A Simulation Study: Population Distribution Function Estimation Using Dual Auxiliary Information under Stratified Sampling Scheme

<div>Efficiency result under populations I, II, and III when <span class="nowrap"><svg height="15.7059pt" id="M525" style="vertical-align:-3.4294pt" version="1.1" viewbox="-0.0498162 -12.2765 78.1005 15.7059" width="78.1005pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.511,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.778,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.921,0)"><path d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.082,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><rect height="0.65243" width="11.6264" x="39.3453" y="-11.5742"></rect><g transform="matrix(.013,0,0,-0.013,39.345,0)"><path d="M571 489C571 598 494 687 374 687C259 687 169 605 169 498C169 421 218 372 294 372C365 372 419 422 442 503L423 508C402 449 368 402 297 402C237 402 210 444 210 497C210 590 282 654 371 654C456 654 495 603 495 506C495 437 458 340 407 249C336 125 246 15 167 15C119 15 90 30 73 69C78 66 91 63 98 63C126 63 145 89 145 116C145 141 121 163 94 163C59 163 36 130 36 91C36 40 74 -15 150 -15C253 -15 324 44 392 141H394C387 126 385 98 385 86C385 19 422 -14 486 -14C568 -14 622 51 691 139L671 154C596 67 552 30 515 30S465 56 465 95C465 162 505 261 532 323C620 520 713 654 786 654C774 642 769 626 769 614C769 582 794 568 819 568C846 568 873 588 873 624C873 669 839 687 799 687C730 687 653 632 573 489H571Z"></path></g><g transform="matrix(.013,0,0,-0.013,50.972,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><rect height="0.65243" width="12.0832" x="56.1148" y="-11.5742"></rect><g transform="matrix(.013,0,0,-0.013,56.115,0)"><path d="M825 658L642 407C494 204 398 112 329 112C309 112 300 126 300 141C300 166 316 203 386 292C491 424 526 479 526 541C526 625 446 687 347 687C219 687 91 607 91 487C91 426 141 380 205 380C277 380 351 424 380 535H360C329 452 280 410 211 410C166 410 132 442 132 485C132 589 221 652 340 652C399 652 466 614 466 569C466 534 451 503 352 375C274 273 245 224 245 175C245 115 278 82 330 82C387 82 468 139 538 236L541 232C497 156 479 117 438 36C323 -1 123 -39 123 -161C123 -200 156 -226 204 -226C336 -226 470 -82 531 38C585 62 632 81 698 142L683 157C639 120 589 93 546 77C637 261 748 431 916 658H825ZM419 -2C355 -118 290 -196 204 -196C181 -196 165 -178 165 -159C165 -84 276 -50 419 -2Z"></path></g><g transform="matrix(.013,0,0,-0.013,68.198,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,73.341,0)"><path d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

tab5

Table 5

Table 5: A Simulation Study: Population Distribution Function Estimation Using Dual Auxiliary Information under Stratified Sampling Scheme 