Asymptotic distribution, sampling

The log-likelihood function at the maximum likelihood estimates is -28.993171. For the model with only a constant term, the value is -31.19884. The t statistic for testing the hypothesis that (3 equals zero is 5.16577/2.51307 = 2.056. This is a bit larger than the critical value of 1.96, though our use of the asymptotic distribution for a sample of 10 observations might be a bit optimistic. The chi squared value for the likelihood ratio test is 4.411, which is larger than the 95% critical value of 3.84, so the hypothesis that 3 equals zero is rejected on the basis of these two tests. 

Based on a sample of 65 observations from a normal distribution, you obtain a median of 34 and a standard deviation of 13.3. Form a confidence interval for the mean. (Hint Use the asymptotic distribution. See Example 4.15.) Compare your confidence interval to the one you would have obtained had the estimate of 34 been the sample mean instead of the sample median. 

Fig. 6. Comparison of simulated and asymptotical densities of the estimated parameter 62 for the mono-exponential model >/(x 0) = ()t cxpi—x/dn) for two SNR (a) Q /a = 10 and (b) 0] /(j — 100. The exponential curve was sampled at two points x = [10, 80]T and the parameters of the model were obtained by LS estimator from Rician distributed samples. Simulated density results from 105 data vectors, the asymptotical density is a Gaussian distribution centered on the true value 02 = 70 with a variance equal to the CRBs.

However, as noted in the discussion on the LRT (Section 12.6.1), the test tends to be conservative for fixed effects, suggesting that the actual critical value for the LRT statistic may be larger than 3.84. Moreover, the number of samples per subject and the sample size may also affect the theoretical critical value, as the likelihood ratio is asymptotically distributed. Previous work also indicated that the likelihood... 

The results from the first study suggest rather clearly that conversation group sizes are limited at about four individuals (one speaker and three listeners) (Dunbar et al. 1995). Fig. 3 plots the cumulative frequency distributions for the number of individuals that a speaker can reach (i.e. conversation group size less one, since there is always only one speaker at any given moment per conversation Dunbar et al. 1995). All three datasets in the sample suggest that the number of listeners rapidly approaches an asymptotic value at around three. 

Use the delta method to obtain the asymptotic variances and covariance of these two functions assuming the data are drawn from a normal distribution with mean ju and variance o2. (Hint Under the assumptions, the sample mean is a consistent estimator of //, so for purposes of deriving asymptotic results, the difference between X and // may be ignored. As such, no generality is lost by assuming the mean is zero, and proceeding from there. Obtain V, the 3x3 covariance matrix for the three moments, then use the delta method to show that the covariance matrix for the two estimators is... 

For random sampling from a normal distribution with nonzero mean u and standard deviation ct, find the asymptotic joint distribution of the maximum likelihood estimators of ct/ ii and u2/ct2. 

It can be shown that nonlinear LS is at least asymptotically an unbiased MVB estimator in the limit of large sample sizes if the samples are from a population with a normal distribution [60,61]. As pointed out above, the... 

It has already been mentioned that certain distributions can be approximated to a normal one. As the size of the sample increases, the binomial distribution asymptotically approaches a normal distribution. This is a useful approximation for large samples. 

A few values of the t-distribution are given in an accompanying table. We note that t values are considerably higher than corresponding standard normal values for small sample size but as n increases, the t-distribution asymptotically approaches the standard normal distribution. Even at a sample size as small as 30, the deviation from normality is small, so that it is possible to use the standard normal distribution for sample sizes larger than 30 (n>30) and in most cases, for n<30 t-distribution is used. This is equivalent to assuming that Sx is an exact estimate of ox at large sample sizes (n>30). 

There are often data sets used to estimate distributions of model inputs for which a portion of data are missing because attempts at measurement were below the detection limit of the measurement instrument. These data sets are said to be censored. Commonly used methods for dealing with such data sets are statistically biased. An example includes replacing non-detected values with one half of the detection limit. Such methods cause biased estimates of the mean and do not provide insight regarding the population distribution from which the measured data are a sample. Statistical methods can be used to make inferences regarding both the observed and unobserved (censored) portions of an empirical data set. For example, maximum likelihood estimation can be used to fit parametric distributions to censored data sets, including the portion of the distribution that is below one or more detection limits. Asymptotically unbiased estimates of statistics, such as the mean, can be estimated based upon the fitted distribution. Bootstrap simulation can be used to estimate uncertainty in the statistics of the fitted distribution (e.g. Zhao Frey, 2004). Imputation methods, such as... 

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