By E.J.G. Pitman

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**Some basic theory for statistical inference**

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U n have the same distribution function F0 . r u 1 , . . , u n are independent. It follows that x1 , . . —independent and identically distributed—random variables . The assumption that there are no systematic errors can be formalized as r u i and −u i have the same distribution, and consequently F0 (x) = 1 − F0 (−x). An estimate μ is a function of the observations: μ = μ(x1 , . . , xn ) = μ(x). We are looking for estimates such that in some sense μ ≈ μ with high probability. One Robust Statistics: Theory and Methods Ricardo A.

Z n + c) and (cz 1 , . . , cz n ). If x is a random variable with distribution F, the mean and median of a function g(x) will be denoted by E F g(x) and Med F g(x), dropping the subscript F when there is no ambiguity. 12). We shall henceforth study estimates of this form, which need not be MLEs for any distribution. The function ρ will be chosen in order to ensure the goals (A) and (B) above. Assume ψ is monotone nondecreasing, with ψ(−∞) < 0 < ψ(∞). 12)—always has a solution. If ψ is continuous and increasing, the solution is unique, otherwise the set of solutions is either a point or an interval (throughout this book, we shall call any function g increasing (nondecreasing) if a < b implies g(a) < g(b) (g(a) ≤ g(b))).

8 shows a gamma Q–Q plot based on the robustly fitted gamma distribution. This plot reveals that the bulk of the data is well fitted by the robust method, while approximately 30 of the largest values of LOS appear to come from a sub-population of the patients characterized by longer LOS values that is properly modeled separately by another distribution, possibly another gamma distribution with different values of the parameters α and σ . 1 Show that if a value x0 is added to a sample x = {x1 , .