Skewness and kurtosis unbiased by Gaussian uncertainties

Transcription

1 Skewness and kurtoss unbased by Gaussan uncertantes Lorenzo Rmoldn Observatore astronomque de l Unversté de Genève, chemn des Mallettes 5, CH-9 Versox, Swtzerland ISDC Data Centre for Astrophyscs, Unversté de Genève, chemn d Ecoga 6, CH-9 Versox, Swtzerland lorenzo@rmoldn.nfo arxv:4.675v [astro-ph.im] Mar 4 Draft verson: Aprl 9, Abstract Nose s an unavodable part of most measurements whch can hnder a correct nterpretaton of the data. Uncertantes propagate n the data analyss and can lead to based results even n basc descrptve statstcs such as the central moments and cumulants. Expressons of nose-unbased estmates of central moments and cumulants up to the fourth order are presented under the assumpton of ndependent Gaussan uncertantes, for weghted and unweghted statstcs. These results are expected to be relevant for applcatons of the skewness and kurtoss estmators such as outler detectons, normalty tests and n automated classfcaton procedures. The comparson of estmators corrected and not corrected for nose bases s llustrated wth smulatons as a functon of sgnal-to-nose rato, employng dfferent sample szes and weghtng schemes. Introducton Measurements generally provde an approxmate descrpton of real phenomena, because data acquston compounds many processes whch contrbute, to a dfferent degree, to nstrumental errors e.g., related to senstvty or systematc bases and uncertantes of statstcal nature. Whle nstrumental effects are addressed before data analyss, statstcal uncertantes propagate n subsequent processng and can affect both precson and accuracy of results, especally at low sgnal-to-nose S/N ratos. Correctng for bases generated by nose can help the characterzaton and nterpretaton of weak sgnals, and n some cases mprove a sgnfcant fracton of all data e.g., the number of astronomcal sources ncreases dramatcally near the fant detecton threshold, snce there are many more sources far away than nearby. In ths paper, nose-unbased estmates of central moments and cumulants up to the fourth order, whch are often employed to characterze the shape of the dstrbuton of data, are derved analytcally. Some of the advantages of these estmators nclude the ease of computaton and the ablty to encapsulate mportant features n a few numbers. Skewness and kurtoss measure the degree of asymmetry and peakedness or weght of the tals of the dstrbuton, respectvely, and they are useful for the detecton of outlers, the assessment of departures from normalty of the data D Agostno, 986, the classfcaton of lght varatons of astronomcal sources Rmoldn, a and many other applcatons. Varous estmators of skewness and kurtoss are avalable n the lterature e.g., Moors et al., 996; Hoskng, 99; Groeneveld & Meeden, 984; Bowley, 9, some of whch am at mtgatng the senstvty to outlers of the conventonal formulatons. On the other hand, robust measures mght mss mportant features of sgnals, especally when these are characterzed by outlers as n astronomcal tme seres where stellar bursts or eclpses from bnary systems represent rare events n the lght curve and weghtng mght help dstngush true outlers from spurous data employng addtonal nformaton such as the accuracy of each measurement, so the tradtonal forms of weghted central moments and cumulants are employed n ths work. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

2 Moments are usually computed on random varables. Heren, ther applcaton s extended to data generated from determnstc functons and randomzed by the uneven samplng of a fnte number of measurements and by ther uncertantes, whereas the correspondng populaton statstcs are defned n the lmt of an nfnte regular samplng wth no random or systematc errors. Ths scenaro s common n astronomcal tme seres, where measurements are typcally non-regular due to observatonal constrants, they are unavodably affected by nose, and sometmes also not very numerous: all of these aspects ntroduce some level of randomness n the characterzaton of the underlyng sgnal of a star. Whle the effects of samplng and sample sze on tme seres are studed n Rmoldn a,b, ths work addresses the bas, precson and accuracy of estmators when measurements are affected mostly by Gaussan uncertantes. Bas s defned as the dfference between expectaton and populaton values and thus expresses a systematc devaton from the true value. Precson s descrbed by the dsperson of measurements, whle accuracy s related to the dstance of an estmator from the true value and thus combnes the bas and precson concepts e.g., accuracy can be measured by the mean square error, defned by the sum of bas and uncertanty n quadrature. Nose-unbased expressons are provded for the varance, skewness and kurtoss central moments and cumulants, weghted and unweghted, assumng Gaussan uncertantes and ndependent measurements. The dependence of nose-unbased estmators on S/N s llustrated wth smulatons employng dfferent sample szes and two weghtng schemes: the common nverse-squared uncertantes and nterpolatonbased weghts as descrbed n Rmoldn a. The latter demonstrated a sgnfcant mprovement n the precson of weghted estmators at the hgh S/N end. Ths paper s organzed as follows. The notaton employed throughout s defned n Sec., followed by the descrpton of the method to estmate Gaussan-nose unbased moments n Sec.. Nose-unbased estmates of moments and cumulants based and unbased by sample-sze are presented n Sectons 4 and 5, n both weghted and unweghted formulatons, and the specal case of error-weghted estmators s presented n Sec. 6. The nose-unbased estmators are compared wth the uncorrected nose-based counterparts wth smulated sgnals as a functon of S/N rato n Sec. 7, ncludng weghted and unweghted schemes and two dfferent sample szes. Conclusons are drawn n Sec. 8, followed by detaled dervatons of the nose-unbased estmators n App. A. Notaton For a set of n measurements x = x, x,..., x n, the followng quanttes are defned. Populaton central moments µ r = x µ r wth mean µ = x, where. denotes expectaton, and cumulants κ = µ, κ = µ, κ 4 = µ 4 µ e.g., Stuart & Ord, 969. The sum of the p-th power of weghts s defned as V p = n = wp. The mean θ of a generc set of n elements θ assocated wth weghts w s θ = n = w θ /V. v Sample central moments m r = n = w x x r /V and correspondng cumulants k r. v Sample-sze unbased estmates of central moments M and cumulants K,.e., M = µ and K = κ. v The standardzed skewness and kurtoss are defned as g = k /k /, g = k 4 /k, G = K /K /, and G = K 4 /K, wth populaton values γ = κ /κ / and γ = κ 4 /κ. G and G satsfy consstency for n but are not unbased n general e.g., see Hemans, 999, for exceptons. v Nose-unbased estmates of central moments and cumulants are denoted by an astersk superscrpt. v No systematc errors are consdered heren and random errors are smply referred to as errors or uncertantes. x Statstcs weghted by the nverse-squared uncertantes are called error-weghted for brevty and nterpolaton-based weghts computed n phase Rmoldn, a are named phase weghts. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

3 Method The goal s to derve an estmator T x, ɛ as a functon of observables measurements x wth correspondng uncertantes ɛ whch s unbased by the nose n the data,.e., such that the expectaton T x, ɛ equals the estmator T ξ n terms of the true unknown values ξ amed at by the measurements. The nose-unbased estmator T x, ɛ s obtaned wth the followng procedure and assumptons. If n ndependent measurements x are assocated wth ndependent Gaussan uncertantes ɛ, the expected value T x of the estmator T x s evaluated from measurements x and the ont probablty densty px ξ, ɛ, for gven true values ξ and measurement uncertantes ɛ: T x = T x px ξ, ɛ d n x, R n where px ξ, ɛ = n = π ɛ exp [ x ξ ɛ ]. As shown n App. A, the expectaton T x of the estmators consdered heren can be decomposed as T x = T ξ + fξ, ɛ. Thus, the nose-free estmator T ξ = T x fξ, ɛ can be estmated n terms of measurements x and uncertantes ɛ by the nose-unbased estmator T x, ɛ = T x f x, ɛ, where f x, ɛ = fξ, ɛ and, by defnton, T x, ɛ = T ξ. The f x, ɛ term s derved frst by computng fξ, ɛ = T x T ξ and then by replacng terms dependng on ξ n fξ, ɛ wth terms as a functon of x whch satsfy the requrement f x, ɛ = fξ, ɛ see App. A. A property often used n the followng sectons s that a nose-unbased lnear combnaton of N estmators s equvalent to the lnear combnaton of noseunbased estmators: [ N N c T x] = c T x, 4 where the coeffcents c are ndependent of the measurements x. = 4 Gaussan-nose unbased sample moments and cumulants Weghted sample central moments unbased by Gaussan uncertantes, such as the varance m, skewness m, kurtoss m 4 and the respectve cumulants about the weghted mean x = x are derved assumng ndependent measurements x, uncertantes ɛ and weghts w, as descrbed n full detal n App. A. They are defned as follows: m = m n w ɛ w = k 5 V V = m = m n w ɛ x x w = k 6 V V m 4 = m 4 6 V = n = w ɛ m = m 4 V k 4 = m 4 m. [x x w ɛ V n w ɛ = [ x x ɛ By defnton, the above expressons satsfy = w + m w V V w ] V + ] n V 4 w ɛ 7 = n w ɛ 8 = m r = n w ξ V ξ r. = 9 Skewness and kurtoss unbased by Gaussan uncertantes page of 6

4 The unweghted forms can be obtaned by substtutng w = for all and V p = n for all p n all terms, leadng to: m = m n n m = m m 4 = m 4 n ɛ = k = n n 6n n n ɛ x x = k = n = ɛ x x 6 m n m = m 4 n n ɛ x x + k 4 = m 4 m. = n ɛ + = n n n n n ɛ 4 n n 4 ɛ = = n ɛ 4 + n n 4 ɛ 4 5 Gaussan-nose and sample-sze unbased moments and cumulants The estmates of weghted central moments whch are unbased by both sample-sze and Gaussan uncertantes, such as the varance M, skewness M, kurtoss M 4 and the respectve cumulants, are defned n terms of the nose-unbased sample estmators as follows: = M = V V V m = K 6 M = M 4 = K 4 = V = 5 V V m = K 7 V + V V V V + V V + V V V 6V V + 8V V + V 6 m 4 + V V V V V V + V V 6V V + 8V V + V 6 m 8 V 4V V + V V V 6V V + 8V V + V 6 m 4 + V V V + 4V V V V V 6V V + 8V V + V 6 m. 9 The dervaton of the sample-sze unbased weghted estmators s descrbed n Rmoldn b. The correspondng unweghted forms can be acheved by drect substtuton V p = n for all p, leadng to: M = M = M 4 = K 4 = n n m = M n n ɛ = K = n n n m = M n n ɛ x x = K = nn n + nn n n n m 4 n n n m n n + n n n n m 4 n n m. Skewness and kurtoss unbased by Gaussan uncertantes page 4 of 6

5 6 Specal cases If weghts are related to measurement errors as w = /ɛ, the nose-unbased weghted sample moments and cumulants reduce to the followng expressons: m = m n = k 4 V m = m V m 4 = m 4 6 V n = n = x x = k 5 [ ] x x ɛ + 6m V V 6 m = m m + m V 7 k4 = m 4 m. 8 In the case of constant errors,.e., ɛ = ɛ for all, some of the unweghted estmators are equvalent or smlar to ther nose-unbased counterparts: Skewness: k = k and K = K also m = m and M = M, 9 Kurtoss: k 4 k 4 and K 4 = K 4, where the approxmaton k 4 k 4 holds for large values of n or S/N ratos snce k 4 k 4 k = 6 ɛ k + k n k 6 [ + S/N ] n [ + S/N ], consderng that, for constant errors, ɛ /n = ɛ and S/N k/ɛ k /ɛ. For sample cumulants up to the fourth order, only the varance depends strongly on nose. However, ths s an mportant estmator because t s often nvolved n defntons of standardzed skewness g and G and kurtoss g and G as follows: g = k /k /, G = K /K /, g = k 4 /k, G = K 4 /K. For consstency wth the above defntons, the nose-unbased equvalents are defned as g = k /k /, G = K /K /, 4 g = k 4/k, G = K 4 /K, 5 although the truly nose-unbased expressons should have been computed on the ratos n Eqs. The applcaton of Eqs 4 5 should generally be restrcted to larger samples e.g., n > 5 wth S/N ratos greater than a few, n order to avod non-postve values of k or K. 7 Estmators as a functon of sgnal-to-nose rato Nose-based and unbased estmators are compared as a functon of sgnal-to-nose rato S/N wth smulated data and dfferent weghtng schemes for specfc sgnals, samplng and error laws. The values of the populaton moments of the contnuous smulated perodc true sgnal ξφ are computed averagng n phase φ as follows: µ r = π π [ξφ µ] r dφ, where µ = π π ξφ dφ. 6 Skewness and kurtoss unbased by Gaussan uncertantes page 5 of 6

6 7. Smulaton Smulated sgnals are descrbed by a snusodal functon to the fourth power, whch has a non-zero skewness and thus makes t possble to evaluate the precson and accuracy of the skewness standardzed by the estmated varance wthout smply reflectng the accuracy of the varance. The S/N level s evaluated by the rato of the standard devaton µ of the true sgnal ξφ and the root of the mean of squared measurement uncertantes ɛ assumed ndependent of the sgnal. The sgnal ξφ s sampled n = and tmes at phases φ randomly drawn from a unform dstrbuton, whle the S/N rato vares from to and determnes the uncertantes ɛ of measurements x as follows: ξφ = A sn 4 φ 7 x N ξ, ɛ for ξ = ξφ and φ U, π 8 ɛ = + ρ µ / S/N for ρ U.8,.8, 9 where the -th measurement x s drawn from a normal dstrbuton N ξ, ɛ of mean ξ and varance ɛ. The latter s defned n terms of a varable ρ randomly drawn from a unform dstrbuton U.8,.8 so that measurement uncertantes vary by up to a factor of for a gven µ and S/N rato. Smulatons were repeated 4 tmes for each S/N rato for n = and. The dependence of weghted estmators on sample sze and the correspondng unbased expressons were presented n Rmoldn b. Heren, only large sample szes are employed so that sample-sze bases are neglgble wth respect to the ones resultng from small S/N ratos. A sample sgnal and smulated data are llustrated n Fg. for n = and S/N =. The reference populaton values of the mean, varance, skewness and kurtoss of the smulated sgnal are lsted n table of Rmoldn b. Error weghts are defned by w = /ɛ, whle mxed error-phase weghts follow Rmoldn a, assumng phase-sorted data: w = hs/n a, b w n + [ hs/n a, b] = w ɛ n = ɛ w = φ + φ, n w = φ φ n + π w n = φ φ n + π hs/n a, b = for a, b >. + e S/N a/b, n Weghtng effectvely decreases the sample sze, snce more mportance s gven to some data at the expense of other ones and results depend mostly on fewer relevant measurements e.g., weghtng by the nverse-squared uncertantes can worsen precson at hgh S/N levels. Weghted procedures are desrable when the dsperson and bas of estmators from an effectvely reduced sample sze are smaller than the mprovements n precson and accuracy e.g., weghtng by nverse-squared uncertantes can mprove both precson and accuracy at low S/N ratos. Also, weghtng mght explot correlatons n the data to mprove precson, as t s shown employng phase weghts Rmoldn, a. Snce correlated data do not satsfy the assumptons of the expressons derved heren, ther applcaton mght return based results. However, small bases could be ustfed f mprovements n precson are sgnfcant and, dependng on the extent of the applcaton, larger bases could be mtgated wth mxed weghtng schemes, such as the one descrbed by Eqs Estmators derved heren assume a sngle weghtng scheme and combnatons of estmators lke the varance and the mean n the standardzed skewness and kurtoss are expected to apply the same weghts to terms assocated wth the same measurements. The functon hs/n a, b consttutes ust an example to acheve a mxed weghtng scheme: tunng parameters a, b offer the possblty to control the transton from error-weghted to phase-weghted estmators n the lmts of low and hgh S/N, respectvely and thus reach a compromse soluton between precson and accuracy for all values of S/N, accordng to the specfc estmators, sgnals, samplng, errors, sample szes and ther dstrbutons n the data Skewness and kurtoss unbased by Gaussan uncertantes page 6 of 6

7 Sgnal and smulated data S/N =, n = Phase φ / π Fgure : A smulated sgnal of the form of sn 4 φ blue curve s rregularly sampled by measurements denoted by trangles wth S/N =. 7. Results The results of smulatons are llustrated for sample estmators, snce the conclusons n Rmoldn b suggested that phase-weghted sample estmators can be more accurate and precse than the sample-sze unbased counterparts n most cases, especally for large sample szes as consdered heren. Fgure llustrates the sample mean n the varous scenaros consdered n the smulatons: sample szes of n = and, unweghted and wth dfferent weghtng schemes error-weghted, phaseweghted and combned error-phase weghted. Whle accuracy s the same n all cases, the best precson of the mean s acheved employng phase weghts ncludng the low S/N end, unlke other estmators. Fgures 6 compare nose-based uncorrected and nose-unbased corrected estmators as a functon of S/N, evaluatng the followng devatons from the populaton values: m /µ vs m /µ, 45 m /µ / γ vs m /µ / γ, g γ vs g γ, 46 m 4 /µ γ vs m 4/µ γ, m 4 /m γ vs m 4/m γ, 47 k 4 /µ γ vs k 4/µ γ, g γ vs g γ, 48 n both weghted and unweghted cases, for n = and. The dependence on n s descrbed n more detals n Rmoldn b. Estmators standardzed by both true and estmated varance are presented to help nterpret the behavour of the ratos from ther components. All fgures confrm that corrected and uncorrected estmators have smlar precson and accuracy at hgh S/N levels typcally for S/N >. Nose-unbased estmators are found to be the most accurate n all cases and over the whole S/N range tested. Ther precson s generally smlar to the nose-uncorrected counterparts, apart from estmators standardzed by the estmated varance, such as g, g and m 4 /m, for whch the uncorrected verson can be much more precse although based for S/N <, typcally. As expected, the precson of estmators employng n = measurements per sample was greater than the one obtaned wth sample szes of n =. Weghtng by the nverse of squared measurement errors made the estmators slghtly less precse at hgh S/N ratos, but more precse and accurate at low S/N levels except for the mean. Weghtng by phase ntervals led to a sgnfcant mprovement n precson of all estmators n the lmt of large S/N ratos and a reducton of precson at low S/N apart from the case of the mean. Tunng parameters such as a = and b =. n Eq. 4 were able to mtgate the mprecson at low S/N Skewness and kurtoss unbased by Gaussan uncertantes page 7 of 6

8 reducng to the error-weghted results, whch appeared to be the most accurate and precse n the lmt of low S/N ratos n these smulatons. Ths soluton mght provde a reasonable compromse between precson and accuracy of all estmators, at least for S/N >. Fgures 5 8 show that the skewness moment m s qute unbased by nose, whle the standardzed verson g s underestmated at hgh S/N because of the overestmated varance m as shown n Fgs 4. Whle the accuracy of g deterorates at low S/N, ts precson s much less affected by nose. The kurtoss moment m 4 Fgs 9 s less precse and accurate than the nose-unbased equvalent, and ts normalzaton by the squared varance reduces dramatcally ts naccuracy and mprecson snce m and m 4 exhbt a smlar trend as a functon of S/N. The kurtoss cumulant k 4, nstead, s much closer to ts nose-unbased counterpart, as shown n Fgs 6. The normalzaton of k 4 by the squared varance mproves ts precson at the cost of lower accuracy for S/N < : the bas of g s smlar to greater than the precson of g for n = n =. The lower the S/N level s, the less precse estmators are and the nose-unbased varance can be underestmated and even become non-postve. Thus, the skewness and kurtoss estmators standardzed by k or K, as n Eqs 4 5, should be avoded n crcumstances that combne small sample szes up to a few dozens of elements and low S/N ratos of the order of a few or less. Fgures related to moments and cumulants of rregularly sampled snusodal sgnals are very smlar to the ones presented heren, wth the excepton of g, whch would have a smlar precson but wth no bas, as a consequence of the null skewness of a snusodal sgnal snce the mean of k estmates s zero, they are not based by the standardzaton wth an overestmated nose-based varance. From the comparson of nose-based and unbased estmators wth dfferent weghtng schemes, t appears that, for large sample szes, nose-unbased phase-weghted estmators are usually the most accurate for S/N > apart from the specal cases of standardzed skewness and kurtoss when ther true value s zero. For nosy sgnals e.g., S/N <, error weghtng seems the most approprate, at least wth Gaussan uncertantes, thus nose-unbased error-phase weghted estmators can provde a satsfactory compromse n general. Further mprovements mght be acheved by tunng parameters better ftted to estmators and sgnals of nterest, n vew of specfc requrements of precson and accuracy. 8 Conclusons Exact expressons of nose-unbased skewness and kurtoss were provded n the unweghted and weghted formulatons, under the assumpton of ndependent data and Gaussan uncertantes. Such estmators can be partcularly useful n the processng, nterpretaton and comparson of data characterzed by low S/N regmes. Smulatons of an rregularly sampled skewed perodc sgnal were employed to compare nose-based and unbased estmators as a functon of S/N n the unweghted, nverse-squared error weghted and phase-weghted schemes. Whle nose-unbased estmators were found more accurate n general, they were less precse than the uncorrected counterparts at low S/N ratos. The applcaton of a mxed weghtng scheme nvolvng phase ntervals and uncertantes was able to balance precson and accuracy on a wde range of S/N levels. The effect of nose-unbased estmators and dfferent weghtng schemes on the characterzaton and classfcaton of astronomcal tme seres s descrbed n Rmoldn a. Acknowledgments The author thanks M. Süveges for many dscussons and valuable comments on the orgnal manuscrpt. References Bowley A.L., 9, Elements of Statstcs, Charles Scrbner s Sons, New York D Agostno R.B., 986, Goodness-of-ft technques, D Agostno & Stephens eds., Marcel Dekker, New York, p. 67 Groeneveld R.A., Meeden G., 984, The Statstcan,, 9 Skewness and kurtoss unbased by Gaussan uncertantes page 8 of 6

9 Hemans R., 999, Statstcal Papers, 4, 7 Hoskng J.R.M., 99, J. R. Statst. Soc. B, 5, 5 Moors J.J.A., Wagemakers R.Th.A., Coenen V.M.J., Heuts R.M.J., Janssens M.J.B.T., 996, Statstca Neerlandca, 5, 47 Rmoldn L., a, preprnt arxv:4.666 Rmoldn L., b, preprnt arxv: Stuart A., Ord J., 969, Kendall s Advanced Theory of Statstcs, Charles Grffn & Co. Ltd, London Skewness and kurtoss unbased by Gaussan uncertantes page 9 of 6

10 Mean n =, Unweghted Error Weghted.4.4 mean / µ.... n = n = mean / µ.... n = n = S/N S/N Phase Weghted a, b Error-Phase Weghted a =, b =..4.4 mean / µ.... n = n = mean / µ.... n = n = S/N S/N Fgure : Sample mean for S/N > and n =, : unweghted on the top-left hand sde, weghted by the nverse of squared measurement errors on the top-rght hand sde, and weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above the lower panels. Shaded areas encompass one standard devaton from the average of the dstrbuton of the mean employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

11 Varance n = Unweghted Error Weghted.5.5 / µ..5 / µ..5 m *. m * S/N S/N Phase Weghted a, b Error-Phase Weghted a =, b =..5.5 / µ..5 / µ..5 m *. m * S/N S/N Fgure : Nose-based uncorrected versus nose-unbased corrected sample varance for S/N > and n = : unweghted on the top-left hand sde, weghted by the nverse of squared measurement errors on the top-rght hand sde, and weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the varance employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

12 Varance n = Unweghted Error Weghted.5.5 / µ..5 / µ..5 m *. m * S/N S/N Phase Weghted a, b Error-Phase Weghted a =, b =..5.5 / µ..5 / µ..5 m *. m * S/N S/N Fgure 4: Nose-based uncorrected versus nose-unbased corrected sample varance for S/N > and n = : unweghted on the top-left hand sde, weghted by the nverse of squared measurement errors on the top-rght hand sde, and weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the varance employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

13 Skewness n = Unweghted Unweghted.. γ.5.5 / µ m *..5 g * γ S/N S/N Error Weghted Error Weghted.. γ.5.5 / µ m *..5 g * γ S/N S/N Fgure 5: Nose-based uncorrected versus nose-unbased corrected sample skewness for S/N > and n = : unweghted n the upper panels and weghted by the nverse of squared measurement errors n the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the skewness employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

14 Skewness n = Phase Weghted a, b Phase Weghted a, b.. γ.5.5 / µ m *..5 g * γ S/N S/N Error-Phase Weghted a =, b =. Error-Phase Weghted a =, b =... γ.5.5 / µ m *..5 g * γ S/N S/N Fgure 6: Nose-based uncorrected versus nose-unbased corrected sample skewness for S/N > and n =, weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above each panel. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the skewness employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page 4 of 6

15 Skewness n = Unweghted Unweghted.. γ.5.5 / µ m *..5 g * γ S/N S/N Error Weghted Error Weghted.. γ.5.5 / µ m *..5 g * γ S/N S/N Fgure 7: Nose-based uncorrected versus nose-unbased corrected sample skewness for S/N > and n = : unweghted n the upper panels and weghted by the nverse of squared measurement errors n the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the skewness employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page 5 of 6

16 Skewness n = Phase Weghted a, b Phase Weghted a, b.. γ.5.5 / µ m *..5 g * γ S/N S/N Error-Phase Weghted a =, b =. Error-Phase Weghted a =, b =... γ.5.5 / µ m *..5 g * γ S/N S/N Fgure 8: Nose-based uncorrected versus nose-unbased corrected sample skewness for S/N > and n =, weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above each panel. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the skewness employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page 6 of 6

17 Kurtoss n = Unweghted Unweghted 5 5 m 4 * / µ γ 5 γ m 4 * / m * S/N S/N Error Weghted Error Weghted 5 5 m 4 * / µ γ 5 γ m 4 * / m * S/N S/N Fgure 9: Nose-based uncorrected versus nose-unbased corrected sample kurtoss moment for S/N > and n = : unweghted n the upper panels and weghted by the nverse of squared measurement errors n the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page 7 of 6

18 Kurtoss n = Phase Weghted a, b Phase Weghted a, b 5 5 γ / µ m 4 * 5 γ / m * m 4 * S/N S/N Error-Phase Weghted a =, b =. Error-Phase Weghted a =, b =. 5 5 γ / µ m 4 * 5 γ / m * m 4 * S/N S/N Fgure : Nose-based uncorrected versus nose-unbased corrected sample kurtoss moment for S/N > and n =, weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above each panel. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page 8 of 6

19 Kurtoss n = Unweghted Unweghted 5 5 m 4 * / µ γ 5 γ m 4 * / m * S/N S/N Error Weghted Error Weghted 5 5 m 4 * / µ γ 5 γ m 4 * / m * S/N S/N Fgure : Nose-based uncorrected versus nose-unbased corrected sample kurtoss moment for S/N > and n = : unweghted n the upper panels and weghted by the nverse of squared measurement errors n the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page 9 of 6

20 Kurtoss n = Phase Weghted a, b Phase Weghted a, b 5 5 γ / µ m 4 * 5 γ / m * m 4 * S/N S/N Error-Phase Weghted a =, b =. Error-Phase Weghted a =, b =. 5 5 γ / µ m 4 * 5 γ / m * m 4 * S/N S/N Fgure : Nose-based uncorrected versus nose-unbased corrected sample kurtoss moment for S/N > and n =, weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above each panel. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

21 k-kurtoss n = Unweghted Unweghted k 4 * / µ γ g * γ S/N S/N Error Weghted Error Weghted k 4 * / µ γ g * γ S/N S/N Fgure : Nose-based uncorrected versus nose-unbased corrected sample kurtoss cumulant for S/N > and n = : unweghted n the upper panels and weghted by the nverse of squared measurement errors n the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

22 k-kurtoss n = Phase Weghted a, b Phase Weghted a, b k 4 * / µ γ g * γ S/N S/N Error-Phase Weghted a =, b =. Error-Phase Weghted a =, b =. k 4 * / µ γ g * γ S/N S/N Fgure 4: Nose-based uncorrected versus nose-unbased corrected sample kurtoss cumulant for S/N > and n =, weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above each panel. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

23 k-kurtoss n = Unweghted Unweghted k 4 * / µ γ g * γ S/N S/N Error Weghted Error Weghted k 4 * / µ γ g * γ S/N S/N Fgure 5: Nose-based uncorrected versus nose-unbased corrected sample kurtoss cumulant for S/N > and n = : unweghted n the upper panels and weghted by the nverse of squared measurement errors n the lower panels. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page of 6

24 k-kurtoss n = Phase Weghted a, b Phase Weghted a, b k 4 * / µ γ g * γ S/N S/N Error-Phase Weghted a =, b =. Error-Phase Weghted a =, b =. k 4 * / µ γ g * γ S/N S/N Fgure 6: Nose-based uncorrected versus nose-unbased corrected sample kurtoss cumulant for S/N > and n =, weghted by phases and errors, accordng to Eq. 4, wth dfferent parameter values, as specfed above each panel. Shaded areas encompass one standard devaton from the mean of the dstrbuton of the kurtoss employng smulatons defned by Eqs 7 9. Skewness and kurtoss unbased by Gaussan uncertantes page 4 of 6

25 A Dervaton of nose-unbased moments The dervatons presented n ths Appendx nvolve weghted estmators under the assumpton of ndependent measurements, uncertantes and weghts. Defntons and some of the relatons often employed heren are lsted below. For brevty, m r = m r x, and and are mpled to nvolve all from the -st to the n-th terms, unless explctly stated otherwse. The followng ntegral solutons are often employed: x s = x s exp π ɛ [ x ξ ɛ ] dx = ξ for s = ξ + ɛ for s = ξ + ξ ɛ for s = ξ 4 + 6ξ ɛ + ɛ4 for s = 4. The expected value m of a generc estmator mx = a x s b x t k, c kx u k l,,k d lx v l of ndependent data wth Gaussan uncertantes s computed as follows n m = mx exp [ x ξ ] R n = π ɛ ɛ d n x 5 = mx exp [ x h ξ h ] h π ɛh ɛ dx h 5 h = x s a exp [ x ξ ] [ π ɛ ɛ dx x t b exp x ξ ] π ɛ ɛ dx x u k c k exp [ x k ξ k ] x v l d l exp [ x l ξ l ] dx l. π ɛk π ɛl k, ɛ k dx k l,,k The results of the followng expressons are employed: w x = = w x w x = w x w x + w x k w kx k = w x + w x w x + w x w x k, w kx k. w x 4 = = w x w x k w k x k + k w kx k l k w lx l = w x w x + w x k w kx k + w x k w kx k l,k w lx l = w4 x4 + 4 w x w x + 6 w x w x k, w kx k + + w x w x + w x w x k, w kx k l,,k w lx l. w x w x = = w x w x + w x k w kx k = w x4 + w x w x + w x w x + w x w x k, w kx k. w ξ k, w kξ k = = w ɛ w ξ k w kξ k w ɛ w ξ w ξ ɛ w ξ w ξ ɛ = V ξ w ɛ w ɛ w ξ + w ξ ɛ V ξ w ξ ɛ + w ξ ɛ w ξ ɛ = V ξ w ɛ w ɛ w ξ V ξ w ξ ɛ + w ξ ɛ. w ɛ ɛ l 49 5 Skewness and kurtoss unbased by Gaussan uncertantes page 5 of 6

26 w ɛ w ξ k, w kξ k = = w ɛ w ξ k w kξ k w ɛ w ξ w ξ ɛ w ξ w4 ξ ɛ = V ξ w ɛ w ɛ w ξ + w4 ξ ɛ V ξ w ξ ɛ + w4 ξ ɛ w4 ξ ɛ = V ξ w ɛ w ɛ w ξ V ξ w ξ ɛ + w4 ξ ɛ. A. Outlne of results The expressons of the elements pursued along the dervaton of nose-unbased estmators detaled n Sec. A. are summarzed below, followng the notaton ntroduced n Sectons and. m = w ξ V ξ + w ɛ w = k 5 V V m = w ξ V ξ + w ɛ ξ V ξ w = k 54 V [ ξ ξ w + ɛ V w ] + w V V + m 4 = w ξ V ξ w ɛ V + 6 V w ɛ w ξ ξ + w ɛ [ m = w ξ V ξ ] + 4 w ɛ + V w ɛ 4 k 4 = m 4 m w V V + V [ w ɛ w 55 V ξ ξ + V w V w ξ ξ ] w ɛ w + V + V 4 w ɛ If fξ, ɛ = c ξ ξ, then f x, ɛ = c x x. 58 If fξ, ɛ = c ξ ξ, then f x, ɛ = c x x ɛ w V V w ɛ. 59 m = m w ɛ V w V m = m w ɛ x x w V m 4 = m 4 6 w ɛ V m = m 4 V k 4 = m 4 m = k 6 V [x x w ɛ V w ɛ [ x x ɛ = k 6 w + m w V V w ] V + ] V 4 w ɛ 6 w ɛ 6 64 Skewness and kurtoss unbased by Gaussan uncertantes page 6 of 6

27 A. Detaled computatons m = w x x 65 V = w x x V V = w x x V w x + x 66 = w x w x 68 V V = w x V V w x V w x w x 69 m = w x x 7 V = w x x w x + x w x x 7 V V V = w x x w x + x 7 V V = w x V V w x w x + w x 7 V = w x V V w x + w x w x + + V w x + w x w x + w x w x w k x k 74 k, m 4 = w x x 4 75 V = w x 4 4 x w x + 6 x w x 4 x w x x 4 76 V V V V = w x 4 4 x w x + 6 x w x x 4 77 V V V = w x 4 4 V V w x w x + 6 V w x 4 w x w x 78 = w x 4 4 V V w x 4 + w x w x + 6 V w x w x w x + w x w x + w x w x w k x k + k, 67 Skewness and kurtoss unbased by Gaussan uncertantes page 7 of 6

28 V 4 w 4 x w x w x + 6 w x w x w k x k + k, + w x w x + w x w x w k x k w l x l 79 k, l,,k [ ] m = w x x 8 V = w x w x 8 V V = V w x V w x 4 w x + w x 8 = V w x V w x 4 w x + w x 8 = V w x 4 + w x w x V w x 4 + w x w x + + w x w x + w x w x w k x k + k, + V 4 w 4 x w x w x + 6 w x w x w k x k + k, + w x w x + w x w x w k x k w l x l 84 k, l,,k m = m x = w ξ V + ɛ π ɛ exp V = w ξ + V V = w ξ + w ɛ V V [ x ξ w w ɛ V ɛ w = w ξ V ξ + w ɛ V ] dx 85 ξ + ɛ V w ξ w ξ 86 w ɛ V w ξ + V w ξ w ξ 87 V ξ 88 w V = k 89 Skewness and kurtoss unbased by Gaussan uncertantes page 8 of 6

29 m = = V + V m x π ɛ exp w ξ + ξ ɛ w V [ x ξ ξ + ξ ɛ + = w ξ + w ξ ɛ V V V V + 6 V w ɛ w ɛ w ξ + V w ξ + V = w ξ + w ξ ɛ V V ɛ w w w V ] dx 9 ξ + ξ ɛ + w ξ + ɛ w ξ + ξ + ɛ w ξ 9 V w ξ + 6 V w ξ w ξ + w ξ ɛ V w ξ ɛ + 6 V w ξ w ξ w k ξ k 9 k, w ξ w ξ + w ξ w ξ + w ξ w k ξ k 9 k, ξ V w ξ V w ɛ w ξ + + ξ + 6 ξ V w ɛ 9 = w ξ V ξ + w ɛ ξ w ξ + w ξ w ɛ ξ w ξ 94 V V V V V = w ξ V ξ [ + w ɛ ξ V ξ w ξ V ξ ] 95 = w ξ V ξ + w ɛ ξ V ξ w = k 96 V Skewness and kurtoss unbased by Gaussan uncertantes page 9 of 6

30 m 4 = m 4 x exp [ x ξ ] π ɛ ɛ dx 97 = w ξ 4 V + 6ξ ɛ + ɛ 4 4 V w ξ 4 + 6ξ ɛ + ɛ 4 + w ξ + ξ ɛ w ξ V w ξ 4 + 6ξ ɛ + ɛ 4 + w ξ + ɛ w ξ + ɛ + + w ξ + ξ ɛ w ξ + w ξ + ɛ w ξ w k ξ k + k, V 4 w 4 ξ 4 + 6ξ ɛ + ɛ w ξ + ξ ɛ w ξ w ξ + ɛ w ξ w k ξ k + w ξ + ɛ w ξ + ɛ + k, + w ξ w ξ w k ξ k w l ξ l 98 k, l,,k = w ξ w ξ ɛ + w ɛ 4 4 V V V V w ξ 4 4 V w ξ ɛ V w ɛ V + 8 V + 6 V + 6 V w ξ w ɛ V w ɛ w ξ w 4 ξ ɛ 9 V 4 w ξ w ξ w ɛ w ξ V w ξ w ɛ + V w ξ k, w ξ k, w ξ 9 w ɛ w ξ ɛ w ξ + 6 V w ξ w k ξ k + 6 V w 4 ɛ 4 w ξ + 6 V w ξ w ξ + 6 V w ɛ w ξ w k ξ k 8 V 4 w ɛ w ξ w ξ V w ɛ + 6 V w ɛ w ξ ɛ w ξ + k, w ξ 6 V 4 w ξ k, w k ξ k V 4 w ξ ɛ + w ξ + w 4 ξ 4 + w ξ ɛ w ξ + w k ξ k + w ξ w ɛ 9 w ɛ w ξ + w ξ w ξ w k ξ k w l ξ l 99 k, l,,k Skewness and kurtoss unbased by Gaussan uncertantes page of 6

31 = w ξ V ξ w ɛ V + V + 6 V + 6 V 8 w ɛ 4 V w ξ V ξ w ɛ w ɛ ξ V w ɛ + 6 V ξ ξ 6 V ξ w ɛ w ɛ w ξ 9 V 4 w ɛ w ξ ɛ + V w ξ V w ɛ w ξ ɛ + V w ɛ V w ξ ɛ + w ɛ w ξ V ξ w ξ ɛ + 6 ξ V w ξ ɛ V V ξ 4 w ɛ w ɛ w ξ V ξ w ξ ɛ + = w ξ V ξ w ɛ ξ V ξ V w ξ ɛ + + w ɛ 4 V 4w V + 4w V + 6 V w ξ = w ξ V ξ w ɛ V + w ɛ 4 V + 6 V + 4 ξ V w ɛ 9 V 4 w ɛ w w ɛ V + 6 V w ξ V ξ 9 w ξ ɛ + 6 V 8 ξ V ξ ξ V w ɛ w ɛ + = w ξ V ξ w ɛ ξ V ξ + 6 V w ɛ w ξ ξ + w ɛ w ɛ w ɛ w ɛ + w ξ ɛ + w 4 ξ ɛ + w ɛ w ɛ + w 4 ξ ɛ w ɛ ξ ξ ξ + ξ + w ɛ w V w V + w ɛ 4 V w + V Skewness and kurtoss unbased by Gaussan uncertantes page of 6

32 m = = V m x π ɛ exp [ x ξ ɛ ] dx 4 ξ 4 + 6ξ ɛ + ɛ 4 + w ξ + ɛ w ξ + ɛ + w V w ξ 4 + 6ξ ɛ + ɛ 4 + w ξ + ɛ w ξ + ɛ + + w ξ + ξ ɛ w ξ + w ξ + ɛ w ξ w k ξ k + k, + V 4 w 4 ξ 4 + 6ξ ɛ + ɛ w ξ + ξ ɛ w ξ w ξ + ɛ w ξ w k ξ k + w ξ + ɛ w ξ + ɛ + k, + w ξ w ξ w k ξ k w l ξ l 5 k, l,,k = V w ξ V w ξ ɛ + V w ɛ 4 + V w ξ w ξ + V w ξ w ɛ + + V V 4 V V w ɛ w ξ w ξ w ɛ w ξ w ɛ w ɛ w ξ + V w ξ V w ξ V w ξ k, w ξ + V 4 w ξ k, w ξ + w ɛ w ξ w ξ ɛ w k ξ k + V 4 w ξ ɛ w k ξ k + V 4 w ɛ w ɛ V w ɛ V w ξ 4 V w ɛ w ξ V w 4 ξ w ξ + 6 V 4 w ξ w ɛ + w ξ w ξ V w ξ w ξ ɛ 6 V k, w ɛ 4 + w ɛ w ɛ + w k ξ k + w 4 ξ ɛ + V 4 w 4 ɛ 4 + w ξ w ξ w k ξ k + w ξ + V 4 w ξ k, w ξ w ɛ + w ξ w k ξ k k, l,,k w l ξ l 6 Skewness and kurtoss unbased by Gaussan uncertantes page of 6

33 [ = w ξ V ξ ] + 6 V + V V V V 4 w ɛ w ɛ V ξ w ɛ V w ξ V w ɛ w 4 ξ ɛ + w ξ ɛ + V w ξ ɛ 6 V w ɛ w ɛ V w ɛ 4 + V w ɛ 4 V w ξ w ɛ + w ξ w ɛ + w ξ ɛ w ξ + w ɛ w ξ V ξ w ξ ɛ + w 4 ɛ 4 + V 4 w ξ ɛ w ξ + w ξ ɛ + V ξ w ɛ w ɛ w ξ V ξ w ξ ɛ + w 4 ξ ɛ + w ξ w ɛ + w ɛ w ɛ 7 [ = w ξ V ξ ] V 8 V [ = V w ɛ + V V ξ 4 V V 4 V w ξ ɛ V ξ w ξ ξ ] + 4 V w ξ ξ w ξ ξ [ = w ξ V ξ ] V w ɛ 4 V w V w ξ ɛ + V w ɛ 4 V w ξ w ɛ + 6 ξ V w ɛ w ɛ + V w ɛ V w ɛ + V w ɛ 4 + V w ξ w ɛ + w ɛ V w ɛ + ξ ξ ξ + ξ + w ɛ w ɛ [ w ɛ 4 V ξ ξ + V w V V w ɛ w ɛ + w ɛ 8 w ɛ 4 + w ɛ 4 + w ɛ + w ɛ 9 w ξ ξ ] w ɛ w + V + V 4 w ɛ k 4 = m 4 m Skewness and kurtoss unbased by Gaussan uncertantes page of 6

34 m x m ξ = w ɛ V w V m x m ξ = w ɛ ξ V ξ w m 4 x m 4 ξ = 6 w ɛ V w ɛ w ɛ + 6 V m x m ξ = 4 V + V w ɛ 4 V ξ ξ w V + w ɛ 4 V w ξ ξ + w ɛ ξ ξ + V w V w ξ ξ [ + V w ɛ w + V w V w ɛ w ] V 4 w V + + V 4 w ɛ 5 Snce the rght-hand sde of Eq. does not depend on ξ, f x, ɛ = fξ, ɛ = m x m ξ and the expresson of the nose-unbased sample varance m = m x f x, ɛ s found mmedately: m = m w ɛ w = k V V. 6 In order to remove the dependence on ξ n Eqs 5, f x, ɛ s derved from fξ, ɛ such that f x, ɛ = fξ, ɛ. In the case of skewness, fξ, ɛ has the followng form: fξ, ɛ = c ξ ξ, 7 where c denotes the coeffcent of the -th term. The computaton of c x x leads to: c x x = x c exp [ x ξ ] π ɛ ɛ dx + [ x c w exp x ξ ] V π ɛ ɛ dx 8 = c ξ c w ξ 9 V = c ξ ξ. Snce Eq. equals Eq. 7, t follows f x, ɛ = c x x, and the nose-unbased sample skewness m = m x f x, ɛ s m = m w ɛ x x w = k V V. Skewness and kurtoss unbased by Gaussan uncertantes page 4 of 6

35 For the kurtoss moment and cumulant, fξ, ɛ nvolves ξ-dependent terms of the form c ξ ξ. The computaton of c x x leads to: c x x = = c x exp π ɛ c w V c w V + V + V c w [ x ξ ɛ x exp π ɛ x x π ɛ ɛ exp x exp π ɛ c w w k k ] dx + [ x ξ ɛ ] dx + [ x ξ x ξ ] dx dx + ɛ ɛ [ x ξ ] dx + x x k π ɛ ɛ k exp ɛ [ x ξ ɛ c ξ + ɛ c w ξ V + ɛ c ξ w ξ + V + V c w ξ + ɛ + V ] x k ξ k ɛ dx dx k k c w ξ w k ξ k 4 = c ξ + c ɛ ξ c ξ c w ɛ + V + V c w ɛ + ξ c 5 = c ξ ξ + c ɛ w + V V c w ɛ. 6 Thus, each of the terms of the form c ξ ξ n Eqs 4 5 can be replaced by the expresson c x x ɛ w V V k w ɛ, 7 Skewness and kurtoss unbased by Gaussan uncertantes page 5 of 6

36 and the nose-unbased sample kurtoss moment m 4 and cumulant k4 are found as follows: m 4 = m 4 6 w ɛ x x w ɛ w V V V V w w ɛ + V w ɛ 4 V w ɛ + V = m 4 6 V 6 V w ɛ = m 4 6 w ɛ V m = m 4 V V V w 6 V V w ɛ w ɛ x x w w ɛ w x x w ɛ w + V w 8 V V w x x [x x w w ɛ x x ɛ w w ɛ 4 x x ɛ = m m w ɛ V 4 V + V w ɛ = m 4 V + 4 V + 4 w ɛ w V w V [ x x ɛ w ɛ w ɛ w ɛ = m 4 V k 4 = m 4 m. w ɛ w V V + w ɛ 4 V w ɛ ɛ w V w V [ V + V w ɛ V [ w ɛ w V w V V w + V V 4 w ɛ 9 ] + m w V w ɛ + w ɛ ] w V w ] V V + V [ x x ɛ w ɛ + V + V w ɛ w w ɛ w ɛ [ x x ɛ V 4 V w V w ɛ w ɛ V 4 w ɛ w + V V 4 w ɛ ] V w ɛ w ɛ + w ɛ w ɛ V ] V w ɛ + w ɛ w ɛ V w ɛ w ɛ + w ɛ V 4 w ɛ w ] V + w ɛ 4 Skewness and kurtoss unbased by Gaussan uncertantes page 6 of 6 5