Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
5.1. Convergence in Probability 325 0 500 1000 1500 2000 −0.2 −0.1 0.0 0.1 0.2 0.3 n xbar Figure 5.1.1:Realizations of the point ...
326 Consistency and Limiting Distributions Hence the pdf ofYnis fYn(t)= { n θnt n− (^10) <t≤θ 0elsewhere. (5.1.2) Based on it ...
5.2. Convergence in Distribution 327 5.1.5.Consider the R functionconsistmeanwhich produces the plot shown in Figure 5.1.1. Obta ...
328 Consistency and Limiting Distributions the asymptotics of certain situations. Moreover, for illustration, instead of saying ...
5.2. Convergence in Distribution 329 Clearly,FXn(x)=FX(x) for allxin the support ofX,sothatXn D →X.Onthe other hand, the sequenc ...
330 Consistency and Limiting Distributions The last example shows in general that we cannot determine limiting distribu- tions b ...
5.2. Convergence in Distribution 331 Remark 5.2.2(Stirling’s Formula).In advanced calculus the following approxi- mation is deri ...
332 Consistency and Limiting Distributions Theorem 5.2.1.IfXnconverges toXin probability, thenXnconverges toXin distribution. Pr ...
5.2. Convergence in Distribution 333 Theorem 5.2.4.SupposeXnconverges toXin distribution andgis a continuous function on the sup ...
334 Consistency and Limiting Distributions As the following example shows, the converse of this theorem is not true. Example 5.2 ...
5.2. Convergence in Distribution 335 Thelittle-onotation is used in terms of convergence in probability, also. We often writeop( ...
336 Consistency and Limiting Distributions whereopis interpreted as in (5.2.10). Rearranging, we have √ n(g(Xn)−g(θ)) =g′(θ) √ n ...
5.2. Convergence in Distribution 337 Example 5.2.6.LetYnhave a distribution that isb(n, p). Suppose that the mean μ=npis the sam ...
338 Consistency and Limiting Distributions If this sum is substituted foret √ 2 /nin the last expression forMY n(t), it is seen ...
5.2. Convergence in Distribution 339 Sample Size 5 y Density 0246 0.0 0.1 0.2 0.3 0.4 0.5 Sample Size 10 y Density − 2 − 101234 ...
340 Consistency and Limiting Distributions 5.2.10.Modify the R functioncdistpltto show histograms of the valueswn discussed in E ...
5.3. Central Limit Theorem 341 5.2.19.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample (see Section 5.2) f ...
342 Consistency and Limiting Distributions Theorem 5.3.1(Central Limit Theorem).LetX 1 ,X 2 ,...,Xndenote the observa- tions of ...
5.3. Central Limit Theorem 343 Sincem′′(t) is continuous att=0andsinceξ→0asn→∞,wehave lim n→∞ [m′′(ξ)−σ^2 ]=0. The limit proposi ...
344 Consistency and Limiting Distributions Example 5.3.2.LetXdenote the mean of a random sample of size 75 from the distribution ...
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