n the context of statistical inferences need to postulate both probability as well as a sampling model and thus we define a statistical model as comprising A statistical model is defined as comprising (a). a probability model重={f(x;6),∈e};and (b). a sampling model x≡(X1,X2,…,Xn) It must be emphasized that the two important components of a statistical model, the probability and sampling models, are clearly interrelated. For ex ample we cannot postulate the probability modelΦ={f(x;),θ∈e} if the sample x is non-random. This is because if the r.v. 's X1, X2,. Xn are not independent the probability model must be defined in terms of their joint distri- bution,ie.重={f(x1,x2,…,xn;日),θ∈}( for example, stock price).More over, in the case of an independent but not identically distributed sample we need to specify the individual density functions for each r.v. in the sample, i.e. 重={(xk;日),6∈,k=1,2,…,n}. The most important implication of this relationship is that when the sampling model postulated is found to be inappro- priate it means that the probability model has to be re-specified as well. 1. 2 An overview of statistical inference The statistical model in conjunction with the observed data enable us to consider the following question (A). Are the observed data consistent with the postulated statistical model (model misspeci fication) (B). Assuming that the postulated statistical model is consistent with the ob served data, what can we infer about the unknown parameter bEe? (a). Can we decrease the uncertainty about 8 by reducing the parameters space from e to Oo where Oo is a subset of e.(confidence estimation (b). Can we decrease the uncertainty about 8 by choosing a particular value in 8, say 8, as providing the most representative value of 0?(point estimationIn the context of statistical inferences need to postulate both probability as well as a sampling model and thus we define a statistical model as comprising both. Definition 6: A statistical model is defined as comprising (a). a probability model Φ = {f(x; θ), θ ∈ Θ}; and (b). a sampling model x ≡ (X1, X2, ..., Xn) ′ . It must be emphasized that the two important components of a statistical model, the probability and sampling models, are clearly interrelated. For ex￾ample we cannot postulate the probability model Φ = {f(x; θ), θ ∈ Θ} if the sample x is non-random. This is because if the r.v.’s X1, X2, ..., Xn are not independent the probability model must be defined in terms of their joint distri￾bution,.i.e. Φ = {f(x1, x2, ..., xn; θ), θ ∈ Θ} (for example, stock price). More￾over, in the case of an independent but not identically distributed sample we need to specify the individual density functions for each r.v. in the sample, i.e. Φ = {fk(xk; θ), θ ∈ Θ, k = 1, 2, ..., n}. The most important implication of this relationship is that when the sampling model postulated is found to be inappro￾priate it means that the probability model has to be re-specified as well. 1.2 An overview of statistical inference The statistical model in conjunction with the observed data enable us to consider the following question: (A). Are the observed data consistent with the postulated statistical model ? (model misspecif ication) (B). Assuming that the postulated statistical model is consistent with the ob￾served data, what can we infer about the unknown parameter θ ∈ Θ ? (a). Can we decrease the uncertainty about θ by reducing the parameters space from Θ to Θ0 where Θ0 is a subset of Θ. (conf idence estimation) (b). Can we decrease the uncertainty about θ by choosing a particular value in θ, say θˆ, as providing the most representative value of θ ? (point estimation) 4