## Introduction to Mathematical Statistics |

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Page 244

Thus the 1 1 statistic 1 & - 6 = u(X1, X2,..., X.) =#xx = x is the unique

likelihood statistic for the mean 6. In this instance the

is unbiased, sufficient, and efficient. - Example 2. Let sø, o] = } 0 < a. 3 8, 0 < 0 <

oo ...

Thus the 1 1 statistic 1 & - 6 = u(X1, X2,..., X.) =#xx = x is the unique

**maximum**likelihood statistic for the mean 6. In this instance the

**maximum**likelihood statisticis unbiased, sufficient, and efficient. - Example 2. Let sø, o] = } 0 < a. 3 8, 0 < 0 <

oo ...

Page 246

u,(r, y, ..., z) which maximize this likelihood function with respect to 61, 62, ..., 0,...,

respectively, define

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 286

Suppose, however, we modify this ratio in the following manner: We shall find the

shall find the

, ...

Suppose, however, we modify this ratio in the following manner: We shall find the

**maximum**of L(w) in w; that is, the**maximum**of L(a) with respect to 62. And weshall find the

**maximum**of L(Q) in Q; that is, the**maximum**of L(Q) with respect to 0, ...

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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