Statistical Models
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چکیده
Statistical Models " Y is a random variable with density function f (y; θ) or f (y) ". This is the starting point for most of the material to be covered. Typically Y will be a scalar random variable or a vector random variable of length n, and f (y; θ) will be a density function with respect to counting measure or Lebesgue measure. The problem is to reason from observed data y back to θ or f (·). In more complicated problems, such as observations taken in continuous time, the definitions of Y and its density may not be obvious. In advanced texts this is often formulated from a measure-theoretic point of view. We have a so-called " probability triple " (Y, A, P), where Y is the sample space, typically identified with R n or R, A is the Borel σ-field, and P is a probability measure which is absolutely continuous with respect to Lebesgue or counting measure. A very concise but helpful account is given in Chapter 1.2 and 1.3 of TPE. (I found two typos: on p.8, l.-2 " Example 2.1 " should be " Example 2.2 " , p.16, 2nd display should be E(T) = T (x)dP Y (x).) A simple set of examples of such models might be the following: are assumed to be independent, identically distributed, with each following a N (0, 1) distribution. Then f (y; θ) = 1 √ (2π) n exp − 1 2 (y i − α − βz i) 2 and θ = (α, β). are assumed to be independent, identically distributed, with each following the density f 0 (e) or f 0 (e; ν) where the form of f 0 is known (possibly up to an unknown number of parameters). are assumed to be independent, identically distributed, with each following an unknown density f that satisfies some smoothness conditions 1
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