Arduino PWM 1kHz and 10-bit resolution
I see you are talking about using the 16-bit timer, which is Timer 1 on the Uno.

Assuming you are using a Uno this code will do it:

// Clock frequency divided by 1 kHz frequency desired
const long timer1_OCR1A_Setting = F_CPU / 1000;

const byte outputPin = 10;  // Timer 1 "B" output: OC1B

void setup() 
 {
  pinMode (outputPin, OUTPUT);

  // set up Timer 1
  // Fast PWM top at OCR1A
  TCCR1A = bit (WGM10) | bit (WGM11) | bit (COM1B1); // fast PWM, clear OC1B on compare
  TCCR1B = bit (WGM12) | bit (WGM13) | bit (CS10);   // fast PWM, no prescaler
  OCR1A =  timer1_OCR1A_Setting - 1;                 // zero relative
  OCR1B =  (timer1_OCR1A_Setting / 2) - 1;           // 50 % duty cycle
  }  // end of setup

void loop()
  {
  }


You can change the 1000 in the calculation for timer1_OCR1A_Setting, within reason, to get other frequencies.

See my page about timers for more background information.