## Introduction to mathematical statistics |

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Page 123

... x2 = y2 define a one-to-one transformation from sf = {(x1, x2); 0 < x1 < oo, 0 <

x2 < oo} onto 3S = {(y1, y2); -2y1 < y2 and 0 < y2, -oo < y1 < oo}. The Jacobian of

the transformation is 12 II / = 0 1 = 2;

... x2 = y2 define a one-to-one transformation from sf = {(x1, x2); 0 < x1 < oo, 0 <

x2 < oo} onto 3S = {(y1, y2); -2y1 < y2 and 0 < y2, -oo < y1 < oo}. The Jacobian of

the transformation is 12 II / = 0 1 = 2;

**hence**the joint p.d.f. of Y1 and Y2 is g(yi.Page 263

However, by the hypothesis of the theorem, L(d") > (l/k)L(8') at each point of C,

and

each point of C*, and

However, by the hypothesis of the theorem, L(d") > (l/k)L(8') at each point of C,

and

**hence**at each point oiC C\ A*; thus f L(d") > \ f L(8'). But L(8") < (l/k)L(8') ateach point of C*, and

**hence**at each point of A O C* ; accordingly f L(fl') < I f I(flO.Page 337

But the statistics W2 = Ta2)3) W3 = Ta23)i, . . ., Wb_1 = Tl12...(b_inb are functions

only of these joint complete sufficient statistics. Thus W1 is stochastically

independent of W2, . . ., Wb_1 when ^ = jj.2, a2 = a2, and

= erf = ...

But the statistics W2 = Ta2)3) W3 = Ta23)i, . . ., Wb_1 = Tl12...(b_inb are functions

only of these joint complete sufficient statistics. Thus W1 is stochastically

independent of W2, . . ., Wb_1 when ^ = jj.2, a2 = a2, and

**hence**when ^ = fj.2, a2= erf = ...

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Accordingly best critical region binomial distribution cent confidence interval Chapter chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges stochastically critical region decision function defined degrees of freedom denote a random discrete type distribution having p.d.f. Equation Example EXERCISES F distribution function of Y1 given H0 is true independent random variables inequality integral joint p.d.f. Let the random Let X1 limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral order statistics Poisson distribution positive integer power function Pr X1 probability density functions probability set function quadratic form random experiment random interval random sample random variables X1 reject H0 respectively sample space Show significance level simple hypothesis H0 statistic for 9 statistic Y1 stochastically independent random subset testing H0 theorem type of random unbiased statistic variance a2 X1 and X2 Xn denote zero elsewhere