aa r X i v : . [ s t a t . O T ] M a y Linear Regression under Special Relativity
Si Hyung Joo([email protected])Department of Industrial Engineering,Chonnam National UniversityJuly 24, 2020
Abstract
This study investigated the problem posed by using ordinary least squares (OLS)to estimate parameters of simple linear regression under a specific context of specialrelativity, where an independent variable is restricted to an open interval, ( − c, c ). Itis found that the OLS estimate for the slope coefficient is not invariant under Lorentzvelocity transformation. Accordingly, an alternative estimator for the parameters oflinear regression under special relativity is proposed. This estimator can be considereda generalization of the OLS estimator under special relativity; when c approaches toinfinity, the proposed estimator and its variance converges to the OLS estimator andits variance, respectively. The variance of the proposed estimator is larger than thatof the OLS estimator, which implies that hypothesis testing using the OLS estimatorand its variance may result in a liberal test under special relativity. Keywords: invariance of estimate, bounded independent variable, open interval, Lorentztransformation, Lorentz invariant, method of moments1
Introduction
Linear regression is one of the most frequently employed models in empirical analysis. Y i = β + β · X i + β · X i + · · · + β p · X pi + ǫ i The β p is a measure of association between the independent variable X p and the depen-dent variable Y , and β is the expected value of Y when all X p s are equal to zero. The ǫ i is the error term.The linear regression model is based on the following assumptions.(1) The independent variables are measured without error.(2) The errors are independent from the independent variables.(3) The errors are independently and identically normally distributed.The unknown parameters in a linear regression model are often estimated using theordinary least squares (OLS) method because the OLS estimator has desirable propertiesas an estimator of parameters, such as unbiasedness, consistency, and efficiency (Greene,2012).The linear regression model and OLS estimator provide accurate inferences and esti-mates only if the assumptions above hold true. The assumption of an error term that isnormally distributed conditional on the independent variables implies that the dependentvariable can be any real number. This assumption is violated if a dependent variable hasa limited range, for example, a discontinuous or bounded dependent variable. Because alinear regression model with a limited dependent variable may lead to serious errors ofinference, alternative nonlinear models and procedures have been developed and employed,2uch as a Tobit model for censored dependent variables and a Poisson regression model forcount (non-negative integer) dependent variables (Long, 1997).In contrast, researchers rarely pay attention to whether independent variables with alimited range exist in the model—as long as they are exogenous and measured withouterror—because no assumption of the linear regression model is violated. Although theyhave limited independent variables, conventional (e.g., OLS) estimators are commonly usedto estimate the unknown parameters of the linear regression model.Are there no problems posed by using the OLS estimator when independent variablesare restricted as long as they are exogenous and measured without error? If there are, whatwould be the proper estimator when independent variables are restricted?To investigate the problem posed by using the OLS estimator when an independentvariable is restricted, this study investigates a simple linear regression model with an ex-ogenous error-free independent variable intrinsically restricted to an open interval. Thelinear regression model emerges when one tries to estimate the scale (slope coefficient) andthe accuracy (intercept coefficient) of a velocity meter under special relativity (see Section2 for details). In this model, the dependent variable is the velocity of an object measuredby an observer with a velocity meter that has a normal error, and the independent variableis the true velocity of the object relative to the observer. In the real world, where thespecial theory of relativity applies, the true velocity of a (massive) object from an observer(independent variable) is restricted to an open interval, ( − c, c ), where c is the speed oflight (Taylor, 1992).The OLS estimate for the slope coefficient is found to depend on the velocity of an3bserver under special relativity. To address the problem, a new estimator for the slopecoefficient, which is independent from the velocity of an observer, is proposed.The proposed estimator is found to be unbiased and converges to the OLS estimatorwhen c approaches to infinity. Its variance is larger than the OLS estimator, which reflectsthe fact that there is larger uncertainty if an independent variable is restricted. Its variancealso converges to that of the OLS estimator when c approaches to infinity.The rest of this paper is organized as follows. Section 2 illustrates the linear regressionmodel under special relativity. Section 3 investigates the inadequacy of using the OLSestimator for the linear regression model under special relativity. Section 4 provides therationale for an alternative estimator. Section 5 proposes an alternative estimator for thelinear regression model under special relativity. Section 6 examines the properties of thealternative estimator. Finally, Section 7 summarizes and provides concluding remarks. In this section, we describe a special relativistic situation in which a simple linear regressionmodel with an independent variable that is intrinsically bounded ( − c, c ) emerges.Suppose an engineer developed a velocity meter. In terms of precision, the velocitymeter has a random error that follows a normal distribution with zero mean and unknownvariance σ . The random error is independent not only from the true velocity of the objectbeing measured but also from the velocity of the observer holding the velocity meter. Theengineer is not sure through which unit—meter/second, mile/hour, or others—their velocitymeter measures the velocity of an object. If the true velocity of an object is measured by4eter/second, the unit of the newly developed velocity meter can be represented by β meter/second. In the worst case, their velocity meter does not reflect the true velocity ofobjects at all, that is, β = 0. In addition, the engineer is not sure if their velocity meteris zero adjusted ( β = 0), that is, if the velocity meter shows zero velocity when measuringthe velocity of a stationary object from the observer. In other words, the velocity meterhas a systematic bias β in terms of accuracy.The engineer took their velocity meter to their fellow researcher who has a velocitymeter that exactly measures the velocity of an object in meter/second. The engineerasked the researcher if their velocity meter can measure the velocity of an object—that is, β = 0—and, if so, which scale the velocity meter is using ( β = 0) and how much theirown velocity meter needs to be adjusted ( β = 0) to ensure zero velocity for a stationaryobject. In the real (relativistic) world, the true velocity of a (massive) object is restrictedto an open interval, ( − c, c ), where c is the speed of light in meter/second (Taylor, 1992).The researcher’s velocity meter therefore always shows a value within a range ( − c, c ).Therefore, the population regression equation can be represented as Y i = β + β · X i + ǫ i (1)where: Y i : the velocity of object i measured by the newly developed velocity meter X i : the true velocity of object i in terms of meter/second, X i ∈ ( − c, c ) β : systematic bias of the velocity meter for a stationary object β : the scale of the velocity meter in terms of meter/second ǫ i : random error of the velocity meter, which follows N (0 , σ )5 Investigating the inadequacy of OLS estimator un-der special relativity
In this section, the inadequacy of using the OLS estimator under special relativity is inves-tigated.
Let us suppose the engineer and researcher were living in the Newtonian universe. Becausethe true velocity of an (massive) object can range from −∞ to ∞ in the Newtonian universe(Taylor, 1992), the regression model becomes a simple linear regression model with anunrestricted independent variable.The researcher may conduct an experiment measuring the velocity of N objects movingalong a straight line with both a newly developed velocity meter and the exact velocitymeter. With the relevant data, they can estimate unknown parameters using the OLSestimator.The OLS sample regression equation corresponding to equation (1) can be written as Y i = ˆ β ,OLS + ˆ β ,OLS · X i + ˆ ǫ i (2)where ˆ β ,OLS and ˆ β ,OLS are the OLS estimator of β and β , respectively, and ˆ ǫ i is theOLS residual for sample i .The first-order conditions for the OLS estimators ˆ β ,OLS and ˆ β ,OLS are P Ni =1 ˆ ǫ i = 0 and P Ni =1 ˆ ǫ i · X i = 0, respectively. 6he OLS estimator can be regarded as a method of moments estimator based on thepopulation moment condition E [ ǫ ] = 0 and E [ ǫ · X ] = 0 (Dray, 2012).The estimates ˆ β ,OLS and ˆ β ,OLS are bivariate normally distributed, and their means,variances, and covariance are as follows.ˆ β ,OLS = P Ni =1 ( Y i − ¯ Y ) · ( X i − ¯ X ) P Ni =1 ( X i − ¯ X ) (3)ˆ β ,OLS = ¯ Y − ˆ β ,OLS · ¯ X (4) V ar ( ˆ β ,OLS ) = 1 P Ni =1 ( X i − ¯ X ) σ (5) V ar ( ˆ β ,OLS ) = P Ni =1 X i P Ni =1 N ( X i − ¯ X ) σ (6) Cov ( ˆ β ,OLS , ˆ β ,OLS ) = − ¯ X P Ni =1 ( X i − ¯ X ) σ (7)where ¯ X = N P Ni =1 X i and ¯ Y = N P Ni =1 Y i .ˆ β ,OLS is the estimate of the scale of the velocity meter and ˆ β ,OLS is the estimate ofthe systematic bias. Their (simultaneous) confidence intervals can be determined by theirvariances and covariance.If the researcher were measuring the velocity of objects with the exact velocity meterwhile moving in a relatively positive direction with a constant velocity v ∗ than before, theywould obtain X ′ i = X i − v ∗ according to the Galilean velocity transformation (Hall, 2005).The relationships between x i and x ′ i can be denoted as follows. X i = ( X i − ¯ X ) + ¯ X = x i + ¯ X (8) X ′ i = X i − v ∗ = x i + ¯ X − v ∗ = x i + ¯ X ′ (9)7here x i = X i − ¯ X and ¯ X ′ = N P Ni =1 X ′ i . X i and X ′ i have the same demeaned velocity x i . In other words, the demeaned velocityis invariant under Galilean velocity transformation. In addition, P Ni =1 x i = 0 and X i and X ′ i and x i have the same variance.If the researcher were measuring the velocity of objects with the newly developed ve-locity meter while moving in a positive direction with a constant velocity v ∗ than before,they would obtain Y ′ i = Y i − β · ( X i − X ′ i ) = Y i − β · v ∗ .Because the error terms need to be independent from the velocity of the researcher aswell as the true velocity of objects, the independence of error terms from the true velocityneeds to be specified by the demeaned velocity x i , which is invariant from the velocity ofthe researcher. Hence, the independence of error terms from the true velocity of objectsand that of the researcher is specified as E [ ǫ · x ] = 0. Because E [ ǫ · x ] = 0 is equivalentto E [ ǫ · X ] = 0 and E [ ǫ · X ′ ] = 0, the error term, which is independent from x , is alsoindependent from X and X ′ .The OLS estimates ˆ β ′ ,OLS and its variance V ar ( ˆ β ′ ,OLS ) based on X ′ i and Y ′ i are thesame as ˆ β ,OLS and V ar ( ˆ β ,OLS ), respectively.It shows that the OLS estimates for β and its variance remain invariant regardless ofthe velocity of the researcher in the Newtonian universe. In other words, OLS estimatesfor β and its variance are invariant under the Galilean velocity transformation.8 .2 Real relativistic world The real world is not like the Newtonian universe. In the real (relativistic) world, the truevelocity of an (massive) object is restricted to an open interval, ( − c, c ), where c is the speedof light (Taylor, 1992). Moreover, if the researcher in the real (relativistic) world were tomeasure the velocity of objects while moving in a positive direction with a constant velocity v ∗ than before, they would obtain the following velocities according to the Lorentz velocitytransformation (Taylor, 1992). X ′′ i = X i − v ∗ − v ∗ · X i c (10) Y ′′ i = y i − β · ( X i − X ′′ i ) (11)Unlike the Newtonian universe case, the OLS estimate ˆ β ′′ ,OLS and its variance V ar ( ˆ β ′′ ,OLS )based on X ′′ i and Y ′′ i are different from ˆ β ,OLS and V ar ( ˆ β ,OLS ). It shows that the OLS esti-mate for β and its variance depend on the velocity of the researcher in the real (relativistic)world.This problem arises because the demeaned velocity is not invariant under the Lorentzvelocity transformation. Because X i and X ′′ i have different demeaned velocity, x i = x ′′ i ,where x ′′ i = X ′′ i − ¯ X ′′ and ¯ X ′′ = N P Ni =1 X ′′ i , E [ ǫ · X ] = 0 is not equivalent to E [ ǫ · X ′′ ] = 0.This result shows that, to obtain an estimate for β and its variance, which is indepen-dent from the velocity of the researcher in the real (relativistic) world, the independenceof the error term from the velocity of objects and from that of the researcher needs to bespecified by a quantity that is invariant under the Lorentz velocity transformation.9 Independence of error term in the relativistic uni-verse
In this section, we search for invariant quantities (Taylor, 1992) under the Lorentz velocitytransformation and suggest specifications for the independence of the error term from thevelocity of objects and that of the researcher; the goal is to properly estimate the parametersof the regression model. Please refer to Dray (2012) for details of the concepts under specialrelativity and the Lorentz invariant quantities.In physics, the rapidity θ of a velocity X is defined as follows. θ = tanh − ( Xc ) (12)The relativistic momentum and energy of an object with rapidity θ and rest mass m aredefined as follows. M omentum = m · c · sinh ( θ ) (13) Energy = m · c · cosh ( θ ) (14)Let θ i , θ ′′ i and θ ∗ be the rapidity of X i , X ′′ i and v ∗ , respectively. θ i = tanh − ( X i c ) (15) θ ′′ i = tanh − ( X ′′ i c ) (16) θ ∗ = tanh − ( v ∗ c ) (17)Then, the following relationship among θ i , θ ′′ i and θ ∗ holds true. θ ′′ i = θ i − θ ∗ (18)10ence, X i and X ′′ i can be represented as follows. X i = c · tanh ( θ i ) (19) X ′′ i = c · tanh ( θ ′′ i ) = c · tanh ( θ i − θ ∗ ) (20)Let θ and θ ′′ be defined as follows. tanh ( θ ) = P Ni =1 sinh ( θ i ) P Ni =1 cosh ( θ i ) (21) tanh ( θ ′′ ) = P Ni =1 sinh ( θ ′′ i ) P Ni =1 cosh ( θ ′′ i ) (22)(23)Then, the following relationship among θ , θ ′′ and θ ∗ holds true. θ ∗ = θ − θ ′′ (24)Let φ i = θ i − θ ; then, θ i and θ ′′ i can be represented as follows. θ i = ( θ i − θ ) + θ = φ i + θ (25) θ ′′ i = θ i − θ ∗ = θ i − ( θ − θ ′′ ) = ( θ i − θ ) + θ ′′ = φ i + θ ′′ (26)This result shows that φ i remains invariant regardless of the velocity of the researcher.Because φ i remains invariant, any function of φ i — especially the relativistic momentum m · c · sinh ( φ i ) and the relativistic energy m · c · cosh ( φ i ) — remain invariant, regardlessof the velocity of the researcher. In addition, the following relationships hold true. N X i =1 sinh ( φ i ) = N X i =1 sinh ( θ i − θ ) = N X i =1 sinh ( θ ′′ i − θ ′′ ) = 0 (27) N X i =1 cosh ( φ i ) = N X i =1 cosh ( θ i − θ ) = N X i =1 cosh ( θ ′′ i − θ ′′ ) (28)1127) shows that if the researcher were measuring the velocity (rapidity) of objects whilemoving in a positive direction with a constant rapidity of θ than before, then the sum ofthe relativistic momentum of objects equals zero if we assume that all the objects have thesame rest mass. Under this assumption, the rapidity θ is associated with the relativisticcenter of momentum. Therefore, φ i can be considered the rapidity of object i measuredfrom the relativistic center of momentum when we assume that all the objects have thesame rest mass.Because the error terms need to be independent not only from the true velocity of anobject but also from that of the researcher, the independence of the error term needs tobe specified by a quantity that is invariant from the velocity of the researcher. Hence, theindependence of the error term can be specified as E [ ǫ · f ( φ )] = 0.The parameters can be estimated using the following sample moment conditions corre-sponding to the population moment conditions E [ ǫ ] = 0 and E [ ǫ · f ( φ )] = 0.¯ Y − ˆ β − ˆ β · ¯ X = 01 N { N X i =1 Y i · f ( φ i ) − ˆ β N X i =1 f ( φ i ) − ˆ β N X i =1 X i · f ( φ i ) } = 0 (29)When β = 0, the β is not uniquely identified if P Ni =1 f ( φ i ) = 0. P Ni =1 f ( φ i ) needsto be equal to zero. Therefore, E [ ǫ · sinh ( φ )] = 0 is selected as the population momentcondition. 12 Special relativistic linear regression estimator
The population regression equation is the same as equation (1). The population momentconditions are E [ ǫ ] = 0 (30) E [ ǫ · sinh ( φ )] = 0 (31)The sample regression equation is Y i = ˆ β + ˆ β · X i + ˆ ǫ i (32)Meanwhile, the sample moment conditions are1 N N X i =1 ˆ ǫ i = 0 (33)1 N N X i =1 ˆ ǫ i · sinh ( φ i ) = 0 (34)From equation (33), 1 N N X i =1 ˆ ǫ i = 1 N N X i =1 ( Y i − ˆ β − ˆ β · X i )= 1 N N X i =1 Y i − ˆ β − ˆ β N N X i =1 X i = ¯ Y − ˆ β − ˆ β · ¯ X = 0 (35)From equation (34),1 N N X i =1 ˆ ǫ i · sinh ( φ i ) = 1 N N X i =1 ( Y i − ˆ β − ˆ β · X i ) · sinh ( φ i ) (36)= 1 N { N X i =1 Y i · sinh ( φ i ) − ˆ β N X i =1 sinh ( φ i ) − ˆ β N X i =1 X i · sinh ( φ i ) } = 1 N { N X i =1 Y i · sinh ( φ i ) − ˆ β N X i =1 X i · sinh ( φ i ) } = 0 by (27)13ote that P Ni =1 X i · sinh ( φ i ) > β = P Ni =1 Y i · sinh ( φ i ) P Ni =1 X i · sinh ( φ i ) (37)ˆ β = ¯ Y − ˆ β · ¯ X = ¯ Y − P Ni =1 Y i · sinh ( φ i ) P Ni =1 X i · sinh ( φ i ) · ¯ X (38) In this section, the properties of the proposed estimator are examined. ˆ β and ˆ β The estimator ˆ β and ˆ β can be written as a linear combination of the sample values of Y ,the Y i ( i = 1 , · · · , N ). Note equation (37) and (38), ˆ β = P Ni =1 k i · Y i and ˆ β = P Ni =1 h i · Y i ,where k i = sinh ( φ i ) P Ni =1 X i · sinh ( φ i ) and h i = sinh ( φ i ) · ¯ X P Ni =1 X i · sinh ( φ i ) .Because ˆ β and ˆ β are linear combination of normally distributed random variables Y i ,ˆ β and ˆ β are normally distributed. ˆ β and ˆ β Note that P Ni =1 k i = P Ni =1 sinh ( φ i ) P Ni =1 X i · sinh ( φ i ) = 0 and P Ni =1 k i · X i = P Ni =1 X i · sinh ( φ i ) P Ni =1 X i · sinh ( φ i ) = 1.14 β = N X i =1 k i · Y i = N X i =1 k i ( β + β · X i + ǫ i )= β N X i =1 k i + β N X i =1 k i · X i + N X i =1 k i · ǫ i = β + N X i =1 k i · ǫ i (39) E [ ˆ β ] = E [ β + N X i =1 k i · ǫ i ] = E [ β ] + E [ N X i =1 k i · ǫ i ] (40)= β + N X i =1 k i · E [ ǫ i | X i ] since β is a constant and the k i are random= β + N X i =1 k i · E [ ǫ i | X i ] = 0 by assumption= β Therefore, ˆ β is an unbiased estimator of β .ˆ β = ¯ Y − ˆ β · ¯ X = ( β + β · ¯ X + ¯ ǫ ) − ˆ β · ¯ X = β + ( β − ˆ β ) · ¯ X + ¯ ǫ (41)15 [ ˆ β ] = E [ β + ( β − ˆ β ) · ¯ X + ¯ ǫ ] (42)= E [ β ] + E [( β − ˆ β ) · ¯ X ] + E [¯ ǫ ]= β + ¯ X · E [( β − ˆ β )] + E [¯ ǫ ] since β is a constant= β + ¯ X · E [( β − ˆ β )] since E [¯ ǫ ] = 0 by assumption= β + ¯ X ( E [( β ] − E [ ˆ β )])= β + ¯ X ( β − β ) since E [ β ] = β and E [ ˆ β ] = β = β Therefore, ˆ β is an unbiased estimator of β . ˆ β and ˆ β Using the assumption that y i are independently distributed, the variance of ˆ β is V ar ( ˆ β ) = E [ { ˆ β − E [ ˆ β ] } ] (43)= E [ { ˆ β − β } ] since E [ ˆ β ] = β Note equation (39)( ˆ β − β ) = ( N X i =1 k i ǫ i ) = N X i =1 k i ǫ i + 2 N − X i =1 N X j = i +1 k i k j ǫ i ǫ j (44)Hence, 16 [ { ˆ β − β } ] = E [ N X i =1 k i ǫ i + 2 N − X i =1 N X j = i +1 k i k j ǫ i ǫ j ] (45)= N X i =1 k i E [ ǫ i | X i ] + 2 N − X i =1 N X j = i +1 k i k j E [ ǫ i ǫ j | X i X j ]= N X i =1 k i E [ ǫ i | X i ] since E [ ǫ i ǫ j | X i X j ] = 0 by assumption= N X i =1 k i · σ = σ N X i =1 k i since E [ ǫ i | X i ] = σ by assumption N X i =1 k i = 1 { P Ni =1 X i · sinh ( φ i ) } N X i =1 { sinh ( φ i ) } (46) N X i =1 { sinh ( φ i ) } = N X i =1 { cosh (2 · φ i ) − } (47)= 12 N X i =1 cosh (2 · φ i ) − N N { N N X i =1 cosh (2 · φ i ) − } N X i =1 { sinh ( φ i ) } = N T −
1) where T = 1 N N X i =1 cosh (2 · φ i ) (48)17 X i =1 X i · sinh ( φ i ) = N X i =1 c · tanh ( θ i ) · sinh ( θ i − θ ) (49)= c N X i =1 tanh ( θ i ) { sinh ( θ i ) cosh ( θ ) − cosh ( θ i ) sinh ( θ ) } = c N X i =1 { sinh ( θ i ) cosh ( θ i ) · sinh ( θ i ) cosh ( θ ) − sinh ( θ i ) cosh ( θ i ) · cosh ( θ i ) sinh ( θ ) } = c N X i =1 { cosh ( θ ) sinh ( θ i ) cosh ( θ i ) − sinh ( θ ) sinh ( θ i ) } = c N X i =1 [ cosh ( θ ) { cosh ( θ i ) − cosh ( θ i ) } − sinh ( θ ) sinh ( θ i )]= c [ cosh ( θ ) { N X i =1 cosh ( θ i ) − N X i =1 cosh ( θ i ) } − sinh ( θ ) N X i =1 sinh ( θ i )]Let C = N P Ni =1 cosh ( θ i ), S = N P Ni =1 sinh ( θ i ), and H = N P Ni =1 1 cosh ( θi ) . tanh ( θ ) = SC (50) cosh ( θ ) = 1 p − tanh ( θ ) = 1 q − S C = C √ S − C (51) sinh ( θ ) = tanh ( θ ) · cosh ( θ ) = S √ S − C (52) N X i =1 cosh ( θ i ) = NH (53) N X i =1 X i · sinh ( φ i ) = c [ C √ S − C { N · C − NH } − S √ S − C · N · S ] (54)= c · N √ S − C [ C − S − CH ]18herefore, V ar ( ˆ β ) = σ N ( T − S − C c · N ( C − S − CH ) = ( S − C )( T − c · N ( C − S − CH ) σ (55)The variance of ˆ β is V ar ( ˆ β ) = V ar ( ¯ Y − ˆ β · ¯ X ) = V ar ( ¯ Y ) + ¯ X V ar ( ˆ β ) (56)= V ar ( 1 N N X i =1 ( β + β · X i + ǫ i )) + ¯ X V ar ( ˆ β ) (57)= 1 N · N · σ + ¯ X V ar ( ˆ β ) (58)= 1 N (cid:18) S − C )( T −
1) ¯ X c ( C − S − CH ) (cid:19) σ (59) ˆ β and ˆ β The covariance between ˆ β and ˆ β is Cov ( ˆ β , ˆ β ) = E [( ˆ β − E [ ˆ β ])( ˆ β − E [ ˆ β ])] (60)= E [ { ( ¯ Y − ˆ β ¯ X ) − E [ ˆ β ] } ( ˆ β − E [ ˆ β ])] from equation (38)= E [ { ( ¯ Y − ˆ β ¯ X ) − ( ¯ Y − β ¯ X ) } ( ˆ β − E [ ˆ β ])] since E [ ˆ β ] = ¯ Y − E [ ˆ β ] ¯ X = ¯ Y − β ¯ X = E [ { ( ¯ Y − ˆ β ¯ X ) − ( ¯ Y − β ¯ X ) } ( ˆ β − β )] from equation (40)= E [ − ¯ X · ( ˆ β − β ) ]= − ¯ X · E [( ˆ β − β ) ]= − ¯ X · V ar ( ˆ β ) 19 .5 Convergence of ˆ β and ˆ β to ˆ β ,OLS and ˆ β ,OLS when c → ∞ Let X = c · tanh ( θ ). X = c · tanh ( θ ) = c · P Ni =1 sinh ( θ i ) P Ni =1 cosh ( θ i ) = P Ni =1 c · sinh ( θ i ) P Ni =1 cosh ( θ i ) (61)lim c →∞ c · sinh ( θ i ) = lim c →∞ c · sinh ( tanh − ( X i c )) = X i (62)lim c →∞ cosh ( θ i ) = lim c →∞ cosh ( tanh − ( X i c )) = 1 (63)lim c →∞ X = lim c →∞ P Ni =1 c · sinh ( θ i ) P Ni =1 cosh ( θ i ) = lim c →∞ P Ni =1 X i P Ni =1 N N X i =1 X i = ¯ X (64) c · sinh ( φ i ) = c · sinh ( θ i − θ ) = c · sinh ( tanh − ( X i c ) − tanh − ( X c )) (65)= c · sinh ( tanh − ( X i c )) · cosh ( tanh − ( X c )) − c · cosh ( tanh − ( X i c )) · sinh ( tanh − ( X c )) (66)lim c →∞ c · sinh ( φ i ) = lim c →∞ c · sinh ( tanh − ( X i c )) · lim c →∞ cosh ( tanh − ( X c )) − lim c →∞ cosh ( tanh − ( X i c )) · lim c →∞ c · sinh ( tanh − ( X c )) (67)= X i − ¯ X (68)20 X i =1 c · X i · sinh ( φ i ) = N X i =1 c · { ( X i − X ) + X } · sinh ( φ i ) (69)= N X i =1 c · ( X i − X ) · sinh ( φ i ) + N X i =1 c · X · sinh ( φ i ) (70)= N X i =1 c · ( X i − X ) · sinh ( φ i ) (71) c · ( X i − X ) · sinh ( φ i ) = c · ( X i − X ) · sinh ( θ i − θ ) (72)= c · ( X i − X ) · sinh ( tanh − ( X i c ) − tanh − ( X c )) (73)= c · ( X i − X ) · sinh ( tanh − ( X i c )) · cosh ( tanh − ( X c )) − c · ( X i − X ) · cosh ( tanh − ( X i c )) · sinh ( tanh − ( X c )) (74)lim c →∞ c · ( X i − X ) · sinh ( φ i )= lim c →∞ ( X i − X ) · lim c →∞ c · sinh ( tanh − ( X i c )) · lim c →∞ cosh ( tanh − ( X c )) − lim c →∞ ( X i − X ) · lim c →∞ cosh ( tanh − ( X i c )) · lim c →∞ c · sinh ( tanh − ( X c ))= ( X i − ¯ X ) · X i − ( X i − ¯ X ) · ¯ X = ( X i − ¯ X ) (75)lim c →∞ ˆ β = lim c →∞ P Ni =1 Y i · sinh ( φ i ) P Ni =1 X i · sinh ( φ i ) = lim c →∞ P Ni =1 Y i · c · sinh ( φ i ) P Ni =1 X i · c · sinh ( φ i )= lim c →∞ P Ni =1 Y i · c · sinh ( φ i ) P Ni =1 ( X i − X ) · c · sinh ( φ i )= P Ni =1 lim c →∞ Y i · c · sinh ( φ i ) P Ni =1 lim c →∞ ( X i − X ) · c · sinh ( φ i )= P Ni =1 Y i · ( X i − ¯ X ) P Ni =1 ( X i − ¯ X ) = ˆ β ,OLS (76)21herefore, ˆ β converges to ˆ β ,OLS when c → ∞ .lim c →∞ ˆ β = lim c →∞ ( ¯ Y − ˆ β · ¯ X ) from equation (38)= ¯ Y − ¯ X · lim c →∞ ˆ β = ¯ Y − ˆ β ,OLS · ¯ X = ˆ β ,OLS (77)Therefore, ˆ β converges to ˆ β ,OLS when c → ∞ . V ar ( ˆ β ) and V ar ( ˆ β ) to V ar ( ˆ β ,OLS ) and V ar ( ˆ β ,OLS ) when c → ∞ V ar ( ˆ β ) = σ N X i =1 k i = σ P Ni =1 { sinh ( φ i ) } { P Ni =1 X i · sinh ( φ i ) } = σ P Ni =1 { c · sinh ( φ i ) } { P Ni =1 X i · c · sinh ( φ i ) } = σ P Ni =1 { c · sinh ( φ i ) } { P Ni =1 ( X i − X ) · c · sinh ( φ i ) } (78)lim c →∞ V ar ( ˆ β ) = σ P Ni =1 { lim c →∞ c · sinh ( φ i ) } { P Ni =1 lim c →∞ ( X i − X ) · c · sinh ( φ i ) } = σ P Ni =1 ( X i − ¯ X ) ( P Ni =1 ( X i − ¯ X ) ) = σ P Ni =1 ( X i − ¯ X ) = V ar ( ˆ β ,OLS ) (79)Therefore, V ar ( ˆ β ) converges to V ar ( ˆ β ,OLS ) when c → ∞ .lim c →∞ V ar ( ˆ β ) = lim c →∞ ( V ar ( ¯ Y ) + ¯ X · V ar ( ˆ β ))= V ar ( ¯ Y ) + ¯ X · lim c →∞ V ar ( ˆ β )= V ar ( ¯ Y ) + ¯ X · V ar ( ˆ β ,OLS )= V ar ( ˆ β ,OLS ) (80)22herefore, V ar ( ˆ β ) converges to V ar ( ˆ β ,OLS ) when c → ∞ . C ov ( ˆ β , ˆ β ) to C ov ( ˆ β ,OLS , ˆ β ,OLS ) when c → ∞ lim c →∞ Cov ( ˆ β , ˆ β ) = lim c →∞ ( − ¯ X · V ar ( ˆ β )) = − ¯ X · lim c →∞ V ar ( ˆ β ))= − ¯ X · V ar ( ˆ β ,OLS )= Cov ( ˆ β ,OLS , ˆ β ,OLS ) (81)Therefore, Cov ( ˆ β , ˆ β ) converges to Cov ( ˆ β ,OLS , ˆ β ,OLS ) when c → ∞ . V ar ( ˆ β ) and V ar ( ˆ β ) to V ar ( ˆ β ,OLS ) and V ar ( ˆ β ,OLS ) V ar ( ˆ β ) = σ P Ni =1 { sinh ( φ i ) } { P Ni =1 X i · sinh ( φ i ) } (82) V ar ( ˆ β ,OLS ) = σ P Ni =1 ( X i − ¯ X ) (83) V ar ( ˆ β ) − V ar ( ˆ β ,OLS ) = σ { P Ni =1 X i · sinh ( φ i ) } · P Ni =1 ( X i − ¯ X ) · ( N X i =1 { sinh ( φ i ) } · N X i =1 ( X i − ¯ X ) − { N X i =1 X i · sinh ( φ i ) } ) (84) N X i =1 { sinh ( φ i ) } · N X i =1 ( X i − ¯ X ) − { N X i =1 X i · sinh ( φ i ) } (85)= N X i =1 { sinh ( φ i ) } · N X i =1 ( X i − ¯ X ) − { N X i =1 ( X i − ¯ X ) · sinh ( φ i ) } > V ar ( ˆ β ) > V ar ( ˆ β ,OLS ). V ar ( ˆ β ) = V ar ( ¯ Y ) + ¯ X · V ar ( ˆ β ) > V ar ( ¯ Y ) + ¯ X · V ar ( ˆ β ,OLS ) = V ar ( ˆ β ,OLS ) (86)Therefore, V ar ( ˆ β ) > V ar ( ˆ β ,OLS ). Cov ( ˆ β , ˆ β ) = − ¯ X · V ar ( ˆ β ) > − ¯ X · V ar ( ˆ β ,OLS ) = Cov ( ˆ β ,OLS , ˆ β ,OLS ) (87)Therefore, Cov ( ˆ β , ˆ β ) > Cov ( ˆ β ,OLS , ˆ β ,OLS ). This study investigated the problem posed by using OLS to estimate linear regressionparameters when an independent variable is restricted to an open interval, ( − c, c ), underthe context of special relativity. Our investigation revealed that the OLS estimate for theslope parameter is not invariant under the Lorentz velocity transformation.As an alternative estimator for the parameters of linear regression under special relativ-ity, we proposed an estimator that is invariant under the Lorentz velocity transformation.The proposed estimator was found to be unbiased and converges to the OLS estimatorwhen c approaches to infinity. The variance of the proposed estimator also converges tothat of the OLS estimator when c approaches to infinity. Therefore, the proposed estimatorcan be considered a generalization of the OLS estimator when an independent variable isrestricted to an open interval. 24he variance of the proposed estimator is larger than that of the OLS estimator, whichindicates that there is larger uncertainty when an independent variable is restricted. Itshows that hypothesis testing using the OLS estimator and its variance may result in aliberal test when an independent variable is restricted because the confidence interval con-structed from the OLS estimator and its variance is narrower than the confidence intervalconstructed from the proposed estimator and its variance.There are many circumstances in which independent variables in regression models arerestricted to an open interval. Although the proposed estimator may not be applicableto general cases, our results suggest that one needs to pay attention to the mechanism ofhow and why the independent variables are restricted and reflect the mechanism in theestimation process. Otherwise, one may obtain misleading estimates, which may result inliberal hypothesis tests. 25 ppendix: Proof of P Ni =1 X i · sinh ( φ i ) > Let sinh q ( x ), cosh q ( x ), and tanh q ( x ) be q-deformed hyperbolic functions (de Souza Dutra,2005), as follows. sinh q ( x ) ≡ e x − q · e − x cosh q ( x ) ≡ e x + q · e − x tanh q ( x ) ≡ sinh q ( x ) cosh q ( x ) = e x − q · e − x e x + q · e − x (88)The q-deformed hyperbolic functions have the following properties. cosh q ( x ) − sinh q ( x ) = qcosh q ( x ) ≥ √ qsinh q ( x ) = 0 if x = 12 ln ( q ) (89) sinh ( φ i + θ ) = 1 √ q sinh q ( φ i ) cosh ( φ i + θ ) = 1 √ q cosh q ( φ i ) tanh ( φ i + θ ) = tanh q ( φ i ) (90)where q = e − θ . 26 X i =1 X i · sinh ( φ i ) = N X i =1 c · tanh ( φ i + θ ) · sinh ( φ i )= N X i =1 c · tanh q ( φ i ) · sinh ( φ i )= c N X i =1 sinh q ( φ i ) cosh q ( φ i ) · sinh ( φ i )= c N X i =1 sinh q ( φ i ) q q + sinh q ( φ i ) · sinh ( φ i ) > c X sinh q ( φ i ) =0 sinh q ( φ i ) q sinh q ( φ i ) · sinh ( φ i ) since q = e − θ > sinh q ( φ i ) q sinh q ( φ i ) = φ i > − θ , − φ i < − θ . (92) X sinh q ( φ i ) =0 sinh q ( φ i ) q sinh q ( φ i ) · sinh ( φ i ) = X φ i > − θ sinh ( φ i ) − X φ i < − θ sinh ( φ i ) (93)When θ = 0 , X φ i > − θ sinh ( φ i ) − X φ i < − θ sinh ( φ i ) = X φ i > sinh ( φ i ) − X φ i < sinh ( φ i )= 2 X φ i > sinh ( φ i ) > θ > X φ i > − θ sinh ( φ i ) − X φ i < − θ sinh ( φ i )= X φ i > sinh ( φ i ) + X − θ <φ i < sinh ( φ i ) − X φ i ≤− θ sinh ( φ i )= − X φ i ≤− θ sinh ( φ i ) + { X − θ <φ i < sinh ( φ i ) − X φ i < sinh ( φ i ) } = − X φ i ≤− θ sinh ( φ i ) ≥ θ < X φ i > − θ sinh ( φ i ) − X φ i < − θ sinh ( φ i )= X φ i > − θ sinh ( φ i ) − X <φ i < − θ sinh ( φ i ) − X φ i < sinh ( φ i )= X φ i > − θ sinh ( φ i ) + { X φ i > sinh ( φ i ) − X <φ i < − θ sinh ( φ i ) } = 2 X φ i > − θ sinh ( φ i ) ≥ P Ni =1 X i · sinh ( φ i ) > θ .28 eferences de Souza Dutra, A. (2005). Mapping deformed hyperbolic potentials into nondeformedones. Physics Letters A 339 (3-5), 252–254.Dray, T. (2012).
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