obtain, respective ly , ˆ β ˆ σ 2 δ ( ˆ σ 2 ǫ + ˆ β 2 ˆ σ 2 δ − S Y Y + ˆ β S X Y ) = ( S X Y − ˆ β S X X ) ˆ σ 2 ǫ , (2.6) 5 ˆ σ 2 ǫ + ˆ β 2 ˆ σ 2 δ = S Y Y − 2 ˆ β S X Y + ˆ β 2 S X X , (2.7) k S Y 0 Y 0 ( ˆ σ 2 ǫ ) 2 − k ˆ σ 2 ǫ = n ˆ σ 2 ǫ + ˆ β 2 ˆ σ 2 δ − n ( S Y Y − 2 ˆ β S X Y + ˆ β 2 S X X ) ( ˆ σ 2 ǫ + ˆ β 2 ˆ σ 2 δ ) 2 , (2.8) where S X X = 1 n P n i =1 ( X i − ¯ X ) 2 , S X Y = 1 n P n i =1 ( X i − ¯ X )( Y i − ¯ Y ) , S Y Y = 1 n P n i =1 ( Y i − ¯ Y ) 2 and S Y 0 Y 0 = 1 k P n + k i = n +1 ( Y 0 i − ¯ Y 0 ) 2 , and the relev ant estimator notation has b een introduced. F rom (2.6) and ( 2 .7) we ha ve the following equations: ( ˆ β S X X − S X Y )( S Y Y − 2 ˆ β S X Y + ˆ β 2 S X X ) = 0 , hence ˆ β S X X − S X Y = 0 or (2.9) S Y Y − 2 ˆ β S X Y + ˆ β 2 S X X = 0 . (2.10) Therefore, from (2.9) , we hav e that ˆ β = S X Y /S X X . But, according to the Cauc h y-Sc hw arz inequalit y , S X X S Y Y ≥ S 2 X Y , hence (2.1 0 ) has real ro ots if and only if Y i = c X i , where c is a constan t. The estimator of σ 2 δ can b e obtained from the equation (2.7) ˆ σ 2 δ = ( S Y Y − 2 ˆ β S X Y + ˆ β 2 S X X ) − ˆ σ 2 ǫ ˆ β 2 . Lik ewise, fr o m equations (2.7) and ( 2 .8) we obtain the estimator of the v ariance σ 2 ǫ ˆ σ 2 ǫ = S Y 0 Y 0 . (2.11) In order to find the v ar iance of ˆ X 0 , we need to deriv e the Fisher infor- mation matrix of θ = ( α, β , X 0 , σ 2 δ , σ 2 ǫ ), whic h can be show n to b e given b y I ( θ ) =            n γ + k σ 2 ǫ n ¯ X γ + k X 0 σ 2 ǫ k β σ 2 ǫ 0 0 n ¯ X γ + k X 0 σ 2 ǫ P n i =1 X 2 i γ + 2 nβ 2 σ 4 δ γ 2 + k X 2 0 σ 2 ǫ k β X 0 σ 2 ǫ nβ 3 σ 2 δ γ 2 nβ σ 2 δ γ 2 k β σ 2 ǫ k β X 0 σ 2 ǫ k β 2 σ 2 ǫ 0 0 0 nβ 3 σ 2 δ γ 2 0 nβ 4 2 γ 2 nβ 2 2 γ 2 0 nβ σ 2 δ γ 2 0 nβ 2 2 γ 2 n 2 γ 2 + k 2 σ 4 ǫ            , where γ = β 2 σ 2 δ + σ 2 ǫ . (2.12) When k = q n , q ∈ Q + and n − → ∞ , the estimator ˆ θ is approx imately normally distributed with mean θ and v aria nce I ( θ ) − 1 , thus w e hav e that the appro ximate v ar ia nce to order n − 1 for ˆ X 0 is given b y V 1 ( ˆ X 0 ) = σ 2 ǫ β 2 " 1 k + γ nσ 2 ǫ + γ σ 2 ǫ ( ¯ X − X 0 ) 2 nS X X # . (2.13) 6 Considering k fixed and expanding ˆ X 0 in a T a ylor series a r o und ( α , β ) and ignoring terms of order less than n − 2 , it can b e shown that the bias and v ariance of ˆ X 0 (the pro of is giv en in App endix A), are giv en b y B ias ( ˆ X 0 ) = γ ( ¯ X − X 0 ) nβ 2 S X X , (2.14) V 2 ( ˆ X 0 ) = σ 2 ǫ β 2 " 1 k + γ nσ 2 ǫ + γ ( ¯ X − X 0 ) 2 nσ 2 ǫ S X X + 3 γ nk β 2 S X X # . (2.15) W e can observ e that the estimator of X 0 is biased, but it is asymptotically un biased. With relation to t he v ariance of the estimator ˆ X 0 , let us notice that when k = q n, q ∈ Q + , and ignoring the t erms of order less than n − 1 the v ariance in (2.15) coincide with the v aria nce given in ( 2 .13), whic h w as found through the Fisher inf o rmation. Equation (2.13) consider large sample sizes in the first and second stage ( n and k ), whereas (2.15) consider large sample sizes in t he first stage a nd a fixed sample size in the second stage. Notice that when σ 2 δ = 0, (2.13) and (2.15) coincide with (1.6) and (1.8) of the usual mo del, respectiv ely . Caso 2: k no wn v ar iance σ 2 δ Assuming now that σ 2 δ is kno wn and equating to zero the partial deriv ative of (2.5) with respect to the para meters β and σ 2 ǫ , w e ha v e the follo wing equations, respective ly , ˆ β σ 2 δ ( ˆ σ 2 ǫ + ˆ β 2 σ 2 δ − S Y Y + ˆ β S X Y ) = ( S X Y − ˆ β S X X ) ˆ σ 2 ǫ and (2.16) k S Y 0 Y 0 ( ˆ σ 2 ǫ ) 2 − k ˆ σ 2 ǫ = n ˆ σ 2 ǫ + ˆ β 2 σ 2 δ − S Y Y − 2 ˆ β S X Y + ˆ β 2 S X X ( ˆ σ 2 ǫ + ˆ β 2 σ 2 δ ) 2 . (2.17) The estimates of β and σ 2 ǫ are obtained using some iterativ e method to solve (2.16) and (2.17). Similarly , a s in Case 1, the Fisher infor ma t io n matrix of θ = ( α , β , X 0 , σ 2 ǫ ) is given by I ( θ ) =         n γ + k σ 2 ǫ n ¯ X γ + k X 0 σ 2 ǫ k β σ 2 ǫ 0 n ¯ X γ + k X 0 σ 2 ǫ P n i =1 X 2 i γ + 2 nβ 2 σ 4 δ γ 2 + k X 2 0 σ 2 ǫ k β X 0 σ 2 ǫ nβ σ 2 δ γ 2 k β σ 2 ǫ k β X 0 σ 2 ǫ k β 2 σ 2 ǫ 0 0 nβ σ 2 δ γ 2 0 n 2 γ 2 + k 2 σ 4 ǫ         , (2.18) where γ is defined in (2.1 2). The large sample v ariance of ˆ X 0 follo ws by in v erting the Fisher info rma- tion matrix and is given by V ( ˆ X 0 ) = σ 2 ǫ β 2 " 1 k + γ nσ 2 ǫ + γ σ 2 ǫ E # , (2.19) 7 where, E = nX 2 0 σ 4 ǫ + k X 2 0 γ 2 − 2 nX 0 ¯ X σ 4 ǫ − 2 k X 0 ¯ X γ 2 + n ¯ X 2 σ 4 ǫ + k ¯ X 2 γ 2 ( nσ 4 ǫ + k γ 2 ) P n i =1 X 2 i + 2 nk β 2 γ σ 4 δ − n 2 ¯ X 2 σ 4 ǫ − nk ¯ X 2 γ 2 . Notice that if σ 2 δ = 0, the expression (2.19) is reduced to (1.6). T o construct a confidence in terv al for X 0 , for b oth cases σ 2 δ unkno wn and kno wn , we consider the interv al (1.1 0), where ˆ V ( ˆ X 0 C ) is t he estimated v ariance that follows from (2.13), (2.15) o r (2.19). 3 Sim ulation s tudy In this section we presen t a sim ulation study for b oth cases of the ho- moscedastic con trolled calibrat ion mo del: σ 2 δ kno wn and unkno wn. The ob jetiv e of this section is to study the p erformance of the estimators o f the prop osed mo del (Prop osed-M) and v erify the impact by considering errati- cally the usual mo del (Usual-M). It was considered 50 0 0 samples g enerated from the homoscedastic con- trolled calibration mo del. In all samples, the v alue of the pa r ameters α and β w ere 0.1 and 2, resp ective ly . The rang e of v alues for the controlled v ari- able w as [0 ,2]. The fixed v alues for the con trolled v ariable w ere x 1 = 0 , x i = x i − 1 + 2 n − 1 , i = 2 , · · · , n, and the parameter v alues X 0 w ere 0 .01 (extreme in- ferior v alue), 0.8 (near to the cen tral v alue) a nd 1.9 (extreme sup erior v a lue). It was considered σ 2 ǫ = 0 . 04 a nd the parameter v a lues of σ 2 δ w ere 0.01 and 0.1, whic h ar e named, resp ectiv ely , as small and large v ariances. F or the first and second stages w e consider the sample of sizes n = 5 , 20 , 100 a nd k = 2 , 20 , 1 00, resp ectiv ely . The empirical mean bias is given b y P 5000 j =1 ( ˆ X 0 − X 0 ) / 5000 and the empir- ical mean squared error (MSE) is giv en by P 5000 j =1 ( ˆ X 0 − X 0 ) 2 / 5000. The mean estimated v ariance of ˆ X 0 is giv en b y P 5000 j =1 ˆ V ( ˆ X 0 ) / 5000, with ˆ V ( ˆ X 0 ) = ˆ V 1 ( ˆ X 0 ) or ˆ V 2 ( ˆ X 0 ), where ˆ V 1 ( ˆ X 0 ) is t he estimated v ariance of (1.6), (2.1 3) or (2.19) and ˆ V 2 ( ˆ X 0 ) is the estimated v aria nce of (2.15). The theoretical v ariances of ˆ X 0 denoted as V 1 ( ˆ X 0 ) and V 2 ( ˆ X 0 ), are referred, resp ectiv ely , to the expres- sions (1.6), (2.13) or (2.1 9) a nd (2.15) ev aluated on the relev ant pa rameter v alues. In App endix B it is presen ted t he sim ulation results. T ables B1, B2, B5 and B6 presen t the empirical bias, the empirical mean squares error, the theoretical v ariance and the estimated v ariance of X 0 . In these tables, it is considered o nly the v a r iance (1.6) of the usual mo del, b ecause based on a simulation study in [15] it w as sho wn that the v aria nces (1.6) a nd (1 .8) give similar results. T ables B3, B4 and B7 presen t t he co v ering p ercentages and the confidence in terv al amplitudes constructed with a 95% confidence lev el for the parameter X 0 . In T able B3 , the co v ering p ercen tages % 1 and % 2 and amplitudes A 1 and 8 A 2 are referred to the confidence in terv als constructed using the equations (2.13) and (2.19). T ables B1-B4 consider the ho mo scedastic con trolled calibrat io n mo del assuming that σ 2 δ is unknow n. In T able B1 the empirical bias and MSE of ˆ X 0 are little and an addition in the size of the v aria nce σ 2 δ , describ ed in T able B2, causes an increasing in the bias and MSE. Moreo v er, w e ha v e that the bias and MSE of ˆ X 0 are smaller when X 0 is near to the cen ter v a lue of the v ariation interv al o f the v ariable X . The se tables sho w that for all n, k and X 0 , the theoretical v ar ia nces obtained using the expressions (2.19) and (2.15) are equal. This fact o ccurs also for the mean estimated v ariances. W e v erify also that when n ≥ 20 a nd k ≥ 20 the theoretical v ar iances and the mean estimated v aria nces fr o m the prop osed mo del a r e approximately equal. Observing these tables, we can a lso notice that there exists differences b et w een the mean estimated v aria nces of the usual and prop osed mo dels. Analyzing T a bles B3 and B4, w e observ e that f or all n and X 0 when it is adopted erratically the usual mo del, the amplitudes decrease very muc h as the size of k increases. This causes the cov ering p ercen tage to decrease mo ving aw ay from 95%. Whereas, adopting the prop osed mo del it is observ ed that when k increases the confidence interv al amplitude decreases, but the co v ering p ercen tag es increase appro a c hing 95%. Notice that the cov ering p ercen tage % 1 and % 2 and the amplitudes A 1 and A 2 are appro ximately equal, the amplitudes are v ery small fo r X 0 = 0 . 8. In these tables, w e observ e that when k = 2 0 or 100 and when n increases the amplitudes of the in terv a ls decrease and the co v ering p ercen tages approache s 9 5%. In most cases, the co v ering p ercen tage obta ined through t he prop osed mo del are greater than that for the usual mo del r esults a nd are close to 95%. T ables B5 and B7 describ e t he results for the controlled homoscedastic calibration mo del with σ 2 δ kno wn. The iterative metho d Quasi-Newton [14] has b een used. In T a bles B5 and B6 w e hav e that the empirical bias and SME decrease as the size of n or k increase and they are small when X 0 is near to the cen tral v alue of the v ariation inte rv al, X 0 = 0 . 8. When σ 2 δ is small (T able B5), for all n and k , the empirical v alues o f MSE f r om the usual a nd pr o p osed mo del are close to the theoretical v ariance, but only the mean estimated v ariance from the prop osed mo del is close t o the theoretical v ariance. When σ 2 δ is large (T able B6), in general, the empirical MSE and the mean estimated v ariance from the usual and prop osed mo del are differen t, but the v a lues supplied by the pro p osed mo del are v ery close to the theoretical v ariance. Analyzing T able B7, w e can mak e similar commen ts to the ones w e made ab out T ables B3 and B4. 9 4 Aplication In this section we t est our mo del, considering b oth cases σ δ kno wn and unkno wn, using the da ta supplied by the c hemical lab oratory of the ”Instituto de Pes quisas T ecnol´ ogicas (IPT)” - Brasil. W e also consider the usual mo del in order to observ e the p erformance of t he prop osed mo del. Our main in terest is to estimate the unkno wn concentration v alue X 0 of tw o samples A and B of the c hemical elemen ts cromo and cadmium. T ables 1 and 4 presen t the fixed v alues of concentration of t he standard solutions and the corresp onding in tensities for the cromo and cadmium ele- men t, resp ectiv ely , which are supplied b y the plasma sp ectrometry metho d. This da ta is referred t o as the first stage of the calibration mo del. T ables 2 and 5 presen t the inten sities corresp o nding to 3 sample solutions from the sample A and B. This da t a is referred to as the second stage of the calibration mo del. T ables 3 and 6 describ e the estimates of α , β , X 0 , V ( ˆ X 0 ) , σ 2 δ and the confidence in terv al amplitude U ( X 0 ) from t he ho mo scedastic controlled cal- ibration mo del of the samples A and B for the c hemical elemen ts cromo and cadmium. The v alues of the v ariance σ 2 δ considered as kno wn are obtained from an external study carried out b y the IPT, whic h are σ 2 δ = 2 , 5865 E − 06 for the cromo elemen t and σ 2 δ = 0 . 0017 E + 02 for the cadmium elemen t. As seen in Section 2, in order to obtain the estimates of the parameters β and σ 2 ǫ of the prop osed mo del when σ 2 δ is kno wn, iterativ e metho ds are required. In or der to solv e the system of equations (2.16 ) and (2.17) it w as used the Quasi-Newton iterative metho d. It is also presen ted t he estimates fr o m the usual mo del. The estimates of the v aria nce of ˆ X 0 are computed using the relev an t expres sions (1 .6 ), (2.13) o r (2.19). The amplitude U ( X 0 ) is give n b y the pro duct of the squared ro ot of the estimated v a riance o f ˆ X 0 and 1.96. In T ables 3 and 6 w e can observ e that the estimates of α and β supplied b y the usual mo del is equal to the prop osed mo del when σ 2 δ is unkno wn and they are equal for samples A and B, this o ccurs b ecause the expression of the estimators ˆ α and ˆ β of b oth mo dels a re equal and they o nly dep end on the first stage of the calibration mo del. These estimates are sligh tly differen t when compared with the estimates from the prop osed mo del when σ 2 δ is know n. With resp ect to t he estimate of X 0 , w e o bserv e that there is no difference of the estimates supplied by t he usual and the prop osed mo dels of the cro mo and cadmium elemen t in b oth samples A and B, resp ective ly . The estimates of the concen tratio n of the sample A, of the elemen ts cromo and cadmium, are outside of the v ariation range of the standard solution concentrations. W e v erify tha t, except to the sample B of the cadmium elemen t , the estimates of the v ariance o f ˆ X 0 and the amplitude U ( X 0 ) f r o m the usual mo del are greater than the estimates supplied by the b oth prop osed mo dels. 10 T able 1: Concen trat io n ( mg /g ) and in tensit y of the standard solutions of cromo elemen t. X i In tensit y 0,05 6455,900 0,11 13042,933 0,26 32621,733 0,79 97364,500 1,05 129178,10 0 T able 2: In tensit y o f the sample solutions A and B of cromo elemen t. In tensit y Sample A Sample B 1465,0 10173,6 1351,0 10516,9 1495,6 10352,2 T able 3: Estimates of α, β , X 0 , V ( ˆ X 0 ) a nd the confidence in terv al amplitude U ( X 0 ) from the usual and prop osed mo del for the samples A a nd B of cromo elemen t. Sample A Sample B Pa rameters Usual-M Prop osed-M Usual-M Prop osed-M unkno wn σ 2 δ kno wn σ 2 δ unkno wn σ 2 δ kno wn σ 2 δ α 123,574 123,574 123,889 123,574 123,574 124,021 β 1,23E+05 1,23E+05 1,23E+05 1,23E+05 1,23E+05 1,23E+05 X 0 0,011 0,011 0,011 0,083 0,083 0,083 V ( ˆ X 0 ) 9,80E-07 9,15E-07 1,35E-06 1,16E-06 1,13E-06 1,71E-06 σ 2 δ - 1,60E-06 - - 5,48E-07 - U ( X 0 ) 2,55E-03 2,46E-03 2,99E-03 2,77E-03 2,73E-03 3,36E-03 T able 4: Concen trat io n ( mg /g ) and in tensit y of the standard solutions of cadmium elemen t. X i In tensit y 0,05 4,89733 0,10 9,706 0,25 23,41333 0,73 69,73 1,01 96,85667 11 T able 5: In tensit y o f the sample solutions A and B of cadmium elemen t. In tensit y Sample A Sample B 0,679 5,066 0,6837 5,027 0,6846 5,085 T able 6: Estimates of α, β , X 0 , V ( ˆ X 0 ) a nd the confidence in terv al amplitude U ( X 0 ) from the usual and homoscedastic mo dels for the samples A and B of cadmium elemen t. Sample A Sample B Pa rameters Usual- M Prop osed-M Usual-M Proposed-M unkno wn σ 2 δ kno wn σ 2 δ unkno wn σ 2 δ kno wn σ 2 δ α -0,156 - 0,156 -0,158 -0,156 - 0,156 -0,158 β 95,828 95,828 95,831 95,828 95,828 95 ,831 X 0 8,75E-03 8,75E-03 8,77E-03 0,054 0,054 0,054 V ( ˆ X 0 ) 4,06E-06 3,72E-06 1,26E-06 3,81E-06 3,32E-06 1,17E-06 σ 2 δ - 8,31E-06 - - 8,24E-06 - U ( X 0 ) 5,18E-03 4,96E-03 2,89E-03 5,02E-03 4,68E-03 2,78E-03 5 Conclud ing remarks In general, the sim ulation study rev eals that the prop osed mo del is sensible to the presence of error related to the indep enden t v ariable and giv es b etter results in contrast to the usual mo del results. It w as noticed that when the error v ariance σ 2 δ increases, the mean estimated v ariance of ˆ X 0 obtained us- ing the usual mo del mov es aw ay f r o m t he theoretical v alue. In the example ab ov e, the confidence in terv al amplitude fr o m the prop osed mo dels a r e sup- plied by the incorp o ration of error due to the lecture of equipmen t and the preparation of the standard solutions. It is observ ed that despite the classical mo del only considers the error origina t ed fr om the lecture of the equipmen t, the amplitude is greater t ha n t he obtained b y the new approach. Ac knowle dgemen ts Betsab ´ e G. Blas Ac hic has b een partially supp o rted b y IPT (S˜ ao P aulo). 12 App endix A Bias and v ariance for the maxim um lik e- liho o d est imator In the follo wing w e deriv e the bias (2.14) and the v ariance (2.15) of the estimator ˆ X 0 from the homoscedastic con trolled calibratio n mo del when σ 2 δ is known . Considering the mo del (2.1) and (2.2), the estimator ˆ X 0 = ( ¯ Y 0 − ˆ α ) / ˆ β can b e expressed as ˆ X 0 = ¯ X + β ( X 0 − ¯ X ) + ¯ ǫ 0 − ¯ φ ˆ β , (A.1) where ¯ ǫ 0 = P n + k i = n +1 ǫ i /k a nd ¯ φ = P n i =1 ( ǫ i − β δ i ) /n . Considering k fixed, expanding 1 / ˆ β in a T a ylor series around β and ig - noring terms of order less than n − 2 , w e obtain the exp ected v alue of (A.1), giv en b y E ( ˆ X 0 ) = X 0 + γ ( ¯ X − X 0 ) nβ 2 S X X . (A.2) F rom this last equation w e get the bias (2.14). T o deriv e t he v aria nce ( 2.15) we take the v ariance of (A.1), whic h is giv en b y V ( ˆ X 0 ) = β 2 ( X 0 − ¯ X ) 2 V ( 1 ˆ β ) + V ( ¯ ǫ 0 ˆ β ) + V ( ¯ φ ˆ β ) . (A.3) W e call atten tion to the fact that (A.3) is only express ed as a function o f the related v ariances b ecause the corresp onding cov ariances are zero. The v ar i- ances V (1 / ˆ β ) , V ( ¯ ǫ 0 / ˆ β ) and V ( ¯ φ/ ˆ β ) can b e obta ined b y expanding 1 / ˆ β , ¯ ǫ 0 / ˆ β and ¯ φ/ ˆ β in a T a ylor series aro und β and ignor ing terms of or der less than n − 2 . They are giv en b y V (1 / ˆ β ) = V ( ˆ β ) β 4 , (A.4) V ( ¯ ǫ 0 / ˆ β ) = σ 2 ǫ k β 2 + 3 σ 2 ǫ k β 4 V ( ˆ β ) , (A.5) V ( ¯ φ/ ˆ β ) = γ nβ 2 . (A.6) Substituing (A.4), (A.5) and (A.6) in (A.3), then, the v ariance (2.15) is obtained. 13 B T ables T able B1. Empirical bias and mean squared erro r, theoretical v ar iance a nd the mean estimated v ariance of ˆ X 0 , for σ 2 δ = 0 , 01 and unkno wn. Empirical Theoretical Mean of ˆ V ( ˆ X 0 ) X 0 n k Prop osed-M Usual-M Proposed-M Bias MSE V 1 ( ˆ X 0 ) V 2 ( ˆ X 0 ) ˆ V 1 ( ˆ X 0 ) ˆ V 1 ( ˆ X 0 ) ˆ V 2 ( ˆ X 0 ) 0,01 5 2 -0,0060 0,0180 0,0170 0,0170 0,0120 0,0100 0,0100 20 -0,0087 0,0130 0,0120 0,0120 0,0072 0,0120 0,0120 100 -0,0060 0,0130 0,0120 0,0120 0,0065 0,0120 0,0120 20 2 -0,0038 0,0086 0,0087 0,0087 0,0120 0,0052 0,0053 20 -0,0028 0,0043 0,0042 0,0042 0,0033 0,0040 0,0040 100 -0,0032 0,0038 0,0038 0,0038 0,0022 0,0036 0,0036 100 2 -0,0023 0,0058 0,0058 0,0058 0,0100 0,0027 0,0027 20 -0,0002 0,0013 0,0013 0,0013 0,0016 0,0012 0,0012 100 -0,0007 0,0008 0,0009 0,0009 0,0007 0,0009 0,0009 0,8 5 2 -0,0011 0,0094 0,0093 0,0094 0,0079 0,0045 0,0046 20 -0,0034 0,0050 0,0048 0,0048 0,0029 0,0045 0,0046 100 -0,0007 0,0047 0,0044 0,0044 0,0024 0,0045 0,0045 20 2 0,00 05 0, 0063 0,0061 0,0061 0, 0095 0,0028 0,0029 20 0,0007 0,0016 0,0016 0,0016 0,0015 0,001 5 0,0015 100 -0,0001 0,0012 0,0012 0,0012 0,0008 0,0012 0,0012 100 2 0,0005 0,0050 0,0052 0,0052 0,0099 0,0021 0,0021 20 -0,0001 0,0007 0,0007 0,0007 0,0011 0,0007 0,0007 100 -0,0003 0,0003 0,0003 0,0003 0,0003 0,0003 0,0003 1,9 5 2 0,0041 0,0160 0,01 50 0,0160 0,0120 0,0093 0,0094 20 0,0026 0,0110 0,0110 0,0110 0,0065 0,011 0 0,0110 100 0,0076 0,0110 0,0110 0,0110 0,0058 0,0110 0,0110 20 2 0,00 06 0, 0079 0,0082 0,0082 0, 0110 0,0049 0,0049 20 0,0040 0,0039 0,0037 0,0037 0,0030 0,003 5 0,0035 100 0,0008 0,0033 0,0033 0,0033 0,0019 0,0031 0,0031 100 2 0,0020 0,0057 0,0057 0,0057 0,0100 0,0025 0,0025 20 0,0003 0,0012 0,0012 0,0012 0,0015 0,001 1 0,0011 100 0,0003 0,0008 0,0008 0,0008 0,0006 0,0008 0,0008 14 T able B2. Empirical bias and mean squared error, theoretical v ariance and the mean estimated v ariance of ˆ X 0 , f or σ 2 δ = 0 , 1 and unkno wn. Empirical Theoretical Mean of ˆ V ( ˆ X 0 ) X 0 n k Prop osed-M Usual-M Proposed-M Bias MSE V 1 ( ˆ X 0 ) V 2 ( ˆ X 0 ) ˆ V 1 ( ˆ X 0 ) ˆ V 1 ( ˆ X 0 ) ˆ V 2 ( ˆ X 0 ) 0,01 5 2 -0,0510 0,1000 0,0700 0,0710 0,0770 0,0680 0,0690 20 -0,0500 0,0950 0,0660 0,0660 0,0220 0,0660 0,0660 100 -0,0510 0,0950 0,0650 0,0650 0,0130 0,0730 0,0730 20 2 -0,0180 0,0280 0,0250 0,0250 0,0670 0,0240 0,0240 20 -0,0160 0,0230 0,0210 0,0210 0,0140 0,0210 0,0210 100 -0,0170 0,0230 0,0200 0,0200 0,0055 0,0210 0,0210 100 2 -0,0046 0,0094 0,0093 0,0093 0,0580 0,0069 0,0069 20 -0,0026 0,0048 0,0048 0,0048 0,0082 0,0048 0,0048 100 -0,0033 0,0043 0,0044 0,0044 0,0029 0,0044 0,0044 0,8 5 2 -0,0084 0,0370 0,0290 0,0290 0,0460 0,0250 0,0250 20 -0,0095 0,0270 0,0240 0,0240 0,0071 0,0200 0,0200 100 -0,0072 0,0290 0,0240 0,0240 0,0037 0,0200 0,0200 20 2 -0,0030 0,0120 0,0110 0,0110 0,0530 0,0085 0,0086 20 -0,0040 0,0068 0,0066 0,0066 0,0061 0,0064 0,0064 100 -0,0031 0,0063 0,0062 0,0062 0,0017 0,0060 0,0060 100 2 -0,0011 0,0063 0,0062 0,0063 0,0550 0,0038 0,0038 20 -0,0015 0,0017 0,0017 0,0017 0,0057 0,0017 0,0017 100 -0,0011 0,0014 0,0013 0,0013 0,0013 0,0013 0,0013 1,9 5 2 0,0430 0,1090 0,06 30 0,0630 0,0830 0,0750 0,0750 20 0,0450 0,0860 0,0580 0,0580 0,0210 0,065 0 0,0650 100 0,0410 0,0860 0,0580 0,0580 0,0110 0,0600 0,0600 20 2 0,01 60 0, 0260 0,0230 0,0230 0, 0650 0,0210 0,0210 20 0,0140 0,0190 0,0180 0,0180 0,0130 0,018 0 0,0180 100 0,0170 0,0200 0,0180 0,0180 0,0048 0,0180 0,0180 100 2 0,0050 0,0088 0,0087 0,0088 0,0570 0,0063 0,0063 20 0,0030 0,0043 0,0042 0,0042 0,0078 0,004 2 0,0042 100 0,0020 0,0039 0,0038 0,0038 0,0026 0,0038 0,0038 15 T able B3. Cov ering percentage (%) and amplitude (A) of the in terv als with a 95% confidence lev el f o r the parameter X 0 , when σ 2 δ = 0 , 01 and unkno wn. X 0 n k Usual-M Proposed-M % A % 1 A 1 % 2 A 2 0,01 5 2 83,04 0,40 79,34 0,36 79,37 0,36 20 84,95 0, 32 91,15 0,41 91,15 0,41 100 83,72 0,31 92,24 0,42 92,24 0,42 20 2 96,46 0,42 83,48 0, 28 83,52 0, 28 20 90,19 0, 22 92,47 0,24 92,47 0,24 100 86,16 0,18 92,78 0,23 92,78 0,23 100 2 98,84 0,40 73,16 0,19 73,16 0,19 20 96,71 0, 16 94,14 0,14 94,14 0,14 100 91,90 0,11 94,68 0,12 94,68 0,12 0,8 5 2 85,43 0,32 74,33 0,24 74,39 0,24 20 85,16 0, 21 91,34 0,26 91,34 0,26 100 85,04 0,19 92,50 0,25 92,50 0,25 20 2 97,90 0,38 73,55 0, 20 73,55 0, 20 20 93,55 0, 15 93,89 0,15 93,89 0,15 100 86,53 0,11 93,12 0,13 93,12 0,13 100 2 99,41 0,39 65,05 0,16 65,05 0,16 20 98,54 0, 13 94,05 0,10 94,05 0,10 100 94,56 0,07 94,86 0,07 94,86 0,07 1,9 5 2 82,47 0,39 78,05 0,35 78,13 0,35 20 84,50 0, 31 90,92 0,39 90,92 0,39 100 84,75 0,29 92,83 0,39 92,83 0,39 20 2 96,79 0,41 83,09 0, 26 83,11 0, 26 20 91,32 0, 21 93,29 0,23 93,31 0,23 100 86,43 0,17 93,32 0,22 93,32 0,22 100 2 98,88 0,40 73,06 0,19 73,06 0,19 20 97,12 0, 15 94,31 0,13 94,31 0,13 100 92,56 0,10 94,74 0,11 94,74 0,11 16 T able B4. Cov ering percentage (%) and amplitude (A) of the in terv als with a 95% confidence lev el for the parameter X 0 , when σ 2 δ = 0 , 1 and un- kno wn. X 0 n k Usual-M Prop osed-M % A % 1 A 1 % 2 A 2 0,01 5 2 84,89 0,94 80,73 0,85 80,79 0,86 20 64,10 0,51 82,75 0,86 82,75 0,86 100 52,09 0,39 82,42 0,86 82,42 0,86 20 2 99,40 0,99 91,04 0,58 91,10 0,58 20 87,10 0,45 92,60 0,55 92,62 0,55 100 65,70 0,28 92,16 0,55 92,18 0,55 100 2 100,00 0,94 87,60 0,32 87,66 0,32 20 98,82 0,36 94,88 0,27 94,88 0,27 100 89,12 0,21 94,84 0,26 94,84 0,26 0,8 5 2 90,27 0, 73 80,63 0,52 80,75 0,52 20 64,39 0,31 82,68 0,51 82,74 0,51 100 50,57 0,23 84,08 0,50 84,08 0,50 20 2 99,92 0,88 87,30 0,35 87,50 0,35 20 91,80 0,30 92,78 0,31 92,82 0,31 100 67,46 0,16 92,38 0,30 92,38 0,30 100 2 100,00 0,91 77,88 0,22 77,98 0,22 20 99,94 0,29 94,98 0,16 95,04 0,16 100 94,60 0,14 94,92 0,14 94,92 0,14 1,9 5 2 85,63 0, 91 81,42 0,81 81,42 0,81 20 61,71 0,49 81,89 0,82 81,89 0,82 100 50,71 0,37 81,24 0,81 81,24 0,81 20 2 99,54 0,97 91,12 0,54 91,18 0,55 20 88,10 0,43 92,74 0,52 92,76 0,52 100 66,44 0,26 92,86 0,51 92,86 0,51 100 2 100,00 0,93 86,38 0,30 86,38 0,30 20 99,12 0,35 94,76 0,25 94,76 0,25 100 89,40 0,20 95,00 0,24 95,00 0,24 17 T able B5. Empirical bias and mean squared error, theoretical v ariance and the mean estimated v ariance of ˆ X 0 , f or σ 2 δ = 0 , 01 and kno wn. Empirical The oretical Mean of ˆ V ( X 0 ) X 0 n k Usual-M Prop osed-M Prop osed-M U s ual-M Proposed-M Bias MSE B i as MSE V ( ˆ X 0 ) 0,01 5 2 -0,0290 0,0210 -0,0280 0, 0210 0,0170 0,0180 0,0160 20 - 0,0290 0,0140 -0,0320 0,0140 0,0120 0,0081 0,0130 100 -0,0240 0,0140 -0,0270 0,0140 0, 0120 0,0070 0,0130 20 2 -0,0081 0,0091 -0,0064 0, 0090 0,0086 0,0130 0,0076 20 - 0,0060 0,0043 -0,0072 0,0043 0,0041 0,0034 0,0041 100 -0,0038 0,0038 -0,0060 0,0038 0, 0037 0,0022 0,0038 100 2 -0,0011 0,0056 -0,0005 0,0056 0, 0058 0,0100 0,0053 20 - 0,0002 0,0013 -0,0001 0,0013 0,0013 0,0016 0,0012 100 -0,0009 0,0009 -0,0012 0,0009 0, 0009 0,0007 0,0009 0,8 5 2 -0,0074 0,0110 -0,0072 0,0100 0,0093 0,0120 0,0085 20 - 0,0046 0,0051 -0,0051 0,0052 0,0048 0,0032 0,0049 100 -0,0076 0,0048 -0,0082 0,0048 0, 0044 0,0025 0,0046 20 2 -0,0034 0,0063 -0,0031 0, 0063 0,0061 0,0100 0,0052 20 0,0001 0,001 6 0, 0000 0,0016 0,0016 0,0015 0,0016 100 -0,0009 0,0012 -0,0013 0,0012 0, 0012 0,0008 0,0012 100 2 0,0000 0 ,0053 0,0002 0,0053 0,0052 0,0099 0,0048 20 0,0000 0,000 7 0, 0000 0,0007 0,0007 0,0011 0,0007 100 -0,0001 0,0003 -0,0002 0,0003 0, 0003 0,0003 0,0003 1,9 5 2 0,0200 0,0180 0,0200 0,0180 0,0150 0,0170 0,0140 20 0,0240 0,0120 0,0260 0, 0130 0,0110 0,0071 0,0120 100 0,0200 0,0130 0,0230 0,0130 0,0100 0,0062 0,0110 20 2 0,0037 0, 0082 0,0021 0,0081 0,0082 0,0120 0,0071 20 0,0059 0,0037 0,0066 0, 0037 0,0037 0,0030 0,0036 100 0,0033 0,0032 0,0051 0,0032 0,0032 0,0020 0,0033 100 2 0,0020 0,0058 0,0015 0,0057 0,0057 0,0100 0,0053 20 0,0003 0,0012 0,0002 0, 0012 0,0012 0,0015 0,0011 100 0,0003 0,0008 0,0006 0,0008 0,0008 0,0006 0,0008 18 T able B6. Empirical bias and mean squared error, theoretical v ariance and the mean estimated v ariance of ˆ X 0 , f or σ 2 δ = 0 , 1 and kno wn. Empirical The oretical Mean of ˆ V ( X 0 ) X 0 n k Usual-M Prop osed-M Prop osed-M U s ual-M Proposed-M Bias MSE Bias MSE V ( ˆ X 0 ) 0,01 5 2 -0,4330 0,5590 -0,3830 0, 4580 0,0590 0,3650 0,2310 20 - 0,1140 0,1310 0,4140 1,0880 0,0540 0,0250 0,0760 100 -0,1890 0,1540 -0,0290 0,7730 0, 0540 0,0200 0,1250 20 2 -0,0930 0,0430 -0,0770 0, 0360 0,0210 0,0950 0,0380 20 - 0,0490 0,0210 -0,0510 0,0210 0,0160 0,0150 0,0180 100 -0,0430 0,0190 -0,0250 0,0640 0, 0150 0,0058 0,0170 100 2 -0,0200 0,0110 -0,0160 0,0097 0, 0084 0,0630 0,0110 20 - 0,0077 0,0041 -0,0085 0,0039 0,0037 0,0084 0,0038 100 -0,0066 0,0037 -0,0087 0,0037 0, 0033 0,0029 0,0033 0,8 5 2 -0,0500 0,0620 -0,0430 0,0550 0,0280 0,1150 0,0620 20 - 0,0450 0,0380 0,0820 0,0810 0,0240 0,0086 0,0320 100 -0,0530 0,0510 -0,0069 0,0980 0, 0230 0,0055 0,0360 20 2 -0,0280 0,0150 -0,0240 0, 0140 0,0110 0,0740 0,0210 20 - 0,0079 0,0068 -0,0085 0,0069 0,0064 0,0065 0,0067 100 -0,0055 0,0065 -0,0030 0,0079 0, 0060 0,0018 0,0062 100 2 -0,0030 0,0067 -0,0021 0,0065 0, 0062 0,0600 0,0091 20 - 0,0018 0,0018 -0,0018 0,0017 0,0017 0,0058 0,0017 100 -0,0021 0,0013 -0,0025 0,0013 0, 0013 0,0013 0,0013 1,9 5 2 0,3480 0,3770 0,3070 0,3000 0,0530 0,2850 0,1760 20 0,1400 0,1630 -0,3310 0,9370 0,0490 0,0300 0,0780 100 0,0410 0,0180 0,0270 0,0510 0,0140 0,0051 0,0150 20 2 0,0970 0, 0430 0,0790 0,0350 0,0190 0,0930 0,0350 20 0,1400 0,1630 -0,3310 0,9370 0,0490 0,0300 0,0780 100 0,1670 0,1300 -0,0005 0,7090 0,0480 0,0160 0,1120 100 2 0,0200 0,0093 0,0160 0,0087 0,0080 0,0620 0,0110 20 0,0120 0,0039 0,0130 0, 0037 0,0033 0,0080 0,0035 100 0,0072 0,0031 0,0089 0,0031 0,0029 0,0027 0,0030 19 T able B7. Cov ering percentage (%) and amplitude (A) of the in terv als with a 95% confidence lev el for the parameter X 0 , when σ 2 δ = 0 , 01 . and 0,1 and kno wn. σ 2 δ = 0 , 01 σ 2 δ = 0 , 1 X 0 n k U s ual-M Proposed-M Usual-M Prop osed-M % A % A % A % A 0,01 5 2 92,10 0,51 91,20 0,48 95,06 1,89 92,40 1,49 20 87,32 0, 34 95,18 0,44 63,38 0,59 64,47 1,09 100 84,50 0,32 95,19 0,44 52,69 0,48 89,24 1,18 20 2 97,46 0,43 90,12 0, 33 99,87 1,17 96,27 0,71 20 91,00 0, 23 94,03 0,25 92,76 0,48 94,48 0,52 100 86,73 0,19 95,21 0,24 70,38 0,30 90,35 0,51 100 2 97,01 0,43 90,00 0,33 100,00 0,98 93,41 0,40 20 92,34 0, 23 95,26 0,25 99,77 0,36 95,09 0,24 100 85,93 0,19 94,55 0,24 92,70 0,21 94,42 0,23 0,8 5 2 94,33 0,41 89,28 0,35 98,77 1,31 97,54 0,98 20 86,84 0, 22 95,03 0,27 61,85 0,34 81,25 0,69 100 85,65 0,19 95,56 0,27 49,29 0,27 87,82 0,69 20 2 98,35 0,39 88,44 0, 27 100,00 1,03 94,09 0,53 20 93,84 0, 15 94,41 0,15 94,44 0,32 95,11 0,32 100 85,98 0,11 94,32 0,14 70,38 0,17 92,41 0,31 100 2 98,08 0,39 88,22 0,27 100,00 0,95 92,25 0,34 20 92,75 0, 15 93,96 0,15 99,89 0,30 94,89 0,16 100 87,17 0,11 94,60 0,14 95,00 0,14 94,88 0,14 1,9 5 2 92,30 0,50 90,61 0,46 96,26 1,78 92,14 1,45 20 86,96 0, 33 95,13 0,42 63,58 0,56 67,24 1,03 100 85,84 0,30 94,73 0,42 49,30 0,46 83,75 1,21 20 2 97,61 0,43 89,92 0, 32 100,00 1,17 96,27 0,70 20 91,60 0, 21 94,27 0,23 94,26 0,46 94,95 0,49 100 86,16 0,17 94,99 0,23 72,44 0,28 94,18 0,48 100 2 97,04 0,43 89,70 0,32 100,00 0,97 94,27 0,39 20 91,21 0, 21 93,85 0,23 99,59 0,35 95,18 0,23 100 87,04 0,17 95,11 0,23 93,38 0,20 94,98 0,21 20 References [1] Berkson, J. Estimation of a linear function for a calibration line: con- sideration o f a recen t prop osal. T ec hnometrics 11 (1969) 6 49. [2] Williams, E.J. A note o n r egr ession metho ds in calibra tion. T ec hnome- trics 11 (1969) 192. [3] Lwin, T. Discussion of Hunter and Lamboy’s pap er. T ec hnometrics 14 (1981) 339 . [4] Sh ukla, G.K. On the problem of calibration. T echnometric s 14 (1972) 547. [5] Bro wn, P .J. Measuremen t, regression and calibration. Oxford: Oxford Univ ersit y Press. [6] T ellinghuis en, J. Inv erse vs. classic al calibration for small data sets. F re- senius J Anal Chem 368 (2000) 585. [7] Aitc hison, J. and D unsmore, I. (1975). Statistical Prediction Analysis. Cam bridge Univ ersity Press. [8] Osb ourne, C. Statistical calibration: A r eview. In ternational Statistical Review 59 (1991) 309. [9] Cheng, Chi-Lun, V an Ness J. W. (1999). Statistical regression with mea- suremen t error. London; New Y ork : Arnold: Oxford Univ ersity Press. [10] So ciedade Brasileira de Metrolog ia (SBM) (200 2). Determinando a In- certeza em Medic˜ ao Anal ´ ıtica. 2. ed. Brasileira - T raduc˜ ao do Guia EU- RA CHEM /CIT A C. [11] Berkson, J. Are there tw o regression?. Journal of the American Statis- tical Asso ciation 45 (1950) 164. [12] F uller, W. A. (1987). Measuremen t error models. John Wiley Sons, New Y or k. [13] Lima, C.R.P (1996). Calibra¸ c˜ ao absoluta com erro na s v ari´ av eis. S˜ a o P aulo: IME-USP . T ese de Doutorado. [14] P olak, E. (1 997). Optimization Algo rithms and Consisten t Approxima- tions. New Y ork: Springer-V erlag. [15] Blas, B. G.(20 0 5). Calibra¸ c˜ ao controlada aplicada ` a qu ´ ımica anal ´ ıtica. S˜ ao P aulo: IME-USP . Disserta¸ c˜ ao de Mestrado. 21