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An Efficient, Closed-Form MAP Estimator for Nakagami-m Fading Parameter Pedro Luiz Ramos, Francisco Louzada, and Eduardo Ramos Abstract— A maximum a posteriori (MAP) estimator for the Nakagami-m fading parameter is proposed. The MAP estimator has a simple closed-form expression and can be rewritten as a bias corrected generalized moment estimator. Numerical results demonstrate that the MAP estimation scheme outperforms the existing estimation procedures and produces almost unbiased estimates for the fading parameter even for a small sample size. Index Terms— Nakagami-m fading parameter, maximum a posteriori estimator, closed-form m estimator, almost unbiased estimator.
MAP estimation scheme outperforms the existing estimation procedures and produces almost unbiased estimates for the fading parameter even for small sample size. II. C OMMON E STIMATION P ROCEDURES In this section, we review three useful estimation procedures used to obtain the estimates for μ and of the NK distribution. A. Moments Estimators
I. I NTRODUCTION
T
HE Nakagami-m (NK) distribution is a powerful statistical tool introduced for modeling radio links. A random variable X has the probability density function (p.d.f) given by μ 2 μ μ 2μ−1 (1) t exp f (t|μ, ) = t2 , (μ) for all t > 0, where μ ≥ 0.5 and > 0 are the shape (also known as a fading parameter) and scale parameters, respectively [1]. An unbiased estimator for the scale parameter can be easily obtained through the method of moments [1]. However, considerable effort has been made to derive efficient estimators for the fading parameter. The maximum likelihood (ML) estimators have been discussed earlier [2]. The accuracy of different procedures have been compared numerically and the use of a closed-form approximation to the ML estimator has been suggested [3]. A generalized moments estimator has been presented [4]. Some estimators based on approximations of the transcendental equations that arise in the computation of ML and the generalized moment (GM) estimators have been discussed [5]. However, such estimators are only approximations to the natural methods motivated by fast computation avoiding solving nonlinear equations. A closed-form estimator for the fading parameter obtained as a limiting procedure of the traditional GM estimators have been proposed [6]. In this letter, firstly we prove that for the shape parameter, all estimation procedures lead to the same unbiased estimator. Secondly, we discuss another approach based on the maximum likelihood estimator that leads to the same limiting GM estimator. Then, we present the maximum a posteriori (MAP) estimator for the Nakagami-m fading parameter. We prove that such estimator can be rewritten as a bias corrected GM estimator. Finally, numerical results have shown that the
Manuscript received August 2, 2016; accepted August 24, 2016. Date of publication August 26, 2016; date of current version November 9, 2016. The associate editor coordinating the review of this letter and approving it for publication was F. Gao. The authors are with the Institute of Mathematical Science and Computing, University of São Paulo, São Carlos 05508-900, Brazil (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2016.2603530
The moments estimators [1] are n 2 2 i=1 ti μˆ ME = n 2 2 n ni=1 ti4 − i=1 ti
and
n ˆ ME = 1 ti2 . n i=1
(2) ˆ M E is an Since the second moment for the NK is , then unbiased estimator for . B. Maximum Likelihood Estimators The maximum likelihood estimates for the Nakagami distribution [2] are obtained by maximizing the likelihood function, given by n
n 2n μ nμ 2μ−1 μ 2 L(, μ; t) = ti ti . exp − (μ)n i=1
i=1
(3) ∂ ∂
From the expressions log L(, μ; t) = 0, log L(, μ; t) = 0 and after some algebraic manipulations ˆ MLE = 1 ni=1 t 2 and the MLE the MLE for is given by i n for μ is obtained by solving the non-linear equation n 1 ˆ MLE − log (μˆ MLE ) −ψ (0) (μˆ MLE ) = log log(ti2 ), n ∂ ∂μ
i=1
(4) ∂ m+1 m+1
where ψ (m) (k) = ∂k function of order m.
log (k) = log (k) is the polygamma
C. Generalized Moment Estimators An useful estimation procedure for the fading parameter of the NK distribution was proposed earlier [4] and is given through the fractional moment estimator μˆ 1/ p =
mˆ 1/ p mˆ 2 , 2 p mˆ 2+1/ p − mˆ 1/ p mˆ 2
where the kth-order moment is given by
(μ + k/2) k/2 mk = (μ) μ
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RAMOS et al.: EFFICIENT, CLOSED-FORM MAP ESTIMATOR FOR NAKAGAMI-m FADING PARAMETER
and p is a positive real number. Further, the limiting estimator, μˆ 0 , that combined with the fractional moment estimator [6], is given by ⎧ mˆ 1/ p mˆ 2 ⎪ ⎪ , k>0 ⎨ 2 p mˆ 2+1/ p − mˆ 1/ p mˆ 2 μˆ k = mˆ 2 ⎪ ⎪ 2 1 , k = 0 ⎩ 1 n 2 ˆ 2 ni=1 log ti2 i=1 ti log ti − n m n (6) The authors showed that in the limiting case, μˆ 0 is expected to achieve the best performance among this fractional momentbased estimator family. Hence, we considered only the case when k = 0, namely μˆ G M E . It was also presented for k = 0 the asymptotic variance, Var(μ M A P ), given by V ar (μGME ) = μ2 + μ3 ψ (1) (μ + 1).
(7)
In the next section, we present a justification for μˆ G M E based on the maximum likelihood estimators. III. A G ENERALIZATION OF THE NAKAGAMI -m D ISTRIBUTION
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IV. BAYESIAN A NALYSIS In this section, we consider the Bayesian inference of the unknown parameters of the GNK. Initially, the most common objective priors were considered such as Jeffreys Prior, Reference Prior, MDI prior (see [7] for more details). However, such priors depend on polygamma function which did not allow to obtain MAP estimators in closed form. The chosen objective prior for the parameters is given by π (θ ) ∝
1
(13)
c1 μc2 α c3
where ci ≥ 0, i = 1, 2, 3 are known hyperparameteres. From the product of the likelihood function (9) and the prior distribution (13), the joint posterior distribution for θ is given by μ nμ α n−c3 1 π(θ|t) = c c n d(t) μ 2 1 (μ) n
n αμ μ α × ti ti , exp − (14) i=1
Let X be a random variable with Nakagami-m distribution, then a generalized Nakagami-m (GNK) distribution, with a parameter vector θ = (, μ, α), can be obtained by taking α T = X 2 , where α > 0 and its p.d.f is given by μ α μ μ αμ−1 f (t|θ ) = tα . t exp (8) (μ)
where
i=1
d(t) =
μ nμ α n−c3 μc2 c1 (μ)n A
n n αμ μ α exp − × ti ti dθ i=1
1
i=1
This generalization follows from [4], which considers X p , α p > 0. However, by considering X 2 , the algebraic manipulation necessary to obtain closed form estimates for the parameters are simpler. Moreover, such generalizations are rewritten forms of the generalized Gamma distribution. The likelihood function from (8) is given by n
n μ α α n μ nμ αμ−1 ti ti . (9) exp − L(θ ; t) = (μ)n
and A = {(0, ∞) × (0.5, ∞) × (, M)} is the parameter space of θ, where 0 < < 2 is a small constant and M > 2 is a large constant. We chose (, M) for the interval of α since the only interest is in the case where α = 2. Therefore, any interval (, M) containing α = 2 will be satisfactory for our purposes. The MAP of θ is computed through θˆ M A P = arg max log (π(θ|t)). After some algebraic manipulation we
The maximum likelihood estimates of the parameters are obtained by solving the following likelihood equations n μˆ = (10) n α , n 1 α α i=1 ti log ti − i=1 log ti ˆ
have
i=1
i=1
ˆ =
1 n
n
tiαˆ
(11)
i=1
and the MLE for α is obtained solving the non-linear equation n 1 (0) ˆ log(μ) ˆ − ψ (μ) (12) ˆ = log() − log tiαˆ . n i=1
In case α = 2 then (10) is the same as (6) for k = 0 ˆ = 1 ni=1 t 2 is an unbiased estimator. In fact, the and i n computational results obtained from (4) and (6) are almost the same. However, it is worth noting that in the case of small samples, the maximum likelihood estimator of μ has a systematic bias that decreases as n → ∞. Therefore, in the next section, we propose a maximum a posteriori (MAP) estimator for the fading parameter that is almost unbiased.
θ
ˆ =
μ ni=1 tiαˆ . nμ + c1
(15)
It is important to point out that (15) will be equal (11), if and only if c1 = 0, i.e, is unbiased when α = 2. Therefore, we consider only that c1 = 0. To obtain reliable inference results, we have to check if (14) is a proper posterior distribution, i.e, d(t) < ∞. Theorem 1: The posterior distribution (14) is proper. Proof: See Appendix B. The other MAP estimators are given by (n − c3 ) , α log t α − n log t α t i=1 i i=1 i i
μˆ = n 1 ˆ
(16)
and the MAP for α is obtained by solving the non-linear equation n 1 c2 ˆ − . log μˆ − ψ (0) (μ) ˆ = log log(tiαˆ ) + n n μˆ i=1
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Therefore, for α = 2, a hybrid MAP estimator of is given ˆ M A P = 1 ni=1 t 2 and the Nakagami-m fading parameter by i n can be estimated by (n − c3 ) n1 ni=1 ti2 μˆ MAP = n 2 2 1 n 2 n 2 . (17) i=1 ti log ti − n i=1 ti i=1 log ti Theorem 2: Let μˆ M A P be an estimator of μ , and let t = (t1 , . . . , tn ), with tn ≥ . . . ≥ t1 , not all equal, then for n > c3 we have: (n − c3 ) n1 ni=1 ti2 > 0. μˆ MAP = k 2 n−1 1 n 2− 1 2 2 k t i=1 i i=1 ti log(tk+1 /tk ) k=1 n k
Fig. 1. MREs for μ considering c3 = (2, 2.1, · · · , 4), μ = 6, = 10, n = 120 (left panel) μ = 20, = 2, n = 20 (right panel) and for N = 1, 000, 000 simulated samples and n = 50.
Proof: See Appendix A. Note that the MAP estimator does not depend on c2 . Moreover, the μˆ MAP can be rewritten as a bias corrected generalized moment estimator (n − c3 ) μˆ GME . (18) n Due to this relationship, the asymptotic variance, Var(μ M A P ), can be obtained by 3) V ar (μMAP ) = V ar (n−c ˆ GME n μ 2 2 + μ3 ψ (1) (μ + 1) . 3) μ = (n−c 2 n μˆ MAP =
Fig. 2. MREs, RMSEs for μ and considering μ = 4, = 2 for N = 1, 000, 000 simulated samples and n = (10, 15, . . . , 140).
The variance, Var( M A P ), can also be easily obtained considering 2 1 E[X 4 ] − E[X 2 ]2 = . V ar (MAP ) = n nμ V. R ESULTS In this section, we present Monte Carlo simulation studies to compare the efficiency of our proposed estimation method. The comparison between such procedures is carried out by computing the mean relative error (MRE) and the root-meansquare error (RMSE), given by N N (θˆi, j − θ j )2 1 θˆi, j , and RMSEθi = MREθi = N θi N j =1
j =1
for i = 1, 2, where N = 1, 000, 000 is the number of estimates obtained through the ME, ML, GM and MAP estimators. The MRE and the RMSE of the are the same for the different estimation procedures. Considering this approach, we expect that the most efficient estimators would yield the MREs closer to one with smaller RMSEs. These results were computed using the software R (R Core Development Team). The seed used to generate the pseudo-random samples from the NK distribution was 2016. The MAP estimator depends on c3 . Therefore, we have to find a value for c3 in which the MRE is closer to one. Figure 1 presents the MREs for μMAP considering different values of c3 , for μ = 20, = 2, n = 20 and μ = 20, = 2, n = 20. We omitted the results of the simulation study for different values of μ, and n since they are similar to the one presented here.
Fig. 3. MREs, RMSEs for μ considering μ = 4, = 80 for N = 1, 000, 000 simulated samples and n = (10, 15, . . . , 140).
From the Figure 1, we observed that a good choice is c3 = 3. Therefore, we considered that c3 = 3 in (17). Figures 2-5 show the MREs, RMSEs from the estimates of μ obtained using the MC method. Figure 2 also presents the MREs, RMSEs from the estimates of , we omitted the other graphics since they were similar. The horizontal lines in the figures correspond to MREs and RMSEs being one and zero respectively. From the Figures 2 and 3, we observed that the estimates of the fading parameter are asymptotically unbiased, i.e., the MREs tend to one when n increases and the RMSEs decrease to zero. Moreover, the MAP estimators present extremely efficient estimates for μ even for small sample sizes, for instance, considering n = 15, the errors related to the MAP are of the order 10−3 while for the GME are 10−1 , i.e, the MAP estimator is almost unbiased for small samples. From Figures 4 and 5, we obtained similar results considering different values of μ. Taking into account the results of the
RAMOS et al.: EFFICIENT, CLOSED-FORM MAP ESTIMATOR FOR NAKAGAMI-m FADING PARAMETER
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A PPENDIX B P ROOF OF T HEOREM 1
Fig. 4. MREs, RMSEs for different values of μ = (0.5, 1.0, . . . , 20) considering = 2 for N = 1, 000, 000 simulated samples and n = 50.
Proof: Let B = {(, M)×[0.5, ∞)×(0, ∞)} and consider the change in the coordinates through the transformation θ : B → A, where (α, μ, ) = θ (β, φ, λ) = β, φ, φλ and A = θ(B) Noticing that | det(D θ (φ, λ))| = φλ−2 , denoting = (β, φ, λ) and applying the change of variables on the Lebesgue integral and the Fubini-Tonelli Theorem (see [8]) we have that n
n μ α α n−c3 μ nμ αμ exp − ti ti dθ d(t) ∝ μc2 (μ)n i=1 i=1 A
n−c3 nφ−2 n n β λ βφ β exp −λ = ti ti d φ c2 −1 (φ)n i=1
B
M ∞ ∞ ∝ 0.5 0
β n−c3 φ c2 −1 (φ)n
Fig. 5. MREs, RMSEs for different values of μ = (0.5, 1.0, 1.5, . . . , 20) considering = 80 for N = 1, 000, 000 simulated samples and n = 50.
=
simulation studies, the MAP estimators should be considered for estimating the fading parameter of the NK distribution.
∝
A PPENDIX A P ROOF OF T HEOREM 2 Proof: We have that (n − c3 ) n1 ni=1 ti2 μˆ = n 2 1 n 2 2 k=1 tk − n i=1 ti log(tk ) 1 n (n − c3 ) n i=1 ti2 = n 2 k 2 n (k−1) n 2 − k−1 t 2 − k 2 t k=1 i=1 i i=1 ti + i=1 ti log(tk ) i=1 i n n (n − c3 ) n1 ni=1 ti2 = k 2 n−1 1 n 2− 1 2 2 k t i=1 i i=1 ti log(tk+1 /tk ) k=1 n k Now, since tn ≥ tn−1 ≥ · · · ≥ t1 >0, we have that n k 1 2 /t 2 ) ≥ 0 and 1 2 2 log(tk+1 i=1 ti > k i=1 ti for every k n 1 ≤ k < n. Moreover, since t1 , · · · , tn are not all equal, we have that log(t 2j +1 /t 2j ) > 0 for some 0 ≤ j < n. Then n−1 1 n 2 1 k 2 2 2 i=1 ti − k i=1 ti log(tk+1 /tk ) > 0 which, k=1 k n for n > c3 , implies that μˆ > 0.
β
n−c3
β n−c3
i=1
dλdφdβ β φ n t i=1 i (nφ − 1) dφdβ φ c2 −1 (φ)n n β nφ−1 t i=1 i
n
β ti
i=1
M
VI. D ISCUSSION In this work, we have introduced a MAP estimator for the Nakagami-m fading parameter. Some mathematical properties for this new estimator are presented. We show that such estimator can be rewritten as a bias corrected generalized moment estimators. Numerical results have shown that the MAP estimator outperforms the existing estimator procedures for the μ parameter. In addition, we conclude that the MAP estimator present almost unbiased estimates for the fading parameter even for small sample sizes.
0.5
M
βφ
ti
β i=1 ti
∞ β n−c3
n
n
×λnφ−2 e−λ M
i=1
n
∞ φ
n−1−2c2 2
enp(β)φ dφdβ
0.5 β
2 ( n+1−2c , 0.5n p(β)) 2
dβ < ∞ n+1−2c2 (n p(β)) 2
β β −1 n n 1 n > 0 by the where p(β) = i=1 ti i=1 ti n ∝
ti
i=1
Inequality of the arithmetic and geometric mean.
ACKNOWLEDGMENT We are thankful to the Editorial Board and Referees for their valuable comments and suggestions. Thanks to the Brazilian Organizations CAPES, CNPq and FAPESP for the founding. R EFERENCES [1] N. Nakagami, “The m-distribution—A general formula of intensity distribution of rapid fading,” in Proc. Symp. Statist. Methods Radio Wave Propag., 1960, pp. 3–36. [2] J. Cheng and N. C. Beaulieu, “Maximum-likelihood based estimation of the Nakagami m parameter,” IEEE Commun. Lett., vol. 5, no. 3, pp. 101–103, Mar. 2001. [3] Q. T. Zhang, “A note on the estimation of Nakagami-m fading parameter,” IEEE Commun. Lett., vol. 6, no. 6, pp. 237–238, Jun. 2002. [4] J. Cheng and N. C. Beaulieu, “Generalized moment estimators for the Nakagami fading parameter,” IEEE Commun. Lett., vol. 6, no. 4, pp. 144–146, Apr. 2002. [5] J. Gaeddert and A. Annamalai, “Further results on Nakagami-m parameter estimation,” in Proc. IEEE 60th Veh. Technol. Conf. (VTC-Fall), vol. 6. Sep. 2004, pp. 4255–4259. [6] N. Wang, X. Song, and J. Cheng, “Generalized method of moments estimation of the Nakagami-m fading parameter,” IEEE Trans. Wireless Commun., vol. 11, no. 9, pp. 3316–3325, Sep. 2012. [7] J. M. Bernardo, “Reference analysis,” Handbook Statist., vol. 25, pp. 17–90, 2005. [8] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed. New York, NY, USA: Wiley, 1999.
