Skewness and kurtosis unbiased by Gaussian … and kurtosis unbiased by Gaussian uncertainties...

Skewness and kurtosis unbiased byGaussian uncertainties

Lorenzo Rimoldini

Observatoire astronomique de l’Universite de Geneve, chemin des Maillettes 51, CH-1290 Versoix, SwitzerlandISDC Data Centre for Astrophysics, Universite de Geneve, chemin d’Ecogia 16, CH-1290 Versoix, Switzerland

[email protected]

Draft version: April 29, 2013

Abstract

Noise is an unavoidable part of most measurements which can hinder a correct in-terpretation of the data. Uncertainties propagate in the data analysis and can lead tobiased results even in basic descriptive statistics such as the central moments and cumu-lants. Expressions of noise-unbiased estimates of central moments and cumulants up tothe fourth order are presented under the assumption of independent Gaussian uncertain-ties, for weighted and unweighted statistics. These results are expected to be relevantfor applications of the skewness and kurtosis estimators such as outlier detections, nor-mality tests and in automated classification procedures. The comparison of estimatorscorrected and not corrected for noise biases is illustrated with simulations as a functionof signal-to-noise ratio, employing different sample sizes and weighting schemes.

1 IntroductionMeasurements generally provide an approximate description of real phenomena, because data acquisitioncompounds many processes which contribute, to a different degree, to instrumental errors (e.g., relatedto sensitivity or systematic biases) and uncertainties of statistical nature. While instrumental effectsare addressed before data analysis, statistical uncertainties propagate in subsequent processing and canaffect both precision and accuracy of results, especially at low signal-to-noise (S/N) ratios. Correctingfor biases generated by noise can help the characterization and interpretation of weak signals, and insome cases improve a significant fraction of all data (e.g., the number of astronomical sources increasesdramatically near the faint detection threshold, since there are many more sources far away than nearby).

In this paper, noise-unbiased estimates of central moments and cumulants up to the fourth order,which are often employed to characterize the shape of the distribution of data, are derived analytically.Some of the advantages of these estimators include the ease of computation and the ability to encapsulateimportant features in a few numbers. Skewness and kurtosis measure the degree of asymmetry andpeakedness or weight of the tails of the distribution, respectively, and they are useful for the detection ofoutliers, the assessment of departures from normality of the data (D’Agostino, 1986), the classificationof light variations of astronomical sources (Rimoldini, 2013a) and many other applications. Variousestimators of skewness and kurtosis are available in the literature (e.g., Moors et al., 1996; Hosking, 1990;Groeneveld & Meeden, 1984; Bowley, 1920), some of which aim at mitigating the sensitivity to outliersof the conventional formulations. On the other hand, robust measures might miss important features ofsignals, especially when these are characterized by outliers (as in astronomical time series where stellarbursts or eclipses from binary systems represent rare events in the light curve) and weighting might helpdistinguish true outliers from spurious data (employing additional information such as the accuracy ofeach measurement), so the traditional forms of weighted central moments and cumulants are employedin this work.

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Moments are usually computed on random variables. Herein, their application is extended to datagenerated from deterministic functions and randomized by the uneven sampling of a finite number ofmeasurements and by their uncertainties, whereas the corresponding ‘population’ statistics are defined inthe limit of an infinite regular sampling with no random or systematic errors. This scenario is common inastronomical time series, where measurements are typically non-regular due to observational constraints,they are unavoidably affected by noise, and sometimes also not very numerous: all of these aspectsintroduce some level of randomness in the characterization of the underlying signal of a star.

While the effects of sampling and sample size on time series are studied in Rimoldini (2013a,b), thiswork addresses the bias, precision and accuracy of estimators when measurements are affected (mostly)by Gaussian uncertainties. Bias is defined as the difference between expectation and population valuesand thus expresses a systematic deviation from the true value. Precision is described by the dispersionof measurements, while accuracy is related to the distance of an estimator from the true value and thuscombines the bias and precision concepts (e.g., accuracy can be measured by the mean square error,defined by the sum of bias and uncertainty in quadrature).

Noise-unbiased expressions are provided for the variance, skewness and kurtosis (central moments andcumulants), weighted and unweighted, assuming Gaussian uncertainties and independent measurements.The dependence of noise-unbiased estimators on S/N is illustrated with simulations employing differentsample sizes and two weighting schemes: the common inverse-squared uncertainties and interpolation-based weights as described in Rimoldini (2013a). The latter demonstrated a significant improvement inthe precision of weighted estimators at the high S/N end.

This paper is organized as follows. The notation employed throughout is defined in Sec. 2, followedby the description of the method to estimate Gaussian-noise unbiased moments in Sec. 3. Noise-unbiasedestimates of moments and cumulants (biased and unbiased by sample-size) are presented in Sections 4and 5, in both weighted and unweighted formulations, and the special case of error-weighted estimatorsis presented in Sec. 6. The noise-unbiased estimators are compared with the uncorrected (noise-biased)counterparts with simulated signals as a function of S/N ratio in Sec. 7, including weighted and un-weighted schemes and two different sample sizes. Conclusions are drawn in Sec. 8, followed by detailedderivations of the noise-unbiased estimators in App. A.

2 Notation

For a set of n measurements x = (x1, x2, ..., xn), the following quantities are defined.

(i) Population central moments µr = 〈(x − µ)r〉 with mean µ = 〈x〉, where 〈.〉 denotes expectation,and cumulants κ2 = µ2, κ3 = µ3, κ4 = µ4 − 3µ2

2 (e.g., Stuart & Ord, 1969).

(ii) The sum of the p-th power of weights is defined as Vp =∑ni=1 w

pi .

(iii) The mean θ of a generic set of n elements θi associated with weights wi is θ =∑ni=1 wiθi/V1.

(iv) Sample central moments mr =∑ni=1 wi(xi − x)r/V1 and corresponding cumulants kr.

(v) Sample-size unbiased estimates of central moments Mi and cumulants Ki, i.e., 〈Mi〉 = µi and〈Ki〉 = κi.

(vi) The standardized skewness and kurtosis are defined as g1 = k3/k3/22 , g2 = k4/k

22, G1 = K3/K

3/22 ,

and G2 = K4/K22 , with population values γ1 = κ3/κ

3/22 and γ2 = κ4/κ

22. G1 and G2 satisfy

consistency (for n→∞) but are not unbiased in general (e.g., see Heijmans, 1999, for exceptions).

(vii) Noise-unbiased estimates of central moments and cumulants are denoted by an asterisk superscript.

(viii) No systematic errors are considered herein and random errors are simply referred to as errors oruncertainties.

(ix) Statistics weighted by the inverse-squared uncertainties are called ‘error-weighted’ for brevity andinterpolation-based weights computed in phase (Rimoldini, 2013a) are named ‘phase weights’.


3 Method

The goal is to derive an estimator T ∗(x, ε) as a function of observables (measurements x with correspond-ing uncertainties ε) which is unbiased by the noise in the data, i.e., such that the expectation 〈T ∗(x, ε)〉equals the estimator T (ξ) in terms of the true (unknown) values ξ aimed at by the measurements.

The noise-unbiased estimator T ∗(x, ε) is obtained with the following procedure and assumptions. Ifn independent measurements x are associated with independent Gaussian uncertainties ε, the expectedvalue 〈T (x)〉 of the estimator T (x) is evaluated from measurements x and the joint probability densityp(x|ξ, ε), for given true values ξ and measurement uncertainties ε:

〈T (x)〉 =

∫Rn

T (x′) p(x′|ξ, ε) dnx′, (1)

where

p(x′|ξ, ε) =

n∏i=1

1√2π εi

exp

[− (x′i − ξi)2

2ε2i

]. (2)

As shown in App. A, the expectation 〈T (x)〉 of the estimators considered herein can be decomposed as

〈T (x)〉 = T (ξ) + f(ξ, ε). (3)

Thus, the noise-free estimator T (ξ) = 〈T (x)〉− f(ξ, ε) can be estimated in terms of measurements x anduncertainties ε by the noise-unbiased estimator T ∗(x, ε) = T (x)−f∗(x, ε), where 〈f∗(x, ε)〉 = f(ξ, ε) and,by definition, 〈T ∗(x, ε)〉 = T (ξ). The f∗(x, ε) term is derived first by computing f(ξ, ε) = 〈T (x)〉 − T (ξ)and then by replacing terms depending on ξ in f(ξ, ε) with terms as a function of x which satisfy therequirement 〈f∗(x, ε)〉 = f(ξ, ε) (see App. A). A property often used in the following sections is thata noise-unbiased linear combination of N estimators is equivalent to the linear combination of noise-unbiased estimators: [

N∑i=1

ci Ti(x)

]∗=

N∑i=1

ci T∗i (x), (4)

where the coefficients ci are independent of the measurements x.

4 Gaussian-noise unbiased sample moments and cumulantsWeighted sample central moments unbiased by Gaussian uncertainties, such as the variance m∗2, skewnessm∗3, kurtosis m∗4 and the respective cumulants about the weighted mean x∗ = x are derived assumingindependent measurements xi, uncertainties εi and weights wi, as described in full detail in App. A. Theyare defined as follows:

m∗2 = m2 −1

V1

n∑i=1

wiε2i

(1− wi

V1

)= k∗2 (5)

m∗3 = m3 −3

V1

n∑i=1

wiε2i (xi − x)

(1− 2wi

V1

)= k∗3 (6)

m∗4 = m4 −6

V1

n∑i=1

wiε2i

[(xi − x)

2

(1− 2wi

V1

)− ε2i

2

(1− 2wi

V1

)2

+m∗2wiV1

]− 3

V 41

(n∑i=1

w2i ε

2i

)2

(7)

(m22)∗ = (m∗2)

2 − 4

V 21

n∑i=1

w2i ε

2i

[(xi − x)

2 − ε2i2

(1− 2wi

V1

)]+

2

V 41

(n∑i=1

w2i ε

2i

)2

(8)

k∗4 = m∗4 − 3 (m22)∗. (9)

By definition, the above expressions satisfy

〈m∗r〉 =1

V1

n∑i=1

wi(ξi − ξ

)r. (10)


The unweighted forms can be obtained by substituting wi = 1 (for all i) and Vp = n (for all p) in allterms, leading to:

m∗2 = m2 −n− 1

n2

n∑i=1

ε2i = k∗2 (11)

m∗3 = m3 −3(n− 2)

n2

n∑i=1

ε2i (xi − x) = k∗3 (12)

m∗4 = m4 −6(n− 2)

n2

n∑i=1

ε2i (xi − x)2 − 6m∗2

n2

n∑i=1

ε2i +3(n− 2)2

n3

n∑i=1

ε4i −3

n4

(n∑i=1

ε2i

)2

(13)

(m22)∗ = (m∗2)2 − 4

n2

n∑i=1

ε2i (xi − x)2

+2(n− 2)

n3

n∑i=1

ε4i +2

n4

(n∑i=1

ε2i

)2

(14)

k∗4 = m∗4 − 3 (m22)∗. (15)

5 Gaussian-noise and sample-size unbiased moments and cumu-lants

The estimates of weighted central moments which are unbiased by both sample-size and Gaussian uncer-tainties, such as the variance M∗2 , skewness M∗3 , kurtosis M∗4 and the respective cumulants, are definedin terms of the noise-unbiased sample estimators as follows:

M∗2 =V 21

V 21 − V2

m∗2 = K∗2 (16)

M∗3 =V 31

V 31 − 3V1V2 + 2V3

m∗3 = K∗3 (17)

M∗4 =V 21 (V 4

1 − 3V 21 V2 + 2V1V3 + 3V 2

2 − 3V4)

(V 21 − V2)(V 4

1 − 6V 21 V2 + 8V1V3 + 3V 2

2 − 6V4)m∗4 +

− 3V 21 (2V 2

1 V2 − 2V1V3 − 3V 22 + 3V4)

(V 21 − V2)(V 4

1 − 6V 21 V2 + 8V1V3 + 3V 2

2 − 6V4)(m2

2)∗ (18)

K∗4 =V 21 (V 4

1 − 4V1V3 + 3V 22 )

(V 21 − V2)(V 4

1 − 6V 21 V2 + 8V1V3 + 3V 2

2 − 6V4)m∗4 +

− 3V 21 (V 4

1 − 2V 21 V2 + 4V1V3 − 3V 2

2 )

(V 21 − V2)(V 4

1 − 6V 21 V2 + 8V1V3 + 3V 2

2 − 6V4)(m2

2)∗. (19)

The derivation of the sample-size unbiased weighted estimators is described in Rimoldini (2013b). Thecorresponding unweighted forms can be achieved by direct substitution Vp = n for all p, leading to:

M∗2 =n

n− 1m∗2 = M2 −

1

n

n∑i=1

ε2i = K∗2 (20)

M∗3 =n2

(n− 1)(n− 2)m∗3 = M3 −

3

n− 1

n∑i=1

ε2i (xi − x) = K∗3 (21)

M∗4 =n(n2 − 2n+ 3)

(n− 1)(n− 2)(n− 3)m∗4 −

3n(2n− 3)

(n− 1)(n− 2)(n− 3)(m2

2)∗ (22)

K∗4 =n2(n+ 1)

(n− 1)(n− 2)(n− 3)m∗4 −

3n2

(n− 2)(n− 3)(m2

2)∗. (23)


6 Special cases

If weights are related to measurement errors as wi = 1/ε2i , the noise-unbiased weighted sample momentsand cumulants reduce to the following expressions:

m∗2 = m2 −n− 1

V1= k∗2 (24)

m∗3 = m3 −3

V1

n∑i=1

(xi − x) = k∗3 (25)

m∗4 = m4 −6

V1

n∑i=1

[(xi − x)

2 − ε2i2

]+

6m∗2V1− 3

V 21

(26)

(m22)∗ = (m∗2)2 − 2(m∗2 +m2)

V1(27)

k∗4 = m∗4 − 3 (m22)∗. (28)

In the case of constant errors, i.e., εi = ε0 for all i, some of the unweighted estimators are equivalentor similar to their noise-unbiased counterparts:

Skewness: k∗3 = k3 and K∗3 = K3 (also m∗3 = m3 and M∗3 = M3), (29)

Kurtosis: k∗4 ≈ k4 and K∗4 = K4, (30)

where the approximation k∗4 ≈ k4 holds for large values of n or S/N ratios since

k∗4 − k4k22

=6 ε20 (k∗2 + k2)

nk22≈

6[1 + 2 (S/N)2

]n [1 + (S/N)2]

2 , (31)

considering that, for constant errors,∑i ε

2i /n = ε20 and (S/N)2 ≈ k∗2/ε

20 ≈ k2/ε

20 − 1. For sample

cumulants up to the fourth order, only the variance depends strongly on noise. However, this is animportant estimator because it is often involved in definitions of standardized skewness (g1 and G1) andkurtosis (g2 and G2) as follows:

g1 = k3/k3/22 , G1 = K3/K

3/22 , (32)

g2 = k4/k22, G2 = K4/K

22 . (33)

For consistency with the above definitions, the noise-unbiased equivalents are defined as

g∗1 = k∗3/(k∗2)3/2, G∗1 = K∗3/(K

∗2 )3/2, (34)

g∗2 = k∗4/(k∗2)2, G∗2 = K∗4/(K

∗2 )2, (35)

although the truly noise-unbiased expressions should have been computed on the ratios in Eqs (32)–(33).The application of Eqs (34)–(35) should generally be restricted to larger samples (e.g., n > 50) with S/Nratios greater than a few, in order to avoid non-positive values of k∗2 or K∗2 .

7 Estimators as a function of signal-to-noise ratio

Noise-biased and unbiased estimators are compared as a function of signal-to-noise ratio S/N with sim-ulated data and different weighting schemes for specific signals, sampling and error laws. The values ofthe population moments of the continuous simulated periodic ‘true’ signal ξ(φ) are computed averagingin phase φ as follows:

µr =1

2π

∫ 2π

0

[ξ(φ)− µ]r

dφ, where µ =1

2π

∫ 2π

0

ξ(φ) dφ. (36)


7.1 Simulation

Simulated signals are described by a sinusoidal function to the fourth power, which has a non-zeroskewness and thus makes it possible to evaluate the precision and accuracy of the skewness standardizedby the estimated variance without simply reflecting the accuracy of the variance. The S/N level isevaluated by the ratio of the standard deviation

√µ2 of the true signal ξ(φ) and the root of the mean of

squared measurement uncertainties εi (assumed independent of the signal). The signal ξ(φ) is sampledn = 100 and 1000 times at phases φi randomly drawn from a uniform distribution, while the S/N ratiovaries from 1 to 1000 and determines the uncertainties εi of measurements xi as follows:

ξ(φ) = A sin4 φ

xi ∼ N (ξi, ε2i ) for ξi = ξ(φi) and φi ∼ U(0, 2π)

ε2i = (1 + ρi) µ2 / (S/N)2 for ρi ∼ U(−0.8, 0.8),

(37)

(38)

(39)

where the i-th measurement xi is drawn from a normal distribution N (ξi, ε2i ) of mean ξi and variance ε2i .

The latter is defined in terms of a variable ρi randomly drawn from a uniform distribution U(−0.8, 0.8)so that measurement uncertainties vary by up to a factor of 3 for a given µ2 and S/N ratio. Simulationswere repeated 104 times for each S/N ratio (for n = 100 and 1000).

The dependence of weighted estimators on sample size and the corresponding unbiased expressionswere presented in Rimoldini (2013b). Herein, only large sample sizes are employed so that sample-sizebiases are negligible with respect to the ones resulting from small S/N ratios. A sample signal andsimulated data are illustrated in Fig. 1 for n = 100 and S/N = 2. The reference population values of themean, variance, skewness and kurtosis of the simulated signal are listed in table 1 of Rimoldini (2013b).

Error weights are defined by wi = 1/ε2i , while mixed error-phase weights follow Rimoldini (2013a),assuming phase-sorted data:

wi = h(S/N |a, b) w′i∑nj=1 w

′j

+ [1− h(S/N |a, b)] ε−2i∑nj=1 ε

−2j

∀i ∈ (1, n)

w′i = φi+1 − φi−1 ∀i ∈ (2, n− 1)

w′1 = φ2 − φn + 2π

w′n = φ1 − φn−1 + 2π

h(S/N |a, b) =1

1 + e−(S/N−a)/bfor a, b > 0.

(40)

(41)

(42)

(43)

(44)

Weighting effectively decreases the sample size, since more importance is given to some data at theexpense of other ones and results depend mostly on fewer ‘relevant’ measurements (e.g., weighting by theinverse-squared uncertainties can worsen precision at high S/N levels). Weighted procedures are desirablewhen the dispersion and bias of estimators from an effectively reduced sample size are smaller than theimprovements in precision and accuracy (e.g., weighting by inverse-squared uncertainties can improveboth precision and accuracy at low S/N ratios). Also, weighting might exploit correlations in the data toimprove precision, as it is shown employing phase weights (Rimoldini, 2013a). Since correlated data donot satisfy the assumptions of the expressions derived herein, their application might return biased results.However, small biases could be justified if improvements in precision are significant and, depending onthe extent of the application, larger biases could be mitigated with mixed weighting schemes, such as theone described by Eqs (40)–(44).

Estimators derived herein assume a single weighting scheme and combinations of estimators (like thevariance and the mean in the standardized skewness and kurtosis) are expected to apply the same weightsto terms associated with the same measurements. The function h(S/N |a, b) constitutes just an exampleto achieve a mixed weighting scheme: tuning parameters a, b offer the possibility to control the transitionfrom error-weighted to phase-weighted estimators (in the limits of low and high S/N , respectively) andthus reach a compromise solution between precision and accuracy for all values of S/N , according to thespecific estimators, signals, sampling, errors, sample sizes and their distributions in the data.


0.0 0.2 0.4 0.6 0.8 1.0

−1.

0−

0.5

0.0

0.5

1.0

1.5

Phase (φ / 2π)

Sig

nal a

nd s

imul

ated

dat

a

S/N = 2, n = 100

Figure 1: A simulated signal of the form of sin4 φ (blue curve) is irregularly sampled by 100 measurements(denoted by triangles) with S/N = 2.

7.2 Results

The results of simulations are illustrated for sample estimators, since the conclusions in Rimoldini (2013b)suggested that phase-weighted sample estimators can be more accurate and precise than the sample-sizeunbiased counterparts in most cases, especially for large sample sizes as considered herein.

Figure 2 illustrates the sample mean in the various scenarios considered in the simulations: samplesizes of n = 100 and 1000, unweighted and with different weighting schemes (error-weighted, phase-weighted and combined error-phase weighted). While accuracy is the same in all cases, the best precisionof the mean is achieved employing phase weights (including the low S/N end, unlike other estimators).

Figures 3–16 compare noise-biased (‘uncorrected’ ) and noise-unbiased (‘corrected’ ) estimators as afunction of S/N , evaluating the following deviations from the population values:

m2/µ2 − 1 vs m∗2/µ2 − 1, (45)

m3/µ3/22 − γ1 vs m∗3/µ

3/22 − γ1, g1 − γ1 vs g∗1 − γ1, (46)

m4/µ22 − 3− γ2 vs m∗4/µ

22 − 3− γ2, m4/m

22 − 3− γ2 vs m∗4/(m

∗2)2 − 3− γ2, (47)

k4/µ22 − γ2 vs k∗4/µ

22 − γ2, g2 − γ2 vs g∗2 − γ2, (48)

in both weighted and unweighted cases, for n = 100 and 1000. The dependence on n is described in moredetails in Rimoldini (2013b). Estimators standardized by both true and estimated variance are presentedto help interpret the behaviour of the ratios from their components.

All figures confirm that ‘corrected’ and ‘uncorrected’ estimators have similar precision and accuracy athigh S/N levels (typically for S/N > 10). Noise-unbiased estimators are found to be the most accurate inall cases and over the whole S/N range tested. Their precision is generally similar to the noise-uncorrectedcounterparts, apart from estimators standardized by the estimated variance, such as g1, g2 and m4/m

22,

for which the uncorrected version can be much more precise (although biased) for S/N < 2, typically.As expected, the precision of estimators employing n = 1000 measurements per sample was greater thanthe one obtained with sample sizes of n = 100.

Weighting by the inverse of squared measurement errors made the estimators slightly less precise athigh S/N ratios, but more precise and accurate at low S/N levels (except for the mean).

Weighting by phase intervals led to a significant improvement in precision of all estimators in the limitof large S/N ratios and a reduction of precision at low S/N (apart from the case of the mean). Tuningparameters such as a = 2 and b = 0.3 in Eq. (40) were able to mitigate the imprecision at low S/N


reducing to the error-weighted results, which appeared to be the most accurate and precise in the limitof low S/N ratios (in these simulations). This solution might provide a reasonable compromise betweenprecision and accuracy of all estimators, at least for S/N > 1.

Figures 5–8 show that the skewness moment m3 is quite unbiased by noise, while the standardizedversion g1 is underestimated at high S/N because of the overestimated variance m2 (as shown in Figs 3–4).While the accuracy of g1 deteriorates at low S/N , its precision is much less affected by noise.

The kurtosis moment m4 (Figs 9–12) is less precise and accurate than the noise-unbiased equivalent,and its normalization by the squared variance reduces dramatically its inaccuracy and imprecision (sincem2 and m4 exhibit a similar trend as a function of S/N). The kurtosis cumulant k4, instead, is muchcloser to its noise-unbiased counterpart, as shown in Figs 13–16. The normalization of k4 by the squaredvariance improves its precision at the cost of lower accuracy for S/N < 10: the bias of g2 is similar to(greater than) the precision of g∗2 for n = 100 (n = 1000).

The lower the S/N level is, the less precise estimators are and the noise-unbiased variance can beunderestimated (and even become non-positive). Thus, the skewness and kurtosis estimators standardizedby k∗2 or K∗2 , as in Eqs (34)–(35), should be avoided in circumstances that combine small sample sizes(up to a few dozens of elements) and low S/N ratios (of the order of a few or less).

Figures related to moments and cumulants of irregularly sampled sinusoidal signals are very similarto the ones presented herein, with the exception of g1, which would have a similar precision but with nobias, as a consequence of the null skewness of a sinusoidal signal (since the mean of k3 estimates is zero,they are not biased by the standardization with an overestimated noise-biased variance).

From the comparison of noise-biased and unbiased estimators with different weighting schemes, it ap-pears that, for large sample sizes, noise-unbiased phase-weighted estimators are usually the most accuratefor S/N > 2 (apart from the special cases of standardized skewness and kurtosis when their true valueis zero). For noisy signals (e.g., S/N < 2), error weighting seems the most appropriate, at least withGaussian uncertainties, thus noise-unbiased error-phase weighted estimators can provide a satisfactorycompromise in general. Further improvements might be achieved by tuning parameters better fitted toestimators and signals of interest, in view of specific requirements of precision and accuracy.

8 ConclusionsExact expressions of noise-unbiased skewness and kurtosis were provided in the unweighted and weightedformulations, under the assumption of independent data and Gaussian uncertainties. Such estimatorscan be particularly useful in the processing, interpretation and comparison of data characterized by lowS/N regimes.

Simulations of an irregularly sampled skewed periodic signal were employed to compare noise-biasedand unbiased estimators as a function of S/N in the unweighted, inverse-squared error weighted andphase-weighted schemes. While noise-unbiased estimators were found more accurate in general, theywere less precise than the uncorrected counterparts at low S/N ratios. The application of a mixedweighting scheme involving phase intervals and uncertainties was able to balance precision and accuracyon a wide range of S/N levels. The effect of noise-unbiased estimators and different weighting schemeson the characterization and classification of astronomical time series is described in Rimoldini (2013a).

AcknowledgmentsThe author thanks M. Suveges for many discussions and valuable comments on the original manuscript.

ReferencesBowley A.L., 1920, Elements of Statistics, Charles Scribner’s Sons, New York

D’Agostino R.B., 1986, Goodness-of-fit techniques, D’Agostino & Stephens eds., Marcel Dekker, NewYork, p. 367

Groeneveld R.A., Meeden G., 1984, The Statistician, 33, 391


Heijmans R., 1999, Statistical Papers, 40, 107

Hosking J.R.M., 1990, J. R. Statist. Soc. B, 52, 105

Moors J.J.A., Wagemakers R.Th.A., Coenen V.M.J., Heuts R.M.J., Janssens M.J.B.T., 1996, StatisticaNeerlandica, 50, 417

Rimoldini L., 2013a, preprint (arXiv:1304.6616)

Rimoldini L., 2013b, preprint (arXiv:1304.6564)

Stuart A., Ord J., 1969, Kendall’s Advanced Theory of Statistics, Charles Griffin & Co. Ltd, London


http://xxx.lanl.gov/abs/1304.6616

http://xxx.lanl.gov/abs/1304.6564

Mean(n = 100, 1000)

Unweighted

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

mea

n / µ

− 1

n = 100

n = 1000

Phase Weighted (a, b→ 0)

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

mea

n / µ

− 1

n = 100

n = 1000

Error Weighted

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

mea

n / µ

− 1

n = 100

n = 1000

Error-Phase Weighted (a = 2, b = 0.3)

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

mea

n / µ

− 1

n = 100

n = 1000

Figure 2: Sample mean for S/N > 1 and n = 100, 1000: unweighted on the top-left hand side, weighted bythe inverse of squared measurement errors on the top-right hand side, and weighted by phases and errors,according to Eq. (40), with different parameter values, as specified above the lower panels. Shaded areasencompass one standard deviation from the average of the distribution of the mean employing simulationsdefined by Eqs (37)–(39).


Variance(n = 100)

Unweighted

−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected


−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected

Error Weighted

−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected


−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected

Figure 3: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample variance for S/N > 1and n = 100: unweighted on the top-left hand side, weighted by the inverse of squared measurementerrors on the top-right hand side, and weighted by phases and errors, according to Eq. (40), with differentparameter values, as specified above the lower panels. Shaded areas encompass one standard deviationfrom the mean of the distribution of the variance employing simulations defined by Eqs (37)–(39).


Variance(n = 1000)

Unweighted

−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected


−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected

Error Weighted

−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected


−0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m2( *) /

µ 2 −

1

Corrected

Uncorrected

Figure 4: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample variance for S/N > 1and n = 1000: unweighted on the top-left hand side, weighted by the inverse of squared measurementerrors on the top-right hand side, and weighted by phases and errors, according to Eq. (40), with differentparameter values, as specified above the lower panels. Shaded areas encompass one standard deviationfrom the mean of the distribution of the variance employing simulations defined by Eqs (37)–(39).


Skewness(n = 100)

Unweighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected

Error Weighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected

Unweighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected

Error Weighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected

Figure 5: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample skewness for S/N > 1and n = 100: unweighted in the upper panels and weighted by the inverse of squared measurement errorsin the lower panels. Shaded areas encompass one standard deviation from the mean of the distributionof the skewness employing simulations defined by Eqs (37)–(39).


Skewness(n = 100)


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected

Figure 6: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample skewness for S/N > 1and n = 100, weighted by phases and errors, according to Eq. (40), with different parameter values,as specified above each panel. Shaded areas encompass one standard deviation from the mean of thedistribution of the skewness employing simulations defined by Eqs (37)–(39).


Skewness(n = 1000)

Unweighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected

Error Weighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected

Unweighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected

Error Weighted

−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected

Figure 7: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample skewness for S/N > 1and n = 1000: unweighted in the upper panels and weighted by the inverse of squared measurement errorsin the lower panels. Shaded areas encompass one standard deviation from the mean of the distributionof the skewness employing simulations defined by Eqs (37)–(39).


Skewness(n = 1000)


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m3( *) /

µ 232 −

γ1

Corrected

Uncorrected


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected


−1.0

−0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 1( *) −

γ1

Corrected

Uncorrected

Figure 8: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample skewness for S/N > 1and n = 1000, weighted by phases and errors, according to Eq. (40), with different parameter values,as specified above each panel. Shaded areas encompass one standard deviation from the mean of thedistribution of the skewness employing simulations defined by Eqs (37)–(39).


Kurtosis(n = 100)

Unweighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected

Error Weighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected

Unweighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected

Error Weighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected

Figure 9: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis moment forS/N > 1 and n = 100: unweighted in the upper panels and weighted by the inverse of squared measure-ment errors in the lower panels. Shaded areas encompass one standard deviation from the mean of thedistribution of the kurtosis employing simulations defined by Eqs (37)–(39).


Kurtosis(n = 100)


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected

Figure 10: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis moment forS/N > 1 and n = 100, weighted by phases and errors, according to Eq. (40), with different parametervalues, as specified above each panel. Shaded areas encompass one standard deviation from the mean ofthe distribution of the kurtosis employing simulations defined by Eqs (37)–(39).


Kurtosis(n = 1000)

Unweighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected

Error Weighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected

Unweighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected

Error Weighted

−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected

Figure 11: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis moment forS/N > 1 and n = 1000: unweighted in the upper panels and weighted by the inverse of squared mea-surement errors in the lower panels. Shaded areas encompass one standard deviation from the mean ofthe distribution of the kurtosis employing simulations defined by Eqs (37)–(39).


Kurtosis(n = 1000)


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

µ 22 − 3

− γ

2

Corrected

Uncorrected


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected


−5

0

5

10

15

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

m4( *) /

m2( *)2

− 3

− γ

2 Corrected

Uncorrected

Figure 12: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis moment forS/N > 1 and n = 1000, weighted by phases and errors, according to Eq. (40), with different parametervalues, as specified above each panel. Shaded areas encompass one standard deviation from the mean ofthe distribution of the kurtosis employing simulations defined by Eqs (37)–(39).


k-Kurtosis(n = 100)

Unweighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected

Error Weighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected

Unweighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected

Error Weighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected

Figure 13: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis cumulantfor S/N > 1 and n = 100: unweighted in the upper panels and weighted by the inverse of squaredmeasurement errors in the lower panels. Shaded areas encompass one standard deviation from the meanof the distribution of the kurtosis employing simulations defined by Eqs (37)–(39).


k-Kurtosis(n = 100)


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected

Figure 14: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis cumulant forS/N > 1 and n = 100, weighted by phases and errors, according to Eq. (40), with different parametervalues, as specified above each panel. Shaded areas encompass one standard deviation from the mean ofthe distribution of the kurtosis employing simulations defined by Eqs (37)–(39).


k-Kurtosis(n = 1000)

Unweighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected

Error Weighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected

Unweighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected

Error Weighted

−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected

Figure 15: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis cumulantfor S/N > 1 and n = 1000: unweighted in the upper panels and weighted by the inverse of squaredmeasurement errors in the lower panels. Shaded areas encompass one standard deviation from the meanof the distribution of the kurtosis employing simulations defined by Eqs (37)–(39).


k-Kurtosis(n = 1000)


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

k 4( *) /

µ 22 − γ

2

Corrected

Uncorrected


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected


−3

−2

−1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

(S/N)−1

g 2( *) −

γ2

Corrected

Uncorrected

Figure 16: Noise-biased (‘uncorrected’ ) versus noise-unbiased (‘corrected’ ) sample kurtosis cumulant forS/N > 1 and n = 1000, weighted by phases and errors, according to Eq. (40), with different parametervalues, as specified above each panel. Shaded areas encompass one standard deviation from the mean ofthe distribution of the kurtosis employing simulations defined by Eqs (37)–(39).


A Derivation of noise-unbiased momentsThe derivations presented in this Appendix involve weighted estimators under the assumption of inde-pendent measurements, uncertainties and weights. Definitions and some of the relations often employedherein are listed below.

• For brevity, mr = mr(x), and∑i and

∏i are implied to involve all (from the 1-st to the n-th)

terms, unless explicitly stated otherwise.

• The following integral solutions are often employed:

〈xsi 〉 =

∫ ∞−∞

x′is

√2π εi

exp

[− (x′i − ξi)2

2ε2i

]dx′i =

ξi for s = 1

ξ2i + ε2i for s = 2

ξ3i + 3ξiε2i for s = 3

ξ4i + 6ξ2i ε2i + 3ε4i for s = 4.

(49)

• The expected value 〈m〉 of a generic estimator m(x) =∑i aix

si

∑j 6=i bjx

tj

∑k 6=i,j ckx

uk

∑l 6=i,j,k dlx

vl

of independent data with Gaussian uncertainties is computed as follows

〈m〉 =

∫Rn

m(x′)

n∏i=1

1√2π εi

exp

[− (x′i − ξi)2

2ε2i

]dnx′ (50)

=∏h

∫ ∞−∞

m(x′)1√

2π εhexp

[− (x′h − ξh)2

2ε2h

]dx′h (51)

=∑i

ai

∫ ∞−∞

x′is

√2π εi

exp

[− (x′i − ξi)2

2ε2i

]dx′i

∑j 6=i

bj

∫ ∞−∞

x′jt

√2π εj

exp

[−

(x′j − ξj)2

2ε2j

]dx′j ×

×∑k 6=i,j

ck

∫ ∞−∞

x′ku

√2π εk

exp

[− (x′k − ξk)2

2ε2k

]dx′k

∑l 6=i,j,k

dl

∫ ∞−∞

x′lv

√2π εl

exp

[− (x′l − ξl)2

2ε2l

]dx′l.

(52)

• The results of the following expressions are employed:(∑i wixi)

3 ==∑i wixi (

∑j wjxj)

2

=∑i wixi (

∑j w

2jx

2j +

∑j wjxj

∑k 6=j wkxk)

=∑i w

3i x

3i + 3

∑i w

2i x

2i

∑j 6=i wjxj +

∑i wixi

∑j 6=i wjxj

∑k 6=i,j wkxk.

(∑i wixi)

4 ==∑i wixi

∑j wjxj (

∑k w

2kx

2k +

∑k wkxk

∑l 6=k wlxl)

=∑i wixi (

∑j w

3jx

3j + 3

∑j w

2jx

2j

∑k 6=j wkxk +

∑j wjxj

∑k 6=j wkxk

∑l 6=j,k wlxl)

=∑i w

4i x

4i + 4

∑i w

3i x

3i

∑j 6=i wjxj + 6

∑i w

2i x

2i

∑j 6=i wjxj

∑k 6=i,j wkxk +

+ 3∑i w

2i x

2i

∑j 6=i w

2jx

2j +

∑i wixi

∑j 6=i wjxj

∑k 6=i,j wkxk

∑l 6=i,j,k wlxl.∑

i wix2i (∑j wjxj)

2 =

=∑i wix

2i (∑j w

2jx

2j +

∑j wjxj

∑k 6=j wkxk)

=∑i w

3i x

4i +

∑i wix

2i

∑j 6=i w

2jx

2j + 2

∑i w

2i x

3i

∑j 6=i wjxj +

∑i wix

2i

∑j 6=i wjxj

∑k 6=i,j wkxk.∑

i wiε2i

∑j 6=i wjξj

∑k 6=i,j wkξk =

=∑i wiε

2i

∑j wjξj

∑k wkξk −

∑i wiε

2i

∑j 6=i w

2j ξ

2j − 2

∑i w

2i ξiε

2i

∑j 6=i wjξj −

∑i w

3i ξ

2i ε

2i

= V 21 ξ

2∑i wiε

2i −

∑i wiε

2i

∑j w

2j ξ

2j +

∑i w

3i ξ

2i ε

2i − 2V1ξ

∑i w

2i ξiε

2i + 2

∑i w

3i ξ

2i ε

2i −

∑i w

3i ξ

2i ε

2i

= V 21 ξ

2∑i wiε

2i −

∑i wiε

2i

∑j w

2j ξ

2j − 2V1ξ

∑i w

2i ξiε

2i + 2

∑i w

3i ξ

2i ε

2i .


∑i w

2i ε

2i

∑j 6=i wjξj

∑k 6=i,j wkξk =

=∑i w

2i ε

2i

∑j wjξj

∑k wkξk −

∑i w

2i ε

2i

∑j 6=i w

2j ξ

2j − 2

∑i w

3i ξiε

2i

∑j 6=i wjξj −

∑i w

4i ξ

2i ε

2i

= V 21 ξ

2∑i w

2i ε

2i −

∑i w

2i ε

2i

∑j w

2j ξ

2j +

∑i w

4i ξ

2i ε

2i − 2V1ξ

∑i w

3i ξiε

2i + 2

∑i w

4i ξ

2i ε

2i −

∑i w

4i ξ

2i ε

2i

= V 21 ξ

2∑i w

2i ε

2i −

∑i w

2i ε

2i

∑j w

2j ξ

2j − 2V1ξ

∑i w

3i ξiε

2i + 2

∑i w

4i ξ

2i ε

2i .

A.1 Outline of results

The expressions of the elements pursued along the derivation of noise-unbiased estimators (detailed inSec. A.2) are summarized below, following the notation introduced in Sections 2 and 3.

〈m2〉 =1

V1

∑i

wi(ξi − ξ

)2+

1

V1

∑i

wiε2i

(1− wi

V1

)= 〈k2〉 (53)

〈m3〉 =1

V1

∑i

wi(ξi − ξ

)3+

3

V1

∑i

wiε2i

(ξi − ξ

)(1− 2wi

V1

)= 〈k3〉 (54)

〈m4〉 =1

V1

∑i

wi(ξi − ξ

)4+

6

V1

∑i

wiε2i

[(ξi − ξ

)2(1− 2wi

V1

)+ ε2i

(1

2− 2wi

V1+

2w2i

V 21

)]+

+6

V 31

∑i

w2i ε

2i

∑j

wj(ξj − ξ

)2+∑j

wjε2j

(1− 3wj

2V1

) (55)

〈m22〉 =

[1

V1

∑i

wi(ξi − ξ

)2]2+

4

V 21

∑i

w2i ε

2i

(ξi − ξ

)2+

2

V 21

∑i

wi(ξi − ξ

)2∑j

wjε2j

(1− wj

V1

)+

+2

V 21

∑i

w2i ε

4i

(1− 2wi

V1

)+

1

V 21

[∑i

wiε2i

(1− wi

V1

)]2+

2

V 41

(∑i

w2i ε

2i

)2

(56)

〈k4〉 = 〈m4〉 − 3 〈m22〉 (57)

If f(ξ, ε) =∑i

ci(ξi − ξ

), then f∗(x, ε) =

∑i

ci (xi − x) . (58)

If f(ξ, ε) =∑i

ci(ξi − ξ

)2, then f∗(x, ε) =

∑i

ci

(xi − x)2 − ε2i

(1− 2wi

V1

)− 1

V 21

∑j

w2j ε

2j

. (59)

m∗2 = m2 −1

V1

∑i

wiε2i

(1− wi

V1

)= k∗2 (60)

m∗3 = m3 −3

V1

∑i

wiε2i (xi − x)

(1− 2wi

V1

)= k∗3 (61)

m∗4 = m4 −6

V1

∑i

wiε2i

[(xi − x)

2

(1− 2wi

V1

)− ε2i

2

(1− 2wi

V1

)2

+m∗2wiV1

]− 3

V 41

(∑i

w2i ε

2i

)2

(62)

(m22)∗ = (m∗2)

2 − 4

V 21

∑i

w2i ε

2i

[(xi − x)

2 − ε2i2

(1− 2wi

V1

)]+

2

V 41

(∑i

w2i ε

2i

)2

(63)

k∗4 = m∗4 − 3 (m22)∗ (64)


A.2 Detailed computations

m2 =1

V1

∑i

wi (xi − x)2

(65)

=1

V1

∑i

wix2i −

2

V1x∑i

wixi + x2 (66)

=1

V1

∑i

wix2i − x2 (67)

=1

V1

∑i

wix2i −

(1

V1

∑i

wixi

)2

(68)

=1

V1

∑i

wix2i −

1

V 21

∑i

w2i x

2i −

1

V 21

∑i

wixi∑j 6=i

wjxj (69)

m3 =1

V1

∑i

wi (xi − x)3

(70)

=1

V1

∑i

wix3i −

3

V1x∑i

wix2i +

3

V1x2∑i

wixi − x3 (71)

=1

V1

∑i

wix3i −

3

V1x∑i

wix2i + 2x3 (72)

=1

V1

∑i

wix3i −

3

V 21

∑i

wix2i

∑j

wjxj + 2

(1

V1

∑i

wixi

)3

(73)

=1

V1

∑i

wix3i −

3

V 21

∑i

w2i x

3i +

∑i

wix2i

∑j 6=i

wjxj

+

+2

V 31

∑i

w3i x

3i + 3

∑i

w2i x

2i

∑j 6=i

wjxj +∑i

wixi∑j 6=i

wjxj∑k 6=i,j

wkxk

(74)

m4 =1

V1

∑i

wi (xi − x)4

(75)

=1

V1

∑i

wix4i −

4

V1x∑i

wix3i +

6

V1x2∑i

wix2i −

4

V1x3∑i

wixi − x4 (76)

=1

V1

∑i

wix4i −

4

V1x∑i

wix3i +

6

V1x2∑i

wix2i − 3x4 (77)

=1

V1

∑i

wix4i −

4

V 21

∑i

wix3i

∑j

wjxj +6

V 31

∑i

wix2i

∑j

wjxj

2

− 3

V 41

(∑i

wixi

)4

(78)

=1

V1

∑i

wix4i −

4

V 21

∑i

w2i x

4i +

∑i

wix3i

∑j 6=i

wjxj

+6

V 31

(∑i

w3i x

4i+

+∑i

wix2i

∑j 6=i

w2jx

2j + 2

∑i

w2i x

3i

∑j 6=i

wjxj +∑i

wix2i

∑j 6=i

wjxj∑k 6=i,j

wkxk

+


− 3

V 41

∑i

w4i x

4i + 4

∑i

w3i x

3i

∑j 6=i

wjxj + 6∑i

w2i x

2i

∑j 6=i

wjxj∑k 6=i,j

wkxk+

+3∑i

w2i x

2i

∑j 6=i

w2jx

2j +

∑i

wixi∑j 6=i

wjxj∑k 6=i,j

wkxk∑l 6=i,j,k

wlxl

(79)

m22 =

[1

V1

∑i

wi (xi − x)2

]2(80)

=

1

V1

∑i

wix2i −

(1

V1

∑i

wixi

)22

(81)

=1

V 21

(∑i

wix2i

)2

− 2

V 31

∑i

wix2i

∑j

wjxj

2

+1

V 41

(∑i

wixi

)4

(82)

=1

V 21

(∑i

wix2i

)2

− 2

V 31

∑i

wix2i

∑j

wjxj

2

+1

V 41

(∑i

wixi

)4

(83)

=1

V 21

∑i

w2i x

4i +

∑i

wix2i

∑j 6=i

wjx2j

− 2

V 31

∑i

w3i x

4i +

∑i

wix2i

∑j 6=i

w2jx

2j+

+2∑i

w2i x

3i

∑j 6=i

wjxj +∑i

wix2i

∑j 6=i

wjxj∑k 6=i,j

wkxk

+

+1

V 41

∑i

w4i x

4i + 4

∑i

w3i x

3i

∑j 6=i

wjxj + 6∑i

w2i x

2i

∑j 6=i

wjxj∑k 6=i,j

wkxk+

+3∑i

w2i x

2i

∑j 6=i

w2jx

2j +

∑i

wixi∑j 6=i

wjxj∑k 6=i,j

wkxk∑l 6=i,j,k

wlxl

(84)

〈m2〉 =∏i

∫ ∞−∞

m2(x′)1√

2π εiexp

[− (x′i − ξi)2

2ε2i

]dx′i (85)

=1

V1

∑i

wi(ξ2i + ε2i

)− 1

V 21

∑i

w2i

(ξ2i + ε2i

)− 1

V 21

∑i

wiξi∑j 6=i

wjξj (86)

=1

V1

∑i

wiξ2i +

1

V1

∑i

wiε2i −

1

V 21

∑i

w2i ε

2i −

1

V 21

∑i

w2i ξ

2i +

1

V 21

∑i

wiξi∑j 6=i

wjξj

(87)

=1

V1

∑i

wiξ2i +

1

V1

∑i

wiε2i

(1− wi

V1

)− ξ 2

(88)

=1

V1

∑i

wi(ξi − ξ

)2+

1

V1

∑i

wiε2i

(1− wi

V1

)= 〈k2〉 (89)


〈m3〉 =∏i

∫ ∞−∞

m3(x′)1√

2π εiexp

[− (x′i − ξi)2

2ε2i

]dx′i (90)

=1

V1

∑i

wi(ξ3i + 3ξiε

2i

)− 3

V 21

∑i

w2i

(ξ3i + 3ξiε

2i

)+∑i

wi(ξ2i + ε2i

)∑j 6=i

wjξj

+

+2

V 31

∑i

w3i

(ξ3i + 3ξiε

2i

)+ 3

∑i

w2i

(ξ2i + ε2i

)∑j 6=i

wjξj +∑i

wiξi∑j 6=i

wjξj∑k 6=i,j

wkξk

(91)

=1

V1

∑i

wiξ3i +

3

V1

∑i

wiξiε2i −

3

V 21

∑i

w2i ξ

3i −

9

V 21

∑i

w2i ξiε

2i −

3

V 21

∑i

wiξ2i

∑j 6=i

wjξj+

− 3

V 21

∑i

wiε2i

∑j 6=i

wjξj +2

V 31

∑i

w3i ξ

3i +

6

V 31

∑i

w3i ξiε

2i +

6

V 31

∑i

w2i ξ

2i

∑j 6=i

wjξj+

+6

V 31

∑i

w2i ε

2i

∑j 6=i

wjξj +2

V 31

∑i

wiξi∑j 6=i

wjξj∑k 6=i,j

wkξk (92)

=1

V1

∑i

wiξ3i +

3

V1

∑i

wiξiε2i

(1− 3wi

V1

)− 3

V1ξ∑i

wiξ2i −

3

V 21

∑i

wiε2i

∑j 6=i

wjξj+

+ 2 ξ3

+6

V 21

ξ∑i

w2i ε

2i (93)

=1

V1

∑i

wi(ξi − ξ

)3+

3

V1

∑i

wiε2i

(ξi −

3wiV1

ξi +2wiV1

ξ

)− 3

V1

∑i

wiε2i

(ξ − wi

V1ξi

)(94)

=1

V1

∑i

wi(ξi − ξ

)3+

3

V1

∑i

wiε2i

[ξi − ξ −

2wiV1

(ξi − ξ

)](95)

=1

V1

∑i

wi(ξi − ξ

)3+

3

V1

∑i

wiε2i

(ξi − ξ

)(1− 2wi

V1

)= 〈k3〉 (96)


〈m4〉 =∏i

∫ ∞−∞

m4(x′)1√

2π εiexp

[− (x′i − ξi)2

2ε2i

]dx′i (97)

=1

V1

∑i

wi(ξ4i + 6ξ2i ε

2i + 3ε4i

)− 4

V 21

∑i

w2i

(ξ4i + 6ξ2i ε

2i + 3ε4i

)+∑i

wi(ξ3i + 3ξiε

2i

)∑j 6=i

wjξj

+

+6

V 31

∑i

w3i

(ξ4i + 6ξ2i ε

2i + 3ε4i

)+∑i

wi(ξ2i + ε2i

)∑j 6=i

w2j

(ξ2j + ε2j

)+

+2∑i

w2i

(ξ3i + 3ξiε

2i

)∑j 6=i

wjξj +∑i

wi(ξ2i + ε2i

)∑j 6=i

wjξj∑k 6=i,j

wkξk

+

− 3

V 41

∑i

w4i

(ξ4i + 6ξ2i ε

2i + 3ε4i

)+ 4

∑i

w3i

(ξ3i + 3ξiε

2i

)∑j 6=i

wjξj+

+ 6∑i

w2i

(ξ2i + ε2i

)∑j 6=i

wjξj∑k 6=i,j

wkξk + 3∑i

w2i

(ξ2i + ε2i

)∑j 6=i

w2j

(ξ2j + ε2j

)+

+∑i

wiξi∑j 6=i

wjξj∑k 6=i,j

wkξk∑l 6=i,j,k

wlξl

(98)

=1

V1

∑i

wiξ4i +

6

V1

∑i

wiξ2i ε

2i +

3

V1

∑i

wiε4i −

4

V 21

∑i

w2i ξ

4i −

24

V 21

∑i

w2i ξ

2i ε

2i −

12

V 21

∑i

w2i ε

4i+

− 4

V 21

∑i

wiξ3i

∑j 6=i

wjξj −12

V 21

∑i

wiξiε2i

∑j 6=i

wjξj +6

V 31

∑i

w3i ξ

4i +

36

V 31

∑i

w3i ξ

2i ε

2i+

+18

V 31

∑i

w3i ε

4i +

6

V 31

∑i

wiξ2i

∑j 6=i

w2j ξ

2j +

6

V 31

∑i

wiξ2i

∑j 6=i

w2j ε

2j +

6

V 31

∑i

wiε2i

∑j 6=i

w2j ξ

2j+

+6

V 31

∑i

wiε2i

∑j 6=i

w2j ε

2j +

12

V 31

∑i

w2i ξ

3i

∑j 6=i

wjξj +36

V 31

∑i

w2i ξiε

2i

∑j 6=i

wjξj+

+6

V 31

∑i

wiξ2i

∑j 6=i

wjξj∑k 6=i,j

wkξk +6

V 31

∑i

wiε2i

∑j 6=i

wjξj∑k 6=i,j

wkξk −3

V 41

∑i

w4i ξ

4i +

− 18

V 41

∑i

w4i ξ

2i ε

2i −

9

V 41

∑i

w4i ε

4i −

12

V 41

∑i

w3i ξ

3i

∑j 6=i

wjξj −36

V 41

∑i

w3i ξiε

2i

∑j 6=i

wjξj+

− 18

V 41

∑i

w2i ξ

2i

∑j 6=i

wjξj∑k 6=i,j

wkξk −18

V 41

∑i

w2i ε

2i

∑j 6=i

wjξj∑k 6=i,j

wkξk+

− 9

V 41

∑i

w2i ξ

2i

∑j 6=i

w2j ξ

2j −

9

V 41

∑i

w2i ξ

2i

∑j 6=i

w2j ε

2j −

9

V 41

∑i

w2i ε

2i

∑j 6=i

w2j ξ

2j+

− 9

V 41

∑i

w2i ε

2i

∑j 6=i

w2j ε

2j −

3

V 41

∑i

wiξi∑j 6=i

wjξj∑k 6=i,j

wkξk∑l 6=i,j,k

wlξl (99)


=1

V1

∑i

wi(ξi − ξ

)4+

6

V1

∑i

wiε2i

(ξi − ξ

)2 − 6

V1ξ2∑i

wiε2i −

12

V 21

∑i

w2i ξ

2i ε

2i+

+3

V1

∑i

wiε4i −

12

V 21

∑i

w2i ε

4i +

36

V 21

ξ∑i

w2i ξiε

2i +

12

V 31

∑i

w3i ε

4i +

6

V 31

∑i

wiε2i

∑j

w2j ε

2j+

+6

V 31

∑i

wiξ2i

∑j

w2j ε

2j +

6

V 31

∑i

wiε2i

∑j

w2j ξ

2j −

12

V 31

∑i

w3i ξ

2i ε

2i+

+6

V 31

V 21 ξ

2∑i

wiε2i −

∑i

wiε2i

∑j

w2j ξ

2j − 2V1ξ

∑i

w2i ξiε

2i + 2

∑i

w3i ξ

2i ε

2i

+

− 18

V 41

∑i

w2i ε

2i

∑j

w2j ξ

2j −

9

V 41

(∑i

w2i ε

2i

)2

− 36

V 31

ξ∑i

w3i ξiε

2i +

36

V 41

∑i

w4i ξ

2i ε

2i+

− 18

V 41

V 21 ξ

2∑i

w2i ε

2i −

∑i

w2i ε

2i

∑j

w2j ξ

2j − 2V1ξ

∑i

w3i ξiε

2i + 2

∑i

w4i ξ

2i ε

2i

(100)

=1

V1

∑i

wi(ξi − ξ

)4+

6

V1

∑i

wiε2i

(ξi − ξ

)2 − 12

V 21

∑i

w2i ξ

2i ε

2i+

+3

V1

∑i

wiε4i

(1− 4wi

V1+

4w2i

V 21

)+

24

V 21

ξ∑i

w2i ξiε

2i +

6

V 31

∑i

wiε2i

∑j

w2j ε

2j+

+6

V 31

∑i

wiξ2i

∑j

w2j ε

2j −

9

V 41

(∑i

w2i ε

2i

)2

− 18

V 21

ξ2∑i

w2i ε

2i (101)

=1

V1

∑i

wi(ξi − ξ

)4+

6

V1

∑i

wiε2i

(ξi − ξ

)2 − 12

V 21

∑i

w2i ε

2i

(ξ2i − 2ξiξ + ξ

2)

+

+3

V1

∑i

wiε4i

(1− 2wi

V1

)2

+6

V 31

∑i

w2i ε

2i

∑j

wjε2j+

+6

V 31

∑i

w2i ε

2i

∑j

wjξ2j − V1ξ

2

− 9

V 41

(∑i

w2i ε

2i

)2

(102)

=1

V1

∑i

wi(ξi − ξ

)4+

6

V1

∑i

wiε2i

(ξi − ξ

)2(1− 2wi

V1

)+

3

V1

∑i

wiε4i

(1− 2wi

V1

)2

+

+6

V 31

∑i

w2i ε

2i

∑j

wj(ξj − ξ

)2+∑j

wjε2j

(1− 3wj

2V1

) (103)


〈m22〉 =

∏i

∫ ∞−∞

m22(x′)

1√2π εi

exp

[− (x′i − ξi)2

2ε2i

]dx′i (104)

=1

V 21

∑i

w2i

(ξ4i + 6ξ2i ε

2i + 3ε4i

)+∑i

wi(ξ2i + ε2i

)∑j 6=i

wj(ξ2j + ε2j

)+

− 2

V 31

∑i

w3i

(ξ4i + 6ξ2i ε

2i + 3ε4i

)+∑i

wi(ξ2i + ε2i

)∑j 6=i

w2j

(ξ2j + ε2j

)+

+2∑i

w2i

(ξ3i + 3ξiε

2i

)∑j 6=i

wjξj +∑i

wi(ξ2i + ε2i

)∑j 6=i

wjξj∑k 6=i,j

wkξk

+

+1

V 41

∑i

w4i

(ξ4i + 6ξ2i ε

2i + 3ε4i

)+ 4

∑i

w3i

(ξ3i + 3ξiε

2i

)∑j 6=i

wjξj+

+ 6∑i

w2i

(ξ2i + ε2i

)∑j 6=i

wjξj∑k 6=i,j

wkξk + 3∑i

w2i

(ξ2i + ε2i

)∑j 6=i

w2j

(ξ2j + ε2j

)+

+∑i

wiξi∑j 6=i

wjξj∑k 6=i,j

wkξk∑l 6=i,j,k

wlξl

(105)

=1

V 21

∑i

w2i ξ

4i +

6

V 21

∑i

w2i ξ

2i ε

2i +

3

V 21

∑i

w2i ε

4i +

1

V 21

∑i

wiξ2i

∑j 6=i

wjξ2j +

1

V 21

∑i

wiξ2i

∑j 6=i

wjε2j+

+1

V 21

∑i

wiε2i

∑j 6=i

wjξ2j +

1

V 21

∑i

wiε2i

∑j 6=i

wjε2j −

2

V 31

∑i

w3i ξ

4i −

12

V 31

∑i

w3i ξ

2i ε

2i −

6

V 31

∑i

w3i ε

4i+

− 2

V 31

∑i

wiξ2i

∑j 6=i

w2j ξ

2j −

2

V 31

∑i

wiξ2i

∑j 6=i

w2j ε

2j −

2

V 31

∑i

wiε2i

∑j 6=i

w2j ξ

2j −

2

V 31

∑i

wiε2i

∑j 6=i

w2j ε

2j+

− 4

V 31

∑i

w2i ξ

3i

∑j 6=i

wjξj −12

V 31

∑i

w2i ξiε

2i

∑j 6=i

wjξj −2

V 31

∑i

wiξ2i

∑j 6=i

wjξj∑k 6=i,j

wkξk+

− 2

V 31

∑i

wiε2i

∑j 6=i

wjξj∑k 6=i,j

wkξk +1

V 41

∑i

w4i ξ

4i +

6

V 41

∑i

w4i ξ

2i ε

2i +

3

V 41

∑i

w4i ε

4i+

+4

V 41

∑i

w3i ξ

3i

∑j 6=i

wjξj +12

V 41

∑i

w3i ξiε

2i

∑j 6=i

wjξj +6

V 41

∑i

w2i ξ

2i

∑j 6=i

wjξj∑k 6=i,j

wkξk+

+6

V 41

∑i

w2i ε

2i

∑j 6=i

wjξj∑k 6=i,j

wkξk +3

V 41

∑i

w2i ξ

2i

∑j 6=i

w2j ξ

2j +

3

V 41

∑i

w2i ξ

2i

∑j 6=i

w2j ε

2j+

+3

V 41

∑i

w2i ε

2i

∑j 6=i

w2j ξ

2j +

3

V 41

∑i

w2i ε

2i

∑j 6=i

w2j ε

2j +

1

V 41

∑i

wiξi∑j 6=i

wjξj∑k 6=i,j

wkξk∑l 6=i,j,k

wlξl

(106)


=

[1

V1

∑i

wi(ξi − ξ

)2]2+

6

V 21

∑i

w2i ξ

2i ε

2i +

3

V 21

∑i

w2i ε

4i +

2

V 21

∑i

wiξ2i

∑j 6=i

wjε2j+

+1

V 21

∑i

wiε2i

∑j 6=i

wjε2j −

12

V 31

∑i

w3i ξ

2i ε

2i −

6

V 31

∑i

w3i ε

4i −

2

V 31

∑i

wiξ2i

∑j 6=i

w2j ε

2j+

− 2

V 31

∑i

wiε2i

∑j 6=i

w2j ξ

2j −

2

V 31

∑i

wiε2i

∑j 6=i

w2j ε

2j −

12

V 31

∑i

w2i ξiε

2i

∑j 6=i

wjξj+

− 2

V 31

V 21 ξ

2∑i

wiε2i −

∑i

wiε2i

∑j

w2j ξ

2j − 2V1ξ

∑i

w2i ξiε

2i + 2

∑i

w3i ξ

2i ε

2i

+

+6

V 41

∑i

w4i ξ

2i ε

2i +

3

V 41

∑i

w4i ε

4i +

12

V 41

∑i

w3i ξiε

2i

∑j 6=i

wjξj+

+6

V 41

V 21 ξ

2∑i

w2i ε

2i −

∑i

w2i ε

2i

∑j

w2j ξ

2j − 2V1ξ

∑i

w3i ξiε

2i + 2

∑i

w4i ξ

2i ε

2i

+

+6

V 41

∑i

w2i ξ

2i

∑j 6=i

w2j ε

2j +

3

V 41

∑i

w2i ε

2i

∑j 6=i

w2j ε

2j (107)

=

[1

V1

∑i

wi(ξi − ξ

)2]2+

4

V 21

∑i

w2i ξ

2i ε

2i +

2

V 21

∑i

w2i ε

4i +

2

V 21

∑i

wiξ2i

∑j

wjε2j+

+1

V 21

(∑i

wiε2i

)2

− 4

V 31

∑i

w3i ε

4i −

2

V 31

∑i

wiξ2i

∑j

w2j ε

2j −

2

V 31

∑i

wiε2i

∑j

w2j ε

2j+

− 8

V 21

ξ∑i

w2i ξiε

2i −

2

V1ξ2∑i

wiε2i +

6

V 21

ξ2∑i

w2i ε

2i +

3

V 41

(∑i

w2i ε

2i

)2

(108)

=

[1

V1

∑i

wi(ξi − ξ

)2]2+

4

V 21

∑i

w2i ε

2i

(ξ2i − 2ξiξ + ξ

2)

+2

V 21

∑i

w2i ε

4i+

+2

V 21

∑i

wi

(ξ2i − ξ

2)∑

j

wjε2j +

1

V 21

(∑i

wiε2i

)2

− 4

V 31

∑i

w3i ε

4i+

− 2

V 31

∑i

wi

(ξ2i − ξ

2)∑

j

w2j ε

2j −

2

V 31

∑i

wiε2i

∑j

w2j ε

2j +

3

V 41

(∑i

w2i ε

2i

)2

(109)

=

[1

V1

∑i

wi(ξi − ξ

)2]2+

4

V 21

∑i

w2i ε

2i

(ξi − ξ

)2+

2

V 21

∑i

wi(ξi − ξ

)2∑j

wjε2j

(1− wj

V1

)+

+2

V 21

∑i

w2i ε

4i

(1− 2wi

V1

)+

1

V 21

[∑i

wiε2i

(1− wi

V1

)]2+

2

V 41

(∑i

w2i ε

2i

)2

(110)

〈k4〉 = 〈m4〉 − 3 〈m22〉 (111)


〈m2(x)〉 −m2(ξ) =1

V1

∑i

wiε2i

(1− wi

V1

)(112)

〈m3(x)〉 −m3(ξ) =3

V1

∑i

wiε2i

(ξi − ξ

)(1− 2wi

V1

)(113)

〈m4(x)〉 −m4(ξ) =6

V1

∑i

wiε2i

(ξi − ξ

)2(1− 2wi

V1

)+

3

V1

∑i

wiε4i

(1− 2wi

V1

)2

+

+6

V 31

∑i

w2i ε

2i

∑j

wj(ξj − ξ

)2+∑j

wjε2j

(1− 3wj

2V1

) (114)

〈m22(x)〉 −m2

2(ξ) =4

V 21

∑i

w2i ε

2i

(ξi − ξ

)2+

2

V 21

∑i

wi(ξi − ξ

)2∑j

wjε2j

(1− wj

V1

)+

+2

V 21

∑i

w2i ε

4i

(1− 2wi

V1

)+

1

V 21

[∑i

wiε2i

(1− wi

V1

)]2+

2

V 41

(∑i

w2i ε

2i

)2

(115)

Since the right-hand side of Eq. (112) does not depend on ξ, f∗(x, ε) = f(ξ, ε) = 〈m2(x)〉 −m2(ξ)and the expression of the noise-unbiased sample variance m∗2 = m2(x)− f∗(x, ε) is found immediately:

m∗2 = m2 −1

V1

∑i

wiε2i

(1− wi

V1

)= k∗2 . (116)

In order to remove the dependence on ξ in Eqs (113)–(115), f∗(x, ε) is derived from f(ξ, ε) such that〈f∗(x, ε)〉 = f(ξ, ε). In the case of skewness, f(ξ, ε) has the following form:

f(ξ, ε) =∑i

ci(ξi − ξ

), (117)

where ci denotes the coefficient of the i-th term. The computation of 〈∑i ci(xi − x) 〉 leads to:

〈∑i

ci(xi − x) 〉 =∑i

ci

∫ ∞−∞

x′i√2π εi

exp

[− (x′i − ξi)2

2ε2i

]dx′i +

− 2

V1

∑i

ci∑j

wj

∫ ∞−∞

x′j√2π εj

exp

[−

(x′j − ξj)2

2ε2j

]dx′j (118)

=∑i

ciξi −1

V1

∑i

ci∑j

wjξj (119)

=∑i

ci(ξi − ξ

). (120)

Since Eq. (120) equals Eq. (117), it follows

f∗(x, ε) =∑i

ci (xi − x) , (121)

and the noise-unbiased sample skewness m∗3 = m3(x)− f∗(x, ε) is

m∗3 = m3 −3

V1

∑i

wiε2i (xi − x)

(1− 2wi

V1

)= k∗3 . (122)


For the kurtosis moment and cumulant, f(ξ, ε) involves ξ-dependent terms of the form∑i ci(ξi − ξ

)2.

The computation of 〈∑i ci(xi − x)2〉 leads to:

〈∑i

ci(xi − x)2〉 =∑i

ci

∫ ∞−∞

x′i2

√2π εi

exp

[− (x′i − ξi)2

2ε2i

]dx′i +

− 2

V1

∑i

ciwi

∫ ∞−∞

x′i2

√2π εi

exp

[− (x′i − ξi)2

2ε2i

]dx′i+

− 2

V1

∑i

ci∑j 6=i

wj

∫ ∞−∞

x′i x′j

2π εiεjexp

[− (x′i − ξi)2

2ε2i−

(x′j − ξj)2

2ε2j

]dx′i dx′j+

+1

V 21

∑i

ci∑j

w2j

∫ ∞−∞

x′j2

√2π εj

exp

[−

(x′j − ξj)2

2ε2j

]dx′j+

+1

V 21

∑i

ci∑j

wj∑k 6=j

wk

∫ ∞−∞

x′j x′k

2π εjεkexp

[−

(x′j − ξj)2

2ε2j− (x′k − ξk)2

2ε2k

]dx′j dx′k

(123)

=∑i

ci(ξ2i + ε2i

)− 2

V1

∑i

ciwi(ξ2i + ε2i

)− 2

V1

∑i

ciξi∑j 6=i

wjξj+

+1

V 21

∑i

ci∑j

w2j

(ξ2j + ε2j

)+

1

V 21

∑i

ci∑j

wjξj∑k 6=j

wkξk (124)

=∑i

ciξ2i +

∑i

ciε2i − 2 ξ

∑i

ciξi −2

V1

∑i

ciwiε2i+

+1

V 21

∑i

ci∑j

w2j ε

2j + ξ

2∑i

ci (125)

=∑i

ci(ξi − ξ

)2+∑i

ciε2i

(1− 2wi

V1

)+

1

V 21

∑i

ci∑j

w2j ε

2j . (126)

Thus, each of the terms of the form∑i ci(ξi − ξ

)2in Eqs (114)–(115) can be replaced by the expression

∑i

ci

(xi − x)2 − ε2i

(1− 2wi

V1

)− 1

V 21

∑j

w2j ε

2j

, (127)


and the noise-unbiased sample kurtosis moment m∗4 and cumulant k∗4 are found as follows:

m∗4 = m4 −6

V1

∑i

wiε2i

(xi − x)2

(1− 2wi

V1

)− ε2i

(1− 2wi

V1

)2

− 1

V 21

(1− 2wi

V1

)∑j

w2j ε

2j

+

− 3

V1

∑i

wiε4i

(1− 2wi

V1

)2

− 6

V 31

∑i

w2i ε

2i

∑j

wj (xj − x)2 −

∑j

wjε2j

(1− 2wj

V1

)+

− 1

V1

∑j

w2j ε

2j +

∑j

wjε2j

(1− 3wj

2V1

) (128)

= m4 −6

V1

∑i

wiε2i (xi − x)

2

(1− 2wi

V1

)+

3

V1

∑i

wiε4i

(1− 2wi

V1

)2

+

− 6

V 31

∑i

w2i ε

2i

∑j

wj (xj − x)2 −

∑j

wjε2j

(1− wj

V1

)− 3

V 41

(∑i

w2i ε

2i

)2

(129)

= m4 −6

V1

∑i

wiε2i

[(xi − x)

2

(1− 2wi

V1

)− ε2i

2

(1− 2wi

V1

)2

+m∗2wiV1

]− 3

V 41

(∑i

w2i ε

2i

)2

(130)

(m22)∗ = m2

2 −4

V 21

∑i

w2i ε

2i

(xi − x)2 − ε2i

(1− 2wi

V1

)− 1

V 21

∑j

w2j ε

2j

+

− 2

V 21

∑i

wi

(xi − x)2 − ε2i

(1− 2wi

V1

)− 1

V 21

∑j

w2j ε

2j

∑j

wjε2j

(1− wj

V1

)+

− 2

V 21

∑i

w2i ε

4i

(1− 2wi

V1

)− 1

V 21

[∑i

wiε2i

(1− wi

V1

)]2− 2

V 41

(∑i

w2i ε

2i

)2

(131)

= m22 −

2

V1m2

∑i

wiε2i

(1− wi

V1

)+

1

V 21

[∑i

wiε2i

(1− wi

V1

)]2+

− 4

V 21

∑i

w2i ε

2i

[(xi − x)

2 − ε2i2

(1− 2wi

V1

)]− 2

V 21

(∑i

wiε2i

)2

+4

V 31

∑i

wiε2i

∑j

w2j ε

2j+

+2

V 21

∑i

wi

ε2i (1− 2wiV1

)+

1

V 21

∑j

w2j ε

2j

∑j

wjε2j −

1

V1

∑j

w2j ε

2j

(132)

= (m∗2)2 − 4

V 21

∑i

w2i ε

2i

[(xi − x)

2 − ε2i2

(1− 2wi

V1

)]− 2

V 21

(∑i

wiε2i

)2

+

+4

V 31

∑i

wiε2i

∑j

w2j ε

2j +

2

V 21

(∑i

wiε2i

)2

− 4

V 31

∑i

w2i ε

2i

∑j

wjε2j −

2

V 31

∑i

wiε2i

∑j

w2j ε

2j+

+4

V 41

(∑i

w2i ε

2i

)2

+2

V 31

∑i

w2i ε

2i

∑j

wjε2j −

2

V 41

(∑i

w2i ε

2i

)2

(133)

= (m∗2)2 − 4

V 21

∑i

w2i ε

2i

[(xi − x)

2 − ε2i2

(1− 2wi

V1

)]+

2

V 41

(∑i

w2i ε

2i

)2

(134)

k∗4 = m∗4 − 3 (m22)∗. (135)


Skewness and kurtosis unbiased by Gaussian … and kurtosis unbiased by Gaussian uncertainties...

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Transcript of Skewness and kurtosis unbiased by Gaussian … and kurtosis unbiased by Gaussian uncertainties...