Cramer, H. (1946). Mathematical Methods of Statistics. Princeton:Princetion University Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(x) is any monic polynomial of degree i 1, and the determinants are indexed with i, j # 1, N . We index all determinants in this paper with i, j 1. obtain (2.2) 2a) N(u N) 2 . Let n (x) for n 0 denote the monic Hermite polynomial of degree n in x (see, e.g. [C], p. 133) with generating series P n#0 n (x) w n = e xw w . Thus, from (2.1) we have = V (x) det det [H i 1 (x j ) so, from (1.3) we obtain det [H i 1 (x j ) NN e p2 (x) 2 ##SN sgn (#) det #(j) 1 (x) e x = N det j 1 ....

H. Cramer, "Mathematical methods of statistics," Princeton U. Press, Princeton, 1946.

....can write, Q(a) 51) Q(a) 52) Q(a) a 2 # i T i 1 (a) 53) This results in a closed form expression for computing probability of error. 13 VII. CONVERGENCE PROPERTIES Convergence properties of Gram Charlier expansion is investigated in [24] [29], 30] It is proved in [31] that the expansion is convergent if the expanded function satisfies the following condition, f Y (y)e 4 dy #. 54) Reference [13] mentions that this expansion has good asymptotic behavior as defined in [32] In other words, a few terms will give a close ....

H. Cramer, Mathematical Methods of Statistics, Princeton University press, 1957.

....determine the rate of increase of the information transfer ratio when X has a probability function defined over an interval or over a Cartesian product of intervals, we use the approach shown in figure 3 with Z being the maximum likelihood estimator (MLE) of X. Under certain regularity conditions [8] and because the r s are conditionally independent, the MLE is asymptotically (in the number of systems N) Gaussian with a mean equal to X and variance equal to the reciprocal of the Fisher information FYi x. This result applies when the systems aren t homogeneous so long as the third absolute ....

H. Cramer. Mathematical Methods of Statistics. Princeton University Press, Princeton, New Jersey, 1946.

....of the ratio becomes asymptotically unbiased. It can be also shown that the two estimators are uncorrelated. For a large number of samples N , the two random variables, M and V , are Gaussian distributed. Since the two are uncorrelated and Gaussian, the the two estimators are independent [1, 5]. Using this fact, the probability density function of the ratio Z = M can be found. The numerator is shown to be a Gaussian random variable, N ( 1) and the denominator is found to be a Chi distributed random variable with N degrees of freedom and oe = 1. The corresponding ....

Harald Cram'er, Mathematical Methods of Statistics, Princeton University Press, Princeton, 1999.

....the total to the partial situation. For instance, there is no conceptual need to distinguish between total and partial correlation coefficients since a total correlation coefficient is simply a parcor of order . In this section, we shall first recall (and slightly extend) some results [27] [28] [29] 30] giving the covariance matrix (resp. its inverse) in terms of covariances of the random variables (resp. of the random variables ) We thus get lemmas 3.1 and 3.2, which are generalized to theorem 3.1 by considering Schur complements in and in ) ....

H. Cramer, Mathematical methods of statistics, 
& edition, Princeton University press, Princeton, NJ, 1974.

....i ij x p . 3 The Main Algorithm 3.1 Testing of Hypotheses The algorithm consists in consequent test of hypotheses about belonging of a document to given domains. Considering image of a document and images of domains as attributes such a test is reduced to the test of uniformity. According to [2] it can be completed by c criterion of uniformity with n = n 1 degrees of freedom. The criterion requires a calculation of the series of values j c as is shown in (1a) i N p N x j p x c (1a) i N p N p kl p p k c (1b) Then the inequality j c p c is verified ....

Cramer, H. Mathematical methods of statistics. Cambridge, 1946.

.... K) N co n ra 1 : w ( S , z) r m 1 r I m 2 , u uy) I, u u) s) where f, ff) is the derivative of fm(K) wi respect m K, and can be cmputed using (2) d (3) All momentb ed estimators, K, e consistent, asymptotically nomal, and asymptotically unbiased [1]. Moreover, the AsV is given by (8) Notice at mm = m,n which c be shown usg the fact that f(x) f(1 x) ter chging m and n (8) then, should not change the AsV, which is seen to be the case after computing (8) 11y as a nction of K. In order to compare e AsV expression in (8) with a benchmark, ....

H. Cramer, Mathematical Methods of Statistics, Princeton, N J: Princeton University Press, 1946.

....refer to this predictor as a Gaussian trac predictor. The Gaussian modeling of the trunk behavior is justi ed as the following: Gaussian Approximation When the number of individual ows gets large, the aggregate arrival rate tends to have a Gaussian distribution under the Central Limit Theorem [110]. In a bu erless uid model, losses occur when the total arrival rate exceeds the reserved bandwidth BR . In our analysis, we consider sucient bu er space to hold the largest packet, and approximate the probability of packet loss as p loss P ( BR ) Assume has a Gaussian distribution with ....

....Therefore T mea controls a set of trade o s among QoS performance, overprovisioning, and signaling overhead. The magnitude of prediction error is dependent on the following two factors: Level of aggregation: Gaussian approximation holds when the number of ows aggregated, n, is large ( 50) [110]. Since the modeling accuracy increases with n, Gaussian trac predictor should work well for describing the Internet workload in the backbone, since the value of n involved is large (over thousands of ows) However for low bandwidth links that observe less than 50 ows, our technique will not be ....

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H. Cramer, Mathematical Methods of Statistics, Princeton University Press, 1946

....of ) where the sample standard deviation of was calculated according to . The sample confidence region defined here is useful for examining the variations of the estimators in terms of and . It is known that for large , the bias and variance of a moment based estimator are both proportional to [11]. This is why the bias and variance of both estimators in Fig. 2 decrease as increases. Also note that for any sample size in Fig. 2, the performance of the based and the based estimators are almost the same over a broad range of values. Hence, from a practical point of view, both estimators ....

H. Cramer, Mathematical Methods of Statistics. Princeton, NJ: Princeton Univ. Press, 1946.

....MLE case in which the PDF g( Delta) is fully known to the researcher and the object of interest is the estimation of . Under standard regularity conditions, MLE produces 4 optimal estimators which are consistent and asymptotically efficient since they achieve the CramerRao lower bound (Cramer, [5]) For a sample of length T , the log likelihood function is given by L T ( X t t ( Gamma 1 2 X t log oe 2 X t log g t (fi) oe : 2) The MLE estimator is found by maximizing Equation (2) with respect to the vector of parameters . The score function is given ....

H. Cramer. Mathematical Methods of Statistics. Princeton University Press, 1946.

....b h is a set of parameters to be optimized. One of the important properties of them is that they do not satisfy the regularity condition for the asymptotic normality of the maximum likelihood estimator, in general (Hagiwara, Toda, Usui, 1993 ; Fukumizu, 1996) For regular statistical models (Cramer, 1949), the set of true parameters consists of only one point and the Fisher information matrix is positive definite, even if the learning model is larger than necessary to attain the true distribution, which case is called over realizable. On the other hand, if a layered learning machine is in the ....

Cramer, H. (1949) Mathematical methods of statistics, Princeton University Press.

....window, T mes . ERs send regular updates to LCH, which uses these statistics to predict future bandwidth usage along a specific link. Gaussian Predictor: When the number of individual flows gets large, the aggregate arrival rate tends to have a Gaussian distribution under Central Limit Theorem [28]. We estimate the required bandwidth as: B = m ffoe, where ff is a QoS factor that controls the extent to which the bandwidth predictor accommodates variability in the samples. In the buffer less case, the probability of packet loss is approximately Q(ff) where Q( is the complementary ....

H. Cramer, Mathematical Methods of Statistics, Princeton University Press, 1946.

....the denominator is independent of Ip , we can re use the results of that computation to get O(m 2 ) time. Stopping Criterion With omniscience, we would stop when the reconstructed distribution was statistically the same as the original distribution (using, say, the 2 goodness of fit test [Cra46] An alternative is to compare the observed randomized distribution with the result of randomizing the current estimate of the original distribution, and stop when these two distributions are statistically the same. The intuition is that if these two distributions are close to each other, we ....

H. Cramer. Mathematical Methods of Statistics. Princeton University Press, 1946.

....a and b . Therefore it will not be able to distinguish between the ideal prediction score and the worst prediction score and thus it is not a good measure to be used for predictor quality ranking. 4.4. Cramer s 1 coefficient Another association coefficient is the so called 1 coefficient [10] ( 1 0 , 1 2 1 = TN FN FP TP FP TN FN TP FN FP TN TP It ranges from 0 to 1 and does not satisfy conditions a and b mentioned above. As the CTG coefficient, it is not suitable for the predictor score ranking. 4.5. 2 coefficient A better association ....

Cramer, H., Mathematical Methods of Statistics, Princeton University Press, Princeton, N.J., 1946.

....statistical inference takes as its fundamental mechanism the rejection of a statistical hypothesis under certain prespecified conditions. Although in classical statistics probability is emphatically identified with frequencies [ Neyman, 1950 ] or the conceptual counterparts of frequencies [ Cramer, 1951 ] or set measures in a sample space [ Lindgren, 1976 ] and although many statisticians strongly deny that rejecting H amounts to accepting :H , much of classical statistics translates smoothly into the formalism of purely probabilistic acceptance. Furthermore, since statistical knowledge ....

Harald Cramer. Mathematical Methods of Statistics, volume Princeton. Princeton Uniersity Press, 1951.

....an image point x maps as x X = h 11 x h 12 h 21 x h 22 = ffx t x 1 This non linear mapping (on inhomogeneous coordinates) can be expanded in a Taylor series. Statistical moments of X, such as the variance, are then computed in terms of the Taylor coefficients and the moments of x [12, 47]. It is assumed here that the homography is exact (no errors) and the measurement of the image test point x is subject to Gaussian noise with standard deviation oe x . The Taylor series is developed about the point s mean position denoted as x. 4.2.1 First order If the Taylor series is truncated ....

Cramer H. Mathematical Methods of Statistics. Princeton Univ. Press., 1946.

....model. When the response is continuous the neural network non linear function can be used in a non linear regression type setting with the appropriate likelihood maximized. Since the logistic function satisfies the regularity conditions under which the maximum likelihood estimators exist 17 (Cramer 1946), the maximum likelihood estimators of the neural network parameters are consistent, asymptotically normal and asymptotically efficient. Hence, if the relationship between class probabilities and covariates satisfies the neural network logistic models described in section 3, then as the amount of ....

CRAMER, H. (1946). Mathematical Methods of Statistics. Princeton Univ. Press, Princeton NJ.

....is an open subset of R d . The standard theory of asymptotic statistics is based on the condition that the family (P ) is regular. The mathematical definition of regularity has been fixed by many authors in similar but different ways. After the first working concepts by Wald, 12] and Cramer, [2], the most important achievements are due to LeCam, 6] and H ajek, 4] The state of the art is presented by Bickel, Klaassen, Ritov and Wellner, 1] Let (f ) be the densities of the family (P ) Then it is common knowledge that the basic asymptotic expansion argument (local asymptotic ....

H. Cramer. Mathematical Methods of Statistics. Princeton University Press, 1946.

....J Y = Z R k fi fi g Y (v) fi fi dv 1: 19) We recall some simple inequalities for J Y . First, if g Y (v) 0 then J Y = 2 ) k p Y (0) For example, for a Gaussian random vector Y we have J Y = 2 ) k=2 (det ) Gamma1=2 : Furthermore, for all Y such that EjY j 2 1 we get (see Cramer (1946), x 11.12) J Y Z QY 2 i 1 Gamma 1 2 Q Y (v) j dv = Gamma k 2 1 Gamma1 2 k=2 k=2 Gamma det Delta Gamma1=2 ; where Q Y (v) E(v; Y Gamma EY ) 2 : For the functions of random vectors we consider at first two special cases. Case 1. For an integer m 2 ....

Cramer H., "Mathematical Methods of Statistics," Princeton, 1946.

....translation step, and sparse flow field. Furthermore, it appears that the limits of the two frame approach have been reached. Weng, Huang and Ahuja [44] claim (based on simulations) that the performance of their two frame motion algorithm has reached the theoretically possible Cramer Rao [7] [24] lower bounds of optimal estimation of the motion parameters. On the other hand, Dutta and Snyder [9] argue that even small rotation errors (which all two frame motion algorithms suffer from, including the near optimal performance algorithms) cause a large error in structure. Assuming a ....

H.Cramer, Mathematical Methods of Statistics. Princeton Univ. Princeton, New-Jersy, 1946.

....X i g ik q i0 A j (s k Gamma q k0 ) N X k B jk x k ; 5) where B jk = P N i g ik q i0 A j : This expression can be rewritten as: DeltaA = Gamma Gamma1 N = Gamma Gamma1 BX = TX; 6) where T = Gamma Gamma1 B. The covariance matrix M of variables DeltaA is given by [6] M = Tg Gamma1 T t = Gamma Gamma1 Bg Gamma1 B t ( Gamma Gamma1 ) t : 7) In our case Bg Gamma1 B t = Gamma, as shown in [5] and the equation for M becomes: M = Gamma Gamma1 : 8) III. FITTING FUNCTION AND NOISE COVARIANCE MATRIX For simplicity, following [5] we ....

H.Cramer, "Mathematical Method of Statistics" Princeton University Press, 1951 p. 313

....are computable estimates or proxies for this distance. In contrast, ML or GML is in theory optimum if the overall model is correctly specified in a certain statistical sense, except for the parameters being estimated) but may behave badly from a performance oriented criteria if it is not. See Cramer (1954), Wahba(1985a) GCV is the most general performance oriented method, but UBR can be used to advantage when certain error variances are known well (for example, error variances in 500hpa heights from radiosondes) Dee(1993, 1994) discusses ML estimation in a form similar to that which we will ....

Cramer, H. (Sixth Printing, 1954), Mathematical Methods of Statistics, Princeton University Press.

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Cramer, H. (1946). Mathematical Methods of Statistics. Princeton:Princetion University Press.

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H. Cramer, Mathematical Methods of Statistics, Princeton University Press, Princeton, N.J., 1946.

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H. Cramer, Mathematical Methods of Statistics, Princeton University Press, 1946

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H. Cramer, "Mathematical Methods of Statistics," p. 500, Princeton University, Princeton, NJ, 1946.

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H. Cramer, Mathematical Methods of Statistics. Princeton, NJ: Princeton Univ. Press, 1957.

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H. Cramer, Mathematical Methods of Statistics. Princeton, NJ: Princeton Univ. Press, 1946.

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H.Cramer, Mathematical Methods of Statistics, Princeton University Press, 19 th Printing, 1999.

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Cram'er, H. (1946). Mathematical methods of statistics. Princeton University Press, Princeton, NJ.

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Cramer, H. 1946. Mathematical Methods of Statistics, Princeton: Princeton University Press.

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H. Cramer, Mathematical Methods of Statistics, Princeton University press, 1957.

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Cramer, H.: Mathematical Methods Of Statistics. 10 edn. Princeton University Press (1963)

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H. Cramer, Mathematical Methods of Statistics. Princeton, NJ: Princeton Univ. Press, 1957.

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Cramer, H. (1951). Mathematical Methods of Statistics (Princeton University Press, Princeton, NJ).

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H. Cramer. Mathematical Methods of Statistics. Princeton University Press, 1946.

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H. Cramer. Mathematical Methods of Statistics. Princeton University Press, 1946.

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H. Cramer, Mathematical Methods of Statistics, Princeton, NJ: Princeton University Press, 1946.

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H. Cramer. Mathematical Methods Of Statistics. Princeton University Press, tenth edition, 1963.

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H. Cramer. Mathematical methods of statistics. Princeton University Press, 1957.

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H. Cram'er, Mathematical Methods of Statistics, Princeton University Press, 1946.

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Cram'er, H. (1946) Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press.

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Cram'er, H., (1946). Mathematical methods of statistics. Princeton University Press, Princeton.

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Cramer, H. (1946) Mathematical methods of statistics, Princeton University Press, Princeton, NJ.

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H. Cramer, Mathematical methods of statistics, Princeton, Princeton Univ. Press, 1945.

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Cramer, H. (1963): "Mathematical Methods of Statistics", Princeton University Press.

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H. Cramer, Mathematical methods of statistics, Princeton, Princeton University Press, 1945.

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H. Cramer, Mathematical methods of statistics, Princeton, Princeton University Press, 1945.

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H. Cramer, (1949). Mathematical Methods of Statistics, Princeton.

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Cramer, H. (1946) Mathematical Methods of Statistics. Princeton: Princeton University Press.

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