Estimators unbiased

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

The second complication is that the values of z shown in Table 4.11 are derived for a normal distribution curve that is a function of O, not s. Although is an unbiased estimator of O, the value of for any randomly selected sample may differ significantly from O. To account for the uncertainty in estimating O, the term z in equation 4.11 is replaced with the variable f, where f is defined such that f > z at all confidence levels. Thus, equation 4.11 becomes... 

ROBUST AND UNBIASED ESTIMATIONS IN CHEMICAL DATA TREATMENT... 

If the sample is unbiased, estimate the source mean, so that... 

Finally we need to compare the variance of our estimator with the best attainable. It can be shown that The Cramer-Rao lower bound (CRLB) is a lower bound on the variance of an unbiased estimator (Kay, 1993). The quantities estimated can be fixed parameters with unknown values, random variables or a signal and essentially we are finding the best estimate we can possibly make. 

The formula for calculating the budget of a GP practice must offer an unbiased estimator of the expected level of expenditure if each GP practice had a standard response to the needs of its population. Even if the considerable technical difficulties of establishing this formula could be overcome, the actual expenditure of a GP practice would differ from the budgeted amount due to characteristics of the patients not taken into account in the formula (socioeconomic characteristics, chronic diseases, private coverage and so on), variations in clinical practice between GP practices, random variations in the level of disease and price variations. For a population of 10 000 inhabitants (a reasonable mode for a GP practice) there is a one-third probability that the actual expenditure will deviate more than 10 per cent from a well-designed budget.22... 

Instead of estimating -/ 1 ln(exp(-/W(f)) directly using (5.44), one can use cumulant expansion approaches, as in regular free energy perturbation theory (see e.g., [20, 39] for combining cumulant expansions about the initial and final states). Unbiased estimators for cumulants can be used. Probably the most useful relations involve averages and variances of the work ... 

Finally, it is interesting to note that biases can be introduced by data fitting at low counts even with the use of ordinarily unbiased estimators like the maximum likelihood estimator [37],... 

If it is assumed that the measurement errors are normally distributed, the resolution of problem (5.3) gives maximum likelihood estimates of process variables, so they are minimum variance and unbiased estimators. 

The Unbiased Estimation Technique (UBET) was developed by Rollins and Davis (1992). This approach simultaneously provides unbiased estimates and confidence intervals of process variables when biased measurements and process leaks exist. 

Rollins, D., and Davis, J. (1992). Unbiased estimation of gross errors in process measurements. AlChE J. 38,563-572. 

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. 

Unbiased estimation An estimator 9 is called an unbiased estimator for 9 if it satisfies E 9) = 9. If this property holds only for n -> oo, the estimator is said to be asymptotically unbiased. 

Let 1, x2,..., xn be a random sample of N observations from an unknown distribution with mean fi and variance o2. It can be demonstrated that the sample variance V, given by equation A.8, is an unbiased estimator of the population variance a2. 

This shows that V is an unbiased estimator of a2, regardless of the nature of the sample population. 

Sample mean x and variance s2 are convergent and unbiased estimators (e.g., Hamilton, 1964), which implies that the so-called empirical variance a2 given by... 

We therefore make the assumption that the sample data gathered in vector y are only our best estimates of the real (population) values which justifies the bar on the symbol as representing measured values. This notation contradicts the standard usage, but is consistent with the basic definitions of Chapter 4. Indeed, for an unbiased estimate, we can still write that... 

We now proceed to m observations. The ith observation provides the estimates xi of the independent variables Xj and the estimate y, of the dependent variable Y. The n estimates xtj of the variables Xj provided by this ith observation are lumped together into the vector xt. We assume that the set of the (n+1) data (i/,y,) associated with the ith observation represent unbiased estimates of the mean ( yf) of a random (n + 1)-vector distributed as a multivariate normal distribution. The unbiased character of the estimates is equivalent to... 

Note that a scalar behaves as a symmetric matrix.) Because of finite sampling, and P cannot be evaluated exactly. Instead, we will search for unbiased estimates a and P of a and P together with unbiased estimates y( and xtj of yt and xu that satisfy the linear model given by equation (5.4.37) and minimize the maximum-likelihood expression in xt and y,. Introducing m Lagrange multipliers A , one for each linear... 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

Similar to Equation 4.26, an unbiased estimator for the residual variance cr2 is... 

To model the relationship between PLA and PLR, we used each of these in ordinary least squares (OLS) multiple regression to explore the relationship between the dependent variables Mean PLR or Mean PLA and the independent variables (Berry and Feldman, 1985).OLS regression was used because data satisfied OLS assumptions for the model as the best linear unbiased estimator (BLUE). Distribution of errors (residuals) is normal, they are uncorrelated with each other, and homoscedastic (constant variance among residuals), with the mean of 0. We also analyzed predicted values plotted against residuals, as they are a better indicator of non-normality in aggregated data, and found them also to be homoscedastic and independent of one other. 

Although it is beyond the scope of this presentation, it can be shown that if the model yj. = 0 + r, is a true representation of the behavior of the system, then the three sui.. s of squares SS and divided by the associated degrees of freedom (2, 1, and 1 respectively for this example) will all provide unbiased estimates of and there will not be significant differences among these estimates. If y, = 0 + r, is not the true model, the parameter estimate will still be a good estimate of the purely experimental uncertainty, (the estimate of purely experimental uncertainty is independent of any model - see Sections 5.5 and 5.6). The parameter estimate however, will be inflated because it now includes a non-random contribution from a nonzero difference between the mean of the observed replicate responses, y, and the responses predicted by the model, y, (see Equation 6.13). The less likely it is that y, - 0 + r, is the true model, the more biased and therefore larger should be the term Si f compared to 5. ... 

Geostatistical processing is one of the most useful and practical methods for evaluation and estimation of a resource. Its BLUE (Best Liner Unbiased Estimator) Kriging not only can indicate the distribution and amount of ore in a resource, but also, based on variance and error of Kriging can identify some parts of ore body, that have lack of data and need more exploration. For a routine Geostatistical processing some issues should be considered ... 

Block Estimation Kriging Kriging is an unbiased estimator and due to the conditions of mineralization, it has several different methods. On this project, "Simple Kriging" is used. The sizes of blocks are... 

The kriging system of linear equations is derived so that their solution gives kriging weights such that the kriging estimator is a "best linear unbiased estimator." The estimator is linear because the estimator is a weighted sum. It is unbiased because the system... 

Unbiased and Minimum-Variance Unbiased Estimation, Particularly for Variances... 

Bias corrections are sometimes applied to MLEs (which often have some bias) or other estimates (as explained in the following section, [mean] bias occurs when the mean of the sampling distribution does not equal the parameter to be estimated). A simple bootstrap approach can be used to correct the bias of any estimate (Efron and Tibshirani 1993). A particularly important situation where it is not conventional to use the true MLE is in estimating the variance of a normal distribution. The conventional formula for the sample variance can be written as = SSR/(n - 1) where SSR denotes the sum of squared residuals (observed values, minus mean value) is an unbiased estimator of the variance, whether the data are from a normal distribution... 

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