Fundamentals of Probability and Statistics for Engineers

More documents

Recommendations

Info

10Model VerificationThe parameter estimation procedures developed in Chapter 9 presume a distributionfor the population. The validity of the model-building process basedon this approach thus hinges on the substantiability of the hypothesized distribution.Indeed, if the hypothesized distribution is off the mark, the resultingprobabilistic model with parameters estimated by any, however elegant, procedurewould, at best, still give a poor representation of the underlying physical ornatural phenomenon.In this chapter, we wish to develop methods of testing or verifying a hypothesizeddistribution for a population on the basis of a sample taken from thepopulation. Some aspects of this problem were addressed in Chapter 8, inwhich, by means of histograms and frequency diagrams, a graphical comparisonbetween the hypothesized distribution and observed data was made. In thechemical yield example, for instance, a comparison between the shape of anormal distribution and the frequency diagram constructed from the data, asshown in Figure 8.1, suggested that the normal model is reasonable in that case.However, the graphical procedure described above is clearly subjective andnonquantitative. On a more objective and quantitative basis, the problem ofmodel verification on the basis of sample information falls within the frameworkof testing of hypotheses. Some basic concepts in this area of statisticalinference are now introduced.10.1 PRELIMINARIESIn our development, statistical hypotheses concern functional forms of theassumed distributions; these distributions may be specified completely withprespecified values for their parameters or they may be specified with parametersyet to be estimated from the sample.Let X 1 ,X 2 ,...,X n be an independent sample of size n from a population Xwith a hypothesized probability density function (pdf) f(x; q) or probabilityFundamentals of Probability and Statistics for Engineers T.T. Soong © 2004 John Wiley & Sons, LtdISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)TLFeBOOK
316 Fundamentals of Probability and Statistics for Engineersmass function (pmf) p(x; q), where q may be specified or unspecified. We denoteby hypothesis H the hypothesis that the sample represents n values of a randomvariable with pdf f (x; q) or p(x; q). This hypothesis is called a simple hypothesiswhen the underlying distribution is completely specified; that is, the parametervalues are specified together with the functional form of the pdf or the pmf;otherwise, it is a composite hypothesis. To construct a criterion for hypothesestesting, it is necessary that an alternative hypothesis be established againstwhich hypothesis H can be tested. An example of an alternative hypothesis issimply another hypothesized distribution, or, as another example, hypothesisH can be tested against the alternative hypothesis that hypothesis H is not true.In our applications, the latter choice is considered more practical and we shallin general deal with the task of either accepting or rejecting hypothesis H onthe basis of a sample from the population.10.1.1 TYPE-I AND TYPE-II ERRORSAs in parameter estimation, errors or risks are inherent in deciding whether ahypothesis H should be accepted or rejected on the basis of sample information.Tests for hypotheses testing are therefore generally compared in terms of theprobabilities of errors that might be committed. There are basically two typesof errors that are likely to be made – namely, reject H when in fact H is true or,alternatively, accept H when in fact H is false. We formalize the above withDefinition 10.1.D efinition 10. 1. in testing hypothesis H, a Type-I error is committed when His rejected when in fact H is true; a Type-II error is committed when H isaccepted when in fact H is false.In hypotheses testing, an important consideration in constructing statisticaltests is thus to control, insofar as possible, the probabilities of making theseerrors. Let us note that, for a given test, an evaluation of Type-I errors can bemade when hypothesis H is given, that is, when a hypothesized distribution isspecified. In contrast, the specification of an alternative hypothesis dictatesType-II error probabilities. In our problem, the alternative hypothesis is simplythat hypothesis H is not true. The fact that the class of alternatives is so largemakes it difficult to use Type-II errors as a criterion. In what follows, methodsof hypotheses testing are discussed based on Type-I errors only.10.2 CHI-SQUARED GOODNESS-OF-FIT TESTAs mentioned above, the problem to be addressed is one of testing hypothesis Hthat specifies the probability distribution for a population X compared with theTLFeBOOK
Page 3:
TLFeBOOK
Page 10 and 11:
ContentsPREFACExiii1 INTRODUCTION 1
Page 12 and 13:
Contentsix6.1.2 Geometric Distribut
Page 14 and 15:
ContentsxiAPPENDIX A: TABLES 365A.1
Page 16 and 17:
PrefaceThis book was written for an
Page 18 and 19:
1IntroductionAt present, almost all
Page 20 and 21:
Introduction 3may be rejected at th
Page 22 and 23:
Part AProbability and Random Variab
Page 25:
8 Fundamentals of Probability and S
Page 31 and 32:
14 Fundamentals of Probability and
Page 33 and 34:
Page 36:
Basic Probability Concepts 19where
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
TLFeBOOK
Page 55 and 56:
Page 57 and 58:
.40 Fundamentals of Probability and
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
TLFeBOOK
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
TLFeBOOK
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
Page 243 and 244:
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Page 253 and 254:
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
TLFeBOOK
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282: 264 Fundamentals of Probability and
Page 331: 314 Fundamentals of Probability and
Page 351 and 352: TLFeBOOK
Page 361 and 362: Â344 Fundamentals of Probability a
Page 381 and 382: TLFeBOOK
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Page 389 and 390:
Page 391 and 392:
TLFeBOOK
Page 393 and 394:
Page 395 and 396:
TLFeBOOK
Page 397 and 398:
Page 399 and 400:
Page 401 and 402:
Page 403 and 404:
Page 405 and 406:
TLFeBOOK
Page 407 and 408:
390 Subject IndexExtreme-value dist
show all

Fundamentals of Probability and Statistics for Engineers

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?