Biased estimation

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. 

Hoerl, A. E., Kennard, R. W. Ridge regression Biased estimation for nonorthogonal problems. Technometrics 1970, 12, 55-67. 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

Should studies be limited to those which are published It is well known that negative studies that report little or no benefit from following a particular course of action are less likely to be published than are positive studies. Therefore, the published literature may be biased toward studies with positive results, and a synthesis of these studies would give a biased estimate of the impact of pursuing... 

In situ models are to evaluate absorption or membrane permeability under the physiologically relevant tissue condition. While the luminal environment can be modulated by the administered solution, the tissue condition is physiologically controlled. The estimated membrane permeability can be, in most cases, assumed to represent the transport across the epithelial cell layer at steady state or quasisteady state. However, one needs to be aware that the involvement of metabolic degradation, which may occur at the cellular surface or within the cytosol, can be a factor leading to biased estimates of membrane permeability and erroneous interpretation of the transport process. Particularly,... 

A particular situation where bias may be important is in statistical meta-analysis, where statistical estimates are combined across studies. When estimates from individual studies may be averaged arithmetically, it is better to average unbiased estimates (Rao 1973, Section 3a). In case of biases that are consistent across studies, an arithmetic average would have a bias of the same sign, regardless of the number of studies included in the analysis. The average of biased estimates could fail to be consistent (in the statistical sense). 

Haphazardly or arbitrarily select points. This strategy imposes restrictions in sampling—in particular, positions for aesthetic or preservation reasons. Thus, there is the risk of obtaining a biased estimate when this strategy is applied on an object. 

The simplest and most popular biased estimator is due to Hoerl and Kennard (ref. 23), estimating the unknown parameters by... 

A.E. Hoerl and R.W. Kennard, Ridge regression Biased estimation for nonorthogona1 problems, Technometrics, 12 (1970) 55-67. 

In spite of its simplicity the direct integral method has relatively good statistical properties and it may be even superior to the traditional indirect approach in ill-conditioned estimation problems (ref. 18). Good performance, however, can be expected only if the sampling is sufficiently dense and the measurement errors are moderate, since otherwise spline interpolation may lead to severely biased estimates. 

B.R. Hunt, Biased estimation for nonparametric identification of linear systems, Math. Biosciences, 10 (1971) 215-237. 

In this book, we plot analytical response on the y-axis versus concentration on the x-axis. The inverse calibration (y = concentration, x = response) is said to provide a less biased estimate of concentration from a measured response. 

As t—>oo, the ratio goes to one. This would follow from the result that the biased estimator and the unbiased estimator are converging to the same thing, either as a2 goes to zero, in which case the MMSE estimator is the same as OLS, or as x x grows, in which case both estimators are consistent. 

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. 

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 ... 

Biased estimation of pass-through rate if frequency of estimation is higher than frequency of observation of forward electricity prices... 

Similar to (3), but for a small sample of data. The fundamental problem of nonrepresentativeness is the same, and inferences from these data are likely to provide biased estimates of exposure factors. 

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... 

Data are not random but are representative in other ways. This may mean, for example, that the data are a stratified sample applicable to the real-world situation for the assessment scenario of interest. In this case, frequentist methods can be used to make inferences for the strata that are represented by the data (e.g. particular exposed subpopulations), but not necessarily for all aspects of the scenario. However, for the components of the scenario for which the data cannot be applied, there is a lack of representative data. For example, if the available data represent one subpopulation, but not another, frequentist methods can be applied to make inferences about the former, but could lead to biased estimation of the latter. Bias correction methods, such as comparison with benchmarks, use of surrogate (analogous) data or more formal application of expert judgement, may be required for the latter. 

Hoerl, A.E. and Kennard, R.W., Ridge regression biased estimation for nonorthog-onal problems, Technometrics, 12, 55-67, 1970. 

Hoerl, R.W., Schuenemeyer, J.H., and Hoerl, A.E., A simulation of biased estimation and subset selection regression techniques, Technometrics, 28, 369-380, 1986. 

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. 

Although this design is unlikely to produce a biased estimate, it is still very unlikely that our sample will produce a mean half-life that exactly matches the true mean value for the whole population. All samples contain some individuals that have above-average values but their effect is more-or-less cancelled out by others with low values. However, the devil is in that expression more-or-less . In our experiment on half-lives, it would be remarkable if the individuals with shorter than average half-lives exactly counterbalanced the effect of those with relatively long half-lives. Most real samples will have some residual over- or under-estimation. 

For the above table (and those following) (B) indicates the coefficient estimate is a biased estimate. 

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. 

Uncertainty Represents a lack of knowledge about factors affecting exposure or risk and can lead to inaccurate or biased estimates of exposure. The types of uncertainty include scenario uncertainty, parameter uncertainty and model uncertainty (USEPA, 1997c). 

Since most colloidal dispersions are stabilized by particle interactions, the use of equation (10.51) may lead to biased estimates of particle size that are often concentration dependent. The effect may be taken into account by expanding the diffusion coefficient to a concentration power series that, at low concentrations, gives ... 

We will get a biased estimate of the measurement of interest—only low values. 

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