## Introduction to Mathematical Statistics |

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Page 246

u,(r, y, ..., z) which maximize this

respectively, define maximum

Z), ..., 6. – un(X, Y,..., Z) for the m parameters. If joint sufficient statistics for the ...

u,(r, y, ..., z) which maximize this

**likelihood**function with respect to 61, 62, ..., 0,...,respectively, define maximum

**likelihood**statistics 6, = u(X, Y,..., Z), 3, − us(X, Y,...,Z), ..., 6. – un(X, Y,..., Z) for the m parameters. If joint sufficient statistics for the ...

Page 288

Define the

L(Q) = IIf(r. 01, 62, ..., 6m), (61, 62, ..., 6,...) e Q. - Let L(6) and L(Q) be the maxima,

which we assume to exist, of these two

Define the

**likelihood**functions L(a) = IIf(r. 61, 62, ..., 6m), (61, 62, ..., 0,...) e w, andL(Q) = IIf(r. 01, 62, ..., 6m), (61, 62, ..., 6,...) e Q. - Let L(6) and L(Q) be the maxima,

which we assume to exist, of these two

**likelihood**functions. The ratio of L(6) to ...Page 293

Show that the

simple hypothesis Ho against an alternative simple hypothesis H1, as that given

by the Neyman-Pearson theorem. Note that there only two points in Q. 11.4.

Show that the

**likelihood**ratio principle leads to the same test, when testing asimple hypothesis Ho against an alternative simple hypothesis H1, as that given

by the Neyman-Pearson theorem. Note that there only two points in Q. 11.4.

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accept accordance Accordingly alternative approximately assume called cent Chapter complete compute conditional confidence interval Consider constant continuous type critical region decision defined definition degrees of freedom denote a random depend determine discrete type distribution function equal Equation event Example EXERCISES exists expected fact given Hence inequality integral interval joint p.d.f. Let X1 likelihood marginal matrix maximum mean moment-generating function mutually stochastically independent normal distribution Note observed order statistics outcome parameter Pr(X probability density functions problem proof prove random experiment random interval random sample random variable ratio reject respectively result sample space Show significance level simple hypothesis ſº stochastically independent sufficient statistic symmetric matrix Table theorem transformation true unknown variance write X1 and X2 zero elsewhere