Biased variance

The definition of sample variance with an (n-1) in the denominator leads to an unbiased estimate of the population variance, as shown above. Sometimes the sample variance is defined as the biased variance ... 

This is known as the biased variance. It is a good measure of the dispersion of a sample of a flow variable, but according to the statistical definition not the best measure of the dispersion of the whole population of possible observations. A better estimate of the variance (an unbiased variance) of the population, given a sample of data, is ... 

Defining the sample s variance with a denominator of n, as in the case of the population s variance leads to a biased estimation of O. The denominators of the variance equations 4.8 and 4.12 are commonly called the degrees of freedom for the population and the sample, respectively. In the case of a population, the degrees of freedom is always equal to the total number of members, n, in the population. For the sample s variance, however, substituting X for p, removes a degree of freedom from the calculation. That is, if there are n members in the sample, the value of the member can always be deduced from the remaining - 1 members andX For example, if we have a sample with five members, and we know that four of the members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the sample must be... 

Check. Use the Crooks relation (5.35) to check whether the forward and backward work distributions are consistent. Check for consistency of free energies obtained from different estimators. If the amount of dissipated work is large, caution may be necessary. If cumulant expressions are used, the work distributions should be nearly Gaussian, and the variances of the forward and backward perturbations should be of comparable size [as required by (5.35) for Gaussian work distributions]. Systematic errors from biased estimators should be taken into consideration. Statistical errors can be estimated, for instance, by performing a block analysis. 

Several other useful values can be obtained with a complementary procedure, similar to MAXSLOPE except it calculates variances instead of regression slopes (see Grove Meehl, 1993). An investigator can use these values to calculate parameters of latent distributions (e.g., mean and standard deviation of the taxon). These estimates are not biased by nuisance correlations, unlike MAXCOV estimates. Unfortunately, these calculations are fairly arduous, so we will not describe them here (for more details see Grove Meehl, 1993). 

If the errors are normally distributed, the OLS estimates are the maximum likelihood estimates of 9 and the estimates are unbiased and efficient (minimum variance estimates) in the statistical sense. However, if there are outliers in the data, the underlying distribution is not normal and the OLS will be biased. To solve this problem, a more robust estimation methods is needed. 

There are many other distributions used in statistics besides the normal distribution. Common ones are the yl and the F-distributions (see later) and the binomial distribution. The binomial distribution involves binomial events, i.e. events for which there are only two possible outcomes (yes/no, success/failure). The binomial distribution is skewed to the right, and is characterised by two parameters n, the number of individuals in the sample (or repetitions of a trial), and n, the true probability of success for each individual or trial. The mean is n n and the variance is nn(l-n). The binomial test, based on the binomial distribution, can be used to make inferences about probabilities. If we toss a true coin a iarge number of times we expect the coin to faii heads up on 50% of the tosses. Suppose we toss the coin 10 times and get 7 heads, does this mean that the coin is biased. From a binomiai tabie we can find that P(x=7)=0.117 for n=10 and n=0.5. Since 0.117>0.05 (P=0.05 is the commoniy... 

The short estimator, bi is biased E[bi] = Pi + Pi.2P2- It s variance is ct2(X1 X1)"1. It s easy to show that this latter variance is smaller. You can do that by comparing the inverses of the two matrices. The inverse of the first matrix equals the inverse of the second one minus a positive definite matrix, which makes the inverse smaller hence the original matrix is larger - Var[bi.2] > Var[bi]. But, since bi is biased, the variance is not its mean squared eiTor. The mean squared eiTor of bj is Var[bi] + biasxbias. The second term is Pi.2P2P2 Pi.2 - When this is added to the variance, the sum may be larger or smaller than Var[bi 2] it depends on the data and on the parameters, p2. The important point is that the mean squared error of the biased estimator may be smaller than that of the unbiased estimator. 

By far, the predominant methods for determination of amino acids in foods are based on HPLC. However, alternative methods for amino acid analysis do exist. Many of the earliest determinations for certain amino acids were based on microbiological tests (and other bioassays), but these are no longer widely employed. Cost and analysis time are obvious factors in the demise of these type of methods. Also, these types of methods are very prone to biased results and high variance. 

On the other hand, the sample variance defined by Eq. (1.4) is biased ... 

Random variable estimations have, apart from the mean, their own variance. It has been proved that when choosing an estimation it is not sufficient to require an estimation to be consistent and biased. It is easy to cite examples of different estimations for consistent and biased basic population means. The criterion for a better estimation is an estimation is better the smaller dispersion it has. Let us assume that we have two consistent and biased estimations 0i and 02 for a population parameter and let us suppose that 0j has smaller dispersion than 02. Fig. 1.9 presents distributions of the given estimations. 

We have previously introduced the sum of squares due to error as MSE=SSE/(n-2) and said that it is the unbiased estimate of error variance a2 because E(MSe)=o2 no matter whether the null hypothesis H0 Pi=0 is correct or not. It is easy to prove that the expected value of the regression mean square, MSR=SSR/1, is the biased estimate of variance o2 if not Pi=0. This can be written in the form ... 

If the undesigned effect of these covariables is not taken into account, the results of analysis of variance may be biased and serious misinterpretation is possible. 

If we calculate the four nonelementary discriminant functions df we find the following fractions of data variance explained 77.3% (by one function), 98.8% (by two functions), and 99.9% (by three discriminant functions). Hence we do not expect severe biased projections of our data on to the plane. In Fig. 5-23 we find some overlapping laboratories, however. In the 3D-plot of Fig. 5-24 a good, separated display of all laboratories data is indicated. So far, the data projection is satisfactory. 

If a parametric distribution (e.g. normal, lognormal, loglogistic) is fit to empirical data, then additional uncertainty can be introduced in the parameters of the fitted distribution. If the selected parametric distribution model is an appropriate representation of the data, then the uncertainty in the parameters of the fitted distribution will be based mainly, if not solely, on random sampling error associated primarily with the sample size and variance of the empirical data. Each parameter of the fitted distribution will have its own sampling distribution. Furthermore, any other statistical parameter of the fitted distribution, such as a particular percentile, will also have a sampling distribution. However, if the selected model is an inappropriate choice for representing the data set, then substantial biases in estimates of some statistics of the distribution, such as upper percentiles, must be considered. 

Ridge regression analysis is used when the independent variables are highly interrelated, and stable estimates for the regression coefficients cannot be obtained via ordinary least squares methods (Rozeboom, 1979 Pfaffenberger and Dielman, 1990). It is a biased estimator that gives estimates with small variance, better precision and accuracy. 

An extension of the CRB approach can be to minimize not only the minimal uncertainties but also both bias and variance in order to consider the use of biased estimators. Bias and variance result from a trade-off, and so it is possible to reduce the variance of estimates by tolerating an increase in the bias. For this purpose an extension of the CRB has been introduced by Hero et al.,41 which represents the variance of estimates as function of the norm of the bias gradient. This curve shows the achievable trade-off between bias and variance. 

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