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### Keywords:

• Ahrens' law;
• moment estimators;
• maximum likelihood estimators;
• Fisher information matrix

### Abstract

In this article, we study an extension of the power-normal distribution to the case of the more flexible log-power-normal distribution. We find the density function and study some properties of the new distribution, deriving a general expression for its moments. Parameter estimation is implemented by using the moments and the maximum likelihood approaches. We obtain the Fisher information matrix for the new model, which is shown to be nonsingular in the vicinity of symmetry. Results of an application to air contamination data indicate good performance of the proposed model, validating a modification of Ahrens' law. Copyright © 2014 John Wiley & Sons, Ltd.

### INTRODUCTION

The lognormal (LN) distribution obtained as a transformation of the ordinary normal distribution is widely used to model different types of positive data, including income in econometric studies and equipment lifetimes in material engineering, among others. One of its most important applications is in the concentration of chemical elements in the area of geochemistry because of the fundamental law of geochemistry, enunciated by Ahrens (1954): “the concentration of a chemical element in a rock is distributed log-normal.” In many of these situations, the asymmetry of the distribution and its kurtosis are above or below the expected for the LN distribution, which is why it is necessary to think of a more flexible model that achieves such deviation in modeling positive data. It is shown that this is also the case in an application to air pollution data from the city of New York. It is concluded that a modification of Ahrens' law is also called for in the situation of chemical contamination in the atmosphere.

Azzalini (1985) introduced the skew-normal distribution, which includes an asymmetry parameter, for which the density function is

• (1)

where λ is the asymmetry parameter, ϕ( · ) is the density function of the standard normal distribution and Φ( · ) is the corresponding cumulative distribution function (CDF). Clearly, for λ = 0, the standard normal model is obtained. This distribution is widely used for modeling data with an asymmetry coefficient ranging in the interval ( − 0.995,0.995).

As an extension of the model (1) to positive data, Mateu-Figueras et al. 2003, 2004) examined the univariate log-skew-normal (LSN) distribution, which is a generalization of the LN distribution by using the skew-normal distribution. Its density function is given by

• (2)

where is a location parameter and is a scale parameter. We use the notation Y ∼ LSN(ξ,η,λ). Real data applications of this model can be found in Azzalini et al. (2003, in Marchenko and Genton (2010) in a multivariate setting and in Bolfarine et al. (2011) in a bimodal context. Notice that if λ = 0, then the ordinary LN distribution follows. For this model, the Fisher information matrix is singular when the parameter λ = 0, and hence, regularity conditions are not satisfied. One consequence of this fact is that likelihood ratio statistics for testing normality is no longer distributed according to the central chi-square distribution (Arellano-Valle and Azzalini, 2008).

As a second alternative to model data with high asymmetry and kurtosis, Durrans (1992) in a hydrological context introduced a generalization of the normal distribution based on the distribution of the maximum in the sample. Durrans (1992) called it the distribution of fractional order statistics, whereas Gupta and Gupta (2008) called it the power-normal distribution. Because we consider a generalization of both models, we call it the power distribution (Pewsey et al. 2012). The introduction of a new parameter makes the new distribution more flexible, which makes it is possible to model simultaneously the shape and kurtosis of the distribution. Other works have been published in this context, with initial results given by Birnbaum (1950), Lehmann (1953), Roberts (1966) and O'Hagan and Leonard (1976), among others. More results on extensions and properties of the aforementioned models can be found in Azzalini (1986), Henze (1986), Chiogna (1997), Fernández and Steel (1998), Mudholkar and Hutson (2000), Pewsey (2000), Gupta et al. (2002), Arellano-Valle et al. (2004, 2005), Gupta and Gupta (2008) and Gómez et al. (2007, 2011).

We split the article into four sections. In Section 2, we briefly review the main properties of power distributions. Section 3 develops the log-power-symmetric (LPS) distribution and some of its moments and demonstrates that this distribution has no moment-generating function (MGF). Section 4 deals with moment and maximum likelihood estimation approaches in the case of the log-power-normal (LPN) distribution including large-sample results. Results of an application to air pollution data from the city of New York are reported, illustrating the usefulness of the proposed model and its relation to Ahrens' law.

### POWER-SYMMETRIC DISTRIBUTIONS

Durrans 1992) in a hydrological context introduced the distribution of the fractional order statistics (which we call power-symmetric (PS) distribution) with the density function given by

• (3)

where F is an absolutely continuous distribution function with density function f = dF. We use the notation Z ∼ PS(α). We refer to this model as the standard power distribution (also Pewsey et al., 2012. It is an alternative asymmetric model with the asymmetry ranging in the interval ( − 0.611,0.900) and kurtosis in the interval ( 1.717,4.355) (Pewsey et al., 2012). In the particular case where F = Φ( · ), we have the distribution called the power-normal class of distributions (Gupta and Gupta, 2008), denoted PN(α). For F = Φ( · ) and α = 1, model (3) corresponds to the density function of the standard normal distribution, and for α = 2, the skew-normal model (1) with asymmetry parameter λ = 1 follows.

If Z is a random variable from a standard PS(α) distribution, then the location-scale extension of Z, X = ξ + ηZ, where and , has a probability density function (PDF) given by

• (4)

We will denote this extension by using the notation X ∼ PS(ξ,η,α). Notice that this model can be extended to regression models by considering replacing ξ, where β is an unknown vector of regression coefficients and xi a vector of known regressors correlated with the response vector. As can be depicted from Figure 1, parameter α controls also the distribution asymmetry. Moreover, it can be noticed that for α > 1, we have right asymmetry and, for 0 < α < 1, we have left symmetry. Besides being a shape parameter, it will be seen in the following that parameter α affects the mean, variance, asymmetry and kurtosis of the distribution.

Pewsey et al. (2012) derived the Fisher information matrix for the location-scale version of the power-normal model, demonstrating that it is not singular for α = 1. We recall that the Fisher information matrix for the skew-normal distribution (Azzalini, 1985) is singular under the symmetry hypothesis. Hence, with the power-normal model, normality can be tested using ordinary large-sample properties of the likelihood ratio statistics. We recall that the Fisher information matrix for the skew-normal distribution (Azzalini, 1985) is singular under the normality hypothesis.

#### Moments of the power-symmetric distribution

No closed-form expression is available for the moments of the random variable Z ∼ PS(α), which can be written generically as (Gupta and Gupta, 2008)

• (5)

Notice that the last expression corresponds to the expectation of the expression [F − 1(W)]n, where random variable W follows a beta distribution with parameters α and 1. Values for the mean, variance and asymmetry coefficient were obtained by Durrans (1992) using numerical integration.

Result 1. For , it follows that the moments of the random variable Z ∼ PN(α) can be obtained as in Gupta and Nadarajah (2004), where moments of the beta-normal distribution are considered.

Hence, we have that

with

Notice that we can write , where , is the error function (Prudnikov et al., 1990). Therefore,

For the location-scale situation, that is, X = ξ + ηZ, we have that

For the case n = 1, using integration by parts, it follows that

Therefore,

By using results in Prudnikov et al. (1990), it follows that

Therefore, the special cases presented in Table 1 can be derived (Gupta and Nadarajah, 2004).

Table 1. Moments for distribution PN(α)
α2345

Durrans (1992), on the basis of results found by Ruben (1954) and using numerical integration, presented partial derivation for these moments, particularly, the mean, variance and asymmetry coefficient for α between 0.3 and 20.

Result 2. For X ∼ PN(ξ,η,α), we have the MGF for the random variable X given by

• (6)

where

That is, the preceding MGF can be seen as the product of the MGF of a normal random variable and a function that measures departure from normality. Hence, as α = 1, we have that h(α,η) = 1 and , corresponding to the MGF of a normal distribution with parameters ξ and η.

### THE LOG-POWER-SYMMETRIC DISTRIBUTION

A random variable Y, with support in , follows a univariate LPS distribution with parameter α, which we denote Y ∼ LPS(α), if the transformed variable Z = log(Y ) ∼ PS(α). The PDF for the random variable Y ∼ LPS(α) can be written as

• (7)

where F is an absolutely continuous distribution function with density function f = dF. We refer to this model as the standard log-alpha-power distribution. The CDF can be written as

• (8)

According to (8), the inversion method can be used for generating from a random variable with the LPS(α) distribution. That is, if U ∼ U(0,1), the uniform distribution, then the random variable Y = exp(F − 1(U1 ∕ α)) is distributed according to the LPS distribution with parameter α.

In the special case where f = ϕ and F = Φ, the density and distribution functions of the standard normal distribution, respectively, the standard LPN distribution follows, with the density function given by

• (9)

denoted by Y ∼ LPN(α). Notice that LPN is a particular case of model (7) for F = Φ( · ) and z = log(y). Figure 2 depicts PDFs for the LPN distribution for α equal to 0.75, 1, 2 and 7. Kurtosis is greater for α = 1, the LN case, than for α = 2, the LSN case, and similarly for α = 7. On the other hand, the kurtosis is greater for α = 0.75 than for α = 1, the LN model. Asymmetry is always positive and is also controlled by parameter α. Hence, α controls asymmetry as well as kurtosis for the LPN distribution.

In survival analysis, it is common to study the survival function S(t) = 1 − F(t), the risk function and the PDF

For the LPN, the survival and hazard functions are given, respectively, by

Therefore, the cumulative hazard function and the inverse hazard index are

so that the inverse hazard index t is proportional to the inverse hazard index of the LN distribution. The hazard index can be written as

Result 3.

• For α → 0, r(t) becomes a nondecreasing function.

Therefore, the hazard index t is proportional to the hazard index of the LN distribution, and according to Gupta et al. (1997), r(t) is initially an increasing function, attains a maximum value and then decreases to zero as t goes to infinity. A similar behavior is depicted in Figure 3, which also leads to the conclusion that the hazard function is a decreasing function of α.

#### Location-scale extension

Let X ∼ PN(ξ,η,α), where is a location parameter and is a scale parameter. Therefore, the transformation X = log(Y ) leads to the location-scale LPN model, with the PDF given by

• (10)

We use the notation Y ∼ LPN(ξ,η,α), so that LPN(α) = LPN(0,1,α).

This new proposal is an important extension because the inclusion of the shape parameter α increases flexibility, leading to models with smaller or greater kurtosis and/or asymmetry than the corresponding ones for the LN distribution. This is a direct consequence from the fact that LN(ξ,η) ≡ LPN(ξ,η,1).

#### Moments and moment-generating function

As is the case with the power-normal distribution, the r-th moment for the random variable Y ∼ LPN(α) can be written as

• (11)

Similarly, we can use the relation

• (12)

where Z is a standard power-normal random variable. The central moments, , for r = 2,3,4, can be computed using the expressions

By using the preceding results, the variance, coefficients of variation (CV), asymmetry and kurtosis are given by

A small-scale simulation study has been conducted to evaluate the range of possible values for the quantities defined earlier for α ∈ (0,1000]. Calculations were performed using function integrate in R software (R Development Core Team, 2012).

Table 2 depicts ranges of values for the variance and CV, asymmetry, and kurtosis. Figure 4 indicates that the degrees of asymmetry and kurtosis decrease as α increases. The same seems to be the case with the CV, which also decreases with α. It can also be noticed that the relative coefficient of variation is somewhat large. On the other hand, the mean and variance increase for α in the range considered.

Table 2. Values for asymmetry and kurtosis coefficients for random variable LPN(α) and
Statistics  Variance  AsymmetryKurtosisCoefficient of variation (%)
Minimum0.743.0927.3743.13
Maximum139.4013.18517.00305.96

The following result parallels similar ones for the LN and LSN distributions.

Result 4. For all , random variable Y ∼ LPN(ξ,η,α) has no MGF.

Proof. Result is obviously true for α = 1, as it corresponds to the LN case. Moreover, for t > 0,

where for y > 0. By using results from Lin and Stoyanov (2009), it can be shown that for all

and hence, variable Y has no MGF for . The characteristic function , where , exists for , but it does not simplify moment derivation.

### INFERENCE FOR THE LOG-POWER-NORMAL MODEL

Here, we present general results for likelihood-based estimation of the parameters of an LPN distribution. We start by considering standard models with densities of the form given in (9).

#### The standard model

The likelihood function for a random sample of size n,y = (y1, … ,yn) ′ , from the LPN(α) distribution with density (9), is given by

It follows that the maximum likelihood estimator (MLE) of α is given by

which is a function of the complete and sufficient statistics for α, namely

Therefore,

and we can conclude that the asymptotic variance of is given by α2 ∕ n, which agrees with the asymptotic variance in the alpha-power model.

Defining mi = Φ(log(yi)) ∼ U(0,1), we have that the random variable Gi = − log{Φ(log(yi))} ∼ Gamma(1,1) and hence . Consequently, is distributed according to the random variable , for n > 2. Therefore,

so that an unbiased estimator of α is given by

Similarly, we have

where Var and MSE denote variance and mean squared error, respectively. From Chebyshev's inequality, it follows that

Therefore, when n → ∞ , then , which proves that is a consistent estimator for the parameter α. Finally,

where is Cramer–Rao's lower bound for . Moreover, is a pivotal quantity for α, so that a 100(1 − δ)% confidence interval for α is given by

#### Location-scale extension

In this section, we discuss the inference for the location-scale extension of the standard LPN distribution. We start by discussing moment estimation.

##### Estimation using the method of moments

For the location-scale situation of random variable log(Y ) with PN distribution, the mean (μ), variance (σ2) and asymmetry coefficient are given, respectively, by

Hence, by equating sample moments with corresponding population moments given previously, moment estimators for parameters ξ,  η and α can be obtained. In the estimator expressions, quantities Φ1(α),  Φ2(α) and Φ3(α) (mean, variance and asymmetry coefficient, respectively, for the standard power-normal distribution) must be computed numerically.

Note that α is obtained by equating sample and population asymmetry coefficients, which can be used in the estimation of μ and σ2. Results stated in Sen and Singer (2000) can be used to establish the asymptotic distribution for the moment estimators.

##### Maximum likelihood estimation

In this section, we discuss the derivation of the MLEs for model parameters. We start by writing the likelihood equations.

Considering a random sample of size n,y = (y1, … ,yn) ′ from the LPN(ξ,η,α) distribution, the log-likelihood function for θ = (ξ,η,α) ′ given y can be written as

• (13)

where . Therefore, by using an approach similar to that of Pewsey et al. 2012), the first partial derivative of (13) with respect to ξ, η and α can be written as

and

where wi = ϕ(zi) ∕ Φ(zi) and mi = log{Φ(zi)}. Hence, MLEs for parameters ξ, η and α are obtained by solving the system of equations, which follows by equating the preceding derivatives (scores) to zero. The system of equations obtained can be written as , where

and . This system of equations has no analytical solution and has to be solved by numerical methods such as the Newton–Raphson procedure.

Figure 5 depicts the profile log-likelihood function for values of α ∈ (0,5] for samples simulated from the standard distribution LPN(0,1,1) ≡ LN(0,1). The sample sizes used are 50, 150 and 300.

The figure depicts the fact that the profiles are quite regular, providing strong evidence for the existence and unicity (no multiple modes) of the MLEs.

#### Observed information matrix

The elements of the observed information matrix follows from minus the second partial derivatives of the log-likelihood function with respect to the parameters, which are denoted by and given by

Having computed the MLEs and given the relationships , and , the elements of the observed information matrix evaluated at the MLEs are given by

where and .

#### Expected information matrix

The expected (or Fisher) information matrix follows by taking the expectation of the elements of the Hessian matrix. Considering the quantities akj = E{zk(ϕ(z) ∕ Φ(z))j}for k = 0,1,2,3 and j = 1,2,  θ1 = ξ,  θ2 = η and θ3 = α, we have that the elements of the Fisher information matrix, denoted

are given by

The preceding expressions have to be computed numerically. In the particular case of α = 1, that is, , it follows that , so that the Fisher information matrix for θ = (ξ,η,α = 1) ′ is given by

which agrees with the Fisher information matrix of the power-normal distribution (Pewsey et al., 2012). Hence, by using numerical procedures, it can be shown that

so that the Fisher information matrix is not singular at α = 1.0.

However, this is not the case with the LSN model (Mateu-Figueras et al., 2003, 2004), for which the Fisher information matrix is singular for λ = 0. This important feature allows testing (with the LPN model) of normality using the ordinary large-sample property of the likelihood ratio statistics, which states that in a large sample, it follows a chi-square distribution. Another important difference between the two models (LSN and LPN) concerns asymmetry and kurtosis ranges. Whereas the kurtosis range is wider for the LSN model, the asymmetry range is wider for the LPN model. Such differences can help one to select a more appropriate model. The upper-left 2 × 2 submatrix coincides with the Fisher information matrix of the ordinary LN distribution. Therefore, as n is large,

meaning that is consistent and asymptotically normally distributed with I(θ) − 1 as the large-sample variance.

#### An illustration

The data set studied in this illustration was previously analyzed by Nadarajah (2008) and Leiva et al. (2010). It is related to air pollution in the city of New York, USA. For air pollutant concentrations, it is usually assumed that the data are uncorrelated and independent and thus do not require the diurnal or cyclic trend analysis (Gokhale and Khare (2007)). The data correspond to daily measurements of ozone concentration in the atmosphere (in ppb = ppm × 1000) in the city of New York in May–September 1973, from the New York State Department of Conservation.

The concentration of average air pollutants has been used in epidemiological surveillance as an indicator of the atmospheric contamination and its associated adverse effects in humans, causing diseases such as bronchitis. The distribution of this concentration has a bias to the right, as this random variable is always positive. A model that has these characteristics is the LN, and it has been frequently used for modeling the concentration of air contaminants and chemical concentration in soil samples (Ahrens' law), mainly owing to its theoretical arguments. However, the level of air pollution varies depending on factors such as the source of contamination, local weather and topography. Therefore, the actual distribution of the concentration of atmospheric pollutants does not always agree with an LN model, especially at high contamination levels.

Descriptive statistics for the data set are presented in Table 3. Quantities and b2 indicate sample asymmetry and kurtosis coefficients.

Table 3. Descriptive statistics for variables Y and log(Y )
VariablenMeanVarianceb2
Y1164.12931088.20101.20981.1122
log(Y )1163.41850.7490 − 0.54780.7755

Table 3 reveals a positively skewed distribution for the variable Y. Moreover, asymmetry and kurtosis coefficients for log(Y ) are somewhat far from what is expected with the normal distribution, which are 0 and 3, respectively, justifying the use of a more flexible model such as the LPN model discussed in the paper.

To model the amount of ozone-level concentration in the atmosphere, we use LN, LSN and LPN models. We also adjusted the ordinary two-parameter Birnbaum–Saunders (BS) model (denoted BS(γ,β), Birnbaum and Saunders, 1969), which can be used for studying this type of data (Leiva et al. 2010).

To compare model fitting, we use the Akaike information criterion (AIC) (Akaike, 1974), namely We consider also the Bayesian information criterion and the modified AIC, typically called the consistent AIC (CAIC), namely , where k is the number of parameters for the model being considered. The best model is the one with the smallest AIC (or BIC or CAIC). MLEs, estimated standard errors (in parenthesis), for the LN, LSN, BS and LPN models, were computed by maximizing the log-likelihood using the function optim in R. Results are presented in Table 4 together with the AIC, BIC and CAIC. Hence, we have that model LPN presents the best fit to the data set, according to the AIC, BIC or CAIC, where the graphs in Figure 6(a, b) reveal that the LPN model fitting is quite good.

Table 4. Parameter estimates and estimated standard errors for LN, LSN and LPN distributions
ParameterLN LSN BS LPN
1. AIC, Akaike information criterion; BIC, Bayesian information criterion; CAIC, consistent Akaike information criterion; LN, lognormal; LPN, log-power-normal; LSN, log-skew-normal; BS, Birnbaum–Saunders.

Log-likelihood − 543.883  − 541.655  − 549.097  − 540.266
AIC1091.766 1089.310 1102.194 1086.532
BIC1097.273 1097.570 1107.701 1094.792
CAIC1100.273 1100.570 1109.701 1097.792
ξ3.418 (0.079) 4.372 (0.079)  4.986 (0.117)
η0.861 (0.056) 0.7048 (0.075)  0.146 (0.053)
λ 1.5381 (0.478)
α   0.012 (0.009)
γ  0.982 (0.064)
β  28.031 (2.265)

We now consider testing the hypothesis of no difference between the LPN and LN distributions for the data set under study, which corresponds to testing the hypotheses

using the statistics

which is greater than the 5% chi-square critical value, . Hence, the LPN model seems to be a useful alternative to be used for modeling air pollution data, particularly the ozone-level concentration, in the atmosphere of the city of New York, USA.

Figure 7(a–c) shows the qq-plot for the LPN, LN and LSN calculated with the estimates of the parameters in each model. Figure 6(b) contains the empirical CDF for variable Y (solid line), whereas the dotted line corresponds to the CDF for the LPN model.

A test to compare the LPN model against the LSN model requires a non-nested approach. With Fθ and Gβ as two non-nested models and f(yi | xi,θ) and g(yi | xi,β) as the corresponding non-nested densities, the likelihood ratio statistic to compare both models is given by

where

is an estimator for the variance of (Vuong, 1989). This statistic corresponds to the distance between the two models measured in terms of the Kullback–Liebler information criterion. Hence, it was shown that, as n → ∞ ,

under

that is, models are equivalent.

At the δ% critical level, with zδ ∕ 2 as the critical value, we reject that the models are equivalent if | TLR,NN | > zδ ∕ 2.

On the other hand, we reject at the significance level δ the null hypothesis that the models are equivalent in favor of model Fθ (or model Gβ) if TLR;NN > zδ ∕ 2 (or TLR;NN < − zδ ∕ 2).

For the data set under study, with Fθ being the LPN model and Gβ the LSN model, Vuong's approach leads to the observed value TLR,NN = 21.819, which is greater than the critical value z0.025 = 1.96, and hence, the LPN distribution is better than the LSN distribution at the 5% level. The preceding results illustrate the fact that the LPN model is a viable alternative for fitting positive data with asymmetry and kurtosis not contemplated by the LN model. Moreover, results call for an update of Ahrens' law. Parameter estimates for the parameters of the preceding models were computed using library optim in R (R Development Core Team, 2012).

### CONCLUSIONS

In this paper, a new asymmetric distribution was introduced for fitting asymmetric positive data. It extends the LN family of distribution, and hence, it has more flexibility in terms of asymmetry and kurtosis. Maximum likelihood and moment estimation approaches are implemented. A real data illustration reveals that the proposed model can improve over existing alternative models.

### Acknowledgements

We acknowledge two referees for comments and suggestions that substantially improved the presentation. The research of H. Bolfarine was supported by CNPq (Brasil). The research of H. W. Gómez was supported by FONDECYT (Chile) 1130495.

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