Các suy đoán rằng đã được thực hiện trong hình là nhìn thấy rõ ràng hơn trong lô đường viền. Trong Hình , nếu một đường thẳng đứng là rút ra từ trục BMI sau đó một trực quan có thể có được một ý tưởng về sự phân bố DBMI điều kiện về giá trị của chỉ số BMI, nơi mà các đường thẳng đứng đi qua trục BMI. | BIAS ADJUSTMENT TECHNIQUES 141 age until the upper end of the age range and the variance also increases with age until about the age of 40. As noted from Figure BMI and DBMI may be related. This may be pursued further by examining the estimate of the joint density function. Figure shows a 3-D plot and a contour plot for f x y for females where x is BMI and y is DBMI. The speculation that was made from Figure is seen more clearly in the contour plots. In Figure if a vertical line is drawn from the BMI axis then one can visually obtain an idea of the DBMI distribution conditional on the value of BMI where the vertical line crosses the BMI axis. When the vertical line is drawn from BMI 20 then DBMI 20 is in the upper tail of the distribution. At BMI 28 or 29 the upper end of the contour plot falls below DBMI 28. This illustrates that women generally want to weigh less for any given weight. It also shows that average desired weight loss increases with the value of BMI. . BIAS ADJUSTMENT TECHNIQUES The density estimate f y as defined in is biased for f y with the bias given by Bias f y E f y f y . This is a model bias rather than a finite population bias since f y is obtained from a limiting value for a nested sequence of increasing finite populations. If we can obtain an approximation to the bias then we can get a bias-adjusted estimate f y from f y f y Bias f y . An approximation to the bias can be obtained in at least one of two ways obtain either a mathematical or a bootstrap approximation to the bias at each y. The mathematical approximation to the bias of the estimator in is the same as that shown in Silverman 1986 namely . i s2h2 Bias f y I-2- 24f y . Since f y is unknown we substitutef y . Recall that h is the window width of the kernel smoother and b is the bin width of the histogram. When K y is a standard normal kernel then Ơ2 1 and 1X Pi f y y mi 2 h2 1 K FF The bias-corrected estimate is obtained from .
