The derivation of the log-moments of the NL distribution can be carried out by means of its log-cumulants. Therefore, firstly we obtain the log-cumulants of the NL distribution and then the estimators of the three parameters of the NL distribution are calculated by using the log-moments method. Finally, the accuracy and robustness of the fading parameter estimators of the NL distribution are briefly discussed.

### Log-cumulants of the Nakagami-lognormal Distribution

Let *r* be a NL RV corresponding to the received field strength expressed in V/m. The PDF of *r* is given by ([2], (1)):

$$\begin{array}{*{20}l} p_{r} \left(r \right)=&\int_{0}^{\infty} \frac{2}{\Gamma (m)} \left(\frac{m}{\omega}\right)^{m} {r}^{2m-1} \exp \left(-\frac{m {r}^{2}}{\omega} \right) \\ &\times \frac{1}{\sqrt{2 \pi}\sigma \omega} \exp \left(-\frac{\left(\ln (\omega)-\eta \right)^{2}}{2 {\sigma}^{2}} \right) d \omega \quad r\geq 0 \end{array} $$

(1)

where *m* is the shape factor or fading parameter of the Nakagami-*m* fast fading process, \(\Gamma (z)=\int _{0}^{\infty } u^{z-1} e^{-u} du\) is the Gamma function ([21], (6.1.1)), and:

$$\begin{array}{*{20}l} &\eta=K \eta_{d}, \end{array} $$

(2)

$$\begin{array}{*{20}l} &\sigma=K \sigma_{d} \end{array} $$

(3)

where:

$$ K=\frac{\ln(10)}{10} $$

(4)

*η*_{d} and *σ*_{d} are the mean, in dBV/m, and the standard deviation, in dB, respectively, of the associated Gaussian shadowing process.

The cumulative distribution function (CDF) of the NL distribution can be expressed as:

$$\begin{array}{*{20}l} F_{r} (r)=&\int_{0}^{\infty} \frac{1}{\Gamma(m)} \gamma \left(m,\frac{m}{\omega} r^{2} \right) \frac{1}{\sqrt{2 \pi}\sigma \omega} \\ &\times \exp \left(-\frac{\left(\ln (\omega)-\eta \right)^{2}}{2 \sigma^{2}} \right) d \omega \quad r\geq 0 \end{array} $$

(5)

where \(\gamma (a,z)=\int _{0}^{z} u^{a-1} \exp (-u) du\) is the lower incomplete Gamma function ([21], (6.5.2)).

The *n*th moment of *r* is calculated as:

$$ M_{n}^{(r)}=E\left(r^{n} \right)=\frac{\Gamma \left(m+\frac{n}{2} \right)}{m^{\frac{n}{2}}\Gamma(m)} \exp{\left(\frac{n}{2} \eta +\frac{n^{2}}{8} \sigma^{2} \right)} $$

(6)

where *E*(·) denotes the expectation operator. Note that (6) is equivalent to ([22], (2.59)) by substituting 2*n* instead of *n* in (6) which corresponds to the *n*th moment of the gamma-lognormal distribution. Thus, the first three moments of a NL distribution are given by:

$$\begin{array}{*{20}l} M_{1}^{(r)}&=\frac{\Gamma \left(m+\frac{1}{2} \right)}{m^{\frac{1}{2}}\Gamma(m)} \exp{\left(\frac{\eta}{2} +\frac{\sigma^{2}}{8} \right)} \end{array} $$

(7)

$$\begin{array}{*{20}l} M_{2}^{(r)}&=\exp{\left(\eta +\frac{\sigma^{2}}{2} \right)} \end{array} $$

(8)

$$\begin{array}{*{20}l} M_{3}^{(r)}&=\frac{\Gamma \left(m+\frac{3}{2} \right)}{m^{\frac{3}{2}}\Gamma(m)} \exp{\left(\frac{3 \eta}{2} +\frac{9 \sigma^{2}}{8} \right)} \end{array} $$

(9)

In order to derive the log-moments of the NL distribution, we can calculate the first characteristic function of the second kind for the NL distribution by means of the Mellin transform framework of [19]. The first characteristic function of the second kind is defined as ([19], (8)):

$$ \phi_{r}^{(s)}=\int_{0}^{\infty} u^{s-1} p_{r}(u) du $$

(10)

and the first characteristic function of the second kind can alternatively be derived as ([19], (13)):

$$ \phi_{r}^{(s)}=M_{n}^{(r)} \big|_{n=s-1} $$

(11)

Substituting (6) into (11), we obtain the first characteristic function of the second kind of the NL distribution as:

$$ \phi_{r}^{(s)}=\frac{\Gamma \left(m+\frac{s-1}{2} \right)}{m^{\frac{s-1}{2}}\Gamma(m)} \exp{\left(\frac{s-1}{2} \eta +\frac{\left({s-1} \right)^{2}}{8} \sigma^{2} \right)} $$

(12)

The characteristic function of the second kind, defined as the natural logarithm of the first characteristic function of the second kind, can be calculated as:

$$ \Phi_{r}^{(s)}=\ln \left({\phi_{r}^{(s)}} \right) $$

(13)

The log-cumulants of order *n* can be obtained from \(\Phi _{r}^{(s)}\) as ([19], (13)):

$$ \kappa_{n}^{(r)}={\frac{\partial^{n} \Phi_{r}^{(s)}}{\partial s^{n}}}\Big|_{s=1} $$

(14)

From (12) and (13), we can calculate:

$$\begin{array}{*{20}l} \frac{\partial \Phi_{r}^{(s)}}{\partial s}&=\frac{1}{2} \left({\eta+\frac{s-1}{2}\sigma^{2}-\ln(m)+\psi \left({m+\frac{s-1}{2}}\right)}\right) \end{array} $$

(15)

$$\begin{array}{*{20}l} \frac{\partial^{2} \Phi_{r}^{(s)}}{\partial s^{2}}&=\frac{1}{4} \left({\sigma^{2}+\psi^{(1)} \left({m+\frac{s-1}{2}}\right)}\right) \end{array} $$

(16)

$$\begin{array}{*{20}l} \frac{\partial^{n} \Phi_{r}^{(s)}}{\partial s^{n}}&=\left({\frac{1}{2}} \right)^{n} \psi^{(n-1)} \left({m+\frac{s-1}{2}}\right),\ n\geq3 \end{array} $$

(17)

where \(\psi (x)=\frac {\partial \ln \Gamma (x)}{\partial x}\) is the psi (digamma) function ([21], (6.3.1)) and \(\psi ^{(n)}=\frac {\partial ^{n} \psi (x)}{\partial x^{n}}\) is the polygamma function of *n*th order ([21], (6.4.1)).

Hence, substituting *s*=1 in (15), (16) and (17), the log-cumulants of the NL distribution can be derived as:

$$\begin{array}{*{20}l} \kappa_{1}^{(r)}&=\frac{1}{2} \left({\eta-\ln(m)+\psi(m)} \right) \end{array} $$

(18)

$$\begin{array}{*{20}l} \kappa_{2}^{(r)}&=\frac{1}{4} \left(\sigma^{2}+\psi^{(1)} (m) \right) \end{array} $$

(19)

$$\begin{array}{*{20}l} \kappa_{n}^{(r)}&=\left({\frac{1}{2}} \right)^{n} \psi^{(n-1)}(m),\ n\geq3 \end{array} $$

(20)

Applying the additive properties of the log-cumulants ([19], (17)), the log-cumulant of *n*-th order in the NL distribution can also be calculated as the sum of the log-cumulants of *n*-th order of both the Nakagami-*m* and the lognormal distribution. The log-cumulants of the Nakagami-*m* distribution are given by ([19], p. 149) and the first two log-cumulants of the lognormal distribution are given by ([23], (12)). Note that the log-cumulants of the lognormal distribution are equal to 0 for orders higher than 2.

Using the transformation *β*= ln*r*, the PDF of the log NL distribution can be obtained from (1) as:

$$\begin{array}{*{20}l} p_{\beta} (\beta)&=\int_{0}^{\infty} \frac{2}{\Gamma (m)} \left(\frac{m}{\omega}\right)^{m} \exp \left(2 m\beta -\frac{m e^{2 \beta}}{\omega} \right) \\ &\times \frac{1}{\sqrt{2 \pi}\sigma \omega} \exp \left(\,-\,\frac{\left(\ln (\omega)-\eta \right)^{2}}{2 \sigma^{2}} \right) d \omega\!\! \quad \,-\,\infty \!<\!\beta\!<\!\infty \end{array} $$

(21)

### Estimators of the Nakagami-lognormal Distribution

The estimators of the NL distribution based on the log-moments method are derived next. In the log-moments method, the moments of the distribution in logarithmic units are related to the parameters of the distribution. Using the log-cumulants of the NL distribution derived previously, the log-moments estimators are obtained.

The relations between log-cumulants and log-moments are identical to the relations existing between moments and cumulants ([19], Section 2.3). For instance, the three first log-cumulants can be written as:

$$\begin{array}{*{20}l} \kappa_{1}^{(r)}&=M_{1}^{(\beta)}=E \left({\beta}\right) \end{array} $$

(22)

$$\begin{array}{*{20}l} \kappa_{2}^{(r)}&=\mu_{2}^{(\beta)}=E \left({\left({\beta-\bar{\beta}}\right)^{2}}\right) \end{array} $$

(23)

$$\begin{array}{*{20}l} \kappa_{3}^{(r)}&=\mu_{3}^{(\beta)}=E \left({\left({\beta-\bar{\beta}}\right)^{3}}\right) \end{array} $$

(24)

where *β*= ln*r*, \(M_{1}^{(\beta)}\) is the first log-moment and \(\mu _{2}^{(\beta)}\) and \(\mu _{3}^{(\beta)}\) are the second and third central log-moments, respectively.

In the NL distribution, from (20) and (24) with *n*=3, an estimator of *m* can be derived by using a numerical approximation of the inverse function of *ψ*^{(2)}(·) as:

$$ \hat{m}= 1.4 \left(-\hat{v} \right)^{-0.391+0.0096 \ln \left(-\hat{v} \right)},\ -16.8\leq \hat{v} < 0 $$

(25)

where \(\hat {v}\) is given by:

$$ \hat{v}= {8} \hat{\mu}_{3}^{(\beta)} $$

(26)

\(\hat {\mu }_{n}^{(\beta)}\) is the *n*th sample central moment of the log NL distribution defined as:

$$\begin{array}{*{20}l} \hat{\mu}_{n}^{(\beta)}& = \frac{1}{N} \sum\limits_{i=1}^{N} \left(\beta_{i} -\hat{M}_{1}^{(\beta)} \right)^{n} \\ &=\frac{1}{N} \sum\limits_{i=1}^{N} \left(\ln r_{i} - \hat{M}_{1}^{(\beta)} \right)^{n} \end{array} $$

(27)

where:

$$ \hat{M}_{1}^{(\beta)}= {\frac{1}{N}} \sum\limits_{i=1}^{N} \ln r_{i} $$

(28)

is the sample mean of the log NL distribution, *r*_{i},*i*=1,…,*N* are the samples of the realization corresponding to the field strength in linear units, and *N* is the number of samples. Note that in (25), \(\hat {m} \left (\hat {v} =-16.8 \right) \approx 0.5\) and \({{{\lim }_{\hat {v} \to 0^{-}}} \hat {m} \rightarrow \infty }\).

From (18), (19), (22), and (23), we can estimate the other parameters of the NL distribution as:

$$\begin{array}{*{20}l} \hat{\sigma}&=\sqrt{{4} \hat{\mu}_{2}^{(\beta)}-\psi^{(1)} \left(\hat{m}\right)} \end{array} $$

(29)

$$\begin{array}{*{20}l} \hat{\eta}&= {2} \hat{M}_{1}^{(\beta)}+\ln \left({\hat{m}} \right)-\psi\left(\hat{m} \right) \end{array} $$

(30)

Both the *ψ*^{(1)}(·) trigamma and *ψ*(·) digamma functions can be calculated using either the psi or Polygamma functions of Matlab^{Ⓡ} and Mathematica, respectively. These functions have been implemented using Euler-Maclaurin summation, functional equations, and recursion ([24], p. 58).

Figure 1 shows the fading parameter estimator, \(\hat {m}\), of the Nakagami-*m* process as a function of the \(\hat {v}\) estimator in logarithmic units using (25). Figure 1 is split into two plots to illustrate the small variation of \(\hat {m}\) for values of \(\hat {v}\) from − 16.5 to − 1 in the upper plot and the steep increase of \(\hat {m}\) for values of \(\hat {v}\) from − 1 to 0 in the lower plot. The slope of this function is not substantially high for \(-\thinspace 16.5 \leq \hat {v} \leq -0.155\) which corresponds to values of \(\hat {m}\) from 0.5 to 3. For \(\hat {v}\) higher than − 0.155, equivalent to \(\hat {m}>3\), the slope of the function increases significantly, i.e., slight variations of \(\hat {v}\) provide high modifications in the estimated values of the fading parameter. For instance, \(\frac {\partial \hat {m}}{\partial \hat {v}}=22.02\) for \(\hat {m}=4\).

The relative error of the expression (25) as the approximation of the inverse of *ψ*^{(2)}(·) as a function of *m* from 0.5 to 5 is plotted in Fig. 2 using Mathematica. The maximum relative error in this interval of *m* from 0.5 to 5 is 0.55% which occurs for *m* = 1.89 and the relative error remains below 0.97% for *m* until 23.7.

In contrast, it can be shown that the estimators of the NL distribution using the MM (instead of the log-moments method) do not provide stable results. Specifically, in the case of the MM, from (7), (8), and (9), \(\hat {m}\) can be calculated as:

$$ \hat{m}\,=\,\frac{\exp({-1})}{\sqrt{-\hat{u}}}-0.3205^{0.0725+0.0272 \ln(-\hat{u})},\ \!-0.21\!\leq\! \hat{u} < 0 $$

(31)

where \(\hat {u}\) is an estimator which uses the first three sample moments of the NL distribution in linear units as:

$$ \hat{u}=3 \ln \hat{M}_{1}^{(r)}-3 \ln \hat{M}_{2}^{(r)}+ \ln \hat{M}_{3}^{(r)} $$

(32)

where \(\hat {M}_{n}^{(r)}\) is the sample *n*-th moment of the NL distribution defined as:

$$ \hat{M}_{n}^{(r)}=\frac{1}{N} \sum\limits_{i=1}^{N} r_{i}^{n} $$

(33)

The unreliability of \(\hat {m}\) in the MM is due to the considerable slope of \(\hat {m}\) as a function of \(\hat {u}\). For instance, \(\frac {\partial \hat {m}}{\partial \hat {u}}=46.89\) and 141.30 for \(\hat {m}=2\) and \(\hat {m}=3\), respectively.

On the other hand, the ML estimation method of the NL distribution leads to a set of three non-linear equations involving *N*-fold definite integrals which have to be solved by numerical methods without a unique solution. Also, numerical methods based on iterative maximization algorithms, such as those implemented in the mle function of Matlab^{Ⓡ}, require starting points for the parameters and fail to converge if the initial parameter values are far from the ML estimators ([25], p. 323). This is the case for other similar lognormal-based distributions such as the three-parameter lognormal distribution where the search for the local ML estimators must be conducted with great care ([26], p. 122). Furthermore, like the MM estimators, the ML estimators of the NL distribution are also highly sensitive to slight changes of the data.

Based on these encouraging observations, we now move on to the precise evaluation of the suitability of the log-moment estimator of the NL distribution, by using the normalized mean square error (NMSE), the sample mean, and the sample confidence region introduced in the next Section.