Lognormal Sum Approximation with Type IV Pearson Distribution

In this paper, the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables, and the parameters of the Type IV Pearson distribution are derived through matching the mean, variance, skewness and kurtosis of the two distributions. Numerical simulations show that the Type IV Pearson distribution can accurately approximate distribution of the sum of lognormal random variables in a wide probability range.     

Lognormal Sum Approximation with a Variant of Type IV Pearson Distribution

In this paper, a variant of the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables. Numerical and computer simulations show that independent of the statistical characteristics of the lognormal sum distribution, the Type IV Pearson variant outperforms the standard Type IV Pearson distribution and the normal variant distribution in accurately approximating the lognormal sum distribution for a whole probability range.     

An optimal lognormal approximation to lognormal sum distributions

Sums of lognormal random variables occur in many problems in wireless communications because signal shadowing is well modeled by the lognormal distribution. The lognormal sum distribution is not known in the closed form and is difficult to compute numerically. Several approximations to the distribution have been proposed and employed in applications. Some widely used approximations are based on the assumption that a lognormal sum is well approximated by a lognormal random variable. Here, a new paradigm for approximating lognormal sum distributions is presented. A linearizing transform is used with a linear minimax approximation to determine an optimal lognormal approximation to a lognormal sum distribution. The accuracies of the new method are quantitatively compared to the accuracies of some well-known approximations. In some practical cases, the optimal lognormal approximation is several orders of magnitude more accurate than previous approximations. Efficient numerical computation of the lognormal characteristic function is also considered.     

Highly accurate simple closed-form approximations to lognormal sum distributions and densities

Sums of lognormal random variables occur extensively in wireless communications, in part, because a shadowing environment is well modeled by a lognormal distribution. A closed-form expression does not exist for the sum distribution and, furthermore, it is difficult to numerically calculate the distribution. Numerous approximations exist that are based on approximating a sum of lognormal random variables as another lognormal random variable. A new paradigm to calculate an approximation to the lognormal sum distribution, based on curve fitting on lognormal probability paper, is introduced in this letter. Highly accurate, simple closed-form approximations to lognormal sum distributions are presented.     

Outage Probability with Correlated Lognormal Interferers using Log Shifted Gamma Approximation

The computation of outage probability in cellular radio system has been extensively studied. The Signal-to-Interference-plus-Noise Ratio (SINR) distribution involves the sum of lognormal distributions due to dominant effect of shadowing in both the signal and interference components. Since no closed-form expression can be found for the sum of lognormal distributions, many approximation methods and bounds were proposed in the past. In this paper, Log Shifted Gamma (LSG) approximation is proposed to calculate the sum of correlated lognormal random variables (RVs), hence the outage probability, accurately with a wide range of dB spreads, number of interferers M, correlation coefficients r among interference components, and noise power N. Overall, LSG approximation shows consistent accuracy due to its flexibility over the classical lognormal approximation, especially with small correlation coefficients r and/or large dB spreads.     

Bounds on the Distribution of a Sum of Correlated Lognormal Random Variables and Their Application

The cumulative distribution function (cdf) of a sum of correlated or even independent lognormal random variables (RVs), which is of wide interest in wireless communications, remains unsolved despite long standing efforts. Several cdf approximations are thus widely used. This letter derives bounds for the cdf of a sum of 2 or 3 arbitrarily correlated lognormal RVs and of a sum of any number of equally-correlated lognormal RVs. The bounds are single-fold integrals of readily computable functions and extend previously known bounds for independent lognormal summands. An improved set of bounds are also derived which are expressed as 2-fold integrals. For correlated lognormal fading channels, new expressions are derived for the moments of the output SNR and amount of fading for maximal ratio combining (MRC), selection combining (SC) and equal gain combining (EGC) and outage probability expressions for SC.     

Optimal Replacement Rate of Devices with Lognormal Failure Distributions

The time dependence of the replacement rate of devices with lognormal failure distributions is studied. The relationship between the peak replacement rate and a, the standard deviation in the corresponding s-normal distribution, is obtained. The study assumes that each device represents a renewal process and the system of devices represents a superimposed renewal process. The peak replacement rate becomes very large for both extreme values of a. The corresponding replacement rate eventually approaches the conventional asymptotic rate only after many MTTFs (Mean Time To Failure), possibly long after the system becomes obsolete. Ways to cope with this situation are suggested. However, the replacement rate curve becomes almost critically damped with a minimum-peak-factor ?1.03 at ? ?0.63. The study also reveals that asymptotically the lognormal devices with ? << 1 would require fewer replacements than the exponential devices with the same MTTF. The difference is equal to one-half of the population in service. For large a, more replacements are required. The results of this paper apply to a variety of reliability studies and maintenance and inventory strategies for communication systems that employ lognormal devices such as a semiconductor laser, LED, avalanche photo diode, or IMPATT diode. In particular, the present approach is especially helpful when such a device is the least reliable component in the system.     

Approximation for a Sum of on-off Lognormal Processes With Wireless Applications

In this paper, a lognormal approximation is proposed for the sum of lognormal processes weighted by binary processes. The analytical approach moves from the method early proposed by Wilkinson for approximating first-order statistics of a sum of lognormal components, and extends to incorporate second-order statistics and the presence of both time-correlated random binary weights and cross-correlated lognormal components in moments' matching. Since the sum of weighted lognormal processes models the signal-to-interference-plus-noise ratio (SINR) of wireless systems, the method can be applied to evaluate in an effective and accurate way the outage occurrence rate and outage duration for different wireless systems of practical interest. In a frequency-reuse-based cellular system, the method is applied for various propagation scenarios, characterized by different shadowing correlation decay distances and correlations among shadowing components. A further case of relevant interest is related to power-controlled wideband wireless systems, where the random weights are binary random variables denoting the activity status of each interfering source. Finally, simulation results are used to confirm the validity of the analysis.     

A Simple Polynomial Approximation to the Gaussian Q-function and Its Application

A simple polynomial approximation to the Gaussian Q-function is proposed, based on the observation that a Gaussian random variable can be well approximated by a sum of uniform random variables. The approximation can be used to obtain accurate explicit approximations to problems that otherwise do not have explicit solutions or approximate explicit solutions. As an example, an explicit expression for the average symbol error rate of M-ary pulse amplitude modulation in lognormal channels is derived using the new approximation, and the approximate symbol error rate is shown to be very close to the exact value.     

Estimation of typical sum of lognormal random variables using log shifted gamma approximation

The sum of lognormal distributions is a well-known problem that no closed-form expression exists and it is difficult to evaluate numerically. In this paper, log shifted gamma (LSG) approximation method is proposed to represent the sum of lognormal distributions and to derive a closed-form expression of the typical value of the sum. Illustrative results show that the LSG model provides much more accurate approximation than other previous methods for a wide range of lognormal variances.     

Comments, with reply, on `On the hazard rate of the lognormaldistribution' by A.L. Sweet

In the above-named work (see ibid., vol.39, p.325-8, Aug 1990), A.L. Sweet calls attention to some incomplete or inaccurate representation of the hazard rate of lognormal distributions. His principal concern apparently is that these representations may lead modelers to overlook an important property of the lognormal distribution: its hazard rate increases and then decreases as a function of elapsed time. He also provides approximations for the location and value of the hazard rate maximum. The commenter observes that, as theoreticians are likely to have ready access to expanded tables of the normal distribution and unlikely to have need for the approximations, the Sweet paper was aimed at practitioners dealing with actual data. The commenter cautions such practitioners that some serious problems, unrelated to table availability or approximation accuracy, arise if σ_1n(T) is high or low. In his reply, Sweet remarks that the commenter is correct in assuming that Sweet's work was aimed at practitioners-specifically, at those being introduced to the lognormal distribution for the first time by reading about it in the books referenced in Sweet's article     

Capacity of wireless optical communications

We consider the ergodic capacity and capacity-versus-outage probability of direct-detection optical communication through a turbulent atmosphere using multiple transmit and receive apertures. We assume shot-noise-limited operation in which detector outputs are doubly stochastic Poisson processes whose rates are proportional to the sum of the transmitted powers, scaled by lognormal random fades, plus a background noise. In the high and low signal-to-background ratio regimes, we show that the ergodic capacity of this fading channel equals or exceeds that for a channel with deterministic path gains. Furthermore, knowledge of these path gains is not necessary to achieve capacity when the signal-to-background ratio is high. In the low signal-to-background ratio regime, path-gain knowledge provides minimal capacity improvement when using a moderate number of transmit apertures. We also develop expressions for the capacity-versus-outage probability in the high and low signal-to-background ratio regimes, by means of a moment-matching approximation to the distribution for the sum of lognormal random variables. Monte Carlo simulations show that these capacity-versus-outage approximations are quite accurate for moderate numbers of apertures.     

Simple accurate lognormal approximation to lognormal sums

A simple accurate lognormal approximation to the sum of independent non-identical lognormal variates is derived by matching the first two moments of the inverse exact sum with those of the inverse lognormal approximation. Sample examples are given to illustrate the excellent agreement between exact and approximate sum statistics.     

Accurate closed-form approximations to Ricean sum distributions and densities

The statistical distribution of a sum of Ricean random variables occurs extensively in wireless communications. A closed-form expression does not exist for the sum distribution and, furthermore, the Ricean random variable does not have a closed-form characteristic function. For these reasons, it is somewhat difficult to numerically calculate the sum distribution. Highly accurate, closed-form approximations for the sum distributions and densities are presented. These approximations are valid for a wide range of argument values, Rice factors and number of summands.     

对数正态分布下基于MLE的红外LED的寿命预测

为了获得红外LED的寿命信息,采用对数正态分布函数描述了红外LED的寿命分布,利用极大似然法(MLE)估计了对数均值和对数标准差,完成了恒定及步进应力实验数据的统计和分析,并自行开发了寿命预测软件。数值结果表明,红外LED的寿命服从对数正态分布,其加速模型符合逆幂定律。精确计算的加速参数使得快速估算红外LED寿命成为可能。     

对数正态概率纸的自动生成和分布参数的自动提取

采用对数正态概率纸可以直观地判断一组数据是否服从对数正态分布，而且还可以得到对数正态分布参数，即对数均值和对数标准差。介绍对数正态概率纸自动生成和分布参数自动提取的方法以及据此开发的软件。与手工使用概率纸的常规方法相比，采用所开发的软件不但使用简单、方便，而且能提高计算结果的精度。     

Small-Sample Estimation for the Lognormal Distribution With Unit Shape Parameter

Let X be a random variable such that In [(X - ?)/?] has a s-normal distribution with mean zero and variance one. Then X has a 3-parameter lognormal distribution with the third parameter, the shape parameter, fixed at unity. This paper presents the coefficients required to construct the best linear unbiased estimators (BLUEs) of ? and ? for samples of size fifteen and less. The variances and covariances of these estimators are provided. These estimators yield the BLUEs of the mean, standard deviation, and percentiles of X since these quantities are linear functions of ? and ?. Blom's estimators and maximum likelihood estimators compare favorably with the BLUEs.     

Device Failures During Equipment Life for Lognormal Distributions

A simple graph is presented showing the total fraction of devices which will fail over any given time for lognormal failure distributions. The result is a function of the median life and the geometric dispersion.     

Distinguishing between lognormal and Weibull distributions[time-to-failure data]

A previous method for deciding if a set of time-to-fail data follows a lognormal distribution or a Weibull distribution is expanded upon. Pearson's s-correlation coefficient is calculated for lognormal and Weibull probability plots of the time-to-fail data. The test statistic is the ratio of the two s-correlation coefficients. When "standardized", the lognormal and Weibull variables map into 1 of 2 gamma distributions with no dependence on the shape or scaling factors, confirming earlier observations. Using a set of Monte Carlo simulations, the test statistic was found to be s-normally distributed to good approximation. Formulas for estimating the mean and standard deviation of the test statistic were derived, allowing for an estimate of the probability of hypothesis test errors. As anticipated, the test capability increases with increasing sample size, but only if a substantial fraction of the parts actually fail. If less than 10% of the parts are stressed to failure, then it is almost impossible to distinguish between lognormal and Weibull distributions. If all parts are stressed to failure, the probability of making a correct choice is fair for sample sizes as small as 10, and becomes quite good if the sample size is at least 50. The statistical technique for distinguishing lognormal from Weibull distributions is presented. Its theoretical foundation is given at a qualitative level, and the range of useful application is explored. An approximate form for the distribution of the test statistic is inferred from Monte Carlo simulation     

有限贝塔刘维尔混合模型的变分学习及其应用

由于贝塔刘维尔分布的共轭先验分布中存在积分表达式,贝叶斯估计有限贝塔刘维尔混合模型参数异常困难.本文提出利用变分贝叶斯学习模型参数,采用gamma分布作为近似的先验分布并使用合理的非线性近似技术,得到了后验分布的近似解.与常用的EM算法相比,该方法能够同时估计模型参数和确定分量数,且避免了过拟合的问题.在合成数据集及场景分类问题上进行了大量的实验,实验结果验证了本文所提方法的有效性.