外文文獻翻譯數(shù)學(xué)專業(yè)(編輯修改稿)
2025-01-09 12:04 本頁面
【文章內(nèi)容簡介】 exist for the models we are working with. The Newton Raphson algorithm is an iterative procedure that can be used to calculate MLEs. The basic idea behind the algorithm is the following. First, construct a quadratic approximation to the function of interest around some initial parameter value (hopefully close to the MLE). Next, adjust the parameter value to that which maximizes the quadratic approximation. This procedure is iterated until the parameter values stabilize. These notes begin with the easy to visualize case of maximizing a function of one variable. After this case is developed, we turn to the more general case of maximizing a function of k variables. 2 The Newton Raphson Algorithm for Finding the Maximum of a Function of 1 Variable Taylor Series Approximations The first part of developing the Newton Raphson algorithm is to devise a way to approximate the likelihood function with a function that can be easily maximized analytically. To do this we need to make use of Taylor’s Theorem. Theorem 1 (Taylor’s Theorem (1 Dimension)). Suppose the function f is 1?k times differentiable on an open interval I. For any points x and hx? in I there exists a point w between x and hx? such that ? ? ? ? !1239。39。39。 )()!1( 1)(!1)(21)()()( ?????????? kkkk hwfkhxfkhxfhxfxfhxf ?. (1) It can be shown that as h goes to 0 the higher order terms in equation 1 go to 0 much faster than h goes to 0 . This means that (for small values of h ) hxfxfhxf )()()( 39。??? This is referred to as a first order Taylor approximation of f at x . A more accurate approximation to )( hxf ? can be constructed for small values of h as: 239。39。39。 )(21)()()( hxfhxfxfhxf ???? This is known as a second order Taylor approximation of f at x Note that the first order Taylor approximation can be rewritten as: bhahxf ??? )( where )(xfa? and )(39。 xfb? . This highlights the fact that the first order Taylor approximation is a linear function in h . Similarly, the second order Taylor approximation can be rewritten as: 221)( chbhahxf ???? Where )(xfa? , )(39。 xfb? and )(39。39。 xfc? . This highlights the fact that the second order Taylor approximation is a second order polynomial in h Finding the Maximum of a Second Order Polynomial Suppose we want to find the value of x that maximizes 2)( cxbxaxf ??? First, we calculate the first derivative of f : cxbxf 2)(39。 ?? We know that 0)(39。 ??xf , where ?x is the
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