R. J. Larsen and M. L. Marx. An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, 2001.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....object pairs. The ordinate is the number of times subjects confirmed there was no difference between the objects presented. The maximum is 28 subjects multiplied by 3 repetitions for each pair. In this preliminary study, we undertook a nonparametric approach. Using Friedman s test [20] 21] [22] significant treatment (here treatment refers to the particular pairing, e.g. h 3 h 8 ) effects were found (S=72.67, p 0.005) Since in each object pair we compare an object with h 8 , the pairing of h 8 and h 8 acts as a control treatment. We next compare each treatment with the control ....

....Figure 5 shows a screen shot for experiment 3. Figure 5: Graphic display in experiment 3. The right object (h4 ) is deformed. Subjects tasks were the same as for experiment 2. Figure 6 shows experiment 3 results. We note that the height of the fourth column is less than 80 . Friedman s test [20, 21, 22] shows significant treatment effects (S=29.54, p 0.005) Since all objects are compared with h4, the pair (h4, h4) acts as a control treatment. A simultaneous multiple comparison test [21] on columns of data against the control at a significance level of 0.05 results in the grouping in Table 6. ....

Larsen, R.J. and Mark, M.L., An Introduction to Mathematical Statistics and its Applications, published by Prentice-Hall.

....estimators are analogous. Let pL p pU be an exact (see [2] confidence interval for p. For y = 1; 2; n Gamma 1, the lower limit pL satisfies where y is the observed value of the random variable Y and ff is the nominal coverage of the confidence interval (see, for example, [10], page 279) For y = 1; 2; n Gamma 1, the upper limit pU satisfies U (1 Gamma pU ) This confidence interval requires numerical methods to determine pL and pU and takes longer to calculate as n increases. This interval will be used as a basis to check the approximate bounds ....

....n ] pu = FindRoot[ Sum[Binomial[n, k] p k (1 p) n k) k, 0, y] alpha 2, p, y n ] for a given n, y and ff. This code works well for small and moderate sized values of n. Some numerical instability occurred for larger values of n, so the well known relationship (Larsen and Marx [10], page 101) between the successive values of the probability mass function f(x) of the binomial distribution f(x) n Gamma x 1)p x(1 Gamma p) f(x Gamma 1) x = 1; 2; n was used to calculate the binomial cumulative distribution function. The Mathematica code for determining pL ....

R.J. Larsen and M.L. Marx, An Introduction to Mathematical Statistics and Its Applications, Second Edition, Prentice-Hall, 1986.

....the solution to (1.1) is given by ( V ) where N( is the cumulative distribution function (cdf) for a standard normal random variable. The inverse cdf is easy to either compute using mathematics software libraries or to look up in a table for the normal cdf (see for example [37, 52]) 1.2 Generalizing the method from the example The method from the example generalizes [43, 33] Suppose that the vector of relative changes R in the market risk factors is a multivariate normal random variable with the mean and the covariance matrix C. Similar to the example, assume that the ....

.... model, the random variables in the sequence of returns are normal, R i N( i = 1; 2; The probability density function (pdf) for all R i is p(x) 2 ( x ) Given a time series of returns fr i g i=1 , and can be estimated with standard statistical methods [37]: the mean can be estimated by b = r i ; and the variance by (r i b ) Some authors advocate using estimators that give more weight to recent returns than to old ones (see for example [43, 33] To illustrate the performance, we estimate the parameters b and b for ....

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R.J. Larsen and M.L. Marx. An Introduction to Mathematical Statistics and Its Applications. Prentice-Hall, second edition edition, 1986.

....by constructing probability intervals for the events in question, and comparing intervals to see if they are disjoint. The calculation of intervals uses an iterative procedure, but we use the normal distribution as an approximation when the numbers in question are large enough, as described by Larsen and Marx [1986, p. 295] 5.2 Smoothing and More Backo Whether we use a strict cuto or a hypothesis testing method, these techniques are intended to relieve data sparsity problems manifested in the denominator (the conditioning information) But the numerator is is even more susceptible to sparsity ....

Richard J. Larsen and Morris L. Marx, An Introduction to Mathematical Statistics and Its Applications, Prentice-Hall, Englewood Clis, New Jersey, second edition, 1986.

....they are real valued. This limits the algorithms, data structures and definitions available for them. For example, in anomaly detection we cannot meaningfully define the probability of observing any particular set of wavelet coefficients, since the probability of observing any real number is zero [27]. Such limitations have lead researchers to consider using a symbolic representation of time series. While there are literally hundreds of papers on discretizing (symbolizing, tokenizing, quantizing) time series [2, 20] see [11] for an extensive survey) none of the techniques allows a distance ....

....a time series database into PAA, we can apply a further transformation to obtain a discrete representation. It is desirable to have a discretization technique that will produce symbols with equiprobability [3, 28] This is easily achieved since normalized time series have a Gaussian distribution [27]. To illustrate this, we extracted subsequences of length 128 from 8 different time series and plotted a normal probability plot of the data as shown in Figure 4. o 997 099 095 10 0 10 Figure 4: A normal probability plot of the cumulative distribution of values from subsequences of length ....

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Larsen, R. J. & Marx, M. L. (1986). An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, Englewood, Cliffs, N.J. 2 nd Edition.

....Each test induces a fault in the system and the resulting behavior is observed. From these sample tests an inference can be made regarding the total population of faults that can occur in the system. The method used in the following sections is based on Statistical Inference theory [Coch77] Lars81] Hoel62] Yama67] Hoel72] Bern88] Three aspects of the sampling process and statistical inference are applicable: As the sample size increases, the estimate of the parameter of interest generally gets closer to the true value, with complete correspondence reached when the sample size ....

Larsen, R. L. and Marx, M. L., Introduction to Mathematical Statistics and Its Applications, Prentice Hall, 1981.

....4 Thus, the goodness of fit of these models is very high. Table 9. 1 demonstrates that the lifetime distribution measured by Leland and Ott still holds 10 years later, and on a variety 3 See BLSS routine robust , as described in [57] 4 See BLSS routine robust , as described in [57] See also [43] for a more detailed description of the R 2 value. 82 Name Total Number Number Estimated Lifetime Standard R 2 of Processes Processes Distribution Error value Host Studied with Age 1 Curve po1 77440 4107 T Gamma0:97 .016 0.997 po2 154368 11468 T Gamma1:22 .012 0.999 po3 111997 7524 ....

Richard J. Larsen and Morris L. Marx. An introduction to mathematical statistics and its applications. Prentice Hall, Englewood Cliffs, N.J., 2nd edition, 1986.

....L Gamma1 X l=1 w[l] T Q Gamma1 w[l] 2 Assuming the implied w[l] are identically distributed for different l, like their theoretical counterparts. 3 We use a simplified notation for the multiple integral. 4 Throughout the paper, we use standard properties of Gaussian distributions [6,8,10,11]. 5 An efficient estimator if w[l] are statistically independent for different l. 13 as an estimate for K(pjjq) that can be completely determined from the model parameters. Dropping the 1 2 , we arrive at our first consistency hint error function E 1 = log(jQj) Gamma log(j Sigmaj) Gamma ....

....from the solution w[l] 0. This solution is the single most probable solution for w[l] since q assumes its maximum value there. The solution is nonetheless undesirable, since a typical solution for w[l] would have a variety of values that reflect the Gaussian distribution (the goodness of fit [10] seen in figure 5, but not in figure 4) If we generate a single solution, it is likely to be of the typical variety. However, if we actively seek a high probability solution, we will get one, and it may be atypical. The contrast between probable and typical comes up in many contexts, most ....

R. Larsen and M. Marx, An Introduction to Mathematical Statistics and Its Applications, Prentice-Hall, 1986.

....network architectures. Analytical modeling of queuing systems. Simulation methodology. Stochastic process formulation [24, 157] is a key ingredient in both analytical work and simulation output analysis. The latter includes estimation theory and statistical inference procedures [22, 95]. It is generally assumed that the reader has some prior 1 1 knowledge of each of the involved disciplines. Some help is provided by chapter 2 on wormhole switched network architectures. Otherwise, contribution (I) serves as an introduction to analytical modeling and simulation of queuing ....

....debugging features of GMSim also turned out to be handy during model development. 3.4 (V) Paper (V) 116] compares unbuffered wormhole switched networks and buffered packetswitched networks with respect to estimation of expected packet latency in steady state. The ordinary sample mean estimator [95] is subject to evaluation and the discussion is centered around the variance of this estimator. The variance is needed for computation of confidence intervals [15, 127] By intuition, we expect that the different nature of wormhole switched systems and buffered networks will affect the estimator ....

LARSEN, R., AND MARX, M. An Introduction to Mathematical Statistics and Its Applications, second ed. Prentice-Hall, 1986. 38 38

....the a 2 quantile of the F distribution with u 1 and u 2 degrees of freedom. Similarly, the quantity 2 1 , 4 3 a n n F is defined as the 1 a 2 quantile of the F distribution with u 3 and u 4 degrees of freedom. Quantiles of the F distribution are tabulated in many statistics textbooks [e.g. Larsen and Marx, 1981], and are readily available from most commercial statistical software packages. The formulas (4) and (5) work for 0 x n. When x = 0, the bounds: 0 . 0 = L P (6) n U P 1 1 a = 7) are recommended, and when x = n the bounds: 11 n L P 1 a = 8) 0 . 1 = U P (9) are recommended ....

....of the standard normal distribution corresponding to probability 1 a 2. For example, to get a 95 confidence interval, one would choose a = 0.05, and use the value z 0.975 = 1. 960 in equations (10) and (11) Tables of the standard normal distribution are found in many statistics textbooks [e.g. Larsen and Marx, 1981], and are readily available from most commercial statistical software packages. For p away from the endpoints (say, 70 . 0 30 . 0 p ) a simpler formula for approximate 100(1 a ) confidence intervals is appropriate. Here, the bounds are given by: n n p p z p P L 2 1 ) 1 ( 2 1 = ....

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Larsen, R.J., and Marx, M.L. (1981). An Introduction to Mathematical Statistics and Its Applications. Englewood Cliffs, NJ: Prentice-Hall.

....of a hit. The sample hit probability for the anatomical structure, X, is the total number of hits divided by the number of randomly generated trajectories: X # X#n. We use the sample hit probability, X, as an estimate of p because it is an unbiased estimator for p (the true hit probability) [6]. The confidence bounds for a given hit probability are determined using a normal approximation to the binomial distribution. The confidence bounds for X are given by: 0 #X # z ##2 s X#1 #X# n # X # z ##2 s X#1 #X# n 1 A #3# wheren is the number of simulated trajectories and z ##2 is ....

R. J. Larsen and M. L. Marx. An Introduction to Mathematical Statistics and Its Applications . Prentice-Hall, Englewood-Cliffs, New Jersey, 1986.

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R. J. Larsen and M. L. Marx. An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, 2001.

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Larsen, R. J., and Marx, M. L. (2001). An introduction to mathematical statistics and its applications. Prentice Hall. Third edition.

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R. Larsen and M. Marx. An introduction to Mathematical Statistics and Its Applications. Prentice-Hall, NJ, 1981.

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R. Larsen and M. Marx, An Introduction to Mathematical Statistics and its Applications. Prentice Hall, 2000.

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Larsen, R. J., and Marx, M. L. An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, Upper Saddle River, NJ 07458, 2001.

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R. Larsen and M. Marx, Introduction to Mathematical Statistics and its Applications, Prentice Hall (1986).

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R. Larsen and M. Marx. An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, 2001.

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R. Larsen and M. Marx. An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, 2001.

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R. Larsen and M. Marx. An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, 2001.

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R. Larsen and M. Marx. An Introduction to Mathematical Statistics and Its Applications. Prentice Hall, 2001.

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R. Larsen and M. Marx, Introduction to Mathematical Statistics and its Applications, Prentice Hall (1986).

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R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, Prentice-Hall, 1986.

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R. Larsen and M. Marx. An introduction to Mathematical Statistics and Its Applications. Prentice-Hall, NJ, 1981.

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Larsen, R. J. & Max, M. L. (1986). An Introduction to Mathematical Statistics and Its Applications. Prentice Hall.

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