A Course in Mathematical Statistics phần 6 pdf
A Course in Mathematical Statistics phần 6 pdf
... Point Estimationand its variance attains the Cramér–Rao bound.12.4.2 Refer to Exercise 12.3.5 and investigate whether the Cramér–Raobound is attained.12.4.3 Refer to Exercise 12.3 .6 and investigate ... ()∈[]=()∈[]By assuming the existence of an MLE, we have that the maximum at theleft-hand side above is attained at an MLE ˆθθθθθ. Then, clearly, the right-handside attains its maximum at ˆθθθθθ*, ... The Case Where Complete Sufficient Statistics Are Not Available orMay Not Exist: Cramér–Rao InequalityWhen complete, sufficient statistics are available, the problem of finding a UMVU estimator...
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A Course in Mathematical Statistics phần 10 pps
... 2 .65 75 63 0 .67 84 1.2951 1 .66 94 1.9983 2.3870 2 .65 61 64 0 .67 83 1.2949 1 .66 90 1.9977 2.3 860 2 .65 49 65 0 .67 83 1.2947 1 .66 86 1.9971 2.3851 2 .65 36 66 0 .67 82 1.2945 1 .66 83 1.9 966 2.3842 2 .65 24 67 0 .67 82 ... 1.2943 1 .66 79 1.9 960 2.3833 2 .65 12 68 0 .67 81 1.2941 1 .66 76 1.9955 2.3824 2 .65 01 69 0 .67 81 1.2939 1 .66 72 1.9949 2.38 16 2 .64 9070 0 .67 80 1.2938 1 .66 69 1.9944 2.3808 2 .64 7971 0 .67 80 1.29 36 1 .66 66 1.9939 ... 1.2 963 1 .67 16 2.0017 2.3924 2 .66 3359 0 .67 87 1.2 961 1 .67 11 2.0010 2.3912 2 .66 18 60 0 .67 86 1.2958 1 .67 06 2.0003 2.3901 2 .66 03 61 0 .67 85 1.29 56 1 .67 02 1.99 96 2.3890 2 .65 89 62 0 .67 85 1.2954 1 .66 98 1.9990...
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A Course in Mathematical Statistics phần 9 pps
... fertilizers.Comparison of a certain brand of gasoline with and without an additive byusing it in several cars.Comparison of different brands of gasoline by using them in several cars.Comparison of the wearing ... We maysummarize all relevant information in a table (Table 1) which is known as anAnalysis of Variance Table.EXAMPLE 3For a numerical example, take I = 3, J = 5 and letY11= 82 Y21= 61 ... Two-way Layout (Classification) with One Observation Per Cell 447440 17 Analysis of Variance440Chapter 17Analysis of VarianceThe Analysis of Variance techniques discussed in this chapter can...
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A Course in Mathematical Statistics phần 8 doc
... repeated at will, in many other cases this is, indeed, the case. In the latter case, as a rule, it is advantageous not to fix the sample size in advance, but to keep sampling and terminate the ... (as well as in the interval estimation problemsto be dealt with in Chapter 15), sampling according to a stopping time is, in general, profitable, the mathematical machinery involved is well beyond ... Zn alone.REMARK 1 In certain circumstances, a stopping time N is also allowed totake the value ∞ but with probability equal to zero. In such a case and whenforming EN, the term ∞ · 0 appears,...
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A Course in Mathematical Statistics phần 7 ppsx
... H that the manu-facturer’s claim is correct against the appropriate alternative A at level ofsignificanceα= 0.01.13.5.5 The diameters of certain cylindrical items produced by a machine arer.v.’s ... Determine the minimum sample size n required by using the Binomialtables (if possible) and also by using the CLT.13.3.10 The number X of fatal traffic accidents in a certain city during a yearmay ... Testing Hypotheses13.3.4 The rainfall X at a certain station during a year may be assumed to be a normally distributed r.v. with s.d. σ= 3 inches and unknown mean μ. For thepast 10 years,...
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A Course in Mathematical Statistics phần 5 ppsx
... theIntermediate Value Theorem (see, for example, Theorem 3(ii) on page 95 in Calculus and Analytic Geometry, 3rd edition (1 966 ), by George B.Thomas, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts)there ... permutation of the integers 1 through n. Fromthe definition of a determinant and the fact that each row and column containsexactly one 1 and the rest all 0, it follows that the n! Jacobians are ... of a normally distributed r.v. with mean a μ+ b and variance a 2σ2. Thus, if X is N(μ,σ2), then aX + b is N (a μ+ b, a 2σ2).Now it may happen that the transformation h satisfies all...
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A Course in Mathematical Statistics phần 4 docx
... differentiation as follows:178 7 Stochastic Independence with Some ApplicationsTo start with, consider the probability space (S, A, P) and recall that kevents A 1, , A k are said to be independent ... a function defined on the entire real lineand taking values in the complex plane. Those readers who are not well versed in matters related to complex-valued functions may feel uncomfortable in dealing ... respectively. In particular, if X1 and X2 have the BivariateNormal distribution, it can be seen that they are positively quadrant depend-ent or negatively quadrant dependent according to whether...
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A Course in Mathematical Statistics phần 3 docx
... improper integral in which the limitsare taken symmetrically. As an example, for σ= 1, μ= 0, we have, in terms ofthe principal value integral,IxdxxxAA A A A A A A A * lim lim loglim ... over x, as x varies in G, of such intervals is equal to G. The same is true if we consider only thoseintervals corresponding to all rationals x in G. These intervals are countablymany and each ... some preliminary concepts and resultsare needed and will be also discussed. A set G in ޒ is called open if for every x in G there exists an open intervalcontaining x and contained in G. Without...
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A Course in Mathematical Statistics phần 2 pps
... black at least 2 cards in are red exactly 1 card in is an ace the first card in is a diamond,the second is a heart and the third is a club card in is a diamond, 1 is a heart and 1 is a club2.4.30 ... is a generic model for situations often occurring in practice. Forinstance, the urn and the balls may be replaced by a box containing certainitems manufactured by a certain process over a specified ... that of calculating probabilities of events which may be ofimportance to us. Such calculations are facilitated by a transformation of thesample space S, which may be quite an abstract set, into...
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A Course in Mathematical Statistics phần 1 ppt
... .nnjjAA→∞=∞=1UWe recall thatAAAA AAAAAA AAjjccc=∞=+ ∩()+∩∩()+⋅⋅⋅=+ −()+−()+⋅⋅⋅111 2 1 23121 32U,by the assumption that A n↑. HenceP A P A PA PA A PA A PA A PA PA A ... nnnnnnn2321021⎛⎝⎜⎞⎠⎟+⎛⎝⎜⎞⎠⎟+⋅⋅⋅+⎛⎝⎜⎞⎠⎟=−⎛⎝⎜⎞⎠⎟−⎛⎝⎜⎞⎠⎟=−−relationships characterizing the independence of A j, j = 1, , n and they areall necessary. For example, for n = 3 we will have:PA A A PA PA PAPA A PA PAPA A PA PAPA A PA PA123 1 ... =14, and set A 1= {1, 2}, A 2= {1, 3}, A 3= {1, 4}. ThenAAAAA A AA A 121323 12311∩=∩=∩={}∩∩={},.andThusPA A PA A PA A PA A A12 13 23 12314∩()=∩()=∩()=∩∩()= .Next,PA A PA...
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