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Unformatted text preview: M4056 Lecture Notes. Monday, September 20, 2010 Sufficient Statistics The most basic task of inferential statistics is to acquire knowledge about a population from knowledge of a sample. We conceptualize this process by means of a model with two layers. 1) The first layer is the assumption that the population itself is described by a probability distri- bution of some sort. At the present time, we are concerned with cases in which the population distribution is assumed to belong to a parametrized family , and we use to refer to the parameter (which may be a scalar or a vector). Here are some examples: a) The population consists of units that are identical except with respect to a single property, which may have one of several values. (Think M&Ms and colors.) If the population is large, its state is described by the proportion of units with each of the possible values. In this case, is a vector that records the proportion of each kind. A special case occurs when there are exactly two possible values, which for convenience we might take to be 0 and 1. b) The population consists of units that are identical except with respect to a quantitative property that varies from unit to unit, and the proportion of the population in which that quantity does not exceed a given value is described by a cumulative distribution function that belongs to a parametrized family, e.g., normal( , 2...
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