(14) The TV method has the satisfying property of yielding a single set of parameter estimates for any response function, linear or nonlinear, independent of the manner in which the relation or relations among the variables and parameters are expressed. (10) Similar procedures were provided later by Southwell, (11) Jeffries, (12) Lybanon, (13) and Boggs et al. The emphasis on S TV as a minimization target dates back at least to Deming’s work (8) and general-purpose algorithms for accomplishing this have been available since the early 1970s, from Powell and McDonald (9) and Britt and Leucke. (3)where I use the subscript TV for “total variance”, a label which has unfortunately been used in a more general sense at times in the statistical literature.
The most common occurrence of this flaw is likely the use of w i = 1 (unweighted LS) to fit data having nonconstant σ yi (heteroscedastic data). If the weights are not correct to within a constant, the parameter estimates remain normally distributed but are no longer minimum-variance or maximum-likelihood and V is wrong (biased). (2) If the data error variances are known in only a relative sense, the parametric variances (from V post) become χ 2-distributed estimates that include the scale factor S/ν, with degrees of freedom ν equal to the difference in the numbers of data points and adjustable parameters, ν = n – p. Under these conditions and assuming the fit model is correct, the parameter estimates are normally distributed about their true values (e.g., a t, b t), with standard errors (SE) that are given exactly by the covariance matrix V prior when the data error variances are known absolutely and used to compute the weights. If further, the random error in y is normally distributed (Gaussian) about the true values y t of y, the LLS solution is also the maximum likelihood solution. If the weights are taken inversely proportional to the data variances, w i ∝ σ yi –2, the solution is minimum-variance. If the relation between the dependent and independent variables is algebraically linear in the adjustable parameters, for example, the 4-parameter model, f( x) = a + bx + cx 2 + d/ x, the problem is a linear LS one (LLS), solvable in a single computational cycle. This means that in some cases we should not just consider the predicted values of the regression \(\hat(0,\sigma^2)\).(1)where w i is the weight and δ yi the residual in y, δ yi = Y i – y i, with Y i the calculated or “adjusted” value of y for the ith values of the independent variables, for which I will henceforth use just x i. Given the uncertainty of estimates of parameters, the regression line itself and the points around it will be uncertain. 16.4.2 Common confusions related to information criteria.16.3 Calculating number of parameters in models.15.3 The explanatory variables are not correlated with anything but the response variable.14.2 Types of variables transformations.13.2 Categorical variables for the slope.13 Regression with categorical variables.10.2 Residuals of model estimated via OLS.9 Measuring relations between variables.8.1.3 Non-parametric, one-sample Wilcoxon test.7.4 Statistical and practical significance.7.1.1 Common mistakes related to hypothesis testing.6.3.5 Why having biased estimate can be better than having the inefficient one?.5.2.4 Two continuous numerical variables.5.2.3 One numerical continuous variable.5.2.2 Two categorical/discrete variables.5.2.1 One categorical/discrete variable.3.5 Rolling a dice – Discrete Uniform distribution.3.4 Modelling arrivals – Poisson distribution.3.3 Multiple coin tosses – Binomial distribution.3.2 Tossing a coin – Bernoulli distribution.
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