• Title, Summary, Keyword: cumulants

### A NOTE ON SOME HIGHER ORDER CUMULANTS IN k PARAMETER NATURAL EXPONENTIAL FAMILY

• KIM, HYUN CHUL
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.157-160
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• 1999
• We show the cumulants of a minimal sufficient statistics in k parameter natural exponential family by parameter function and partial parameter function. We nd the cumulants have some merits of central moments and general cumulants both. The first three cumulants are the central moments themselves and the fourth cumulant has the form related with kurtosis.

### The Cumulants of the Non-normal t Distribution

• Hwang, Hark
• Journal of the Korean Statistical Society
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• v.5 no.2
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• pp.91-100
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• 1976
• The use of the statistic $t = \sqrt{n} (x-\mu)/S$, where $\bar{X) = \sum X_i/n, \mu = E(X_i), S^2 = \sum(X_i-\bar{X})^2/(n-1)$ in statistical inference is usually done under the assumption of normality of the population. If the population is not normally distributed the tabulated values of student t are no longer valid. The moments of t are obtained as a power series in $1/\sqar{n}$ whose coefficients are functions of the cumulants of X. The cumulants are obtained from the moments in the usual manner. The first eight cumulants of t are given up to terms of order $1/n^3$. The first eight cumulants of t are given up to terms of order $1/n^3$. These results extend those of Geary who gave the first six cumulants of t to order $1/n^2$.

### Fisher's Second k Statistics

• Hwang, Hark
• Journal of the Korean Statistical Society
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• v.6 no.1
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• pp.21-24
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• 1977
• The cumulants of the sample variance are obtained in terms of the population cumulants using ALMAP (Algebraic Manipulation Package) and these results extend those of Fisher who gave the first six cumulants of the sample variance.

### A Study on Blind Channel Equalization Based on Higher-Order Cumulants

• Han, Soo-Whan
• Journal of Korea Multimedia Society
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• v.7 no.6
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• pp.781-790
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• 2004
• This paper presents a fourth-order cumulants based iterative algorithm for blind channel equalization. It is robust with respect to the existence of heavy Gaussian noise in a channel and does not require the minimum phase characteristic of the channel. In this approach, the transmitted signals at the receiver are over-sampled to ensure the channel described by a full-column rank matrix. It changes a single-input/single-output (SISO) finite-impulse response (FIR) channel to a single-input/multi-output (SIMO) channel. Based on the properties of the fourth-order cumulants of the over-sampled channel outputs, the iterative algorithm is derived to estimate the deconvolution matrix which makes the overall transfer matrix transparent, i.e., it can be reduced to the identity matrix by simple reordering and scaling. Both a closed-form and a stochastic version of the proposed algorithm are tested with three-ray multi-path channels in simulation studies, and their performances are compared with a method based on conventional second-order cumulants. Relatively good results are achieved, even when the transmitted symbols are significantly corrupted with Gaussian noise.

### Blind channel equalization using fourth-order cumulants and a neural network

• Han, Soo-whan
• International Journal of Fuzzy Logic and Intelligent Systems
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• v.5 no.1
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• pp.13-20
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• 2005
• This paper addresses a new blind channel equalization method using fourth-order cumulants of channel inputs and a three-layer neural network equalizer. The proposed algorithm is robust with respect to the existence of heavy Gaussian noise in a channel and does not require the minimum-phase characteristic of the channel. The transmitted signals at the receiver are over-sampled to ensure the channel described by a full-column rank matrix. It changes a single-input/single-output (SISO) finite-impulse response (FIR) channel to a single-input/multi-output (SIMO) channel. Based on the properties of the fourth-order cumulants of the over-sampled channel inputs, the iterative algorithm is derived to estimate the deconvolution matrix which makes the overall transfer matrix transparent, i.e., it can be reduced to the identity matrix by simple recordering and scaling. By using this estimated deconvolution matrix, which is the inverse of the over-sampled unknown channel, a three-layer neural network equalizer is implemented at the receiver. In simulation studies, the stochastic version of the proposed algorithm is tested with three-ray multi-path channels for on-line operation, and its performance is compared with a method based on conventional second-order statistics. Relatively good results, withe fast convergence speed, are achieved, even when the transmitted symbols are significantly corrupted with Gaussian noise.

### A Study on the Probabilistic Production Cost Simulation by the Mixture of Cumulants Approximation (Mixture of Cumulants Approximaton 법에 의한 발전 시물레이션에 관한 연구)

• 송길영;김용하
• The Transactions of the Korean Institute of Electrical Engineers
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• v.40 no.1
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• pp.1-9
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• 1991
• This paper describes a new method of calculating expected energy generation and loss of load probability (L.O.L.P) for electric power system operation and expansion planning. The method represents an equivalent load duration curve (E.L.D.C) as a mixture of cumulants approximation (M.O.N.A). By regarding a load distribution as many normal distributions-rather than one normal distribution-and representing each of them in terms of Gram-Charlier expansion, we could improve the accuracy of results. We developed an algorithm which automatically determines the number of distribution and demarcation points. In modeling of a supply system, we made subsets of generators according to the number of generator outage: since the calculation of each subset's moment needs to be processed rapidly, we further developed specific recursive formulae. The method is applied to the test systems and the results are compared with those of cumulant, M.O.N.A. and Booth-Baleriaux method. It is verified that the M.O.C.A. method is faster and more accure than any other method.

### Higber Order Expansions of the Cumulants and the Modified Normalizing Process of Multi-dimensional Maximum Likelihood Estimator

• Jonghwa Na
• Communications for Statistical Applications and Methods
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• v.6 no.1
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• pp.305-318
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• 1999
• In this paper we derive the higher order expansions of the first four cumulants of multi-dimensional Maximum Likelihood Estimator (MLE) under the general parametric model up to and including terms of order O({{{{ {n }^{-1 } }}}}) Also we obtain the explicit form of the expansion of the normalizing trans formation of multi-dimensional MLE and show that the suggested normalizing process is much better than the normal approximation based on central limit theorem through example.

### A STUDY ON THE PROBABILISTIC PRODUCTION COST SIMULATION BY THE MIXTURE OF CUMULANTS APPROXIMATION (MIXTURE OF CUMULANTS APPROXIMATION 법에 의한 발전시뮬레이션에 관한 연구)

• Song, K.Y.;Kim, Y.H.;Cha, J.M.
• Proceedings of the KIEE Conference
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• pp.154-157
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• 1990
• This paper describes a new method of calculating expected energy generation and loss of load probability (L.O.L.P) for electric power system operation and expansion planning. The method represents an equivalent load duration curve (E.L.D.C) as a mixture of cumulants approximation (M.O.C.A), which is the general case of mixture of normals approximation (M.O.N.A). By regarding a load distribution as many normal distributions-rather than one normal distribution-and representing each of them in terms of Gram-Charller expansion, we could improve the accuracy of results. We developed an algorithm which automatically determines the number of distribution and demarcation points. In modelling of a supply system, we made subsets of generators according to the number of generator outage: since the calculation of each subset's moment needs to be processed rapidly, we futher developed specific recursive formulae. The method is applied to the test systems and the results are compared with those of cumulant, M.O.N.A and Booth-Baleriaux method. It is verified that the M.O.C.A method is faster and more accurate than any other methods.

### Edgeworth and Cornish-Fisher Expansion for the Non-normal t

• Hwang, Hark
• Journal of the Korean Statistical Society
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• v.7 no.1
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• pp.3-10
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• 1978
• Let $X_i,...,X_n$ be a random sample from a distribution with cumulants $K_1, K_2,...$. The statistic $t = \frac{\sqrt{x}(\bar{X}-K_1)}{S}$ has the well-known 'student' distribution with $\nu = n-1$ degrees of freedom if the $X_i$ are normally distributed (i.e., $K_i = 0$ for $i \geq 3$). An Edgeworth series expansion for the distribution of t when the $X_i$ are not normally distributed is obtained. The form of this expansion is Prob \$(t

### Korean Single-Vowel Recognition Using Cumulants in Color Noisy Environment (유색 잡음 환경하에서 Cumulant를 이용한 한국어 단모음 인식)

• Lee, Hyung-Gun;Yang, Won-Young;Cho, Yong-Soo
• The Journal of the Acoustical Society of Korea
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• v.13 no.2
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• pp.50-59
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• 1994
• This paper presents a speech recognition method utilizing third-order cumulants as a feature vector and a neural network for recognition. The use of higher-order cumulants provides desirable uncoupling between the gaussian noise and speech, which enables us to estimate the coefficients of AR model without bias. Unlike the conventional method using second-order statistics, the proposed one exhibits low bias even in SNR as low as 0 dB at the expense of higher variance. It is confirmed through computer simulation that recognition rate of korean single-vowels with the cumulant-based method is much higher than the results with the conventional method even in low SNR.