Statistical Distribution Function Research Articles

The influence of the time delay through long trunk fiber on the phase-coding quantum key distribution system

Optical pulse will reach the receiver not at exact time instance due to the transmission through the long trunk fiber, and this phenomenon will effect on the bits error rate in the Quantum key distribution (QKD) system to some extend. The time jitter based on the differential phase shift quantum key distribution system and two asymmetric Mach-Zehnder interferometers is measured outdoors in different transmission distance and the physical model of the relationship between the time jitter and the quantum bit error rate is proposed. According to this relationship, the quantum bit error rate can be estimated for some optical pulse shape and some statistical distribution function.

Investigation of the amplitude features of high-latitude magnetic impulse events

The features of the amplitude distributions of magnetic impulse events have been investigated using the observations from a number of high-latitude observatories in the Northern and Southern hemispheres. It has been shown that the tails of the statistical distribution functions of the impulse amplitudes are approximated by a power law of the form f(A) = A−α, where A is the impulse amplitude and α is the exponent. Therefore, the magnetic-impulse generation regime corresponds to the features of the on-off intermittency model. The distribution of the magnetic impulse amplitudes has been analyzed for various geomagnetic latitudes, local times, seasons, solar activity cycle phases, and interplanetary conditions. It has been found that most statistical distributions of magnetic impulse amplitudes have the exponent α larger than 2, which is typical of the chaotic regimes called “strong turbulence.” In some cases, the exponent α is close to 1, which is typical of the regimes generated in a weakly turbulent medium. Qualitative estimates of the plasma wave turbulence level in the high-latitude magnetosphere have been obtained.

Saddlepoint Approximations to the Limiting Distribution of the Modified Anderson–Darling Test Statistic

The approximation for the distribution function of test statistic is extremely important in statistics. The standard and higher-order saddlepoint approximations are considered in tails of the limiting distribution for the modified Anderson–Darling test. The saddlepoint approximations are compared with the approximation of Sinclair et al. (1990) for upper tail area. An empirical function is derived to estimate the critical values of a saddlepoint approximation.

STATISTICAL GAUSSIAN DISTRIBUTION FUNCTION AS A DISTANCE INDICATOR TO STELLAR GROUPS

In this paper, statistical distribution functions are developed for distance determination of stellar groups. This method depends on the assumption that absolute magnitudes and apparent magnitudes follow a Gaussian distribution function. Due to the limits of the integrands of the frequency function of apparent and absolute magnitudes, we introduce Case A, B, and C Gaussian distributions. The developed approaches have been implemented to determine distances to some clusters and stellar associations. The comparison with the distances derived by different authors reveals good agreement.

EXACT PROPERTIES OF THE CONDITIONAL LIKELIHOOD RATIO TEST IN AN IV REGRESSION MODEL

For a simplified structural equation/IV regression model with one right-side endogenous variable, we derive the exact conditional distribution function of Moreira's (2003) conditional likelihood ratio (CLR) test statistic. This is used to obtain the critical value function needed to implement the CLR test, and reasonably comprehensive graphical versions of this function are provided for practical use. The analogous functions are also obtained for the case of testing more than one right-side endogenous coefficient, but in this case for a similar test motivated by, but not generally the same as, the likelihood ratio test. Next, the exact power functions of the CLR test, the Anderson-Rubin test, and the Lagrange multiplier test suggested by Kleibergen (2002) are derived and studied. The CLR test is shown to clearly conditionally dominate the other two tests for virtually all parameter configurations, but no test considered is either inadmissable or uniformly superior to the other two. The unconditional distribution function of the likelihood ratio test statistic is also derived using the same argument. This shows that both exactly, and under Staiger/Stock weak-instrument asymptotics, the test based on the usual asymptotic critical value is always oversized and can be very seriously so when the number of instruments is large.

Permanent Expansions and Distributions of Order Statistics in the INID Case

Expansions of permanents are applied to derive expressions for the distribution functions of order statistics arising from n INID random variables. For this purpose, we apply two types of expansions. The first one is based on Ryser's method and it leads to a representation of the distribution function as a generalized mixture of distributions of order statistics from IID random variables. The second approach, which is based on the calculation of the permanent of a sum of two matrices, yields an expression in terms of products of both distributions of minima and maxima. It turns out that the most efficient way of calculating the distribution function is the one based on Ryser's expansion.

ON THE CONDITIONAL LIKELIHOOD RATIO TEST FOR SEVERAL PARAMETERS IN IV REGRESSION

For the problem of testing the hypothesis that all m coefficients of the right-hand-side endogenous variables in an instrumental variables (IV) regression are zero, the likelihood ratio (LR) test can, if the reduced form covariance matrix is known, be rendered similar by a conditioning argument. To exploit this fact requires knowledge of the relevant conditional cumulative distribution function (c.d.f.) of the LR statistic, but the statistic is a function of the smallest characteristic root of an (m + 1)-square matrix and is therefore analytically difficult to deal with when m > 1. We show in this paper that an iterative conditioning argument used by Hillier (2009) and Andrews, Moreira, and Stock (2007 Journal of Econometrics 139, 116–132) to evaluate the c.d.f. in the case m = 1 can be generalized to the case of arbitrary m. This means that we can completely bypass the difficulty of dealing with the smallest characteristic root. Analytic results are obtained for the case m = 2, and a simple and efficient simulation approach to evaluating the c.d.f. is suggested for larger values of m.

Melting and thermodynamic properties of rare gas nanocrystals

A self-consistent statistical method [Phys. Rev. B 66, 054302 (2002)] is used to describe the thermodynamic properties of free rare gas nanocrystals, using thin plates as examples. It is shown that the influence of size on the thermodynamic properties of nanocrystals is due to size-dependent quantization of the vibration spectrum affecting the parameters of a statistical distribution function of atomic displacements and, thus, governing the size dependence of the average values of energetic contributions to the Gibbs free energy of the system. For Xe nanocrystals, we present calculated size dependences of the Debye temperature, heat capacity, interatomic distance, melting temperature, etc.

Distribution of ratio of extreme of isotonic estimators of chi-square variables with applications

Let V 1 , … , V k be k independent chi-square variables each with ν degrees of freedom (d.f.) and let W 1 ⩽ ⋯ ⩽ W k be the corresponding isotonic estimators computed by the pool adjacent violator algorithm under simple ordering among Vs . In this paper, a recursive integration algorithm for computing the distribution function of statistic T = W k / W 1 is presented. The test statistic F max * proposed by Fujino (Biometrika 66 (1979) 133-139) to test the homogeneity of variances against ordered alternative follows same null distribution as of the statistic T. Fujino discussed the null F max * distribution of and provided critical constant only for k = 3 , … , 6 , which limited its applications. The recursive procedure given in this article can be implemented efficiently for sufficient higher value of k. Test procedures based on statistic T for testing homogeneity against simple ordered alternative of k normal variances and equality of scale parameters of k exponential distributions are discussed separately with tables of critical constants for their implementation. Correctness of the critical constants is verified through simulation study. The proposed methodology is also extended to address the problem of testing exponentiality (constant failure rate) versus increasing failure rate, using isotonic estimators of normalized spaces between consecutive order statistics, along with tables of relevant critical constants.

Assessment of the statistical distribution of flexural strength of woven-fabric laminates with impact-induced damage

This paper deals with several statistical distribution functions in order to analyze the strength data of composite laminates with impact-induced damage under low temperature. A series of impact tests were performed on the woven-fabric carbon/epoxy laminates under temperatures of 293 K, 263 K and 233 K. Then, three-point flexural tests were carried out so as to generate post-impact strength at the aforementioned temperatures. Two-parameter Weibull, three-parameter Weibull and normal distributions were used to fit the residual strength data of the woven-fabric composite laminates. The two-parameter Weibull distribution most accurately described the experimentally measured strength data for all of the cases considered. Furthermore, the variation of residual strength is simultaneously affected by the impact energy and the temperature.

Robust pitch estimation using a wavelet variance analysis model

In this paper, we present a robust pitch estimation algorithm with the combination of wavelet transform (WT) and variance analysis (VA) for noisy speech. The technique uses the discrete wavelet transform to extract the wavelet parameters of the speech in the fundamental frequency band, and then performs VA to generate a variance statistical distribution function of the wavelet parameters. Finally, peak detection is applied to extract the pitch period. The simulation results show that the proposed method can give a superior accuracy relative to some of the existing methods from high to very low signal-to-noise ratio (SNR) levels in white noise, pink noise and babble noise.

Rearranging Edgeworth–Cornish–Fisher expansions

This paper applies a regularization procedure called increasing rearrangement to monotonize Edgeworth and Cornish-Fisher expansions and any other related approximations of distribution and quantile functions of sample statistics. Besides satisfying the logical monotonicity, required of distribution and quantile functions, the procedure often delivers strikingly better approximations to the distribution and quantile functions of the sample mean than the original Edgeworth-Cornish-Fisher expansions.

The multi-water-bag equations for collisionless kinetic modeling

In this paper we consider the multi-water-bag model for collisionless kinetic equations. The multi-water-bag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations while keeping its kinetic character. After recalling the link of the multi-water-bag model with kinetic formulation of conservation laws, we derive different multi-water-bag (MWB) models, namely the Poisson-MWB, the quasineutral-MWB and the electromagnetic-MWB models. These models are very promising because they reveal to be very useful for the theory and numerical simulations of laser-plasma and gyrokinetic physics. In this paper we prove some existence and uniqueness results for classical solutions of these different models. We next propose numerical schemes based on Discontinuous Garlerkin methods to solve these equations. We then present some numerical simulations of non linear problems arising in plasma physics for which we know some analytical results.

Painlevé V and the distribution function of a discontinuous linear statistic in the Laguerre unitary ensembles

In this paper we study the characteristic or generating function of a certain discontinuous linear statistic of the Laguerre unitary ensembles and show that this is a particular fifth Painlevé transcendent in the variable t, the position of the discontinuity. The proof of the ladder operators adapted to orthogonal polynomial with discontinuous weight announced sometime ago [13] is presented here, followed by the nonlinear difference equations satisfied by two auxiliary quantities and the derivation of the Painlevé equation.

Expansions of multivariate Pickands densities and testing the tail dependence

Multivariate extreme value distribution functions (EVDs) with standard reverse exponential margins and the pertaining multivariate generalized Pareto distribution functions (GPDs) can be parametrized in terms of their Pickands dependence function D with D = 1 representing tail independence. Otherwise, one has to deal with tail dependence. Besides GPDs we include in our statistical model certain distribution functions (dfs) which deviate from the GPDs whereby EVDs serve as special cases. Our aim is to test tail dependence against rates of tail independence based on the radial component. For that purpose we study expansions and introduce a second order condition for the density (called Pickands density) of the joint distribution of the angular and radial component with the Pickands densities under GPDs as leading terms. A uniformly most powerful test procedure is established based on asymptotic distributions of radial components. It is argued that there is no loss of information if the angular component is omitted in the testing problem.

Numerical prediction of statistical masonry wall fragment distribution induced by blast loading

The hazard caused by the fragments of damaged structures is usually significant in accidental explosions or hostile blast events. A reliable and efficient method to estimate probable fragment size, velocity and launch distance will be useful to assess and design countermeasures to mitigate the possible fragment hazards. This paper presents a numerical method for predicting the size and launch distance of the fragments caused by explosive damage of masonry wall. Numerical simulations with different scaled distances are carried out, and the statistical distribution functions of the fragment size and launch distance in terms of the scaled distance are derived.

On copulas and their diagonals

We formulate the necessary and sufficient conditions for a function δ : [ 0 , 1 ] → [ 0 , 1 ] to be the diagonal section of a multivariate absolutely continuous copula. Moreover we provide some simple analytic formulas for copulas having given diagonal section or given distribution functions of order statistics.

Fast Computation by Block Permanents of Cumulative Distribution Functions of Order Statistics from Several Populations

The joint cumulative distribution function for order statistics arising from several different populations is given in terms of the distribution functions of the populations. The computational cost of our formula in the case of two populations is still exponential in the worst case, but it is a dramatic improvement compared to the general formula by Bapat and Beg. In the case when only the joint distribution function of a subset of the order statistics of fixed size is needed, the complexity is polynomial, for the case of two populations.

A Comparison of Statistical Distribution Functions to Predict BGA Attach Reliability

This paper presents a comparison of using different statistical distribution functions to predict ball-grid-array (BGA) solder joint reliability. Four functions are evaluated: two-parameter Weibull, three-parameter Weibull, mixed two-parameter Weibull and Lognormal. Statistical analysis demonstrates that the most commonly used two-parameter Weibull function may not present the best fit to the failure data; three-parameter Weibull and mixed Weibull may be better options in some cases by examining failure data histogram, data fit goodness, and potential failure mechanism change.

Evaluation of the Adequacy of Statistical Distribution Functions for Deriving Unit Hydrograph

The unit hydrograph (UH) is one of the commonly employed techniques for the determination of flood hydrographs. Since the UH satisfies all the properties of a probability distribution function (PDF), it seems logical that PDFs can be employed for deriving the UH. In practice, the gamma distribution function has been commonly employed to derive the UH. In this paper, Beta (Beta), Exponential (EXP), Gamma (GM), Normal, Lognormal (LN), Weibull (WB), Logistic (LG), Generalized logistic (GLG) and Pearson Type 3 (PT 3) distribution functions were employed for the derivation of UH. Parameters of these distribution functions were estimated using the real coded genetic algorithm optimization technique. These distributions were tested on the 13 watersheds of different characteristics and it was observed that except for the EXP distribution function, most other distribution functions produced UHs which were in satisfactory agreement with observed UHs. However, three-parameter distributions GLG, PT 3 and two parameter LG were not capable of reproducing UHs for large watersheds having drainage areas of 3,360 and 4,300 km2. For such large watersheds WB reproduced UHs satisfactorily. Combining the overall performance of the distributions over 13 watersheds, the order of ranking the suitability of distributions were as: GM > PT 3 > Beta ≥ GLG ≥ LN > WB.