Record Values from Exponentiated Pareto Distribution and Associated Inference

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Recod Vaue o Epoetated Paeto Dtbuto ad Aocated Ieece A. I. Shaw * ad Haaa H. Abu-Zadah ** G Coege o Educato Jeddah Scetc Secto Kg Abduazz Uvet P. O. Bo 369 Jeddah 438 Saud Aaba. Abtact I th pape we code the epoetated Paeto EP dtbuto wth two hape paaete ad. We t deve the eact o o ea vaace ad covaace o ecod vaue. The au ehood etato ME' o the two hape paaete ad ae deved. Baed o the oet o owe ecod vaue we obtaed the ecea coecet o the bet ea ubaed etato BUE' o the ocato ad cae paaete o EP dtbuto. The vaace ad covaace o thee etato ae ao gve. The bet ea ubaed pedcto BUP o the utue ecod vaue ao dcued. Ke wod: owe ecod vaue; Epoetated Paeto dtbuto; Moet o ecod vaue; Mau ehood etato; he oato; Bet ea ubaed etato; Bet ea ubaed pedcto.. Itoducto Recod vaue ae atua a ea e appcato vovg data eatg to weathe pot ecooc ad e tetg tude. Ma autho have tuded ecod vaue ad aocated tattc; o eape ee Abd-E-Ha ad Suta Aod et a. 998 Baaha ad Cha 993 Suta ad Mohe Suta et a. ad Suta et a.. Gupta et a. 998 popoed to ode aue data b *tt] whee t the baee dtbuto ucto ad a potve ea ube. Th ebe eough to accoodate both ootoc a we a o- ootoc aue ate eve though the baee aue ate ootoc. Modeg uvva data b oootoc aue ate deabe o eape whe the coue o the deae uch that otat eache a pea ate oe te peod ad the ow dece. Recet Mudhoa ad Huto 996 Mudhoa ad Svatava 993 * Peaet adde: ac. o Eg. at Shouba P.O. Bo 6 E Maad 78 Cao Egpt E-a adde: ahaw @ahoo.co. ** E-a adde: haaa_abuzadah@ahoo.co.

- - Mudhoa et a. 995 Naa ad Ea 3 4 ad Sgh et a. 5 tuded the popoed ode wth Webu a the baee dtbuto ae the epoetated Webu a. I ae wa Gupta ad Kudu999 ab 6 Raqab 4 Raqab ad Ahauah ad Zheg tuded the popoed ode wth epoeta a the baee dtbuto whch the geeazed epoeta a o epoetated- epoeta dtbuto. I th pape we tae the popoed ode wth Paeto dtbuto a the baee dtbuto epoetated Paeto dtbuto EP wth pobabt det ucto pd ] > > >. ad cuuatve dtbuto ucto cd ] > > >. whee ad ae two hape paaete. o oe popete ee Shaw ad Abu- Zadah 6. Whe the above dtbuto coepod to the tadad Paeto dtbuto o the ecod d. et be a equece o..d. ado vaabe wth cd ad pd. Set Y a { }. We a a uppe owe ecod vaue o th equece Y > < Y >. B deto a uppe a we a a owe ecod vaue. Oe ca tao o uppe ecod to owe ecod b epacg the oga equece o { } b { } o P > o a b { }; the owe ecod vaue o th equece w coepod to the uppe ecod vaue o the oga equece. We w coe ou atteto to ut owe ecod vaue. I th pape we deve the eact o o ea vaace ad covaace o owe ecod vaue o the EP dtbuto Secto. I Secto 3 the au ehood etato ME' o the hape paaete ad ae deved. a Secto 4 we obtaed the ecea coecet o the bet ea ubaed etato BUE' o the ocato ad cae paaete o EP dtbuto ad the vaace ad covaace o thee etato baed o the oet o owe ecod vaue. I addto we dcued the bet ea ubaed pedcto BUP o the utue ecod vaue o EP dtbuto. et. Moet o owe ecod vaue... be the t owe ecod vaue o the EP dtbuto. the the ge ad doube oet ae deved a oow:

- 3 -. Sge oet The pd o ca be wtte ee Aod et a. 998] a. ] > Γ. o.. ad. the ge oet o the owe ecod vaue ae Γ ] d E µ. B ettg ] t we d that Γ ]! dt t t µ whee >. ; ;... a a a a B puttg w - t we obta >...! ad µ.. Whe we get >.! µ.3 Whe we have >.! µ.4 o.3 ad.4 the vaace o. Va µ µ σ.5. Doube oet The ot pd o two owe ecod vaue < ca be wtte ee Aod et a. 998] a ] ] Γ Γ. < < <.6

- 4 - o.. ad.6 the doube oet o the owe ecod vaue ad ae gve b d d E ] ] Γ Γ µ ] d I Γ Γ.7 whee. ] d I Settg w we get. ]! Γ I Puttg I.7 we obta ] ]! d A µ whee. A Γ B ettg ] t we d that. ]!!. dt t t A µ Puttg w - t we have!! µ... a > ad..8

- 5 - Whe we get!! > µ..9 The covaace betwee the two owe ecod vaue ad < gve b Cov µ µ µ σ.. 3. Mau ehood etato The ot det ucto o the t owe ecod vaue... o EP dtbuto ehood ucto gve b...... ] ]... >. 3. The og-ehood ucto o 3. gve b ] ]. 3. The oa equato becoe: v 3.3 v v v v v 3.4 whee.... v o 3.3 we obta the au ehood etato ME o a a ucto o a ˆ whee

- 6 - ˆ. v ] Puttg ˆ 3. we obta 3.5 g ˆ v ] v. v ] 3.6 Theeoe ME o a ˆ ca be obtaed b azg 3.6 wth epect to. It ca be how that the au o 3.6 ca be obtaed a a ed pot outo o the oowg equato; h 3.7 whee v v v h. 3.8 v v v Ve pe teatve pocedue ca be ued to d a outo o 3.7 ad t wo ve we. Oce we obta ˆ the ME o a ˆ ca be obtaed o 3.5 a ˆ ˆ ˆ. Now we tate the aptotc oat eut to obta the aptotc vaace o the etate. It ca be tated a oow: ˆ ˆ N I 3.9 R whee I the he oato at ee Ahad ad Agha ].e. I R R E E E E The eeet o the he oato at ae a oow: J J J e e e d! J J J J.

- 7 - J ] e e e d! ]. e e e d e! Aug that the hape paaete ow o 3. we ca obta the ME ˆ o the hape paaete a ˆ. 3. v ] o the aptotc popete o the ME t oow that ˆ ˆ E ad Va J. Now code the ME o whe the hape paaete ow. o ow the ME o a ˆ ca be obtaed b azg u v v v wth epect to. It ca be ea how that u a uoda ucto o ad ˆ whch aze u ca be obtaed a the ed pot outo o 3. whee v v v. v v o the aptotc popete o the ME t oow that ˆ ˆ E ad Va J. I Tabe we have uated baed o Mote Cao u the vaue o ba ad oot ea quae eo RMSE' o ME' o that whe ow o 5.5.54 ad 5. o th tabe we ote that the RMSE' o the ME' o deceae a ceae. We ca ee ao the RMSE' o the ME' o ot o the te ceae a vaue o ceae o ed. Thee o potat eect o the etato whe have deet vaue.

- 8 - Tabe. Suated vaue o ba ad RMSE' o ME o whe ow. 3 4 5.5.896..543 4.3536 4.7564 5.5367 6.5 7.33.4689 4.488 3.5376.383.7547.344 3.4373.348.97884 5.856 5.5687.9 4.666.598.5783.6.5885.68.355.966 4.57648 5.348 5.64653.5574.7.65947 3.378.9.489 3.577 3.9668 5.4346 3.364.76 3.596 4.6667 5.5854 8.4449 4.335 9.8 5.977 3.3 3.554547.39858.8454.5535 3.9469.36 3.89954 5.4474 6.464 9.5867 4.643638.764.58399.5397 3.735.6969.65 3.4748 5.3 6.963 5.7668.3573.3465.783 3.4693.935.355 3.5758 4.456 5.69359 3.5.89.5946 3.9337 4.6 5.4634 3.87965.448 7.688 4.8969 7.8375 3.64486.896.583.938.8554.53446 3.545 6.47 5.3458 7.887 4.59684.46.68.4737 3.458.934953.87978 3.443 6.3596 6.539 5.796.48.545.75 3.766.7899.9683 3.854 3.967 4.8468 4.887.946 3.3564 3.63593 6.654 3.3565 6.88.9 9.835.585 3.6853.9355.83437.69 3.85.5677.96776 4.5955 4.854.5373 4.665.99.73477.696 3.6955.965.45 3.9685 5.7743 6.4666 5.74977.3484.9998.8669 3.388.47.583 3.9 4.4984 5.3 Note: The t et the uated ba. The ecod et the uated RMSE. I Tabe we have uated baed o Mote Cao u the vaue o ba ad RMSE' o ME' o that whe ow o 5.5.54 ad 5. o th tabe we ote that the RMSE' o the ME' o deceae a ceae. We ca ee ao the RMSE' o the ME' o deceae a vaue o ceae o ed. o ed ot o the te the RMSE' o the ME' o ceae a vaue o ceae.

- 9 - Tabe. Suated vaue o ba ad RMSE' o ME o whe ow. 3 4 5.5 7.636.786.9653.6767.588 3.783 4.487.5444.94447.5798 3 4.5793.5844.74635.484.33997.5 3.53698.9745.483.37 4 3..93355.4673.9475.4435 8.73.4779.4883.6488.895974 5 3.39.45969.43.5674 -.863439 7.497.74343.747.84697.757839 3 8.9538.935.343.897666.67397 34.69 5.58654 3.8856.5463.97 3 6.5439.56864.86734.695.38599 6.4 3.66648.583.833.398 4 4.5657.83.43665.55.836.599.9993.698.3.57 5.9878.666463.3365.875 -.43 7.99.434.4447.38.88576 3.5.935.36.36446.88967.73468 45.77 6.64645 3.944.6368.575 3 6.885.74538.9599.57835.368 4.8576 4.348.557.9746.54463 4 5.37.67.65657.398.9595 4.53 3.37.97.59598.595 5 4.48395.64.3.59 -.87547.85.84889.53.95.4585 4.56.88587.745.393.8453 3.758 7.84 4.75895 3.44.5976 3 7.33688.959.998.7575.5784 7.6489 5.346 3.8438.34349.866 4 6.593.4363.696.33569.798 4.953 3.6873.3673.6994.455 5 5.7753.94667.38658.54 -.673.765 3.35.968.46633.8 Note: The t et the uated ba. The ecod et the uated RMSE. I Tabe 3 we have uated baed o Mote Cao u the vaue o ba ad RMSE' o ˆ ad ˆ whe both o the ae uow o 5.5.54 ad 5. o th tabe we ote that the RMSE' o ˆ ad ˆ ot o the te ceae deceae that whe a ceae. We ca ee ao ot o the te the RMSE' o ˆ ceae ad the RMSE' o ˆ deceae

- - that whe the vaue o ceae o ed. o ed ot o the te the RMSE' o ˆ deceae ad the RMSE' o ˆ ceae a vaue o ceae. Tabe 3. Suated vaue o ba ad RMSE' o ME' o ad whe both o the uow. 3 4 5 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ.5 3.899 3.668 8.64497.494976 3.873.97695 4.787.78.85.774 3.337.7477 4.9757.693 43.9579.43984 3.9965 3.5583 9.8483.666 3.394.69 9.535.44765 4.777 39.375 6.5377 3.566 33.9463 3.736 4.7366 3.6933 4.7773 -.94.6.89544 4.595 3.337 7.5938.779 4.687 4.357.54 5.53 7.94 4.9763 35.855 4.8 5.6946 -.58.388 3.9736 8.6 3.686 4.8.97595 4.668 3.6849 9.887 6.6969 33.73 5.75 48.668 4.474 3 3.595 33.45 4.463 -.786 6.646.735 9.744.789 8.47977 44.69.3775 3.3844 9.743.6783 34.394.7536 3.88899 5.5747 6.54.89.773.9866 4.796.834 5.85787 7.764 3.4763 3.56363 3.565 3.6983 7.786 3.64757 4.35496 -.73859 7.37835.34.393 3.366 7.96 3.8798 3.534 6.768 4.9434 4.8969 9.975 5.3848 3.337 4.7589 5.4966 -.884995.885 4.895 6.344 4.4597 9.8797 3.578 3.9845 3.785 9.3776 7.3589 7.8 6.3963 33.854 5.356 3.5.483.38.934 -.8475 3.99787 -.593395 5.337.39 6.9334 8.9.8 4.8466 6.548.88758 7.78.8475 3.6347.974 4.77578.4465 7.757.43 9.56354.559 4.6376 38.789.537 3.5683 5.7434 3.669.7 3.84887 4.85 -.4733 7.43.58.5.8895 3. 3.43685 3.5535 7.3844 8.556 5.77.4994 5.9597.9 5.36358 5.4345 -.8866 9.4948 3.97484 5.836 4.55787 6.9958 3.88634.59785 3.8835 6.775 7.797 7.9558 7.55 8.47 5.886 4.39433 45.893.834344 -.693.3377 -.9594.433 -.89847 7.48 44.794 7.758 7.4935 9.5883 3.69.984 3.46 3.4938 5.48 3.5959 -.339348 5.89.66 7.49698.488 3.7946 46.794. 3.8379.996 3.684 6.6 3.73374 4.65343 -.633 4.4993.47 9.38934.83486.6736 3.9454.6786 8.77.763 4.6674 6.389 5.54577.698 5.39957 5 -.466 -.547 8.49 3.373 4.35 4.6789 7.33 4.5839.876 4.643 6.668 6.84788.446 7.398 7.355 6.834 Note: The t et the uated ba. The ecod et the uated RMSE.

- - 4. BUE' o ocato ad cae Paaete ad BUP o the utue ecod vaue The eact ad epct epeo o the t ad ecod ge oet o owe ecod vaue gve.3 ad.4 aow u to evauate the ea ad vaace o a owe ecod vaue. o the coputato o vaace ad covaace the poduct oet µ wee coputed t. The dagoa eeet µ µ ae obtaed o the eact epeo gve.4. Net the eact ad epct epeo gve.9 wa ued o the coputato o the poduct oet o two owe ecod vaue. Thee vaue wee ued to evauate the covaace.. The vaue o ea vaace ad covaace wee coputed b ug Matheatca 4.. It potat to ea hee that a ou coputato o the tadad Paeto dtbuto o the ecod d ca be obtaed whe. Aue Y... to be a te equece o d ado vaabe o the Y EP dtbuto wth pd µ ; ] µ µ σ σ σ σ 4. µ < < σ >. et Y Y... Y be the t obeved owe ecod vaue o the above equece. The... whee Y µ σ... the vecto o obeved ecod vaue o a popuato wth the tadad EP dtbuto wth pd ad cd gve. ad. epectve. The we ca wte the BUE' o µ ad σ ee Aod et a. 998] a ˆ µ a Y a Y... a Y 4. ˆ σ b Y b Y... b Y 4.3 whee a' ad b' ae the ete o the at T T C A V A A V ;

- - T µ T... µ µ µ... µ A V the vee o the covaace at V etato ae gve b whee σ. Vaace ad covaace o thee Va ˆ µ d σ Va ˆ σ d σ ad Cov ˆ µ ˆ σ d σ 4.4 d d T D. d A V A d Tabe 4: ea o owe ecod vaue 3 4 5.5.66667.8333.4385.6794.93.7.36776.5358.688.835 3.6757.87.3.478.536 4.99.73.9.7355.357 5.397.655.99.9446.5759 3.5.8.5.99.36688.3847.933.444.53.694 3.5533.573.433.3566.46 4.469.879.563.43.895 5.58.5368.588.5859.94 3.5.4.63333.856.9443.696.585.453.34739.436.5355 3.4685.765.35.74.3389 4..7446.38.8597.3596 5.988.4564.8969.3388.76 4.33333.538.6634.7736.86648.9958.7.9539.36949.43358 3.463.6.7493.337.835 4.83.6458.393.68.84 5.86.397.778.583.59

- 3 - Tabe 5: Covaace o owe ecod vaue 3 4 5.5. 3.9936 5.5856 7.76 8.4863.56.757.899.35559.4846 3.68.4853.7.8763.968 4 -.453.45.935.98.59 5 -.4 -.45 -.76 -.38 -.365.675.5438.3599.348.3858 3.845.56.8575.747.4758 4.73.475.4376.68.7978 5.33.346.578.387.4994 3 3.8.379.6344.8886.33 3 4.44.737.35.468.6 3 5.89.59.89.863.388 4 4.94.36.535.3783.54 4 5.5.7.48.35.33 5 5.84.539..9.645 3.75.6.6859.677.4885.583.3.584.885.55 3.64.349.438.547.576 4 -.75.584..75.93 5 -.53 -.469 -.466 -.644 -.995.3943.865.75.645.974 3.54.335.4979.664.86 4.45.56.6.363.4563 5.96.843.57.67.95 3 3.756.9.38.594.6478 3 4.89.3.987.87.36 3 5.5.66.88.76.35 4 4.95.84.594.38.3 4 5.83.459.949.446.93 5 5.57.356.777.9.65

- 4 - Tabe 5: Cotued 3 4 5 3.5.37333.59889.7763.98.6365.376.696.939.746.939 3.8.38.857.337.366 4.6.49.8.95.899 5 -.84 -.87 -.36 -.4 -.38.59.55.796.9966.79 3.789.83.33.43.59 4.3.33.736.365.98 5.37.576.53.496.896 3 3.55.55.5.3386.46 3 4.4.76.346.883.37 3 5.89.43.84.89.535 4 4.39.587.93.57. 4 5.6.34.658.989.99 5 5.4.5.543.84.5 4..34467.43663.535.57973.573.4656.69.6986.7669 3.6.438.979.33.476 4.37.385.65.69.675 5 -.76 -.86 -.35 -.7 -.79.833.3796.5353.6648.7769 3.573.48.6.99.3489 4.9.744.3.657.8 5..48.754.58.37 3 3.386..79.374.885 3 4.5.557.97.34.67 3 5.67.3.59.855.93 4 4.4.433.795.3.433 4 5.45.4.483.78.934 5 5.3.88.4.64.85

- 5 - Tabe 6: Coecet o the BUE o µ 3 4 5.5 -.348 -.5394 -.669 -.68758 -.745.348.5394.669.68758.745 3 -.664 -.58 -.6655 -.7859 -.884 -.4937 -.7854 -.9873 -.5 -.3596.534.83569.488.37.3399 4 -.34 -.857 -.54 -.8 -.446 -.754 -.354 -.359 -.37746 -.46 -.37 -.474 -.5967 -.68465 -.75786.435.76.937.839.844 5.538 -.7848 -.5 -.38 -.6 -.58 -.949 -.474 -.7673 -.45 -.93 3.76436 -.4696 -.399 -.3447 -.6897 4.56 -.4384 -.4937 -.546.467-5.979.8359.979.9693 3 -.383 -.57583 -.6986 -.78574 -.8579.383.57583.6986.78574.8579 3 -.47 -.79 -.566 -.5 -.45 -.44587 -.739 -.935 -.459 -.5487.48693.793.7.677.95 4 -.96 -.57 -.684 -.3554 -.455 -.39 -.399 -.335 -.3895 -.443 -.89 -.436 -.5438 -.67 -.6945.48.699.89363.56.797 5.74. -.35 -.776 -.66 -.677 -.3 -.58 -.9593 -.767 -.947 -.7 -.359 -.8853 -.3356 -.353 -.349 -.459 -.4659 -.565.38466.63.895.9548.7355

- 6 - Tabe 6: Cotued 3 4 5 3.5 -.477 -.66 -.7587 -.8588 -.93643.477.66.7587.8588.93643 3 -.55 -.99 -.33 -.58 -.773 -.449 -.67489 -.85775 -.9993 -.44.47.774.9986.574.97 4 -.35 -. -.55 -.4745 -.565 -.385 -.478 -.79473 -.3986 -.45584 -.787 -.463.6983 -.589 -.654.436.6788.345.359.6645 5.84.93 -.653 -.4 -.6 -.65 -.87 -.749 -.966 -.448 -.895 -.665 -.3494 -.797 -.384 -.5 -.349 -.8 -.4439 -.4876.39779.6468.5383.949.657 4 -.46 -.65449 -.8499 -.9533 -.57.46.65449.8499.9533.57 3 -.573 -.378 -.5348 -.865 -.535 -.4466 -.65 -.838 -.9748 -.93.4697.7639.98466.5693.9666 4 -.46 -.564 -.493 -.5645 -.673 -.353 -.465 -.335 -.4444 -.4659 -.756 -.3999 -.49565 -.56969 -.6345.475.66748.878.359.685 5.86.56 -.675 -.37 -.97 -.67 -.8 -.7599 -.989 -.5777 -.9563 -.6897 -.393 -.7355 -.3768 -.3335 -.374 -.383 -.436 -.468.38738.694.78897.9383.5544

- 7 - Tabe 7: Coecet o the BUE o σ 3 4 5.5..39749.59.996.954 -. -.39749 -.59 -.996 -.954 3.57758.348.695.338.97 6.5569 3.8354 3.67.5843.367-6.8337-4.7769-3.863 -.838 -.573 4.3893.69.537.3.656 3.63394.99396.53.75.33 6.53 3.54764.68486.466.97354 -.756-5.7449-4.34943-3.64834-3.34 5.3368.644.76.9335.895.8876.684.87.838.7663 4.688 -.68.6658.3883.693 6.8779 -.6983.59695.7.836-4.6 9.476-5.37537-4.4455-3.84 3.766.96978.65684.47693.35549 -.766 -.96978 -.65684 -.47693 -.35549 3.656.6565.559.4587.459 6.386 4.8537 3.3436.8846.634-7.394-4.7488-3.836-3.3478-3.85 4.76457.4497.3833.755.459 3.939.7987.7858.539.3747 6.796 3.73635.8865.4493.7466 -.948-6.439-4.98696-4.595-3.794 5.66856.349.388.5.7848 3.747.6479.8.596.999 4.84.457.8448.5669.39365 6.5375 3.7797.84.3877.99737-4.634-8.53-6.95-5.87-4.4787

- 8 - Tabe 7: Cotued 3 4 5 3.5 3.597.55885.843.9688.8345-3.597 -.55885 -.843 -.9688 -.8345 3.5336.96995.7877.6959.639 6.64638 4.435 3.6576 3.48.975-8.7944-5.456-4.44489-3.934-3.633 4.389.6355.753.4533.3849 4.6.56968 3.4669.7958.6347 6.53 4.79-3.543.677.3998 -.88-7. -.6769-4.89773-4.4363 5.67.5555.3856.3657.8333 3.4665.8839.489.98.64 4.3748.657.96739.75.57 7.5654 4.85.664.5396.64-5.9-9.79-4.9366-5.898-5.6 4 4.787 3.5856.757.4777.3-4.787-3.5856 -.757 -.4777 -.3 3.9943.8556.539.939.8569 7.63 4.846 4.47 3.643 3.3586-9.556-6.7-5.935-4.5534-4.355 4.5983.867.6748.5846.598 4.5499.8658.3747.557.88599 6.984 4.964 3.4777.9395.64358 -.986-8.854-6.495-5.57936-5.5937 5.34765.6933.5864.43938.3944 3.75874.46.6584.436.9656 4.7398.86498.3877.93643.75433 7.979 4.355 3.87.75869.39868-7. -9.983-7.6965-6.565-5.84398

- 9 - Tabe 8: Vaace ad Covaace o the BUE' o µ ad σ te o σ 3 4 5.5.9474.8554.59 3.4745 4.77638 8.5 7.897 6.94598 6.8546 6.745 -.39475 -.663-3.6559-4.5896-5.444 3.883.875.973.379.467.9473.759.88.7735.73 -.5466 -.33393 -.499 -.6484 -.78475 4.36.6.56.8967.3653.489.3688.5789.637.63 -.435 -.3 -.787 -.4349 -.339 5.69.946.86.3479.5539.88856 3.7346.87983.78894.7385 -.79 -.5348 -.8437 -.69 -.553 3.367.4879.896.8635.785 5.4759 4.3599 4.47 4.647 4.35 -.6656 -.69 -.7538 -.3583 -.588 3.3.494.576.7343.485.3678.69983.57.4749.36565 -.44 -.84 -.3464 -.496 -.5534 4..57.333.534.7978.8738.6699.98397.95.85438 -.33 -.783 -.6 -.75 -.6 5.48.45.46.6.3397.64447.95454.76799.68597.6466 -.398 -.36 -.65 -.88 -.67

- - Tabe 8: Cotued 3 4 5 3.5.765.358.44753.68586.949 4.86 3.37457 3.88 3.54 3.3476 -.46 -.7897 -.876 -.359 -.678 3.75.3.67.88.543.69.4846.359.3395.8873 -.7486 -.697 -.385 -.3738 -.3697 4.5.858.665.353.58.64558.4989.653.8834.7655 -.445 -.63 -.65 -.35 -.63 5.34.9.84.476.9.49585.8756.7873.6398.5876 -.9 -.848 -.537 -.6897 -.8789 4.4634.57.84.4878.5853 3.467.8883.7447.6996.6896 -.396 -.573 -.7933 -.9889 -.6847 3.539.5.466.747.57.8895.34983.9357.85.847 -.593 -.79 -.879 -.46 -.8985 4..66.463.58.3693.578.9743.8784.747.7755 -.996 -.49 -.789 -.56 -.3 5.6.6.58.7.653.39979.8655.65475.586.546 -.95 -.364 -.438 -.5656 -.798 The ecea coecet 4. ad 4.3 equed o the BUE' o µ ad σ epectve ca be ea obtaed baed o the coputato o the ea vaace ad covaace o owe ecod vaue o the EP dtbuto. Tabe 4 peet the ea o 5 o 5 ad.5.54 to ve deca pace. Tabe 5 povde the vaace ad covaace o owe ecod vaue to ve deca pace o 5 5 5 ad.5.5 4. Tabe 6 ad 7 povde the coecet o the BUE' o the ocato ad cae paaete epectve o 5 5 ad.5.5 4. The accuac o the cacuato have bee checed though the dette

- - a ad b. I Tabe 8 we copute the vaace ad covaace o the BUE' gve 4.4. o th tabe we ote that the vaace o the BUE' o µ ad σ deceae ad the covaace o thee etato ceae whe the vaue o o o ed ceae. o ed the vaace o the BUE' o µ ceae ad the BUE' o σ ad the covaace o thee etato deceae a the vaue o ceae. Eape: et u code the cae whee the copoet have aue te whch oow a EP dtbuto 4. wth 3 4 µ 6 ad σ. Suppoe that we obeve the oowg uated obeved aue te o EP µ σ : 6.64374 6.4459 6.6775 6.86 6.939 8.959 7.668 6.5648 6.99 6.5797. Theeoe we obeve the owe ecod vaue o the obeved data a oow: 6.64374 6.4459 6.939 6.5648. Hee the o the ecoded data aa wth 4 3 ad 4 µ ad V ae obtaed o Tabe 4 ad 5 epectve. The coecet 4. ad 4.3 ae peeted Tabe 6 ad 7 epectve. Theeoe the BUE' o µ ad σ ae coputed to be µˆ 5.84478 ad σˆ.695. The coepodg vaace ad covaace o µˆ ad σˆ ee Tabe 8 ae coputed to be Va ˆ µ.463 σ Va ˆ σ.8784 σ ad Cov ˆ µ ˆ σ.789 σ. et u code the tue popuato ea ζ E Y µ 3B33 4 σ. Had we ued ζ * Y Y Y Y 4 the ea o the obeved ecod. We woud 3 4 have ζ * 6.3697 ad tadad eo S.E. ζ *.398. The BUE o ζ ˆ ζ ˆ µ 3 B 33 4 ˆ σ 6.96745. The tadad eo o ζˆ coputed to be S.E.ζˆ.995. Theeoe the BUE' peo bette tha the ea o obeved ecod the ee o tadad eo. I the cotet o pedcto o the utue ecod obevato uppoe we obeve o the t ecoded obevato Y Y Y... Y ad the goa to pedct Y whee <. The ot we-ow pedcto the bet ea ubaed pedcto BUP ee Aod Baaha ad Nagaaa 998 o Y gve b ˆ Y ˆ ˆ T µ σ µ w Y ˆ µ ˆ V σµ

- - whee µ the ea o the t ecod vaue ad w the vecto o the covaace betwee the th utue ecod vaue ad the t ecoded obevato. The ea quae pedcto eo MSPE o Y ˆ oud to be ee Raqab MSPE Yˆ T T σ { E V E σ E w} whee T T T T T T E µ A V A A V w V I A A V A A V. I ou data etup we have obeved ou ecod vaue. Tabe 4-7 ae ued to copute the BUP o the utue ecod vaue Y baed o the t ou obeved ecod vaue. The vaue coputed to be Y ˆ 5.997 5 5 ad the coepodg MSPE gve b ˆ MSPE Y.7539σ. 5 Acowedgeet. We ae cee thau to the eeee o the vauabe ad cotuctve coet that had poved ou oga pape. Reeece ] Abd-E-Ha N. S. ad Suta K. S.. Mau ehood etate o Webu paaete baed o ecod vaue. J. Egpt. Math. Soc. 9 79-89. ] Ahad J. ad Agha N. R.. O the he oato ecod vaue. Meta 53 95-6. 3] Aod B. C. Baaha N. ad Nagaaa H. N. 998. Recod. Joh We New Yo. 4] Baaha N. ad Cha P. S. 993. Recod vaue o Raegh ad Webu dtbuto ad aocated eece. Natoa Ittute o Stadad ad Techoog Joua o Reeach Speca Pubcato 866 4-5. 5] Gupta R. C. Gupta R. D. ad Gupta P.. 998. Modeg aue te data b eha ateatve. Cou. Statt. - Theo Meth. 74 887-94. 6] Gupta R. D. ad Kudu D. 999. Geeazed epoeta dtbuto. New Zeaad J. Statt. 473-88. 7] Gupta R. D. ad Kudu D. a. Geeazed epoeta dtbuto: Deet ethod o etato. J. Statt. Coput. Su. 69 35-337. 8] Gupta R. D. ad Kudu D. b. Epoetated epoeta a: A ateatve to gaa ad Webu dtbuto. Boetca Joua 7-3.

- 3-9] Gupta R. D. ad Kudu D. 6. O the copao o he oato o the Webu ad GE dtbuto. Joua o Stattca Pag ad Ieece I Pe. ] Mudhoa G. S. ad Huto A. D. 996. The epoetated Webu a: Soe popete ad a ood data appcato. Cou. Statt. - Theo Meth. 5 359-383. ] Mudhoa G. S. ad Svatava D. K. 993. Epoetated Webu a o aazg bath tub aue data. IEEE Ta. Reabt 4 99-3. ] Mudhoa G. S. Svatava D. K. ad ee M. 995. The epoetated Webu a: A eaa o the bu -oto- aue data. Techoetc 374 436-445. 3] Naa M. M. ad Ea. H. 3. O the epoetated Webu dtbuto. Cou. Statt. - Theo Meth. 37 37-336. 4] Naa M. M. ad Ea. H. 4. Baea etato o the epoetated Webu ode. Cou. Statt.-Theo Meth. 33 343-36. 5] Raqab M. Z.. Ieece o geeazed epoeta dtbuto baed o ecod tattc. Joua o Stattca Pag ad Ieece 4 339-35. 6] Raqab M. Z. 4. Geeazed epoeta dtbuto: Moet o ode tattc. Stattc 38 9-4. 7] Raqab M. Z. ad Ahauah M.. Etato o ocato ad cae paaete o geeazed epoeta dtbuto baed o ode tattc. J. Statt. Coput. Su. 69 9-3. 8] Shaw A. I. ad Abu-Zadah H. H. 6. Soe chaactezato o the epoetated Paeto dtbuto. Subtted. 9] Sgh U. Gupta P. K. ad Upadha S. K. 5. Etato o paaete o epoetated- Webu a ude tpe-ii ceog chee. Coputatoa Stattc & Data Aa 48 59-53. ] Suta K. S. ad Mohe M. E.. Recod vaue o geeazed Paeto dtbuto ad aocated eece. Meta 5 5-6. ] Suta K. S. Mohe M. E. ad Abd-E-Ha N. S.. Etato o paaete o oa dtbuto baed o ecod vaue. The Egpta Stattca Joua 45 35-4. ] Suta K. S. Mohe M. E. ad Chd A.. Recod vaue o geeazed powe ucto dtbuto ad aocated eece. Joua o Apped Stattca Scece 43-56. 3] Zheg G.. O the he oato at tpe II ceoed data o the epoetated epoeta a. Boetca Joua 443 353-357.

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