Фактор ошибки логнормального распределения

A log-normal distribution is a continuous distribution of random variable y whose natural logarithm is normally distributed. For example, if random variable y=exp{y} has log-normal distribution then x=log(y) has normal distribution. Log-normal distributions are most often used in finance to model stock prices, index values, asset returns, as well as exchange rates, derivatives, etc.

Log-Normal Distribution Graph

The log-normal distribution has been used quite extensively in analyzing lifetime data. If X has a normal distribution then e^X has a log-normal distribution. Therefore, a log-normal distribution with the scale parameter 0<λ<∞ and the shape parameter σ > 0 has the following CDF:

F(t;λ,σ)=0ift<0Φln(t)−ln(λ)σift≥0.

The corresponding PDF and hazard function become

f(t;λ,σ)=0ift<01σtϕln(t)−ln(λ)σift≥0,

and

h(t;λ,σ)=ϕln(t)−ln(λ)σσtΦ−ln(t)+ln(λ)σ;t>0,

respectively. The PDF and the hazard function of a log-normal distribution are always unimodal functions. The PDF of a log-normal distribution is very similar to the PDFs of gamma, Weibull or generalized exponential distributions when the shape parameters of gamma, Weibull and generalized exponential distributions are greater than one. It has been shown by Kundu and Manglick [85, 86] and Kundu et al. [87] that it is very difficult to discriminate between log-normal and gamma, log-normal and Weibull and log-normal and generalized exponential distributions. For different properties of a log-normal distribution and for its various applications, one is referred to Johnson et al. [59].

Alhadeed [88] considered in his PhD thesis the analysis of the log-normal step-stress model, see also Alhadeed and Yang [34], when the complete data are available. Balakrishnan et al. [55] considered the same problem when the data are Type-I censored. It is assumed that the lifetime distribution of the experimental units at the two different stress levels follow log-normal distributions with different scale parameters, λ₁ and λ₂, but the same shape parameter σ. Based on the CEM assumption, the CDF of the lifetime of an experimental unit from a simple step-stress model can be written as

(2.40)F(t)=0ift<0Φln(t)−ln(λ1)σif0≤t<τ1Φlnt+τ1λ2λ1−τ1−ln(λ2)σifτ1≤t<∞.

Hence, the PDF corresponding to Eq. (2.40) becomes

(2.41)f(t)=0ift<01σtϕln(t)−ln(λ1)σif0≤t<τ11σt+λ2λ1τ1−τ1ϕlnt+τ1λ2λ1−τ1−ln(λ2)σifτ1≤t<∞.

In this case it is more convenient to work with the log-transformation of the data than the original data. Now if a random variable T has the PDF (2.41), then Y=ln(T) has the PDF

fY(y)=0ift<01σϕy−μ1σif0<t<lnτ1eyσey+eμ2−μ1τ1−τ1ϕlney+τ1eμ2−μ1−τ1−μ2σiflnτ1≤y<∞.

Here μ1=lnλ1 and μ2=lnλ2. Therefore, if we denote the log of the observed lifetimes as yi:n=ln(ti:n) for i = 1, …, n, then the log-likelihood function based on the complete observations {y_1:n, …, y_n:n} is

(2.42)l(μ1,μ2,σ)=−n2ln(π)−nlnσ−12∑i=1n1yi:n−μ1σ2−∑i=n1+1nln(eyi:n+τ1eμ2−μ1−τ1)−12∑i=n1+1nln(eyi:n+τ1eμ2−μ1−τ1)−μ2σ2.

Here it is assumed that 1 ≤ n₁ ≤ n − 1 and n ≥ 3; otherwise it is known that the MLEs of σ, μ₁, and μ₂ do not exist. Therefore, the conditional MLEs of the unknown parameters conditioning on 1 ≤ N₁ ≤ n − 1 can be obtained by maximizing Eq. (2.42) with respect to the unknown parameters. In this case the normal equations become

(2.43)l.μ1=∑i=n1+1nτ1eμ2−μ1eyi:n+τ1eμ2−μ1−τ1+1σ2∑i=1n1(yi:n−μ1)+1σ2∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)τ1eμ2−μ1eyi:n+τ1eμ2−μ1=0,

(2.44)l.μ2=−1σ2∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)τ1eμ2−μ1eyi:n+τ1eμ2−μ1−1−∑i=n1+1nτ1eμ2−μ1eyi:n+τ1eμ2−μ1−τ1=0,

(2.45)l.σ=−nσ+1σ3∑i=1n1(yi:n−μ1)2+1σ3∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)2=0.

Clearly, Eqs. (2.43)–(2.45) cannot be solved explicitly. One needs to use the Newton-Raphson type iterative algorithm to solve Eqs. (2.43)–(2.45) numerically. Some initial guesses of the parameters are needed to start the iteration. If μ₁ and μ₂ are known, the MLE of σ² can be obtained from Eq. (2.45) as

(2.46)σ^2(μ1,μ2)=1n∑i=1n1(yi:n−μ1)2+∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)2.

We can obtain the profile log-likelihood function of μ₁ and μ₂ by using Eq. (2.46) in Eq. (2.42). The profile log-likelihood function of μ₁ and μ₂ without the additive constants can be written as

(2.47)p(μ1,μ2)=−n2ln∑i=1n1(yi:n−μ1)2+∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)2−∑i=n1+1nln(eyi:n+τ1eμ2−μ1−τ1).

A contour plot of p(μ₁, μ₂) as in Eq. (2.47) may provide good starting values of μ₁ and μ₂. Once we obtain the starting values of μ₁ and μ₂, the starting value of σ can be easily obtained from Eq. (2.46). Although we have presented the results here for the complete sample, similar results can be developed for different censoring schemes. Balakrishnan et al. [55] performed an extensive simulation study to compare the performances of different confidence intervals. It is observed that the biased corrected bootstrap method works very well in this case. Most of the results have been extended by Lin and Chou [56] for the multiple step-stress model.

The use of log-normal distributions for approximating the PSD has important advantages:

(1)

there is no need to “break” the power-law approximation to reflect changes in the log-log slope of the size distribution;

(2)

the log-normal approximation assumes finite values for all particle diameters, except in the limit B₂ > 0, when the log-normal function becomes a power-law function;

(3)

mass and area distributions resulting from the log-normal distribution are also log-normal (Kerker 1969);

(4)

the power-law function is a limiting case of the log-normal function, so it is naturally included as an approximation of the size distribution.

We should also mention certain pitfalls of choosing the log-normal distribution after Halley and Inchausti (2002), who cite comments by Mandelbrot (1997) (incidentally these problems also relate to other “long-tailed” distributions):

(1)

extreme sensitivity of all moments of the distribution to even small departures from log-normality, which makes the calculation of moments from the parameters of a fitted log-normal distribution unreliable

(2)

slow convergence of the approximate values of the moments to their asymptotic value. This feature is due to an extremely “long tail” of the log-normal distribution at the large-size end of the particle size scale. We have experienced this problem ourselves (Jonasz and Fournier 1996) when trying to evaluate correlations between the total particle surface and volume and the peak diameter of the distribution as fitted to several hundred size distributions of marine particles.

The log-normal size distribution of the zero-th order can be expressed as follows (e.g., Ross 1978, Casperson 1977):

(5.169)n(D)=nmaxexp[−(lnD−lnDpeak)22σ2]

where n_max is expressed as follows:

(5.170)nmax=Ntot2πσDpeakexpσ22

N_tot is the total number of particles [equal to unity in the case of n(D) being the probability distribution], D_peak is the particle diameter corresponding to the peak of the size distribution, and σ is the standard deviation of lnD, i.e., the geometric standard deviation of D.

The width parameter, σ, is related to the ratio of the maximum and minimum diameters, D_max and D_min, of the full-width-at-half-maximum of the log-normal n(D) through the following equation (Jonasz and Fournier 1996):

(5.171)DmaxDmin=exp(2fσ)

where

(5.172)f=2ln2Dmin=Dpeakexp(−fσ)Dmin=Dpeakexp(fσ)

In contrast to the power-law distribution, the log-normal size distribution yields a finite total number of particles, N_tot, total particle cross-sectional area, A_tot, and volume, V_tot (Heintzenberg 1994):

(5.173)Atot=π[12Dpeakexp(2σ2)]2Ntot×exp{−12σ2[(lnDmaxexp(2σ2))−lnDmax]2}

and

(5.174)Vtot=43π[12Dpeakexp(3σ2)]3Ntot×exp{−12σ2[(lnDmaxexp(3σ2))−lnDmax]2}

The cell size distributions reported in the literature frequently appear to be log-normal at visual inspection (see Table A.5 for sources of the relevant PSD data). Interestingly, mathematical models of cell growth and evolution of isolated cell populations do not lead to a log-normal distribution of cell sizes (e.g., Tyson and Hannsgen 1985). However, the agreement between the models and the experimental data is questionable. In fact, Tyson and Hannsgen note that cell size distributions with log-normal size distribution have been reported (ibid. Scherbaum and Rasch 1957, Collins and Richmond 1962). Analysis of the biomass spectrum in aquatic ecosystems composed of organism groups linked via a prey–predator relationship led to the log-normal function as a natural descriptor of the contribution of an organism to the total ecosystem biomas spectrum (Thiebaux and Dickie 1993, Boudreau et al. 1991). The derivation of the log-normal form of the size distributions of the individual organism groups was based on the fact that the production, P (w), is proportional to a power, b, of the body mass, W (allometric relationship):

(5.175)P(W)=aWb

The observation of (1) a persistent curvature of the size distribution of marine particles, when plotted in a log-log scale, (2) the multimodal appearance of many such size distributions, as well as (3) previous suggestions in the literature that complex size distributions of geological material can be well modeled by a sum of log-normal functions (van Andel 1973) led us to develop an automated algorithm of the decomposition of a marine PSD into a sum of log-normal components (Jonasz and Fournier 1999, 1996). In that work, we postulated that the size distribution of marine particles is essentially a linear combination of a cascade of log-normal components, according to the following equation:

(5.176)n(D)=∑1kmaxnk(D)

where index k numbers the log-normal components n_k(D). Each of these components is approximated with a zero-th order log-normal distribution function:

(5.177)nk(D)=nmax,kexp[−(lnD−lnDpeak,k)22σk2]

where n_{max, k} is the maximum value of the component, D_peak,k [μm] is the peak diameter, and σ_k is the width parameter. By taking the logarithm of both sides of (5.177) and performing simple algebraic transformations, one obtains (we omit the component index for simplicity):

(5.178)logn(D)=B0+B1logD+B2(logD)2

where:

(5.179)B0=lognmax−(logDpeak)2ln102σ2B1=logDpeakln10σ2B2=−ln102σ2

Note that (5.178) reduces to a power law when B₂ vanishes. Equations (5.179) can be solved for n_max, D_peak, and σ to yield:

(5.180)lognmax=B0−B124B2logDpeak=−B12Bσ2=−ln102B2

Sample results are shown in Figure 5.34 and Figure 5.35. Log-normal components identified by the algorithm just described range in shape from the power-law-like function to Gaussian-normal-like function. Note that the failure to account for the instrumental error (see section 5.7.1.6) in evaluating the γ² results in the γ² value in excess of 200 per degree of freedom (!) in the case of data shown in Figure 5.35. that include very high particle count values.

Figure 5.34. A log-normal approximation of the size distribution of Figure 5.31: log n (D) = 4.41 − 2.50 log D − 1.02 (log D)² (γ² per degree of freedom = 0.24). The fit parameters were obtained via the logarithmic transform. All weights were set to unity. The γ² was calculated by assuming only the counting error.

Figure 5.35. A multi-component log-normal approximation (thick black curve) of a PSD measured in the Northwest Atlantic waters (•, unpublished data: courtesy of K. Kranck and T. Milligan, file KRAATL86.P08 in Jonasz 1992). The approximation coefficients from equation (5.178) : first component (thin solid curve) B₀₁ = 4.05, B₁₁ = −1.16, B₂₁ = −2.12, second component (dashed curve) B₀₂ = −17.04, B₁₂ = 22.52, B₂₂ = −7.75, third component (gray solid curve) B₀₃ = 4.97, B₁₃ = 7.04, B₂₃ = 7.30, fourth component (gray dashed curve) B₀₄ = −4.27, B₁₄ = −9.60, B₂₄ = −4.81. The first log-normal component removes the greatest amount of the approximation error. Each of the following components removes a progressively smaller amount of that error (χ² per degree of freedom = 0. 021). The fit parameters were obtained via the logarithmic transform. All fitting weights were set to unity. The χ² was calculated by assuming both the counting and instrumental errors.

The statistics of and correlations between the parameters of 853 log-normal components of the 412 PSDs determined using the Coulter technique by different researchers in different areas and seasons are shown in Table 5.10. The average values of coefficients B₀, B₁, and B₂ represent a geometrically averaged component characteristics, i.e., n_avg(D) such that n_avg(D) = [n₁(D) n₂(D) … n_m(D)]^1/m. The average values of n_max, D_peak, and σ do not have simple meanings and are given here for the sake of completeness only. The two sets of averages do not yield the same function of the diameter, D, because they are related via non-linear functions (5.179) and (5.180). The average error of approximation of the size distribution with the sum of log-normal components was 0.057±0.030. The number of components per size distribution varied from 1 to 6, with an average of 2.18 ± 1.22. The value of 1 standard deviation (SD) is shown following the ± sign.

(5.181)lnnmax=(8.070±2.799)−(2.446±0.032)lnDpeak

and

(5.182)σ=(0.626±0.186)−(0.111±0.002)lnDpeak

where the value of 1 SD of the respective parameter is shown following each ± sign.

The log-normal components, which range in shape from the power-law-like function to Gaussian-normal-like function, may be interpreted as the size distributions of the various classes of marine particles, for example, populations of various phytoplankton species. Indeed, Jonasz and Fournier (1996) found two “standard” components (Figure 5.38 and Table 5.11) in 412 size distributions measured in various seasons and regions of the world ocean by different researchers. Bradtke (2004) who used the algorithm just described to analyze 970 PSDs measured with a Coulter counter in the coastal waters of the Baltic Sea (Gdansk Bay) also noted the existence of several “standard” components and linked other transitional components to the occurrences of phytoplankton species.

Figure 5.38. “Standard” components (Table 5.11) of the marine particle size distribution identified by Jonasz and Fournier (1996), who analyzed 412 particle size distributions measured in various seasons and regions of the world ocean by different researchers (as compiled by Jonasz 1992): first component (solid black curve) B₀₁ = 4.038, B₁₁ = −0.9511, B₂₁ = −2.542, second component (dashed curve) B₀₂ = −8.447, B₁₂ = −7.55, B₂₂ = −8.595, compared with the size distribution from Figure 5.35 (symbols) and its retrieved components (gray solid lines).

The classical inferential issues of the TRVM under the assumption that T has a log-normal distribution with the PDF

fT(t)=1σt2πe−12lnt−μσ21(0,∞)(t),

where μ∈R and σ > 0, was addressed by Bai et al. [115]. The authors also assumed that the data are Type-I censored. Under the log-normal distribution, the PDF of T~ is given by

fT~(t)=1σt2πe−12lnt−μσ2if0<t≤τ1βστ1+β(t−τ1)2πe−12ln(τ1+β(t−τ1))−μσ2ift>τ10otherwise.

Given a Type-I censored data t1:n<⋯<tn1:n<τ1<tn1+1:n<⋯<tn1+n2:n<η, the log-likelihood function can be written as

l(μ,σ2,β)=−n1+n22lnσ2−∑i=n1+1n1+n2lnτ1+β(ti:n−τ1)−12σ2∑i=1n1lnti:n−μ2−12σ2∑i=n1+1n1+n2ln(τ1+β(ti:n−τ1))−μ2+(n−n1−n2)ln1−Φln(τ1+β(η−τ1))−μσ,

where Φ(⋅) is the CDF of the standard normal distribution. The MLEs of μ, σ², and β can be found by maximizing the log-likelihood function with respect to the parameters. Note that in this case the MLEs of the unknown parameters do not exist in explicit form and one needs to use a numerical technique to maximize the log-likelihood function. The asymptotic variance of the MLEs and the confidence intervals of the unknown parameters can be obtained using the observed Fisher information matrix.

Pore size and PSD of each membrane were determined from AFM-determined topography. The lognormal distribution was chosen to represent the pore size data for each of the membranes. This was found to give a good fit to the PSD. All of these distributions were fitted to lognormal distributions given by frequencies (%f):

(5.1)%f=%fmaxexp[-12σ2ln(dpX0)2]

where d_p is the measured pore size, σ the standard deviation of the measurements, %f_max the maximum frequency and χ₀ the modal value of d_p.

Mean pore sizes and PSDs of initial membranes determined from AFM images are shown in Table 5.3. This table shows that with a mean pore size of 0.535±0.082 μm, the PVDF membrane has slightly larger pores than the PES membrane (mean pore size 0.470±0.188 μm). PSDs are detailed in Figure 5.3, along with fits described by equation (5.1). The measured pore sizes are larger than the nominal pore size of 0.22 μm as specified by the manufacturers. The AFM data confirm that the pore sizes of studied membranes are of approximately the same size. The topographical images give a clear perception of a notable difference in the surface morphology of the membranes used for the modification. A quantification of the surface parameters (Table 5.3) provides an insight into morphological particularities of these membranes which influence both the membrane separating properties and the process of modification by graft copolymerisation.

Before detailed discussion of the quantitative characteristics of surface morphology, it is worth noting that the chosen lognormal pattern for PDS described by equation (5.1) gave a correlation coefficient of at least 0.95 for all fitted curves. The most probable pore sizes estimated from fitted curves were very close to the mean pore diameter calculated from corresponding sets of pore sizes for the initial and modified membranes (Tables 5.4 and 5.5).

According to Figure 5.5(a), the initial PES membrane has a very wide PSD with a σ value of 0.56 μm. However, grafting of poly-qDMAEMA resulted in narrowing of the PSD and shifting the whole curve towards smaller pore sizes. As a result, mean pore size is gradually decreasing with the increase in the amount of poly-qDMAEMA grafted to the membrane surface. Significant improvement of the PSD was observed even for the modified PES membranes with the smallest DM (Table 5.4).

Figure 5.5. PSDs of initial (a) and modified with qDMAEMA (b)–(d) PES membranes; (b) DM = 202 μg cm⁻², (c) DM = 367 μg cm⁻² and (d) DM = 510 μg cm⁻².

Solution: Robust estimates μ^M, variances Dμ^M and the limits of the 95% confidence interval of the mean are listed in Table 3.6. For the symmetric distributions N, R and L the robust analysis gives accurate estimates quite near to the true values, and the confidence interval is narrow. Worse results were achieved with the asymmetric skewed distributions: for the exponential and log-normal distributions the 95% confidence interval does not contain theoretical value μ.

If a random variable V has a normal distribution with mean μ and variance σ², then e^V has a lognormal distribution with parameters μ and σ². In other words, if a variable has a lognormal distribution, then its logarithm has a normal distribution. The pdf of the distribution is given by

fx;μ,σ2=1xσ2πe−12logx−μσ2

where x and σ are both positive. If X follows a lognormal distribution with parameters μ and σ², then Y = e^a X^b follows a lognormal distribution with parameters a + bμ and b²σ².

The expectation and variance of the lognormal distribution are given by

Ex=eμ+σ22,Vx=e2μ+σ2eσ2−1.

Using its relationships to the normal distribution, the parameters of the lognormal distribution from a sample x₁, x₂,…, x_n of draws from the distribution can be estimated as

μˆ=1n∑log(x1),σˆ=1n−1∑(logxi−μˆ)2.

The normal distribution with parameters μ and σ² > 0 is given by the familiar bell-shaped probability density function

(1.32)ϕ(x;μ,σ2)=12πσe−(x−μ)2/2σ2     −∞<x<∞.

The density function is symmetric about the point μ, and the parameter σ² is the variance of the distribution. The case μ = 0 and σ² = 1 is referred to as the standard normal distribution. If X is normally distributed with mean μ and variance σ², then Z = (X − μ)/σ has a standard normal distribution. By this means, probability statements about arbitrary normal random variables can be reduced to equivalent statements about standard normal random variables. The standard normal density and distribution functions are given respectively by

(1.33)φ(ξ)=12πe−ξ2/2,    −∞<ξ<∞,

and

(1.34)Φ(x)=∫−∞xφ(ξ)dξ,       −∞<x<∞.

The central limit theorem explains in part the wide prevalence of the normal distribution in nature. A simple form of this aptly named result concerns the partial sums S_n = ξ₁ + · ·· + ξ_n of independent and identically distributed summands ξ₁, ξ_2…. having finite means μ = E[ξ_k] and finite variances σ² = Var[ξ_k]. In this case, the central limit theorem asserts that

(1.35)limn→∞ Pr{Sn−nμσn≤x}=Φ(x)    for all x.

Suppose that a sample of 10 grains of metallic sand taken from a large sand pile have respective lengths (in millimeters):

2.2, 3.4, 1.6, 0.8, 2.7, 3.3, 1.6, 2.8, 2.5, 1.9

It helps to first describe problems associated with Student’s t. When testing hypotheses or computing confidence intervals for μ, it is assumed that

(4.1)T=n(X¯−μ)s

has a Student’s t-distribution with ν = n − 1 degrees of freedom. This implies that E (T) = 0, and that T has a symmetric distribution. From basic principles, this assumption is correct when observations are randomly sampled from a normal distribution. However, at least three practical problems can arise. First, there are problems with power and the length of the confidence interval. As indicated in Chapters 1 and 2Chapter 1Chapter 2, the standard error of the sample mean, σ/n, becomes inflated when sampling from a heavy-tailed distribution, so power can be poor relative to methods based on other measures of location, and the length of confidence intervals, based on Eq. (4.1), become relatively long – even when σ is known. (For a detailed analysis of how heavy-tailed distributions affect the probability coverage of the t-test, see Benjamini, 1983.) Second, the actual probability of a type I error can be substantially higher or lower than the nominal α level. When sampling from a symmetric distribution, generally the actual level of Student’s t-test will be less than the nominal level (Efron, 1969). When sampling from a symmetric, heavy-tailed distribution the actual probability of type I error can be substantially lower than the nominal α level, and this further contributes to low power and relatively long confidence intervals. From theoretical results reported by Basu and DasGupta (1995), problems with low power can arise even when n is large. When sampling from a skewed distribution with relatively light tails, the actual probability coverage can be substantially less than the nominal 1 − α level resulting in inaccurate conclusions and this problem becomes exacerbated as we move toward (skewed) heavy-tailed distributions. Third, when sampling from a skewed distribution, T also has a skewed distribution, it is no longer true that E (T) = 0, and the distribution of T can deviate enough from a Student’s t-distribution so that practical problems arise. These problems can be ignored if the sample size is sufficiently large, but given data it is difficult knowing just how large n has to be. When sampling from a lognormal distribution, it is known that n > 160 is required (Westfall & Young, 1993). As we move away from the lognormal distribution toward skewed distributions where outliers are more common, n > 300 might be required. Problems with controlling the probability of a type I error are particularly serious when testing one-sided hypotheses. And this has practical implications when testing two-sided hypotheses because it means that a biased hypothesis testing method is being used, as will be illustrated.

Problems with low power were illustrated in Chapter 1, so further comments are omitted. The second problem, that probability coverage and type I error probabilities are affected by departures from normality, is illustrated with a class of distributions obtained by transforming a standard normal distribution in a particular way. Suppose Z has a standard normal distribution, and for some constant h ≥ 0, let

X=Zexph Z22.

Then X has what is called an h distribution. When h = 0, X = Z, so X is standard normal. As h gets large, the tails of the distribution of X get heavier, and the distribution is symmetric about 0. (More details about the h distribution are described in Section 4.2)

For convenience, assume Y is standard normal in which case E(X)=e, where e = exp(1) ≈ 2.71828, and the standard deviation is approximately σ = 2.16. Then Eq. (4.1) assumes that T=n(X¯−e)/s has a Student’s t-distribution with n − 1 degrees of freedom. The left panel of Figure 4.1 shows a (kernel density) estimate of the actual distribution of T when n = 20; the symmetric distribution is the distribution of T under normality. As is evident, the actual distribution is skewed to the left, and its mean is not equal to 0. Simulations indicate that E (T) = −0.54, approximately. The right panel shows an estimate of the probability density function when n = 100. The distribution is more symmetric compared to n = 20, but it is clearly skewed to the left.

Figure 4.1. Nonnormality can seriously affect Students t. The left panel shows an approximation of the actual distribution of Students t when sampling from a lognormal distribution and n = 20 and the right panel is when n = 100.

Generally, as we move toward a skewed distribution with heavy tails, the problems illustrated by Figure 4.1 become exacerbated. As an example, suppose sampling is from a squared lognormal distribution that has mean exp(2). (i.e., if X has a lognormal distribution, E(X²) = exp(2).) Figure 4.2 shows plots of T values based on sample sizes of 20 and 100. (Again, the symmetric distributions are the distributions of T under normality.)

Figure 4.2. The same as Figure 4.1, only now sampling is from a squared lognormal distribution. This illustrates that as we move toward heavy-tailed distributions, problems with nonnormality are exacerbated.

A log-normal distribution is a continuous distribution of random variable y whose natural logarithm is normally distributed. For example, if random variable y=exp{y} has log-normal distribution then x=log(y) has normal distribution. Log-normal distributions are most often used in finance to model stock prices, index values, asset returns, as well as exchange rates, derivatives, etc.

Log-Normal Distribution Graph

The log-normal distribution has been used quite extensively in analyzing lifetime data. If X has a normal distribution then e^X has a log-normal distribution. Therefore, a log-normal distribution with the scale parameter 0<λ<∞ and the shape parameter σ > 0 has the following CDF:

F(t;λ,σ)=0ift<0Φln(t)−ln(λ)σift≥0.

The corresponding PDF and hazard function become

f(t;λ,σ)=0ift<01σtϕln(t)−ln(λ)σift≥0,

and

h(t;λ,σ)=ϕln(t)−ln(λ)σσtΦ−ln(t)+ln(λ)σ;t>0,

respectively. The PDF and the hazard function of a log-normal distribution are always unimodal functions. The PDF of a log-normal distribution is very similar to the PDFs of gamma, Weibull or generalized exponential distributions when the shape parameters of gamma, Weibull and generalized exponential distributions are greater than one. It has been shown by Kundu and Manglick [85, 86] and Kundu et al. [87] that it is very difficult to discriminate between log-normal and gamma, log-normal and Weibull and log-normal and generalized exponential distributions. For different properties of a log-normal distribution and for its various applications, one is referred to Johnson et al. [59].

Alhadeed [88] considered in his PhD thesis the analysis of the log-normal step-stress model, see also Alhadeed and Yang [34], when the complete data are available. Balakrishnan et al. [55] considered the same problem when the data are Type-I censored. It is assumed that the lifetime distribution of the experimental units at the two different stress levels follow log-normal distributions with different scale parameters, λ₁ and λ₂, but the same shape parameter σ. Based on the CEM assumption, the CDF of the lifetime of an experimental unit from a simple step-stress model can be written as

(2.40)F(t)=0ift<0Φln(t)−ln(λ1)σif0≤t<τ1Φlnt+τ1λ2λ1−τ1−ln(λ2)σifτ1≤t<∞.

Hence, the PDF corresponding to Eq. (2.40) becomes

(2.41)f(t)=0ift<01σtϕln(t)−ln(λ1)σif0≤t<τ11σt+λ2λ1τ1−τ1ϕlnt+τ1λ2λ1−τ1−ln(λ2)σifτ1≤t<∞.

In this case it is more convenient to work with the log-transformation of the data than the original data. Now if a random variable T has the PDF (2.41), then Y=ln(T) has the PDF

fY(y)=0ift<01σϕy−μ1σif0<t<lnτ1eyσey+eμ2−μ1τ1−τ1ϕlney+τ1eμ2−μ1−τ1−μ2σiflnτ1≤y<∞.

Here μ1=lnλ1 and μ2=lnλ2. Therefore, if we denote the log of the observed lifetimes as yi:n=ln(ti:n) for i = 1, …, n, then the log-likelihood function based on the complete observations {y_1:n, …, y_n:n} is

(2.42)l(μ1,μ2,σ)=−n2ln(π)−nlnσ−12∑i=1n1yi:n−μ1σ2−∑i=n1+1nln(eyi:n+τ1eμ2−μ1−τ1)−12∑i=n1+1nln(eyi:n+τ1eμ2−μ1−τ1)−μ2σ2.

Here it is assumed that 1 ≤ n₁ ≤ n − 1 and n ≥ 3; otherwise it is known that the MLEs of σ, μ₁, and μ₂ do not exist. Therefore, the conditional MLEs of the unknown parameters conditioning on 1 ≤ N₁ ≤ n − 1 can be obtained by maximizing Eq. (2.42) with respect to the unknown parameters. In this case the normal equations become

(2.43)l.μ1=∑i=n1+1nτ1eμ2−μ1eyi:n+τ1eμ2−μ1−τ1+1σ2∑i=1n1(yi:n−μ1)+1σ2∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)τ1eμ2−μ1eyi:n+τ1eμ2−μ1=0,

(2.44)l.μ2=−1σ2∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)τ1eμ2−μ1eyi:n+τ1eμ2−μ1−1−∑i=n1+1nτ1eμ2−μ1eyi:n+τ1eμ2−μ1−τ1=0,

(2.45)l.σ=−nσ+1σ3∑i=1n1(yi:n−μ1)2+1σ3∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)2=0.

Clearly, Eqs. (2.43)–(2.45) cannot be solved explicitly. One needs to use the Newton-Raphson type iterative algorithm to solve Eqs. (2.43)–(2.45) numerically. Some initial guesses of the parameters are needed to start the iteration. If μ₁ and μ₂ are known, the MLE of σ² can be obtained from Eq. (2.45) as

(2.46)σ^2(μ1,μ2)=1n∑i=1n1(yi:n−μ1)2+∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)2.

We can obtain the profile log-likelihood function of μ₁ and μ₂ by using Eq. (2.46) in Eq. (2.42). The profile log-likelihood function of μ₁ and μ₂ without the additive constants can be written as

(2.47)p(μ1,μ2)=−n2ln∑i=1n1(yi:n−μ1)2+∑i=n1+1n(ln(eyi:n+τ1eμ2−μ1−τ1)−μ2)2−∑i=n1+1nln(eyi:n+τ1eμ2−μ1−τ1).

A contour plot of p(μ₁, μ₂) as in Eq. (2.47) may provide good starting values of μ₁ and μ₂. Once we obtain the starting values of μ₁ and μ₂, the starting value of σ can be easily obtained from Eq. (2.46). Although we have presented the results here for the complete sample, similar results can be developed for different censoring schemes. Balakrishnan et al. [55] performed an extensive simulation study to compare the performances of different confidence intervals. It is observed that the biased corrected bootstrap method works very well in this case. Most of the results have been extended by Lin and Chou [56] for the multiple step-stress model.

The use of log-normal distributions for approximating the PSD has important advantages:

(1)

there is no need to “break” the power-law approximation to reflect changes in the log-log slope of the size distribution;

(2)

the log-normal approximation assumes finite values for all particle diameters, except in the limit B₂ > 0, when the log-normal function becomes a power-law function;

(3)

mass and area distributions resulting from the log-normal distribution are also log-normal (Kerker 1969);

(4)

the power-law function is a limiting case of the log-normal function, so it is naturally included as an approximation of the size distribution.

We should also mention certain pitfalls of choosing the log-normal distribution after Halley and Inchausti (2002), who cite comments by Mandelbrot (1997) (incidentally these problems also relate to other “long-tailed” distributions):

(1)

extreme sensitivity of all moments of the distribution to even small departures from log-normality, which makes the calculation of moments from the parameters of a fitted log-normal distribution unreliable

(2)

slow convergence of the approximate values of the moments to their asymptotic value. This feature is due to an extremely “long tail” of the log-normal distribution at the large-size end of the particle size scale. We have experienced this problem ourselves (Jonasz and Fournier 1996) when trying to evaluate correlations between the total particle surface and volume and the peak diameter of the distribution as fitted to several hundred size distributions of marine particles.

The log-normal size distribution of the zero-th order can be expressed as follows (e.g., Ross 1978, Casperson 1977):

(5.169)n(D)=nmaxexp[−(lnD−lnDpeak)22σ2]

where n_max is expressed as follows:

(5.170)nmax=Ntot2πσDpeakexpσ22

N_tot is the total number of particles [equal to unity in the case of n(D) being the probability distribution], D_peak is the particle diameter corresponding to the peak of the size distribution, and σ is the standard deviation of lnD, i.e., the geometric standard deviation of D.

The width parameter, σ, is related to the ratio of the maximum and minimum diameters, D_max and D_min, of the full-width-at-half-maximum of the log-normal n(D) through the following equation (Jonasz and Fournier 1996):

(5.171)DmaxDmin=exp(2fσ)

where

(5.172)f=2ln2Dmin=Dpeakexp(−fσ)Dmin=Dpeakexp(fσ)

In contrast to the power-law distribution, the log-normal size distribution yields a finite total number of particles, N_tot, total particle cross-sectional area, A_tot, and volume, V_tot (Heintzenberg 1994):

(5.173)Atot=π[12Dpeakexp(2σ2)]2Ntot×exp{−12σ2[(lnDmaxexp(2σ2))−lnDmax]2}

and

(5.174)Vtot=43π[12Dpeakexp(3σ2)]3Ntot×exp{−12σ2[(lnDmaxexp(3σ2))−lnDmax]2}

The cell size distributions reported in the literature frequently appear to be log-normal at visual inspection (see Table A.5 for sources of the relevant PSD data). Interestingly, mathematical models of cell growth and evolution of isolated cell populations do not lead to a log-normal distribution of cell sizes (e.g., Tyson and Hannsgen 1985). However, the agreement between the models and the experimental data is questionable. In fact, Tyson and Hannsgen note that cell size distributions with log-normal size distribution have been reported (ibid. Scherbaum and Rasch 1957, Collins and Richmond 1962). Analysis of the biomass spectrum in aquatic ecosystems composed of organism groups linked via a prey–predator relationship led to the log-normal function as a natural descriptor of the contribution of an organism to the total ecosystem biomas spectrum (Thiebaux and Dickie 1993, Boudreau et al. 1991). The derivation of the log-normal form of the size distributions of the individual organism groups was based on the fact that the production, P (w), is proportional to a power, b, of the body mass, W (allometric relationship):

(5.175)P(W)=aWb

The observation of (1) a persistent curvature of the size distribution of marine particles, when plotted in a log-log scale, (2) the multimodal appearance of many such size distributions, as well as (3) previous suggestions in the literature that complex size distributions of geological material can be well modeled by a sum of log-normal functions (van Andel 1973) led us to develop an automated algorithm of the decomposition of a marine PSD into a sum of log-normal components (Jonasz and Fournier 1999, 1996). In that work, we postulated that the size distribution of marine particles is essentially a linear combination of a cascade of log-normal components, according to the following equation:

(5.176)n(D)=∑1kmaxnk(D)

where index k numbers the log-normal components n_k(D). Each of these components is approximated with a zero-th order log-normal distribution function:

(5.177)nk(D)=nmax,kexp[−(lnD−lnDpeak,k)22σk2]

where n_{max, k} is the maximum value of the component, D_peak,k [μm] is the peak diameter, and σ_k is the width parameter. By taking the logarithm of both sides of (5.177) and performing simple algebraic transformations, one obtains (we omit the component index for simplicity):

(5.178)logn(D)=B0+B1logD+B2(logD)2

where:

(5.179)B0=lognmax−(logDpeak)2ln102σ2B1=logDpeakln10σ2B2=−ln102σ2

Note that (5.178) reduces to a power law when B₂ vanishes. Equations (5.179) can be solved for n_max, D_peak, and σ to yield:

(5.180)lognmax=B0−B124B2logDpeak=−B12Bσ2=−ln102B2

Sample results are shown in Figure 5.34 and Figure 5.35. Log-normal components identified by the algorithm just described range in shape from the power-law-like function to Gaussian-normal-like function. Note that the failure to account for the instrumental error (see section 5.7.1.6) in evaluating the γ² results in the γ² value in excess of 200 per degree of freedom (!) in the case of data shown in Figure 5.35. that include very high particle count values.

Figure 5.34. A log-normal approximation of the size distribution of Figure 5.31: log n (D) = 4.41 − 2.50 log D − 1.02 (log D)² (γ² per degree of freedom = 0.24). The fit parameters were obtained via the logarithmic transform. All weights were set to unity. The γ² was calculated by assuming only the counting error.

Figure 5.35. A multi-component log-normal approximation (thick black curve) of a PSD measured in the Northwest Atlantic waters (•, unpublished data: courtesy of K. Kranck and T. Milligan, file KRAATL86.P08 in Jonasz 1992). The approximation coefficients from equation (5.178) : first component (thin solid curve) B₀₁ = 4.05, B₁₁ = −1.16, B₂₁ = −2.12, second component (dashed curve) B₀₂ = −17.04, B₁₂ = 22.52, B₂₂ = −7.75, third component (gray solid curve) B₀₃ = 4.97, B₁₃ = 7.04, B₂₃ = 7.30, fourth component (gray dashed curve) B₀₄ = −4.27, B₁₄ = −9.60, B₂₄ = −4.81. The first log-normal component removes the greatest amount of the approximation error. Each of the following components removes a progressively smaller amount of that error (χ² per degree of freedom = 0. 021). The fit parameters were obtained via the logarithmic transform. All fitting weights were set to unity. The χ² was calculated by assuming both the counting and instrumental errors.

The statistics of and correlations between the parameters of 853 log-normal components of the 412 PSDs determined using the Coulter technique by different researchers in different areas and seasons are shown in Table 5.10. The average values of coefficients B₀, B₁, and B₂ represent a geometrically averaged component characteristics, i.e., n_avg(D) such that n_avg(D) = [n₁(D) n₂(D) … n_m(D)]^1/m. The average values of n_max, D_peak, and σ do not have simple meanings and are given here for the sake of completeness only. The two sets of averages do not yield the same function of the diameter, D, because they are related via non-linear functions (5.179) and (5.180). The average error of approximation of the size distribution with the sum of log-normal components was 0.057±0.030. The number of components per size distribution varied from 1 to 6, with an average of 2.18 ± 1.22. The value of 1 standard deviation (SD) is shown following the ± sign.

(5.181)lnnmax=(8.070±2.799)−(2.446±0.032)lnDpeak

and

(5.182)σ=(0.626±0.186)−(0.111±0.002)lnDpeak

where the value of 1 SD of the respective parameter is shown following each ± sign.

The log-normal components, which range in shape from the power-law-like function to Gaussian-normal-like function, may be interpreted as the size distributions of the various classes of marine particles, for example, populations of various phytoplankton species. Indeed, Jonasz and Fournier (1996) found two “standard” components (Figure 5.38 and Table 5.11) in 412 size distributions measured in various seasons and regions of the world ocean by different researchers. Bradtke (2004) who used the algorithm just described to analyze 970 PSDs measured with a Coulter counter in the coastal waters of the Baltic Sea (Gdansk Bay) also noted the existence of several “standard” components and linked other transitional components to the occurrences of phytoplankton species.

Figure 5.38. “Standard” components (Table 5.11) of the marine particle size distribution identified by Jonasz and Fournier (1996), who analyzed 412 particle size distributions measured in various seasons and regions of the world ocean by different researchers (as compiled by Jonasz 1992): first component (solid black curve) B₀₁ = 4.038, B₁₁ = −0.9511, B₂₁ = −2.542, second component (dashed curve) B₀₂ = −8.447, B₁₂ = −7.55, B₂₂ = −8.595, compared with the size distribution from Figure 5.35 (symbols) and its retrieved components (gray solid lines).

The classical inferential issues of the TRVM under the assumption that T has a log-normal distribution with the PDF

fT(t)=1σt2πe−12lnt−μσ21(0,∞)(t),

where μ∈R and σ > 0, was addressed by Bai et al. [115]. The authors also assumed that the data are Type-I censored. Under the log-normal distribution, the PDF of T~ is given by

fT~(t)=1σt2πe−12lnt−μσ2if0<t≤τ1βστ1+β(t−τ1)2πe−12ln(τ1+β(t−τ1))−μσ2ift>τ10otherwise.

Given a Type-I censored data t1:n<⋯<tn1:n<τ1<tn1+1:n<⋯<tn1+n2:n<η, the log-likelihood function can be written as

l(μ,σ2,β)=−n1+n22lnσ2−∑i=n1+1n1+n2lnτ1+β(ti:n−τ1)−12σ2∑i=1n1lnti:n−μ2−12σ2∑i=n1+1n1+n2ln(τ1+β(ti:n−τ1))−μ2+(n−n1−n2)ln1−Φln(τ1+β(η−τ1))−μσ,

where Φ(⋅) is the CDF of the standard normal distribution. The MLEs of μ, σ², and β can be found by maximizing the log-likelihood function with respect to the parameters. Note that in this case the MLEs of the unknown parameters do not exist in explicit form and one needs to use a numerical technique to maximize the log-likelihood function. The asymptotic variance of the MLEs and the confidence intervals of the unknown parameters can be obtained using the observed Fisher information matrix.

Pore size and PSD of each membrane were determined from AFM-determined topography. The lognormal distribution was chosen to represent the pore size data for each of the membranes. This was found to give a good fit to the PSD. All of these distributions were fitted to lognormal distributions given by frequencies (%f):

(5.1)%f=%fmaxexp[-12σ2ln(dpX0)2]

where d_p is the measured pore size, σ the standard deviation of the measurements, %f_max the maximum frequency and χ₀ the modal value of d_p.

Mean pore sizes and PSDs of initial membranes determined from AFM images are shown in Table 5.3. This table shows that with a mean pore size of 0.535±0.082 μm, the PVDF membrane has slightly larger pores than the PES membrane (mean pore size 0.470±0.188 μm). PSDs are detailed in Figure 5.3, along with fits described by equation (5.1). The measured pore sizes are larger than the nominal pore size of 0.22 μm as specified by the manufacturers. The AFM data confirm that the pore sizes of studied membranes are of approximately the same size. The topographical images give a clear perception of a notable difference in the surface morphology of the membranes used for the modification. A quantification of the surface parameters (Table 5.3) provides an insight into morphological particularities of these membranes which influence both the membrane separating properties and the process of modification by graft copolymerisation.

Before detailed discussion of the quantitative characteristics of surface morphology, it is worth noting that the chosen lognormal pattern for PDS described by equation (5.1) gave a correlation coefficient of at least 0.95 for all fitted curves. The most probable pore sizes estimated from fitted curves were very close to the mean pore diameter calculated from corresponding sets of pore sizes for the initial and modified membranes (Tables 5.4 and 5.5).

According to Figure 5.5(a), the initial PES membrane has a very wide PSD with a σ value of 0.56 μm. However, grafting of poly-qDMAEMA resulted in narrowing of the PSD and shifting the whole curve towards smaller pore sizes. As a result, mean pore size is gradually decreasing with the increase in the amount of poly-qDMAEMA grafted to the membrane surface. Significant improvement of the PSD was observed even for the modified PES membranes with the smallest DM (Table 5.4).

Figure 5.5. PSDs of initial (a) and modified with qDMAEMA (b)–(d) PES membranes; (b) DM = 202 μg cm⁻², (c) DM = 367 μg cm⁻² and (d) DM = 510 μg cm⁻².

Solution: Robust estimates μ^M, variances Dμ^M and the limits of the 95% confidence interval of the mean are listed in Table 3.6. For the symmetric distributions N, R and L the robust analysis gives accurate estimates quite near to the true values, and the confidence interval is narrow. Worse results were achieved with the asymmetric skewed distributions: for the exponential and log-normal distributions the 95% confidence interval does not contain theoretical value μ.

If a random variable V has a normal distribution with mean μ and variance σ², then e^V has a lognormal distribution with parameters μ and σ². In other words, if a variable has a lognormal distribution, then its logarithm has a normal distribution. The pdf of the distribution is given by

fx;μ,σ2=1xσ2πe−12logx−μσ2

where x and σ are both positive. If X follows a lognormal distribution with parameters μ and σ², then Y = e^a X^b follows a lognormal distribution with parameters a + bμ and b²σ².

The expectation and variance of the lognormal distribution are given by

Ex=eμ+σ22,Vx=e2μ+σ2eσ2−1.

Using its relationships to the normal distribution, the parameters of the lognormal distribution from a sample x₁, x₂,…, x_n of draws from the distribution can be estimated as

μˆ=1n∑log(x1),σˆ=1n−1∑(logxi−μˆ)2.

The normal distribution with parameters μ and σ² > 0 is given by the familiar bell-shaped probability density function

(1.32)ϕ(x;μ,σ2)=12πσe−(x−μ)2/2σ2     −∞<x<∞.

The density function is symmetric about the point μ, and the parameter σ² is the variance of the distribution. The case μ = 0 and σ² = 1 is referred to as the standard normal distribution. If X is normally distributed with mean μ and variance σ², then Z = (X − μ)/σ has a standard normal distribution. By this means, probability statements about arbitrary normal random variables can be reduced to equivalent statements about standard normal random variables. The standard normal density and distribution functions are given respectively by

(1.33)φ(ξ)=12πe−ξ2/2,    −∞<ξ<∞,

and

(1.34)Φ(x)=∫−∞xφ(ξ)dξ,       −∞<x<∞.

The central limit theorem explains in part the wide prevalence of the normal distribution in nature. A simple form of this aptly named result concerns the partial sums S_n = ξ₁ + · ·· + ξ_n of independent and identically distributed summands ξ₁, ξ_2…. having finite means μ = E[ξ_k] and finite variances σ² = Var[ξ_k]. In this case, the central limit theorem asserts that

(1.35)limn→∞ Pr{Sn−nμσn≤x}=Φ(x)    for all x.

Suppose that a sample of 10 grains of metallic sand taken from a large sand pile have respective lengths (in millimeters):

2.2, 3.4, 1.6, 0.8, 2.7, 3.3, 1.6, 2.8, 2.5, 1.9

It helps to first describe problems associated with Student’s t. When testing hypotheses or computing confidence intervals for μ, it is assumed that

(4.1)T=n(X¯−μ)s

has a Student’s t-distribution with ν = n − 1 degrees of freedom. This implies that E (T) = 0, and that T has a symmetric distribution. From basic principles, this assumption is correct when observations are randomly sampled from a normal distribution. However, at least three practical problems can arise. First, there are problems with power and the length of the confidence interval. As indicated in Chapters 1 and 2Chapter 1Chapter 2, the standard error of the sample mean, σ/n, becomes inflated when sampling from a heavy-tailed distribution, so power can be poor relative to methods based on other measures of location, and the length of confidence intervals, based on Eq. (4.1), become relatively long – even when σ is known. (For a detailed analysis of how heavy-tailed distributions affect the probability coverage of the t-test, see Benjamini, 1983.) Second, the actual probability of a type I error can be substantially higher or lower than the nominal α level. When sampling from a symmetric distribution, generally the actual level of Student’s t-test will be less than the nominal level (Efron, 1969). When sampling from a symmetric, heavy-tailed distribution the actual probability of type I error can be substantially lower than the nominal α level, and this further contributes to low power and relatively long confidence intervals. From theoretical results reported by Basu and DasGupta (1995), problems with low power can arise even when n is large. When sampling from a skewed distribution with relatively light tails, the actual probability coverage can be substantially less than the nominal 1 − α level resulting in inaccurate conclusions and this problem becomes exacerbated as we move toward (skewed) heavy-tailed distributions. Third, when sampling from a skewed distribution, T also has a skewed distribution, it is no longer true that E (T) = 0, and the distribution of T can deviate enough from a Student’s t-distribution so that practical problems arise. These problems can be ignored if the sample size is sufficiently large, but given data it is difficult knowing just how large n has to be. When sampling from a lognormal distribution, it is known that n > 160 is required (Westfall & Young, 1993). As we move away from the lognormal distribution toward skewed distributions where outliers are more common, n > 300 might be required. Problems with controlling the probability of a type I error are particularly serious when testing one-sided hypotheses. And this has practical implications when testing two-sided hypotheses because it means that a biased hypothesis testing method is being used, as will be illustrated.

Problems with low power were illustrated in Chapter 1, so further comments are omitted. The second problem, that probability coverage and type I error probabilities are affected by departures from normality, is illustrated with a class of distributions obtained by transforming a standard normal distribution in a particular way. Suppose Z has a standard normal distribution, and for some constant h ≥ 0, let

X=Zexph Z22.

Then X has what is called an h distribution. When h = 0, X = Z, so X is standard normal. As h gets large, the tails of the distribution of X get heavier, and the distribution is symmetric about 0. (More details about the h distribution are described in Section 4.2)

For convenience, assume Y is standard normal in which case E(X)=e, where e = exp(1) ≈ 2.71828, and the standard deviation is approximately σ = 2.16. Then Eq. (4.1) assumes that T=n(X¯−e)/s has a Student’s t-distribution with n − 1 degrees of freedom. The left panel of Figure 4.1 shows a (kernel density) estimate of the actual distribution of T when n = 20; the symmetric distribution is the distribution of T under normality. As is evident, the actual distribution is skewed to the left, and its mean is not equal to 0. Simulations indicate that E (T) = −0.54, approximately. The right panel shows an estimate of the probability density function when n = 100. The distribution is more symmetric compared to n = 20, but it is clearly skewed to the left.

Figure 4.1. Nonnormality can seriously affect Students t. The left panel shows an approximation of the actual distribution of Students t when sampling from a lognormal distribution and n = 20 and the right panel is when n = 100.

Generally, as we move toward a skewed distribution with heavy tails, the problems illustrated by Figure 4.1 become exacerbated. As an example, suppose sampling is from a squared lognormal distribution that has mean exp(2). (i.e., if X has a lognormal distribution, E(X²) = exp(2).) Figure 4.2 shows plots of T values based on sample sizes of 20 and 100. (Again, the symmetric distributions are the distributions of T under normality.)

Figure 4.2. The same as Figure 4.1, only now sampling is from a squared lognormal distribution. This illustrates that as we move toward heavy-tailed distributions, problems with nonnormality are exacerbated.